Survey of Maneuvering
Target Tracking.
Part V: Multiple-Model
Methods
X. RONG LI, Fellow, IEEE
VESSELIN P. JILKOV, Member, IEEE
University of New Orleans
This is the fifth part of a series of papers that provide a
comprehensive survey of techniques for tracking maneuvering
targets without addressing the so-called measurement-origin
uncertainty. Part I and Part II deal with target motion models.
Part III covers measurement models and associated techniques.
Part IV is concerned with tracking techniques that are based
on decisions regarding target maneuvers. This part surveys the
multiple-model methods–the use of multiple models (and filters)
simultaneously–which is the prevailing approach to maneuvering
target tracking in recent years. The survey is presented in a
structured way, centered around three generations of algorithms:
autonomous, cooperating, and variable structure. It emphasizes
the underpinning of each algorithm and covers various issues in
algorithm design, application, and performance.
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
CONTENTS
Nomenclature
Acronyms
Introduction
Hybrid Estimation
Overview of MM Approach
First Generation: Autonomous MM
Second Generation: Cooperating MM
Third Generation: Variable-Structure MM
MM Algorithm Design Issues
Nonstatistical Techniques
Concluding Remarks
References
Part I: Dynamic Models. IEEE Transactions on Aerospace and
Electronic Systems, 39, 4 (Oct. 2003), 1333—1364.
Part II: Ballistic Target Models. In Proceedings of the 2001 SPIE
Conference on Signal and Data Processing of Small Targets, vol.
4473, 559—581.
Part III: Measurement Models. In Proceedings of the 2001 SPIE
Conference on Signal and Data Processing of Small Targets, vol.
4473, 423—446.
Part IV: Decision-Based Methods. In Proceedings of the 2002 SPIE
Conference on Signal and Data Processing of Small Targets, vol.
4728, 511—534.
Manuscript received December 8, 2003; revised August 5 and
October 11, 2004; released for publication April 13, 2005.
IEEE Log No. T-AES/41/4/860796.
Refereeing of this contribution was handled by W. Koch.
This work was supported in part by ARO Grant
W911NF-04-1-0274, NASA/LEQSF Grant (2001-4)-01 and
NNSF Grants 60374025 and 60328306.
Authors’ address: Dept. of Electrical Engineering, University
of New Orleans, Lakefront, New Orleans, LA 70148, E-mail:
(xli@uno.edu).
c 2005 IEEE
0018-9251/05/$17.00 °
NOMENCLATURE
arg maxx g(x)
E[y]
f(x)
(i)
(ik )
Argument x that maximizes g(x)
Expectation of random variable y
pdf of continuous random variable x
Quantity pertaining to model m(i)
Quantity pertaining to mode/model
sequence m(ik k )
k
Time index (subscript for quantity at
time k; superscript for sequence
through time k)
Gaussian (normal) density function of
x with mean x̄ and covariance P
Model (mathematical model at certain
accuracy level)
Model set (set of models used)
Number of models in the model set M
= fsk = m(i) g = event that model m(i)
matches the true mode at time k
= fm1(i1 ) , : : : , mk(ik ) g = sequence of events
that models match the true mode
= E[(x ¡ x̂)(x ¡ x̂)0 ]
= E[(x ¡ x̂)(x ¡ x̂)0 j z]
Probability of event A
pmf of discrete random variable m
= f(x j A)PfAg = mixed pdf
probability of random variable x and
event A
= f(x j m)p(m) = mixed pdf-pmf of
random variables x and m
Mode (true behavior pattern, system
structure, or exact mathematical
model)
Mode space (set of possible modes)
Base state (continuous valued)
Estimate of y
Data (measurement)
= x0 Ax.
N (x; x̄, P)
m
M
M
mk(i)
m(ik k )
MSE(x̂)
MSE(x̂ j z)
PfAg
p(m)
P(x, A)
p(x, m)
s
S
x
ŷ
z
kxk2A
ACRONYMS1
2D
3D
AMM
ATC
CA
CMM
CT
CV
EKF
EM
FSMM
GPBn
Two dimensional
Three dimensional
Autonomous multiple model
Air traffic control
(Nearly) constant acceleration
Cooperating multiple model
(Nearly) constant turn
(Nearly) constant velocity
Extended Kalman filter
Expectation-maximization
Fixed-structure multiple model
Generalized pseudo-Bayesian of order n
1 The list includes acronyms used throughout the paper, but not those
that are used locally within a subsection.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
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HMM
IMM
LMMSE
MAP
MHT
MJLS
ML
MM
MMSE
MSE
PDA
pdf
pmf
PMHT
VS
VSMM
Hidden Markov model
Interacting multiple model
Linear minimum mean-square error
Maximum a posteriori
Multiple hypothesis tracking
Markov jump-linear system
Maximum likelihood
Multiple model
Minimum mean-square error
Mean-square error (matrix)
Probabilistic data association
Probability density function
Probability mass function
Probabilistic multiple hypothesis tracking
Variable structure
Variable-structure multiple model.
I. INTRODUCTION
This is the fifth part of a series of papers that
provide a comprehensive survey of techniques for
tracking maneuvering targets without addressing
the so-called measurement-origin uncertainty.
Part I [209] and Part II [205] deal with general
target motion models and ballistic target motion
models, respectively. Part III [206] covers
measurement models, including measurement
model-based techniques. Part IV [208] surveys various
decision-based methods. Part VI surveys various
nonlinear filtering methods for target tracking, a part
of which is [210].
In the absence of measurement-origin uncertainty,
maneuvering target tracking faces two interrelated
main challenges: target motion-mode uncertainty and
nonlinearity. Multiple-model (MM) methods have
been generally considered the mainstream approach
to maneuvering target tracking under motion-mode
uncertainty. This part surveys such methods, that
is, methods in which multiple models are used
simultaneously at a time for maneuvering target
tracking. Nonlinearity is best handled by nonlinear
filtering techniques, to be surveyed in subsequent
parts. MM methods and nonlinear filters are clearly
complementary to each other and their integration is
certainly appealing.
Estimation of random quantities can be classified
as point estimation and density estimation. Density
estimation aims at approximating the entire density
(distribution) of the estimatee (i.e., quantity to be
estimated–the target state), while point estimation
approximates the estimatee directly. MM methods
are applicable to density estimation as well as point
estimation. To be more focused, however, this part
only handles MM methods for point estimation,
leaving those for density estimation to subsequent
parts. For the same reason, this part also focuses
on the interplay of model-based filters, rather
than individual model-based filtering. As such, it
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may be helpful for the reader to assume that each
single-model-based filter is a Kalman filter. The more
general case of MM nonlinear filtering is covered in
subsequent parts.
This survey is structured. It emphasizes the
underlying ideas, concepts, and assumptions of the
methods, rather than particular implementations
for specific applications. This should help the
reader understand not only how these methods
work but also their pros and cons. It is hoped that a
distinctive feature of this survey is that it reveals the
interrelationships among various methods. However,
the reader should keep in mind that such statements
are based on our personal views and preferences, not
always accurate or unbiased, although a good deal of
effort has been made toward this goal. In addition to
such discussions, a considerable amount of material
included in this survey has not appeared elsewhere,
including a significant number of open problems and
ideas for further research. Also, more recent results
are discussed in greater detail.
Within maneuvering target tracking, MM methods
probably have the vastest literature, as evidenced
by the reference list and the length of this paper.
Regrettably, many important issues associated with
application of the MM methods, particularly those
of implementation and tuning of the MM algorithms
for specific applications, cannot be discussed in
greater detail as many readers would hope for.
We hope the reader will accept our apology for
omission or oversight of any work that deserves to
be mentioned or discussed at a greater length. As
stated repeatedly in the previous parts, we appreciate
receiving comments, corrections, and missing material
that should be included in this part. While we may not
be able to respond to each input, information received
will be considered seriously for the refinement of this
part for its final publication in a book.
This paper is organized as follows. Section II
formulates the problem of hybrid estimation,
which is what the MM methods are good for.
Section III introduces and provides an overview of
the MM approach, including its strengths, structures,
underlying criteria, and three generations. The
three generations (autonomous, cooperating, and
variable-structure MM estimation) are surveyed
in Sections IV, V, and VI, respectively, including
their tracking applications. Section VII covers MM
algorithm design issues, including model-set design
and transition probability determination. Section VIII
is dedicated to nonstatistical MM techniques.
Concluding remarks are given in the final section.
II.
HYBRID ESTIMATION
In simple terms, hybrid estimation is the estimation
of a quantity (a parameter or a process) that has both
continuous and discrete components [192, 326]. It is
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
particularly good for process estimation with structural
uncertainty. Target tracking is a hybrid estimation
problem [192]. In the prevailing approaches to target
tracking, the modeling of the target motion/dynamics
and the sensory system is essential. It is customary in
these approaches to use the process/plant noise and
measurement noise to cover the continuous-valued
uncertainties in both the target trajectory and the
measurement system. However, the major challenge
of maneuvering target tracking arises from the target
motion-mode uncertainty, which is discrete valued,
not to mention the discrete-valued uncertainties in the
measurement origins and the number of targets.
The target motion-mode uncertainty exhibits
itself in the situations where a target may undergo
a known or unknown maneuver during an unknown
time period. In general, a nonmaneuvering motion
and different maneuvers can be described only in
different motion models (see, e.g., Part I [209]). The
use of an incorrect model often leads to unacceptable
results.
A primary approach to target tracking in the
presence of motion-mode uncertainty is the so-called
MM method, which is one of the most natural
approaches to hybrid estimation. While the MM
method can be applied to hybrid estimation of a
parameter, in this paper we are concerned with its
application to estimation of the target state process
as a hybrid process. Generally speaking, a hybrid
process is one with both continuous and discrete
components, such as the state of a hybrid system. A
continuous-time (stochastic) hybrid system is described
by the following dynamic and measurement equations
_ = f[x(t), s(t), w(t), t]
x(t)
z(t) = h[x(t), s(t), v(t), t]
along with a law that governs the evolution of s,
which is often given in probabilistic terms. Here
x is called the base state, which (usually) varies
continuously, just like the state of a conventional
system; s is known as the system mode or modal state,
which has a staircase-type trajectory, that is, it may
either jump or stay unchanged; z is the measurement;
and w, v are the process and measurement noise,
respectively. We refer to the set of possible modes as
the mode space and denote it by S. In simple terms,
we say that x is continuous valued and s is discrete
valued (even though S may be a continuum in some
cases). In this sense, the whole state » = (x, s) of a
hybrid system is a hybrid process. For such a system,
hybrid estimation refers to the problem of estimating x
and s, or ». Such a system (in continuous or discrete
time) is a Markov jump system if s is a Markov
(j)
chain, that is, if Pfsk+1
j sk(i) g = pij,k , 8i, j, k (for a
discrete-time system with discrete-valued s), where
sk(i) signifies that mode i is in effect at time k. Often,
s is assumed a homogeneous Markov chain, that is,
the transition probabilities pij,k = pij , 8i, j are not a
function of time index k.
One of the simplest discrete-time hybrid systems is
the so-called jump-linear system, given by
xk+1 = Fk (sk+1 )xk + Gk (sk+1 )wk (sk+1 )
(1)
zk = Hk (sk )xk + vk (sk ):
(2)
This system is nonlinear because, for example, x
or z does not depend on the state » of the system
in a linear fashion. Were the system mode s given,
however, the system would be linear. In fact, s may
actually jump at unknown time instants, hence the
name. This system is known as a Markov jump-linear
system (MJLS) if s is a Markov chain. If s is
unknown but time invariant, it can be viewed or
argued as an unknown parameter of the system and
the system is linear in this perspective.
A variant of (1), also with first-order dependence
on fsk g, is xk+1 = Fk (sk )xk + Gk (sk )wk (sk ). It has pros
and cons compared with (1). For a discrete-time
model obtained by sampling a continuous-time
system, they rely on different underlying assumptions
concerning how sk is obtained from s(t). Caution
should be taken when dealing with this subtle issue
in some applications (see, e.g., [368]). More generally,
the following second-order fsk g-dependence model
xk+1 = Fk (sk+1 , sk )xk + Gk (sk+1 , sk )wk (sk+1 , sk )
(3)
was proposed and advocated by Blom [57, 59]. It can
describe jumps of x that occur simultaneously with
and due to jumps of s. In other words, this model is
capable of characterizing the system’s behavior at the
instant of a jump in s and thus eliminates the need
for introducing such a model explicitly, as has been
done in many designs for MM estimation. Important
algorithms for the first-order dependence models can
be generalized to this second-order dependence model
of a greater modeling power. For example, it has
been shown [59] that a time-reversed version of (the
autonomous part of) (3), xk+1 = Fk (sk+1 , sk )xk , has in
general a second-order dependence on time-reversed
fsk g, even if Fk (sk+1 , sk ) is actually fsk g-invariant. Such
time-reversed models are useful in, for example, the
well-known backward filters for smoothing.
III. OVERVIEW OF MULTIPLE-MODEL APPROACH
A. Basic Idea of MM Approach
Conventional solutions to hybrid estimation
problems follow the strategy that can be characterized
as “estimation after decision,” “decision followed
by estimation,” or simply “decision-estimation.”
At any time, it first decides on a (best) model and
then runs a single filter based on the model as if
it were the true one. The decision-based methods
for maneuvering target tracking, surveyed in Part
IV [208], belong to this class. This approach has
LI & JILKOV: SURVEY OF MANEUVERING TARGET TRACKING. PART V: MULTIPLE-MODEL METHODS
1257
several obvious drawbacks. First, possible errors
in deciding on the model are not accounted for in
the estimation. Second, decision is done irrevocably
before estimation, although estimation results are
often beneficial to decision making. Although these
drawbacks have been well perceived, their remedies
are hard to come by within this conventional strategy.
For example, accounting for decision errors would
require estimation in the presence of an unknown
model-truth mismatch, which is very challenging and
is still an open problem. Also, traditional model-based
estimation cannot be done before the decision since it
relies on the use of a single model. One possibility
here is to use a model-free (i.e., nonparametric)
estimation method, which appears an overkill for
maneuvering target tracking, where although uncertain
the true mode is only over a fairly limited set.
Rather, the semi-parametric methodology seems more
attractive here in general (see, e.g., [33]). Another
possible improvement is to have an iterated version
with several decision-estimation cycles at each time to
take advantage of the estimation results in the decision
step. This actually amounts to a degenerated MM
approach and its benefit is probably not commensurate
with the increased complexity.
The MM approach gets around the difficulty
due to the model uncertainty by using more than
one model. Its basic idea is to assume a set M of
models as possible candidates of the true mode in
effect at the time; run a bank of elemental filters, each
based on a unique model in the set; and generate the
overall estimates by a process based on the results
of these elemental filters. As such, the MM method
provides an integrated approach to the joint decision
and estimation problem of maneuvering target
tracking. It can be classified as a semi-parametric
approach since its model coverage is in between
parametric and nonparametric approaches. In the
optimization theoretic parlance, the MM method has
the potential of arriving at a globally optimal solution,
which is inherently superior in performance to the
two-stage optimization strategy of the conventional
decision-estimation approach.
In this survey, we maintain that MM and non-MM
estimation methods are separated as follows.
At any single time, the latter actually runs only
one (model-based) filter, possibly out of a set of
candidates, but filters at different times may differ,
while the former runs multiple model-based filters
at least at some time. One may think that a better
name for MM estimation approach is “multiple-filter”
approach, but the MM approach is not limited to
estimation. For example, it has been applied to
control, modeling, and identification.
For simplicity here we only describe the MM
method for MJLSs for two main reasons. Almost all
MM algorithms are theoretically valid only for this
class of systems, and our description here can be
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extended to other hybrid systems in theory, although
the development of the corresponding MM algorithms
is not necessarily straightforward. For a Markov
jump-linear system, the ith model in the MM method
obeys the following equations:
xk+1 = Fk(i) xk + Gk(i) wk(i)
zk =
Hk(i) xk
(4)
+ vk(i)
(5)
where E[wk(i) ] = w̄k(i) , cov(wk(i) ) = Qk(i) , E[vk(i) ] = v̄k(i) ,
cov(vk(i) ) = Rk(i) . Superscript (i) denotes quantities
pertinent to model m(i) in M, and the jumps, if any,
of the system mode are assumed 8m(i) , m(j) , k to have
the following homogeneous transition probabilities2
(j)
Pfmk+1
j mk(i) g = ¼ij
(6)
where mk(i) denotes the event that model m(i) matches
the system mode s in effect at time k:
¢
mk(i) =fsk = m(i) g:
(7)
Similarly, their finite sequences are denoted as sk =
fs1(i1 ) , : : : , sk(ik ) g and mk = fm1(i1 ) , : : : , mk(ik ) g, respectively.
Mode versus Model: To be more precise, in this
paper a mode refers to a pattern of behavior or a
phenomenon, or a structure of a system, and a model
is a mathematical representation or description of the
phenomenon pattern (system structure) at a certain
accuracy level. It is models, not modes, on which
an estimator is based. For example, the behavior
pattern of an aircraft during a specific turn motion is
a mode of turning. Many mathematical models are
available at different accuracy levels that describe
such a mode [209]. Equivalently, we may think that
a mode describes the truth precisely and a model is
an approximation of the mode. Such a distinction is
necessary whenever the mismatch between the model
and mode is of concern. The model set M differs in
general from the mode space S in two aspects: 1) they
have different numbers of elements–M usually
has much fewer elements than S; and 2) a model
is usually a simplified description of a mode. For
example, one may use a small set of models, such as
a nonmaneuver model plus several constant-turn (CT)
models for tracking a target that may undergo various
(complex) maneuvers (modes).
B. Underlying Structures of MM Algorithms
In general, four key components of MM estimation
algorithms can be identified as follows.
1) Model-set determination: This includes both
offline design and possibly online adaptation of the
model set. An MM estimation algorithm distinguishes
2 However,
homogeneous transition probabilities are not suitable for
systems sampled at a nonuniform rate (see Section VIIB).
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
C.
Fig. 1. General structure of MM estimation algorithms (with 2
model-based filters).
itself from non-MM estimators by the use of a set of
models, instead of a single model. The performance
of an MM estimator depends largely on the set of
models used. The major task in the application of MM
estimation is the design (and possibly adaptation) of
the set of multiple models.
2) Cooperation strategy: This refers to all
measures taken to deal with the discrete-valued
uncertainties within the model set, particularly
those for hypotheses about the model sequences.
It includes not only pruning of unlikely model
sequences, merging of “similar” model sequences,
and selection of (most) likely model sequences, but
also iterative strategies, such as those based on the
expectation-maximization (EM) algorithm.
3) Conditional filtering: This is the recursive
(or batch) estimation of the continuous-valued
components of the hybrid process conditioned on
some assumed mode sequence. It is conceptually the
same as state estimation of a conventional system with
only continuous-valued state.
4) Output processing: This is the process that
generates overall estimates using results of all filters
as well as measurements. It includes fusing/combining
estimates from all filters and selecting the best ones.
The operation of MM estimation algorithms has
a general structure, as depicted in Fig. 1 with only
two models. In the figure, the outer loops between the
filters and the cooperation strategy represent (possibly
multiscan) recursions; the vertical arrows between
the filters and the cooperation strategy represent their
cooperation/interaction within one recursion. The three
components (i.e., exclusive of conditional filtering) are
not present in a non-MM algorithm essentially. Output
processing is covered in Section IVC. Cooperation
strategies are the topic of Section VB. Design and
adaptation of model sets are addressed in detail in
Sections VIIA and VIB, respectively. In some MM
algorithms conditional filtering and cooperation
strategies are tightly coupled and can hardly be
separated. Output processing, cooperation strategies,
and model-set adaptation, respectively, are the
cornerstones of three generations of MM algorithms
developed so far, which are discussed next.
Three Generations of MM Algorithms
Three generations of MM algorithms have
been identified in [195]. This identification is
very beneficial: Different generations have their
fundamental differences in operations, structures,
and limitations/capabilities/potential; the three
generations came into existence sequentially; and
more convincingly, later generations do inherit
superior characteristics of the earlier generations. This
identification also helps reveal possible directions for
further development.
The first generation MM method was pioneered
by Magill [235] and Lainiotis (see, e.g., [179],
[180], [347]), and widely applied and promoted by
Maybeck [242] and others. The second generation,
represented unquestionably by Blom’s interacting MM
(IMM) algorithm [56, 55, 58], has earned an enviable
reputation for MM estimation via a significant number
of successful applications in target tracking. Its
popularization and further development have been
spearheaded by Bar-Shalom (see, e.g., [13], [14],
[18], [19], [21]). Its practical value in tracking has
been strongly advocated and well demonstrated by
Bar-Shalom and others, notably Blom and Blair.
The third generation, characterized by its variable
structure, is gaining momentum rapidly and is
becoming the state of the art of MM estimation. Its
initiation [198, 191, 201] and advancement have been
led by Li and his team (see, e.g., [196], [224], [215],
[192], [195]).
There are two types of estimation problems for
hybrid systems. The first one involves an unknown
(random or nonrandom) but time-invariant mode.
This is the case for estimating the state of a system
with an unknown model that does not change over
time or with a known model involving an unknown
(time-invariant) parameter. In contrast to this, the
mode in the second type may jump sometimes.
The first generation is characterized by the fact
that each of its elemental filters operates individually
and independently of all the other elemental filters.
Its advantage over many non-MM approaches stems
from its superior output processing of results from
elemental filters to generate the overall estimate. It
would be optimal if the true mode were time invariant
but unknown over a set that is identical to the model
set used. These MM algorithms have been known
under various names. A better name is autonomous
MM (AMM) algorithms for several reasons, to
become clear later.
The second generation inherits the first
generation’s superior output processing, and its
elemental filters work together as a team via effective
internal cooperation, rather than work independently
as in the first generation. The cooperation includes
all measures taken to achieve a better performance,
such as individualized reconditioning of each filter
(e.g., reinitialization as in the IMM algorithm,
LI & JILKOV: SURVEY OF MANEUVERING TARGET TRACKING. PART V: MULTIPLE-MODEL METHODS
1259
see Section VB1), performance enhancement via
interactive iterations and competitions among
filters (e.g., those based on the EM algorithms,
see Section VB5), joint parameter adaptation
(e.g., cohesive online identification of transition
probabilities, see Section VIIB), and other hypothesis
reduction strategies, discussed in Section VB. This
generation has the potential of optimal performance
when the true mode may jump among members of
a set that is identical to the model set used. Many of
the algorithms in this generation have been called
“switching” or “dynamic” MM algorithms before,
which is not quite accurate or representative. We refer
to the second generation as cooperating MM (CMM)
algorithms.
The model groups or teams in the first two
generations have a fixed membership over time and
thus have a fixed structure. They are allowed to
have a variable membership in the third generation,
leading to a variable structure, that is, a variable set of
models. Still under active development, this generation
is potentially much more advanced in the sense of
having an open architecture than its ancestors, which
have a closed architecture. Not only does it inherit
the second generation’s effective internal cooperation
and the first generation’s superior output processing,
but it also adapts to the outside world by producing
new elemental filters if the existing ones are not good
enough and by eliminating elemental filters that are
harmful. This generation has been known as VSMM
algorithms. It is most suitable in the case where there
is a significant truth-model mismatch: the model set
used does not match the set of possible true modes.
Using a human-team analogy, a non-MM
algorithm relies entirely on the performance of
a single “best” individual decided prior to his
performance. In contrast, an MM algorithm asks all
individuals in a group to perform simultaneously
and produces the overall estimate after their
performance. In the first generation, these individuals
work independently. Its superiority to non-MM
algorithms stems from its flexibility in generating
its output reports based on the individual results
a posteriori. The second generation focuses on internal
cooperation. Its individuals form a cooperative team.
It outperforms the first generation by team work.
The third generation explores the best team makeup.
It determines an adaptive, cooperative team with a
variable membership–it may recruit new members
and fire bad or incompetent members. Each generation
is more capable than its predecessors at the price of
increased sophistication/complexity. It is interesting
to note that the development of MM algorithms has
been along the direction from the final product to the
underlying structure through the internal mechanisms;
that is, from the output report of the team, to the
internal cooperation of the team, and then to the
makeup of the team.
1260
This paper is limited to MM approach to point
estimation for maneuvering target tracking. MM
approach has been applied successfully in many
other areas, including control and identification,
as well as to target tracking in the presence of
measurement-origin uncertainty, and in recent years
to density estimation for maneuvering target tracking,
such as particle filtering. Outside of the target tracking
area, almost all MM research and development have
dealt only with the first generation so far; knowledge
of the second generation is limited; and the third
generation is hardly known.
D. Optimality Criteria
Consider a hybrid random variable » = (x, m),
where x is continuous valued and m is discrete valued
with M possible values. The complete Bayesian
solution of estimating (x, m) using data z is the mixed
(joint) probability density function/probability mass
function (pdf-pmf) p(x, m j z) = f(x j m, z)p(m j z).
This clearly involves a density estimation problem.
Since a density function requires in general infinitely
many numbers to describe it completely, this solution
generally has an infinite dimension. For a hybrid
process f»k g, this solution requires in general
recursive estimation of the density function, known as
nonlinear filtering. This is the topic of more than one
subsequent part of this survey, which covers various
exact and approximate nonlinear filtering methods.
In this part, we deal only with point estimation; that
is, estimators that have the same, finite dimension as
the estimatee (i.e., the quantity to be estimated, which
could be the base state, modal state, or hybrid state in
our case).
Least squares, maximum likelihood (ML),
minimum mean-square error (MMSE), maximum
a posteriori (MAP), and the method of moments
are probably most widely used methods for point
estimation. Virtually all point estimation algorithms
using MMs developed so far are in essence based
on either the MMSE or MAP criterion. This is
understandable since MMSE and MAP are two
primary Bayesian criteria for estimating a random
quantity and the state of a hybrid system is more
naturally treated as random than as deterministic.
For convenience, we adopt the following
notation for estimators of continuous-value x and
discrete-valued m:
x̂MMSE = E[x j z]
m̂MMSE = E[m j z]
(x̂JMAP , m̂JMAP ) = arg max p(x, m j z)
(x,m)
x̂
x̂
MAP
m̂
MAP
= arg max f(x j z)
x
MAP
= arg max p(m j z)
m
(m̂) = arg max f(x j z, m̂)
x
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
Fig. 2. Illustration of MMSE and MAP estimators. (a) (p1 , p2 , p3 ) = (0:21, 0:41, 0:38). (b) (p1 , p2 , p3 ) = (0:25, 0:51, 0:24), x̂ = x̂2 = x̂4 .
¢
where f(¢) is pdf, p(¢) is pmf, p(x, m) = f(x j m)p(m)
is a mixed (joint) pdf-pmf, and arg maxx g(x) stands
for “the argument x that maximizes g(x),” meaning
the maximizer (i.e., the location of the largest peak) of
g(x). Note the following.
MAP
1) In x̂
(m̂), m̂ can be any mode estimator in
general, but is almost always taken to be m̂MAP .
2) As defined above, m̂MMSE does not exist
whenever the convex sum ®1 m1 + ¢ ¢ ¢ + ®M mM is
meaningless or does not exist, which is the case, for
(j)
example, if m(i)
different dimensions.
Pand m have
(i)
(i)
3) f(x) = M
f(x
j
m
)p(m
) is p(x, m) averaged
i=1
over possible values of m.
Fig. 2 illustrates the differences among x̂1 = x̂MMSE ,
x̂2 = x̂JMAP , x̂3 = x̂MAP , x̂4 = x̂MAP (m̂MAP ), and x̂5 =
x̂MAP (m̂) for a Gaussian mixture density:
f(x) = p1 N (x; x̄1 , ¾12 ) + p2 N (x; x̄2 , ¾22 ) + p3 N (x; x̄3 , ¾32 )
p
where N (x; x̄i , ¾i2 ) = exp[¡(x ¡ x̄i )2 =(2¾i2 )]=( 2¼¾i )
with (x̄1 , x̄2 , x̄3 ) = (¡3, 0, 5), (¾12 , ¾22 , ¾32 ) =
(1:62 , 2:252 , 22 ), and pi = Pfm = m(i) g. In the figure,
the thicker line is the mixture density f(x). Note the
following.
1) x̂1 = x̂MMSE = p1 x̄1 + p2 x̄2 + p3 x̄3 is the center
of probability mass, that is, the balance point at which
the line with f(x) as the mass density function would
not tip to the left or right if a pivot is placed.
2) x̂2 = x̂JMAP is the location of the peak with
the largest weighted peak value of allpcomponent
densities: pi £ maxx N (x; x̄i , ¾i2 ) = pi =( 2¼¾i ).
3) x̂3 = x̂MAP is the location of the largest peak
of the mixture density f(x). It is always within the
interval between the leftmost and the rightmost peaks
of all component densities.
4) x̂4 = x̂MAP (m̂MAP ) is the location of the largest
peak of the component density N (x; x̄i , ¾i2 ) with
largest pi .
5) x̂5 = x̂MAP (m̂ = m(i) ) is the location of the largest
peak of the component density N (x; x̄i , ¾i2 ) (in the
figures, i = 1).
6) Fig. 2(a) and (b) are for two mixture densities
with the same component densities but different sets
of weights: x̂MMSE , x̂JMAP , x̂MAP , x̂MAP (m̂MAP ) are
relatively close to each other in (b), but quite different
in (a), although the sets of weights differ not much.
For the two cases, x̂MMSE changes least and x̂JMAP and
x̂MAP (m̂MAP ) change most.
These estimators minimize expectations of Bayes
cost functions Ci , i = 1, 2, 3, 4, 5, respectively (see,
e.g., [94], [99]). For example, C1 (x ¡ x̂) = (x ¡ x̂)2
and C3 (x ¡ x̂) = lim²!0 1(jx ¡ x̂j ¡ ²), where 1(x) is the
unit-step function and
½
0
jx ¡ x̂j < ²
1(jx ¡ x̂j ¡ ²) =
1
jx ¡ x̂j > ²
describes a “golf hole” of radius ².
Parallel to MAP estimation, ML estimators can
also be obtained with the involved posterior pdf
or pmf replaced by the corresponding likelihood
functions, although they have not been proposed
systematically in MM estimation. It turns out,
however, that the application of the likelihood
principle to estimation of a nonrandom hybrid
quantity differs significantly from the MAP
case. Assume that » = (x, m) is nonrandom with
continuous-valued x and M possible values
fm(1) , : : : , m(M) g for m. Then f(z j x, m) represents a set
of likelihood functions ff(z j x, m(1) ), : : : , f(z j x, m(M) )g.
The joint ML estimator of x and m and x̂ML (m̂) given
m̂ are well defined as
»ˆML = (x̂JML , m̂JML ) = arg max f(z j x, m)
(x,m)
(1)
= arg max ff(z j x, m ), : : : , f(z j x, m(M) )g (8)
(i)
(x,m )
x̂ML (m̂) = arg max f(z j x, m(i) )
x
if m̂ = m(i)
but the ML estimators of x and m separately are not
well defined, because neither f(z j x) nor f(z j m)
has a generally accepted definition, although several
definitions are available. One such definition of the
likelihood functions L(x j z) and L(m j z) is based on
the so-called generalized likelihood principle, leading
to
L(x j z) = maxff(z j x, m(1) ), : : : , f(z j x, m(M) )g
m(i)
L(m j z)jm=m(i) = max f(z j x, m(i) ),
x
LI & JILKOV: SURVEY OF MANEUVERING TARGET TRACKING. PART V: MULTIPLE-MODEL METHODS
i = 1, 2, : : : , M
1261
is no mode-model mismatch and thus we use m to
denote both the mode and the model and treat a model
(which is deterministic) as a realization of a mode
(which is random) in this section. Note that Sk = M
implies, but is not implied by, sk 2 M (i.e., Sk µ M).
A1 implies that m(i) is the true model for all times
if and only if it is so at some single time and thus
conditioned on anything A, the mode probability and
mode-sequence probability are equal:
and the corresponding maximizers are then taken
to be the generalized ML (GML) estimators
x̂GML and m̂GML , respectively, which are however
equal to x̂JML and m̂JML because maxx L(x j z) =
max(x,m(i) ) ff(z j x, m(1) ), : : : , f(z j x, m(M) )g, which is
(8). We choose the continuous-valued part of »ˆML
as x̂ML , namely, x̂ML := x̂JML . As such, x̂ML is the
ML counterpart of x̂JMAP , not x̂MAP . The likelihood
functions can also be defined by expectation or
marginalization (see, e.g., [127])
Pfmk(i) j A, A1g = Pfm1(i) , : : : , mk(i) j A, A1g
L(x j z) = f(z j x) = E[f(z j x, m) j x]
X
=
f(z j x, m(i) )p(m(i) j x)
i
L(m j z) = f(z j m) = E[f(z j x, m) j m]
Z
= f(z j x, m)f(x j m)dx
(9)
which, however, requires treating the quantity being
averaged out as random, in violation of the previous
assumption that x and m are nonrandom. Nevertheless,
for MM estimation f(z j m) = E[f(z j x, m) j m] is a
sensible way to go because it is natural to treat the
base state x as random and the model as nonrandom.
Therefore, we use m̂ML based on (9). With this,
x̂ML (m̂ML ) = arg maxx f(z j x, m̂ML ) is well defined.
where mk(i) denotes that model m(i) matches the true
mode at time k. Denote by M = jMj the number of
models used. Then, all possible model sequences
(through time k) are constant (by A1) and there are
exactly M of them (by A2), given by
k
m(i)
= fm1(i) , : : : , mk(i) g,
Fundamental Assumptions: The first generation,
AMM algorithms were developed based on the
following two fundamental assumptions.
A1. The true mode s is time invariant (i.e., sk = s,
8k).
A2. The true mode s at any time has a mode
space S that is time invariant and identical to the
time-invariant finite model set M used (i.e., Sk = M,
8k).
A2 can be decomposed into three components:
A2a: Mk = M, 8k; A2b: Sk = S, 8k; A2c: S = M.
A2b can be viewed to be implied by A1, but A2
may be invoked without A1, as the second generation
does.
Assumption A2a is the defining assumption of
both first and second generations. A1 and A2 allow
in principle the true mode s to be deterministic or
random. Almost all AMM algorithms developed so far
assume a random s, although assuming an unknown
but nonrandom s appears, in our opinion, somewhat
more natural for many applications. Furthermore,
neither A2b nor A2c is in fact needed if s is assumed
nonrandom, such as for the ML estimation alluded
to above and in Section IVD. According to A2, there
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(10)
Note that Assumptions A1 and A2 are embedded in
k
.
this definition of m(i)
MMSE-AMM: As first proposed in [235], the
MMSE-optimal AMM base-state estimator is given
by the total expectation theorem as (see, e.g., [21])
x̂kjk = E[xk j z k , A1, A2]
=
IV. THE FIRST GENERATION: AUTONOMOUS MM
ESTIMATION
A. Optimal Autonomous MM Estimation
m(i) 2 M:
M
X
i=1
k
k
E[xk j z k , m(i)
]Pfm(i)
j z k , A1, A2g =
M
X
(i) (i)
¹k
x̂kjk
i=1
(11)
where z k = (z1 , : : : , zk ) is measurements through time
(i)
k
k, ¹(i)
k = Pfmk j z , A1, A2g is the posterior mode
probability under A1 and A2 that the mode in effect
is constant and equal to one and only one but possibly
(i)
k
anyone of the models in M, and x̂kjk
= E[xk j z k , m(i)
]
is the MMSE estimate from the ith elemental filter
assuming m(i) is true throughout time.
MMSE
Under A1 and A2, x̂kjk
is unbiased in the sense
MMSE
E[xk ¡ x̂kjk ] = 0; its conditional mean-square error
(MSE) matrix (often called error covariance loosely)
is minimum of all base-state estimators x̂kjk , given
originally in [183] (see also [21]) by3
Pkjk = MSE(x̂kjk j z k , A1, A2)
=
M
X
(i)
(i)
(i)
[Pkjk
+ (x̂kjk
¡ x̂kjk )(x̂kjk
¡ x̂kjk )0 ]¹(i)
k
(12)
i=1
(i)
=
where MSE(x̂ j z) = E[(x ¡ x̂)(x ¡ x̂)0 j z], and Pkjk
(i)
MSE(x̂kjk
j z k , A1, A2) is the conditional MSE matrix
(i)
of the MMSE estimator x̂kjk
under A1 and A2. The
3 In
fact, this equation holds true for any estimator x̂kjk , not just the
(i)
k ].
= E[xk j z k , m(i)
optimal AMM estimator, provided x̂kjk
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
TABLE I
One Cycle of AMM Algorithm
Gaussian system (4)—(5) where the Kalman filter is
optimal given the mode.
MAP-AMM: The mixed pdf-pmf of the base state
and mode at time k is
1. Model-conditioned filtering (for i = 1, 2, : : : , M):
Predicted state:
(i)
(i) (i)
(i)
(i)
x̂kjk¡1
= Fk¡1
x̂k¡1jk¡1 + Gk¡1
w̄k¡1
Predicted covariance:
(i)
(i)
(i)
(i) 0
Pkjk¡1
= Fk¡1
Pk¡1jk¡1
(Fk¡1
)
Measurement residual:
Residual covariance:
Filter gain:
Updated state:
Updated covariance:
p(xk , mk j z k , A1, A2) = f(xk j z k , mk , A1)p(mk j z k , A1, A2)
(i)
(i)
(i) 0
+Gk¡1
Qk¡1
(Gk¡1
)
(i)
(i) (i)
z̃k = zk ¡ Hk x̂kjk¡1 ¡ v̄k(i)
(i)
Sk(i) = Hk(i) Pkjk¡1
(Hk(i) )0 + Rk(i)
(i)
(i)
(i) 0 (i) ¡1
Kk = Pkjk¡1 (Hk ) (Sk )
(i)
(i)
x̂kjk
= x̂kjk¡1
+ Kk(i) z̃k(i)
(i)
(i)
Pkjk
= Pkjk¡1
¡ Kk(i) Sk(i) (Kk(i) )0
= ff(i) (xk j z k )¹(i)
k , i · Mg
k
where f(i) (xk j z k ) = f(xk j z k , m(i)
) is the density
assuming the mode sequence is m1(i) , : : : , mk(i) (i.e.,
m(i) is the true model). It thus follows from the
total probability theorem that the base state has the
posterior mixture density
2. Mode probability update (for i = 1, 2, : : : , M):
¢
k , z k¡1 ] assume
L(i)
= p[z̃k(i) j m(i)
= N (z̃k(i) ; 0, Sk(i) )
k
¹(i)
L(i)
k¡1 k
¹(i)
=
k
(j)
(j)
¹ L
j k¡1 k
Model likelihood:
Mode probability:
P
3. Estimate fusion:
Overall estimate:
Overall covariance:
P
P
=
E[mk j z
M
X
m(i) ¹(i)
k
i=1
=
f(i) (xk j z k )¹(i)
k :
k
k
, m(i)
]Pfm(i)
(15)
xk
MAP
JMAP
JMAP
»ˆkjk
= (x̂kjk
, m̂kjk
)
= arg max ff(i) (xk j z k )¹(i)
k , i · Mg
(xk ,m(i) )
k
m̂MAP
= (m̂1 , : : : , m̂k ),
k
j z , A1, A2g
(13)
i=1
0
k
M
X
i=1
MAP
x̂kjk
= arg max f(xk j z k , A1, A2)
m̂kjk = E[mk j z k , A1, A2]
k
k
f(xk j z k , m(i)
)Pfmk(i) j z k , A1, A2g
The corresponding MAP-AMM estimators are given
by
(i)
¡ x̂kjk
)
corresponding mode estimator is given by
M
X
=
M
X
i=1
x̂(i) ¹(i)
i kjk k
Pkjk =
[P (i) + (x̂kjk
i kjk
(i) 0 (i)
) ]¹k
£ (x̂kjk ¡ x̂kjk
x̂kjk =
f(xk j z k , A1, A2) =
k
MSE(m̂kjk j z ) = E[(mk ¡ m̂kjk )(mk ¡ m̂kjk ) j z , A1, A2]
M
X
(m(i) ¡ m̂kjk )(m(i) ¡ m̂kjk )0 ¹(i)
=
k :
(14)
i=1
This mode estimator exists and is meaningful
only if the convex sum (13) is well defined and
meaningful, which is not the case if m(i) is only an
index of the system structure or behavior pattern,
e.g., M = f1, 2, : : : , Mg. For instance, if m(1) = 1 is a
booster model of a missile and m(2) = 2 represents
a climbing motion of an aircraft, their weighted
sum is meaningless. Even if m(2) = 2 is also for the
missile (say, a reentry model), their weighted sum
is still hard to interpret: what does it mean in this
MMSE
case if m̂kjk
= 1:63? Also, (m(1) , m(2) ) = (1, 2) and
MMSE
,
(m(1) , m(2) ) = (2, 3) would lead to different m̂kjk
which renders interpretation difficult. This mode
estimator is meaningful in general when all m(i) are
points in a vector space.
Table I gives the MMSE-AMM algorithm (of
the base state) under Assumptions A1 and A2 for a
(16)
m̂1 = ¢ ¢ ¢ = m̂k = m̂kMAP = arg max ¹(i)
k
m(i)
MAP
(m̂k ) = arg max f(j) (xk j z k )
x̂kjk
xk
MAP
k
MAP
(m̂MAP
) = x̂kjk
(m̂k )jm̂k =m̂k
x̂kjk
MAP
k
if m̂k = m(j)
:
MAP
In words, the MAP base-state estimator x̂kjk
is the
k
peak location of the mixture density f(xk j z , A1, A2),
which is a probabilistically weighted sum of
f(i) (xk j z k ); the joint MAP (JMAP) estimator
JMAP
JMAP
(x̂kjk
, m̂kjk
) is the maximizer of the pdf-pmf
k
p(xk , mk j z , A1, A2) or the set of f(i) (xk j z k )¹(i)
k ;
the MAP mode estimator m̂kMAP is the one with
the largest posterior probability and thus the MAP
k
mode-sequence estimator m̂MAP
is the constant
sequence of this mode throughout time (m̂kMAP and
k
m̂MAP
can also be interpreted as outcomes of the
corresponding MAP tests); the model-sequence
MAP
conditioned MAP estimator x̂kjk
(m̂k ) is the peak
location of the component density f(j) (xk j z k )
corresponding to the model m̂ = m(j) .
It should be clear that the special MAP estimators
MAP
k
MAP
x̂kjk
(m̂MAP
) and x̂kjk
(m̂k ) can be obtained from
(i)MAP
=
the set of component MAP estimators x̂kjk
k
MAP
arg maxxk f(i) (xk j z ), but the MAP estimator x̂kjk and
JMAP
the JMAP estimator x̂kjk
cannot in general, although
LI & JILKOV: SURVEY OF MANEUVERING TARGET TRACKING. PART V: MULTIPLE-MODEL METHODS
1263
(i)MAP
JMAP
coincides with the x̂kjk
with the largest value
x̂kjk
(i)
of f(i) (xk j z k )¹k over xk and m(i) .
Some of these MAP-AMM estimators were
presented in [99] in simpler terms and evaluated
along with the MMSE estimator in terms of several
performance measures via computer simulations for a
simplified aircraft tracking example.
The MSE matrices (error covariances) of these
MAP estimators do not have a known, explicit,
analytic form, but they are related to that of the
MMSE
via the following easily
MMSE estimator x̂kjk
obtainable general relationship:
Pkjk = MSE(x̂kjk j z k )
MMSE
MMSE
MMSE 0
j z k ) + (x̂kjk ¡ x̂kjk
)(x̂kjk ¡ x̂kjk
)
= MSE(x̂kjk
(17)
where x̂kjk can be any estimator, including the above
MAP estimators.
B. Autonomous Operations of Conditional Filtering
It is clear from the above that there are two key
functions in the operation of an AMM algorithm:
model-based conditional filtering and output
processing. They are discussed next.
Assumption A1 is a defining assumption of
the AMM algorithms. Under A1, each elemental
filter does conditional filtering based on a constant
model sequence. In other words, each filter works
individually and independently of other filters.
As such, each conditional filtering operation
is autonomous, hence the name “autonomous
multiple-model algorithms.”
For elemental filter i, the goal is to compute the
k
) for the MAP case and
pdf f(i) (xk j z k ) = f(xk j z k , m(i)
(i)
(i)
the estimate (x̂kjk , Pkjk ) for the MMSE case, where
R
(i)
(i)
k
x̂kjk
= E[xk j z k , m(i)
] = xk f(i) (xk j z k )dxk and Pkjk
=
(i)
MSE(x̂kjk
j z k ). If the base state xk and measurements
k
, then f(i) (xk j z k ) =
z k are jointly Gaussian under m(i)
(i)
(i)
N (xk ; x̂kjk , Pkjk ) and thus conditional filtering operations
for MMSE and MAP estimation are theoretically the
same. For the general, non-Gaussian case, neither
class has an explicit, analytic form. However, if
model m(i) is linear (i.e., the system conditioned
k
is linear) and satisfies the whiteness and
on m(i)
uncorrelatedness assumption of the Kalman filter for
the process and measurement noises, recursive linear
(i)
(i)
(i)
(i)
, Pk¡1jk¡1
) ! (x̂kjk
, Pkjk
) is
MMSE estimation (x̂k¡1jk¡1
given by the Kalman filter explicitly regardless of
Gaussianity (see, e.g., [21]). This is not the case for
the conditional filtering for MAP estimation.
Adaptive estimation was studied by many
(see, e.g., [184]), including the case of unknown
time-invariant parameters with independent
1264
measurements. These results were extended by Magill
[235] in a nonrecursive form to scalar, dependent
measurements generated by state-space models
with unknown discrete parameters. Magill’s results
were further extended by Lainiotis, et al. [315, 183,
179, 182, 180] to a recursive form, with an exact
error covariance form, for vector measurements and
arbitrary continuous and discrete parameters. Early
works dealt only with estimation for systems with a
time-invariant mode that is unknown (a nonrandom
constant) or uncertain (a random variable, but not
a random process), which led to the autonomous
operations of conditional filtering [315, 308, 118,
185]. Many reinventions, extensions, and applications
of this generation can be found in the literature under
various names, including the “partition (partitioned or
partitioning) filter” [179, 182, 180, 347], the “multiple
model adaptive filter” [179, 182], the “parallel
processing algorithm” [7], the “multiple model
adaptive estimator” [242], the “static multiple-model
algorithm” [18, 216], the “filter bank method” [65],
the “self-tuning estimator” [65], the “operating
regime approach” [156], and in the same spirit
the “mixture of experts” [254, 73]. These names
suggest the structure, features, and capability of
the first generation, particularly in comparison with
non-MM algorithms. For example, this MM algorithm
was applied recently in [181] to state estimation
of a nonlinear system without model uncertainty.
It runs a bank of perturbed (linearized) Kalman
filters, each with a nominal state trajectory that is a
random realization of the true one, as proposed in
[179]. A large number of nominal trajectories are
generated first and then clustered to achieve better
cost effectiveness [181].
C.
Output Processing for MM Estimation
The output processor generates the overall estimate
using information available from all elemental filters
as well as data. It is in general an information fusion
process. This type of fusion, however, differs from
multisensor data fusion in at least two aspects [195]:
1) at most one (and at least one under Assumption
A2) estimate is correct (but not necessarily precise) in
the MM fusion (but which one is unknown), whereas
more than one estimate may be correct in sensor
fusion, and 2) different filters use the same data in the
MM fusion but different data in sensor fusion. Note,
however, that the second difference is not fundamental
since the same data can be treated artificially as
multiple pieces of data with perfect coupling, that is,
sensor fusion with dependent data.
The principles proposed for output processing of
AMM algorithms turn out to be general, applicable
to later generations as well, although their specific
implementations are tuned for AMM algorithms. In
view of this, the following discussion is directed to
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
the general MM, not just AMM, algorithms. These
principles can be classified into two groups: hard
decision and soft decision [192, 195].
Hard Decision: This approach identifies a
“good” subset B of model sequences by a hard
decision procedure, and then generates the overall
estimate from the estimates conditioned on these
model sequences. The set B may change with respect
to time as more data are collected, although only
constant model sequences are considered in the AMM
algorithms.
The most natural subset B is the set of model
sequences deemed most likely or “not too unlikely.”
Several different decision procedures for this purpose
have been proposed, including what can be called
B-best approach and the Viterbi algorithm. This
hard decision amounts to pruning of unlikely model
sequences. It is applied here to output processing. As
discussed later in Section VB3, the same procedure
has also been applied to hypothesis reduction as a
cooperation strategy. These two applications have
appeared together so far, which is natural, although
each actually stands alone without the other.
The Viterbi algorithm or forward dynamic
programming (see, e.g., [115]) can identify the single
best (i.e., most probable) model history for each
model at any time. It always has M (i.e., number of
models) survived model histories at any time k. Each
of them is the best model history for a model-based
filter at a time. These M histories definitely include
k
but they are not necessarily the M
the best one m̂MAP
best ones (for second and third generations) because,
for example, the second best model history for a
model may be better than the best one for another
model. In short, this procedure selects “the fittest” for
each model; some of these fittest can be quite unfit
to the overall truth, but they are needed to guarantee
“survival of the fittest” for the future.
In the B-best approach, the subset B of the B
most probable model sequences at time k can be
found by MAP hypothesis tests (see, e.g., [129],
[338]) using all data through k in a batch format.
However, recursive implementations are usually
needed. Recursive B-best algorithms rely on MAP
hypothesis tests using data available at the time,
where the test statistic is usually a function of mode
sequence probabilities and conditional measurement
residuals. Many MM algorithms for adaptive control
make decisions in this way. However, they actually do
not guarantee to yield the B most probable sequences
over the entire time horizon because some or all
of these best sequences may have less probable
partial histories and been deleted earlier in the
process. In contrast, the Viterbi algorithm can be
implemented recursively if the model sequence is a
(hidden) Markov chain with mode-history independent
measurements. Another drawback of the B-best
approach is that some or many of the B most probable
model sequences may be quite similar and had better
be merged to reduce processing load. More details and
an illustrative example are given in Section VB3.
Knowing the above pros and cons of these two
approaches, it appears sensible to combine them as
follows. Use the Viterbi algorithm for hypothesis
reduction (i.e., to decide which model sequences
should be maintained), which involves M 2 conditional
filtering operations (see Section VB5); but use the
B-best algorithm to select B most probable ones out
of the M 2 model sequences for output processing
if we can only afford processing B sequences at a
time for output, which may be the case for the MAP
MAP
as the peak location of a
base-state estimator x̂kjk
mixture density of B components.
In the extreme case of the “survival of the fittest”
where the set B has only one sequence, the estimate
based on the single most probable mode sequence
i is taken to be the overall estimate: (x̂kjk , Pkjk ) =
(i)
(i)
, Pkjk
). We emphasize that compared with the
(x̂kjk
conventional “decision-estimation” non-MM approach,
this “estimation-decision” MM approach is superior
even in this extreme case because, as explained
at the beginning of Section IIIA, it makes a more
informed decision since the decision is made after the
completion of conditional filtering (estimation).
Other hard decision procedures are possible,
including heuristic rules, expert systems, neural
networks, and so on (see Section VIII).
The overall estimate given a subset B of more
than one model sequence is usually taken to be a
probabilistically weighted sum of estimates based on
model sequences in B, but other ways as described
next could also be used.
Soft Decision: The output processing does not
need to involve hard decisions. In fact, the most
widely used MMSE-based estimators generate
the overall estimate by a weighted sum of MMSE
(i)
from all elemental filters with
estimates x̂kjk
mode-sequence probabilities ¹k(i) as the weights:
P (i) k
MMSE
MMSE
= i x̂kjk
¹(i) . Likewise for m̂kjk
. This can be
x̂kjk
thought of as a soft decision for convenience.4
This soft decision is often applied in practice to
fuse non-MMSE estimates as well. This is particularly
popular for a hybrid system with non-Gaussian linear
subsystems to which the Kalman filters are applicable.
(i)
are linear MMSE (LMMSE)
In this case, x̂kjk
estimators (i.e., have minimum MSE of all linear
estimators),
not MMSE estimators. As a result,
P (i) but
x̂kjk = i x̂kjk
¹k(i) is neither MMSE nor LMMSE. It is
a nonlinear estimator hopefully close to the MMSE
estimator. Although this hope lacks strong theoretical
support, this x̂kjk can be expected to beat the overall
4 However, this is not to be confused with the soft decision in
communication and decoding, which refers to a decision that is
revocable.
LI & JILKOV: SURVEY OF MANEUVERING TARGET TRACKING. PART V: MULTIPLE-MODEL METHODS
1265
LMMSE
LMMSE estimator x̂kjk
(see the end of Section VA
for evidence).
Like any discrete-valued quantity, for a mode
k
is actually an alias
sequence, MAP estimation m̂MAP
of MAP decision (test). The special MAP estimator
MAP
k
x̂kjk
(m̂MAP
) is hard decision based, equal to the
(i)MAP
corresponding to
component MAP estimator x̂kjk
k
. Likewise for
the most probable mode sequence m̂MAP
MAP
k
JMAP
x̂kjk (m̂ ). The JMAP estimator x̂kjk is also equal
(j)MAP
to one of the component MAP estimators x̂kjk
by
MAP
a hard decision. However, the MAP estimator x̂kjk
relies on a soft decision that requires in general all
the component densities as well as the mode-sequence
probabilities [99].
Still another class of soft decision procedures
is nonprobabilistic, such as those based on
Dempster-Shafer evidence theory, fuzzy logic, neural
networks, as discussed in Section VIII.
Other approaches to output processing are
possible, such as combinations of the above
approaches.
D. Convergence of AMM Estimates
Magill’s original work [235] includes sufficient
conditions on the convergence of the mode
probabilities for a single-output linear system. This
was extended to multiple outputs by others [315,
183, 179, 180]. It was shown in [130] and [25] that
the correct (true) model has a probability that tends
to unity (almost surely) as time increases under
Assumptions A1 and A2 and that [25] the correct
model and any of the incorrect models 1) cannot
be perfectly distinguished in finite time by their
k
likelihood functions f(z k j m(i)
) and 2) do not have
k
identical marginal likelihood functions f(zk j z k¡1 , m(i)
)
ML
MAP
MMSE
are
as k increases. As such, m̂kjk , m̂kjk , and m̂kjk
all consistent estimators in that they converge with
probability one to the true model (m̂MMSE is also
mean-square consistent) under the assumptions stated
ML
above [25]. Here the ML estimator m̂kjk
is the model
with the largest model likelihood. For a set M of
linear time-invariant systems with white, uncorrelated,
stationary Gaussian noise under Assumptions A1
and A2, the conditions 1 and 2 defined above can
be replaced by the easily verifiable condition [25]
(i)
s
½(i)
60, 8i 6
= s, m(i) 2 M with5
s = j½ ¡ ½ j =
½(i) = log det(S (i) ) + tr[(S (i) )¡1 S̃ (i) ]
½s = ½(i) jm(i) =s = log det(S̃) + dim(zk )
(18)
or by the even more easily verifiable condition
[22]: det(S̃ ¡ S (i) ) 6
= 0, 8i 6
= s, m(i) 2 M, where S̃ and
5 Note
that ½(i)
s is always dimensionless regardless of what are used
(i)
for S , S̃ (i) , and S̃.
1266
S (i) are the steady-state measurement prediction
covariances (MSE matrices) of the correct model s
and an incorrect model m(i) , respectively, as calculated
in the corresponding Kalman filters, and S̃ (i) is the
true steady-state measurement prediction MSE matrix
of the incorrect model m(i) . For AMM estimation of
the base state, we then have
s,MMSE
MMSE
x̂kjk
! x̂kjk
s,MAP
MAP JMAP MAP
MAP
, x̂kjk , x̂kjk (m̂kjk
) ! x̂kjk
x̂kjk
s,ML
ML ML
ML
ML
ML
, x̂kjk (m̂kjk
), x̂kjk
(m̂kjk
) ! x̂kjk
x̂kjk
as k ! 1 with probability one under the stated
s,MMSE
s,MAP
s,ML
assumptions. Here x̂kjk
, x̂kjk
, and x̂kjk
are
MMSE, MAP, and ML estimators based on the true
ML ML
ML
ML
ML
model, and x̂kjk
, x̂kjk (m̂kjk
), x̂kjk
(m̂kjk
) were defined in
Section IIID.
The above results hold when the true model s
is in M (Assumption A2). For any model pair m(i)
and m(j) in M, as shown in [24], the likelihood ratio
k
k
f(z k jm(i)
)=f(z k jm(j)
) goes to zero with probability one
(j)
(i)
if ½ < ½ and only if ½(j) · ½(i) under Assumption
A1 and that the corresponding measurement residual
sequences of the linear-Gaussian system are ergodic
and have a finite and positive-definite steady-state
mean-square matrix. It follows that regardless whether
the true model s is in M or not, the probability of the
model m(i) in M with the smallest “distance” ½(i)
s to
the true model s tends to unity almost surely as time
increases under the assumptions stated above (but
without A2) if this smallest “distance” is unique [24].
Consequently, all the above convergence results for
m̂kjk and x̂kjk of the linear-Gaussian systems hold
true if the true model s is replaced by the (assumed
unique) model closest to it.6 Closely related, but
slightly less convenient results were obtained in [130]
6 The “distance” ½(i) was given the following Kullback-type
s
information theoretic interpretation in [24]. Let
k
k
¸k (i, j) = log[f(z k jm(i)
)=f(z k jm(j)
)]
k
k
¡ log[f(z k¡1 jm(i)
)=f(z k¡1 jm(j)
)],
d(i, j) = lim jE[¸k (i, j) j sk ]j,
k!1
d (i) = d(s, i)
where s is the true model and sk is the constant true model
(j)
sequence through time k. Then d (i) ¸ d (j) if and only if ½(i)
s ¸ ½s .
As shown in [24], d(i, j) is a pseudo distance (i.e., satisfies the
triangle inequality and is symmetric and nonnegative definite,
but not positive definite, hence the prefix pseudo and the need
for the extra uniqueness assumption), although it is closely
related to the Kullback-Leibler information measure I(s, j) =
k )] j sk ), which does not satisfy the triangle
E(log[f(zk j sk )=f(zk j m(j)
inequality and is not symmetric. Despite this connection of ½(i)
s
to information “distance,” no general results are available for
nonlinear, non-Gaussian systems based on d (i) directly.
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OCTOBER 2005
based on the Kullback-Leibler information. Note that
in this case MMSE- and MAP-AMM estimators all
(implicitly and incorrectly) assume that the true model
is in M, but ML-AMM estimators need not and would
not benefit from this assumption.
As argued in [234], when the true model s is
not in M, the convergence of the probability of the
closest-to-truth model to unity is not necessarily
desirable because many models in M may capture
distinct characteristics of the true one and thus is
better reflected in the limit. An AMM algorithm with
mode “probabilities” converging to constants other
than zero and one was proposed in [234]. It modifies
Bayes’ rule for mode probability update by replacing
the predicted mode probabilities with constant (e.g.,
uniform) prior probabilities.
E. Tracking Applications
The AMM (first generation) algorithms have
numerous applications outside of the target tracking
area. They are particularly popular for problems
involving unknown parameters (see, e.g., [347]).
They form an important approach for dealing with
systems subject to faults (see, e.g., [26]). However,
several more recent publications [106, 253, 371, 297]
demonstrated that the second- and third-generation
MM algorithms still outperform the AMM algorithms
for such applications, wherein the system mode
does jump, in violation of the basic assumptions of
the AMM estimation. This is easily understandable
for people engaged in target tracking research and
development, in view of the duality between fault
detection and maneuvering target tracking [164].
For obvious reasons, here we only survey limited
AMM applications in maneuvering target
tracking.
Although not a straightforward implementation of
the AMM estimator, the maneuvering target tracking
algorithm proposed in [332] has some features of the
AMM algorithms. It combines a nonmaneuver filter
and a maneuver estimator chosen by a hard decision
logic. The maneuver estimator itself is a two-model
AMM procedure, which generates the overall estimate
by a probabilistically weighted sum.
In [245], a two-model AMM algorithm was
designed, which includes a (nearly) constantacceleration (CA) model plus either a Singer model
or a 3D (nearly) constant-turn (CT) model (see [209]),
for line-of-sight angular tracking of a close-range,
highly maneuverable, airborne target using forward
looking infrared sensor measurements. Further
enhancements of this image tracker were proposed
and analyzed in [334]. Some of the elemental filters
were allowed to have a rectangular field of view;
the algorithm was tuned to harsher target dynamics
by considering both Gauss-Markov acceleration
and constant turn-rate models; and an initial target
acquisition algorithm was devised to remove
significant biases in the estimated target template to
be used in a correlator within the tracker. Further
along these lines, a 3-model AMM configuration
based on a second-order Markov acceleration model
[209] together with the CA and Singer models was
investigated in [357]. For a different application, in
[178] a 3-model AMM algorithm with a first-order
Gauss-Markov acceleration, first-order Gauss-Markov
velocity, and a nearly constant position model of
pilot’s head motion was applied as a predictor for
a virtual environment flight simulator. Also, an
AMM algorithm preceding the above applications
can be found in [252]. These publications, via their
demonstrations of the superiority of the MM approach
to the single-model-based trackers (e.g., extended
Kalman filter (EKF)), have also helped establish
that using well-selected, possibly more sophisticated
models for certain tracking applications can reduce
the number of elemental filters significantly since they
provide a better coverage of possible motion modes
than simple models.
Results of a comprehensive study were presented
recently in [276] on the capabilities of an AMM
algorithm for tracking and interception of a highly
maneuverable fighter aircraft armed with electronic
countermeasures (ECM) by an air-to-air missile
equipped with a monopulse radar seeker. The scenario
involves a sequence of periodical evasive maneuvers
of the aircraft and electronic jinking7 generated
by the aircraft ECM system. The system mode
space in terms of the maneuver-jinking pairs of
evasion strategies is unknown to the tracker (the
missile homing system), and was approximated via
“quantization” by a set of 45 strategies, serving
as the “ground truth” in the simulations. Due to
feasibility considerations, however, the design of
the MM tracker included only a small fixed set
of six most representative models, selected by a
“trial-and-error” process to cover the mode space
reasonably. Each of the six elemental EKFs with
an 11-dimensional state was carefully tuned to a
particular evasion strategy (motion-jinking model).
While demonstrating a significant improvement over
previous, non-MM filters (e.g., single EKF), the
simulation results presented therein again exposed
some typical deficiencies of the AMM algorithms.
These include: failed or delayed identification–the
filter may lock on an erroneous model and fail to
switch to the true one; poor estimation, mainly due
to the mismatch between the dominant model and the
true mode. Overall, besides the important practical
results as well as demonstrating the superiority
of the MM approach to single filters, this study
7 It
is a periodic switching of the aircraft’s apparent radar reflection
center from one wing tip to the other.
LI & JILKOV: SURVEY OF MANEUVERING TARGET TRACKING. PART V: MULTIPLE-MODEL METHODS
1267
Fig. 3. Possible model sequences of CMM and AMM algorithms. (a) CMM. (b) AMM.
illustrates that AMM estimation is not well suited to
situations with frequent modal changes, for which
later generations are more suitable. Also, covering
a large mode space by a small fixed set of models
inevitably causes a large model-truth mismatch that
normally leads to inadequate estimation performance.
Instead, the third-generation, VSMM algorithms
appear much more appealing for this challenging,
practical problem, characterized by a large mode
space and high motion dynamics. Another study of
AMM implementation for a similar air-to-air missile
guidance problem was reported in [269].
The AMM approach was employed for another
important class of problems–ballistic target tracking
[104, 103, 312]. Tracking of a ballistic target was
considered in [104] and [103] for all flight phases:
boost (including postboost), coast (free flight), and
reentry (possibly maneuverable) [205]. The tracker
designed consists of seven autonomous filters running
in parallel, each corresponding to one of the three
specific flight phases. The output of the algorithm
is selected from those of the elemental filters by a
hard decision logic, e.g., from the one based on the
most likely model: x̂(m̂ML ). Using an autonomous
system of filters appears more reasonable here than
in such applications as aircraft tracking since the
modal changes here are infrequent. The use of a
hard decision for output was motivated by two
considerations. First, models of the different flight
phases have state vectors of different dimensions
and fusion of estimates of these vectors by soft
decision for output is nontrivial and lacks theoretical
support. Second, such a fused estimate is statistically
inferior to the single best estimate for interception
and destruction applications where the hit rate is
more important than average errors [220]. These
arguments for hard decision, albeit sensible, are
debatable, though. For example, it can be argued
that a hard decision is in general more prone to false
alarms than a soft decision, possibly resulting in much
poorer estimates. For such reasons soft decision based
schemes (e.g., IMM) have been adopted more often
for similar problems [160, 257, 39, 141, 142, 80].
In [312], the focus is on tracking a tactical ballistic
missile in its incoming phase capable of random
(bang-bang) evasive maneuvers for the purposes of
1268
interception and destruction. From the estimation
point of view, the main uncertainty facing the tracker
in such a scenario, embedded in a differential game
framework, reduces to the unknown maneuver
onset time. Thus the models involved differ only
in the maneuver onset time.8 The choice of the
AMM configuration appears appropriate within this
formulation as far as estimation of the target state
after the maneuver is concerned since a mode here
cannot jump to another.
V.
THE SECOND GENERATION: COOPERATING MM
ESTIMATION
A. Optimal Cooperating MM Estimation
Fundamental Assumptions: In the second
generation, cooperating MM (CMM) algorithms, the
fundamental Assumption A2 of the first generation
is retained without a change, but A1 is relaxed to
allow a time-varying mode sequence fsk g. Similar
to the first generation, this sequence in principle
can be random or deterministic, but for tractability,
it is almost always more conveniently assumed to
be a random process, in particular a Markov (or
semi-Markov) process, as stated formally below.
A10 . The true mode sequence fsk g is Markov (or
semi-Markov).
If fsk g is indeed random, the above Markovian
assumption is justifiable for many applications. A
hybrid system with A10 is known as a Markov jump
system. Like the first generation, under A2 there is
no mode-model mismatch and thus we again use
m to denote both the mode and the model and treat
a model as a realization of a mode in this section.
Under A2, there are M k possible model sequences
(or realizations) through time k, which increases
exponentially with time, where M = jMj is the
number of possible models at each time. This full
tree (more precisely, trellis) is illustrated in Fig. 3(a)
8 This idea of using MMs for the maneuver onset time was not
new–the AMM of [235] was mentioned in [64] as an option for
the input estimation method; however, it was abandoned and a
hard decision (generalized likelihood ratio test approach) was used
instead.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
for a three-model case. Under A1, it reduces to the
corresponding “tree” for the first generation, as shown
in Fig. 3(b). We denote a generic event of a model
sequence through time k as
k
m(ik k ) = m(ik 1 ,:::,ik ) = fm1(i1 ) , : : : , mk(ik ) g = fsk = m(i ) g
(19)
and the set of all such sequences as Mk , where
k
sk = (s1 , : : : , sk ),
m(i ) = (m(i1 ) , : : : , m(ik ) ),
m(in ) 2 M:
i2M
(i)
(i)
+ (x̂kjk
¡ x̂kjk )(x̂kjk
¡ x̂kjk )0 ]¹(i)
k :
(i)
E[xk j z k , m(ik k ) ] and x̂kjk
= E[xk j z k , mk(i) ], respectively.
The corresponding mode estimator at time k is
given by
m̂kjk = E[mk j z k , A10 , A2]
For simplicity we use ik 2 M k to mean m(ik k ) 2 Mk (i.e.,
use M to denote the index set of M) and refer to m(ik k )
as a sequence, rather than an event.
The discussion in the rest of this subsection
is largely parallel to that of the optimal AMM
estimation in Section IVA. It is nonetheless presented
here because of its importance for understanding
suboptimal CMM algorithms presented later.
MMSE-CMM: The MMSE-CMM base-state
estimator at time k is given by (see, e.g., [21])
=
=
E[xk j z k , mk(i) ]Pfmk(i) j z k , A10 , A2g
=
X
(i) (i)
¹k
x̂kjk
i2M
X
E[mk j z k , mk(i) ]Pfmk(i) j z k , A10 , A2g
=
X
m(i) ¹(i)
k
i2M
i2M
=
X
ik 2M k
ik 2M k
X
m(ik ) ¹k(ik )
=
(21)
=
X
ik 2M k
=
(i ) k
¹(ik )
x̂kjk
E[mk j z k , m(ik k ) ]Pfm(ik k ) j z k , A10 , A2g
ik 2M k
ik 2M k
=
X
MSE(m̂kjk j z k ) = E[(mk ¡ m̂kjk )(mk ¡ m̂kjk )0 j z k , A10 , A2]
x̂kjk = E[xk j z k , A10 , A2]
X
=
E[xk j z k , m(ik k ) ]Pfm(ik k ) j z k , A10 , A2g
k
(22)
(m(ik ) ¡ m̂kjk )(m(ik ) ¡ m̂kjk )0 ¹k(ik )
X
i2M
(m(i) ¡ m̂kjk )(m(i) ¡ m̂kjk )0 ¹(i)
k
where m(ik ) and m(ik k ) are relative by (19)—(20).
Although not seen in the literature, the estimators
for the base-state sequence and mode sequence using
a batch of data through k can be obtained as
i2M
¹k(ik )
Pfm(ik k )
0
k
where
=
j z , A1 , A2g is the posterior
mode-sequence probability assuming that the mode
sequence in effect is one and only one but possibly
(ik )
anyone in the set Mk , x̂kjk
= E[xk j z k , m(ik k ) ] is the
conditional MMSE estimate assuming sequence m(ik k )
¹(i)
k
Pfmk(i)
k
0
is true,
=
j z , A1 , A2g is the posterior
mode probability under A10 and A2 that m(i) is in
(i)
effect at time k, x̂kjk
= E[xk j z k , mk(i) ] is the conditional
MMSE estimate assuming model m(i) is in effect at
(ik )
time k. It in effect lumps a lot of x̂kjk
. Like the first
0
MMSE
is unbiased and
generation, under A1 and A2 x̂kjk
with a conditional MSE matrix that is minimum of all
base-state estimators x̂kjk , given by (see, e.g., [21])
Pkjk = MSE(x̂kjk j z k , A10 , A2)
X
(ik )
[MSE(x̂kjk
j z k , A10 , A2)
=
ik 2M k
k
k
(i )
(i )
¡ x̂kjk )(x̂kjk
¡ x̂kjk )0 ]¹k(ik )
+ (x̂kjk
(24)
Equations (23) and (24) hold true for any (optimal
(ik )
or nonoptimal) estimator x̂kjk provided x̂kjk
=
(20)
X
X
(i)
[MSE(x̂kjk
j z k , A10 , A2)
=
(23)
m̂kjk = E[mk j z k , A10 , A2] =
=
X
k
x̂kjk = E[xk j z k , A10 , A2] =
=
ik 2M k
E[mk j z k , m(ik k ) ]¹k(ik )
m(i ) ¹k(ik ) = (m̂1jk , m̂2jk , : : : , m̂kjk )
ik 2M k
X
X
X
ik 2M k
E[xk j z k , m(ik k ) ]¹k(ik )
kjk
x̂(ik ) ¹k(ik ) = (x̂1jk , x̂2jk , : : : , x̂kjk )
ik 2M k
k
where m(i ) , given by (20), stands for a possibly
time-varying model sequence through k with the index
kjk
sequence ik = (i1 , : : : , ik ), x̂(ik ) = E[xk j z k , m(ik k ) ] is the
corresponding estimate of the base-state sequence, and
MMSE
MMSE
and x̂njk
are smoothed MMSE estimates,
m̂njk
given by
m̂njk = E[mn j z k , A10 , A2]
X
X
=
E[mn j z k , m(ik k ) ]¹k(ik ) =
m(in ) ¹k(ik )
ik 2M k
LI & JILKOV: SURVEY OF MANEUVERING TARGET TRACKING. PART V: MULTIPLE-MODEL METHODS
ik 2M k
1269
x̂njk = E[xn j z k , A10 , A2]
X (ik )
X
E[xn j z k , m(ik k ) ]¹k(ik ) =
x̂njk ¹k(ik ) :
=
ik 2M k
Unlike the MMSE estimation, the component m̂njk of
kjk
ik 2M k
MAP-CMM: The mixed pdf-pmf of the base
state at time k and the mode sequence through time
k is
p(xk , mk j z k , A10 , A2) = f(xk j z k , mk )p(mk j z k , A10 , A2)
= ff(ik ) (xk j z k )¹k(ik ) , ik 2 M k g
= f(xk j z k , mk(i) )p(mk j z k , A10 , A2)
6
= ff(i) (xk j z k )¹(i)
k , i 2 Mg
where f(ik ) (xk j z k ) = f(xk j z k , m(ik k ) ) is the density
assuming the mode sequence in effect is m(ik k ) . It
follows that the base state has the posterior mixture
density
f(xk j z k , A10 , A2)
X
=
f(xk j z k , m(ik k ) )Pfm(ik k ) j z k , A10 , A2g
ik 2M k
X
=
ik 2M k
f(ik ) (xk j z k )¹k(ik )
X
=
6
f(xk j z k , mk(i) )Pfmk(i) j z k , A10 , A2g
i2M
=
X
i2M
f(i) (xk j z k )¹(i)
k :
(25)
Note the difference: summation over the model
sequences is the same as over the current models for
MMSE estimation (see (21)—(22)), but not the same
for MAP estimation (see above). Thus the MAP-CMM
estimators are given by
X
MAP
x̂kjk
= arg max
f(ik ) (xk j z k )¹k(ik )
xk
ik 2M k
kjk
JMAP
(x̂kjk
, m̂JMAP ) = arg max ff(ik ) (xk j z k )¹k(ik ) , ik 2 M k g
(xk ,m(ik ) )
kjk
m̂MAP = arg maxf¹k(ik ) , ik 2 M k g
m(ik )
= (m̂1jk , m̂2jk , : : : , m̂kjk )MAP
kjk
x̂MAP = arg max
xk
X
ik 2M k
(26)
f(ik ) (xk j z k )¹k(ik )
= (x̂1jk , x̂2jk , : : : , x̂kjk )MAP
MAP
(m̂k ) = arg max f(ik ) (xk j z k )
x̂kjk
xk
kjk
MAP
MAP
(m̂MAP ) = x̂kjk
(m̂k )jm̂k =m̂kjk :
x̂kjk
MAP
1270
if m̂k = m(i
k
)
MAP
a MAP sequence estimate m̂MAP is not equal to m̂njk
,
(n)
the MAP estimate of the component m of the mode
sequence mk . Likewise for the base state.
Compared with Section IVA, these optimal CMM
estimators clearly correspond to the respective optimal
AMM estimators by replacing the constant mode
k
therein with a possibly time-varying
sequence m(i)
mode sequence m(ik k ) . As a result, all discussions
of the AMM estimators regarding interpretations,
dependence on component MAP estimators,
and MSE matrix apply to the CMM estimators
accordingly.
MMSE versus MAP: As explained in Section IIID,
MMSE
minimizes the MSE,
the MMSE estimator x̂kjk
MAP
while the MAP estimator x̂kjk
maximizes the rate of
hitting a tiny golf hole centered at xk in the base-state
space. They are suitable for different applications, as
kjk
JMAP
, m̂JMAP ) is a joint estimator
elaborated in [220]. (x̂kjk
that maximizes the rate of hitting the “golf hole” and
simultaneously choosing the correct model sequence.
MMSE
is the mean of
As explained in Section IIID, x̂kjk
MAP
the posterior mixture density; x̂kjk is the location
JMAP
of the highest peak of the mixture density; x̂kjk
MAP
k
and x̂kjk (m̂ ) are locations of the highest peaks
of the corresponding component densities. When
the mixture density consists of many components,
as is the case for CMM estimation with a random
mode sequence having an exponentially increasing
JMAP
MAP
or x̂kjk
(m̂k )
number of realizations, the use of x̂kjk
k
for any m̂ is on shaky ground: they are based on a
component density which might have a very small
(say, 1%), albeit largest of all components, probability
MMSE
MAP
and x̂kjk
of being the true density.9 The use of x̂kjk
is on much firmer ground. If, however, the mode
sequence is not random, the mixture density and
MMSE
MAP
and x̂kjk
are meaningless. In this case,
hence x̂kjk
kjk
JML
ML
the ML estimators x̂kjk
and x̂kjk
(m̂ML ) appear to be
reasonable choices.
The above discussion is based on Assumption
A2 that the true mode is exactly one of the models
in the set M used. While this assumption is exact or
very reasonable for many communication or decoding
problems, it is almost never true for maneuvering
target tracking where the motion-mode uncertainty
is almost never resolved exactly by the models used.
For instance, a target almost never takes an exact
constant turn and even if it should, the turn rate
would not be exactly equal to one of those used in
JMAP
MAP
JML
ML
, x̂kjk
(m̂k ), x̂kjk
, and x̂kjk
(m̂k ) rely
the models. x̂kjk
heavily on impossibility of such mismatch between
9 The
kjk
MAP estimator m̂MAP of the mode sequence is still one of the
most reasonable choices.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
modes and models. While assuming no such mismatch
MMSE
MAP
explicitly, x̂kjk
and x̂kjk
are more robust against
(less sensitive to) this mismatch because they rely on
all component densities, rather than put all their eggs
in one basket as the other MAP and ML estimators
do. This in effect “covers” modes between any two
models used (i.e., the convex set formed by the
models used).
As explained in Sections IIID and IVB, given
the probabilistic weights, the entire density function
f(ik ) (xk j z k ) of every component is needed to
MAP
MMSE
, while x̂kjk
relies only on its
compute x̂kjk
k
k
(i )
(i )
, Pkjk
). Note, however, that
first two moments (x̂kjk
MMSE
is
calculation of the moments needed for x̂kjk
an integration problem, as opposed to solving the
MAP
maximization problem needed for x̂kjk
given every
component density, which can usually be reduced
to differentiation and equation solving. Note also
that finding the global maximizer of a mixture
density numerically is much simpler than that of a
general multivariate function. For instance, the global
maximizer of a mixture density is bound to be in the
convex set formed by the outmost peak locations of
all component densities.
While the output processing of an MMSE-based
MM algorithm is usually soft decision based in that
its overall estimate is a weighted sum of results
from each elemental filter, a hard decision based
output processing could also be used. Reference
[98] proposed for certain applications to use the
estimate from a single elemental filter that has the
smallest MSE among all elemental filters. The best
(i)
elemental filter x̂kjk
can be identified as the one with
(i)
MMSE 0 (i)
MMSE
the smallest deviation (x̂kjk
¡ x̂kjk
) (x̂kjk ¡ x̂kjk
)
from the optimal estimate because it can be easily
shown [190] that
(i) 2
(i)
MMSE 2
MMSE 0 (i)
MMSE
kxk ¡ x̂kjk
k = kxk ¡ x̂kjk
k + (x̂kjk
¡ x̂kjk
) (x̂kjk ¡ x̂kjk
)
(i) 2
(i) 0
(i)
where kxk ¡ x̂kjk
k = E[(xk ¡ x̂kjk
) (xk ¡ x̂kjk
) j z k ] is the
conditional MSE. This in general requires knowledge
MMSE
of the MMSE-optimal estimate x̂kjk
.
Linear MMSE: An optimal linear MMSE
estimator was proposed in [81] for an MJLS, defined
similarly by (1)—(2) except that w and v are replaced
by mode-independent w and Dv, with D a matrix.
The cornerstone of this LMMSE estimator is the
introduction of the (M ¢ n)-dimensional stacked vector
yk = [(yk(1) )0 , : : : , (yk(M) )0 ]0 to represent the n-dimensional
base-state subject to the assumed model uncertainty
among M known models, where every yk(i) is an
n-dimensional zero vector except yk(j) = xk if model
m(j) is true. The problem of estimating the hybrid
state (xk , sk ) is thus reduced to the conventional
problem of estimating yk . As a result, the recursive
(1) 0
(M) 0 0
optimal LMMSE estimator ŷkjk = [(ŷkjk
) , : : : , (ŷkjk
)]
of yk is available. The base-state estimator is simply
P (i)
and MSE(x̂kjk ) is equal to sum over all
x̂kjk = i ŷkjk
n £ n blocks of MSE(ŷkjk ). A more informative but
concise description of this LMMSE estimator was
given in [192]. Simulation results given in [81] show
that this optimal linear MMSE estimator performs in
general not as well as the suboptimal nonlinear IMM
estimator, and there was no single case considered in
which the LMMSE estimator outperforms the IMM
estimator significantly. This demonstrates the high
nonlinearity of hybrid estimation problems and the
need for good nonlinear estimators. More recently,
[83] showed that for a mean-square stable MJLS
with an ergodic Markov chain, MSE(ŷkjk ) of this
LMMSE estimator converges to a unique positive
semidefinite solution of an algebraic Riccati equation.
The corresponding matrix in the LMMSE estimator
was replaced by this steady-state solution to arrive
at a steady-state estimator, similar in spirit to the
development of the steady-state Kalman filter. This
steady-state estimator was generalized in [82] to
the case where the system given a mode sequence
involves some uncertain parameters.
Most Probable Trajectory Estimation:
Conceptually similar to ML sequence estimation,
the most probable trajectory (MPT) estimation is a
nonlinear filtering approach that determines a state
sequence fitting best to the data in some sense (see,
e.g., [268]). References [370] and [227] considered
state estimation of a continuous-time hybrid system
with a fixed number M of possible mode process
s(t), where s(t) has N known, possible distributions.
A finite-dimensional hybrid filter in recursive form
was presented in [370] that is optimal in the MPT
sense, which includes optimal base-state sequence
estimation and identification of the most probable
distribution of s(t). The MPT optimality here is with
respect to a cost function defined for a system that
is an average of the hybrid system over possible
modes. This approach is quite robust with regard
to noise characterization: it was applied in [227]
to state estimation of a jump-linear system as a
piecewise-linear approximation of a highly nonlinear
system, including a satellite orbit determination
example.
B. Cooperation Strategies for MM Estimation
Since the number of possible model sequences
(hypotheses) increases exponentially with time
(more precisely, geometrically with discrete time),
brute-force implementations of the above optimal
MMSE
MAP
and x̂kjk
are infeasible.
CMM estimators x̂kjk
Consequently, strategies have been developed to cope
with this difficulty. We refer to them as cooperation
strategies. CMM algorithms are distinct from one
LI & JILKOV: SURVEY OF MANEUVERING TARGET TRACKING. PART V: MULTIPLE-MODEL METHODS
1271
another in the cooperation strategies used. Cooperation
strategies developed so far can be classified into two
general categories: hypothesis reduction strategies
and iterative strategies. Different hypothesis reduction
strategies have been proposed to keep the number of
hypotheses within a certain limit:
1) merge “similar” model sequences, resulting in a
tree with combined branches, which can be achieved
by soft decisions and in effect reinitialization of the
filters with an (approximately) “equivalent” estimate
and covariance;
2) prune “unlikely” model sequences, or select
the best model sequences, including the globally best
single sequence or the best sequence for each current
model, both resulting in a truncated tree, which is
necessarily hard-decision based;
3) randomly select a subset of the possible
hypotheses;
4) others, such as decoupling weakly coupled
mode sequences to form clusters, as briefly reviewed
in [280].
The basic idea of merging and pruning is to
replace the ever growing mixture tree with a simpler
tree that approximates the original tree in some sense.
In general, since the base state under A10 and A2
has a mixture density of an ever growing number of
components, the problem of hypothesis reduction is in
essence that of mixture density reduction, which has
been studied in other areas within target tracking (see,
e.g., [299], [381]) as well as in statistics extensively
(see, e.g., [333], [248]), and thus various mixture
density reduction techniques can be applied here.
Although closely related, hypothesis reduction is not
to be confused with the output processing, discussed
in Section IVC.
Random selection decides on a subset C of
all possible model sequences at random (not
necessarily with an equal probability), performs
the corresponding conditional filtering operations,
and then generates the overall estimate from the
conditional estimates. This approach can be argued
as a special, albeit extreme, selection strategy,
especially when the selection is not equally likely.
It has a straightforward batch implementation and
several possible recursive implementations. For
example, one natural recursive implementation is
to select a set of model histories m¤k¡1 at random
for each elemental filter at time k, where m¤k¡1 =
(i)
(m¤k¡2 , mk¡1
) is formed from some model history
m¤k¡2 selected before and some model m(i) at time
k ¡ 1. Another one is to select a set of model
sequences m¤k = (m¤k¡1 , mk(i) ) at random at time k.
Note that the first implementation runs elemental
filters based on every model at k, but this is not
generally the case in the second implementation.
An early publication of the random selection
1272
approach is [2]. More recently developed MM particle
filtering algorithms also belong to this class. These
results are surveyed in a subsequent part of this
survey.
Another class is iterative strategies. Disengaged
from the tree (mixture) structure of the distribution,
they try to solve the estimation problem with recourse
to the power of iteration. More developed in this class
are those based on the so-called EM algorithm, an
elegant and powerful optimization method particularly
suitable for ML and MAP estimation.
All results in Section VB rely on A10 and A2, but
we drop explicit indications of this dependence for
simplicity.
1) GPB and IMM Merging Strategies: In
these strategies, the ever growing hypothesis tree
is approximated repeatedly by a simpler tree, each
branch of which lumps “close” or “similar” branches
of the original tree. In the following discussion, for
simplicity we omit formulas for MSE matrices (error
covariances).
a) GPB: A straightforward and probably the most
natural implementation of this idea is the so-called
generalized pseudo-Bayesian algorithms of order n
(GPBn) [148, 321, 18]. They reduce the hypothesis
tree by having a fixed memory depth such that all
the hypotheses that are the same in the latest n time
steps are merged and thus each of the M filters runs
M n¡1 times at each recursion. The GPB1 and GPB2
algorithms are the most popular ones in this class
[1, 148, 74, 321, 18]. Although for simplicity our
discussion below is based on the GPB2 algorithm
explicitly, it can be extended to the general GPBn
case straightforwardly. Here the effects of different
(ik )
estimates x̂kjk
= E[xk j z k , m(ik k ) ] with probabilities
¹k(ik ) = Pfm(ik k ) j z k g based on the same model pair
m(i) and m(j) at k ¡ 1 and k, respectively, are lumped
(i,j)
with probability
(merged) by the single estimate x̂kjk
¹(i,j)
k¡1,k :
(i,j)
(i)
x̂kjk
= E[xk j z k , mk¡1
, mk(j) ]
(j)
(i)
k
¹(i,j)
k¡1,k = Pfmk¡1 , mk j z g:
(j)
(i)
k¡2
Since m(ik k ) = (m(ik¡2
k¡2 ) , mk¡1 , mk ), where m(ik¡2 ) =
(i
)
k¡2
) is mode history through time
(m1(i1 ) , m2(i2 ) , : : : , mk¡2
k ¡ 2, it follows from the total probability
(i,j)
and ¹(i,j)
(expectation) theorem that x̂kjk
k¡1,k are actually
k¡2
(i
averages of x̂kjk
,i,j)
(j)
(i)
= E[xk j z k , m(ik¡2
k¡2 ) , mk¡1 , mk ] and
(j)
(i)
¹k(ik¡2 ,i,j) = Pfm(ik¡2
j z k g over m(ik¡2
k¡2 ) , mk¡1 , mk
k¡2 ) :
(i,j)
x̂kjk
=
X
(i
x̂kjk
X
¹k(ik¡2 ,i,j) :
ik¡2 2M k¡2
¹(i,j)
k¡1,k =
k¡2
,i,j)
(j)
(i)
k
Pfm(ik¡2
k¡2 ) j z , mk¡1 , mk g
ik¡2 2M k¡2
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
This lumping or merging is exact for MMSE output
processing in that10
MMSE
=
x̂kjk
X
k
k
(i ) (i )
¹k =
x̂kjk
ik 2M k
X
(i,j) (i,j)
¹k¡1,k
x̂kjk
(27)
(i,j)2M 2
which stems from the linearity of the conditional
expectation. This lumping is, however, approximate
for conditional filtering, which is a nonlinear
(j)
operation in that the output x̂kjk
is nonlinear in
(j)
.
x̂k¡1jk¡1
This is the case even for a
the input
linear-Gaussian system with a known mode sequence,
(j)
(j)
where only one term of x̂kjk
is linear in x̂k¡1jk¡1
and
the other term depends on the measurement at k.
For recursive conditional filtering at k, this lumping
approximates the recursion
This GPB2 approximation can also be applied to
CMM estimation based on non-MMSE criteria. For
example,
f(ik¡3 ,i,j) (xk¡1 j z k¡1 )
(j)
(i)
= f(xk¡1 j z k¡1 , m(ik¡3
k¡3 ) , mk¡2 , mk¡1 )
(j)
(i)
, mk¡1
):
¼ f(i,j) (xk¡1 j z k¡1 ) = f(xk¡1 j z k¡1 , mk¡2
Clearly, this approximation implies (28) but is not
implied by (28) and thus is actually more fundamental
than (28). Similarly as above, f(i,j) (xk¡1 j z k¡1 ) is
actually the probabilistically weighted average of
f(ik¡3 ,i,j) (xk¡1 j z k¡1 ) over m(ik¡3
k¡3 ) :
f(i,j) (xk¡1 j z k¡1 ) =
k¡1
(i )
k¡1
fx̂k¡1jk¡1
, ¹(ik¡1
2 M k¡1 g
k¡1 ) : i
k
(i ) k
=) fx̂kjk
, ¹(ik ) : ik 2 M k g
which consists of M k conditional filtering operations,
by the simplified recursion
X
ik¡3 2M k¡3
[f(ik¡3 ,i,j) (xk¡1 j z k¡1 )
(j)
(i)
k¡1
, mk¡2
, mk¡1
g]:
¢ Pfm(ik¡3
k¡3 ) j z
The (MMSE-based) GPB1 algorithm [1] is based
on approximately lumping (merging) the effects of all
(ik¡2 ,i)
past model histories on x̂k¡1jk¡1
with ¹k¡1
to yield,
(ik¡2 ,i)
for i 2 M:
k¡2
(i,j)
, ¹(i,j)
fx̂k¡1jk¡1
k¡2,k¡1
(i ,i)
(i)
(i)
¼ x̂k¡1jk¡1
= E[xk¡1 j z k¡1 , mk¡1
]
x̂k¡1jk¡1
X
(i)
k¡1
g=
¹k¡1
¹(i)
k¡1 = Pfmk¡1 j z
(ik¡2 ,i)
2
: (i, j) 2 M g
(i,j) (i,j)
=) fx̂kjk
, ¹k¡1,k : (i, j) 2 M 2 g
k¡3
(i ,i,j)
(j)
(i)
= E[xk¡1 j z k¡1 , m(ik¡3
x̂k¡1jk¡1
k¡3 ) , mk¡2 , mk¡1 ]
(i,j)
(j)
(i)
¼ x̂k¡1jk¡1
= E[xk¡1 j z k¡1 , mk¡2
, mk¡1
]
(28)
which can be called GPB2’s fundamental assumption.
The conditional estimates of a jump-linear
system (4)—(5) are given explicitly, for (i, j) 2 M 2 ,
by
(i,j)
(i)
= E[xk j z k , mk¡1
, mk(j) ]
x̂kjk
(j) (i)
(i)
= x̂k¡1jk¡1
+ Kk(j) (zk ¡ Hk(j) Fk¡1
x̂k¡1jk¡1 )
(i)
,
where Kk(j) is the Kalman filter gain at k, and x̂k¡1jk¡1
for i 2 M, is a lumped estimate:
k¡1
which requires only M conditional filtering operations,
one for each model, at each recursion.
The standard GPBn strategy requires M n
conditional filtering operations and storage of M n¡1
(ik¡n ,:::,ik¡1 )
estimates x̂k¡1jk¡1
. References [339] and [347]
proposed to trade computation for storage by storing
(i)
(i)
only the nM estimates fx̂k¡njk¡n
, : : : , x̂k¡1jk¡1
, i 2 Mg,
together with fzk¡n , zk¡n+1 , : : : , zk¡1 g, and recomputing
(ik¡n ,:::,ik¡1 )
all x̂k¡1jk¡1
, which requires (M 2 + M 3 + ¢ ¢ ¢ + M n )
conditional filtering operations.
b) Reinitialization: Hypothesis reduction for
recursive single-scan CMM estimation amounts to
reinitialization of each elemental filter since it is
reflected in the inputs to elemental filters at each
recursion [192]. Consider the recursion at time k and
(i)
denote by X̄k¡1
the set of input quantities (omit MSE
matrices) to the elemental filter based on model m(i) ,
(i)
as depicted in Fig. 4. Then X̄k¡1
is as given by (32)
(i)
, mk¡1
]
= E[xk¡1 j z
X (l,i)
(l)
(i)
=
j z k¡1 , mk¡1
g:
x̂k¡1jk¡1 Pfmk¡2
(31)
ik¡2 2M k¡2
which has only M 2 conditional filtering operations,
M for each model. In simple words, the lumping
amounts to assuming
(i)
x̂k¡1jk¡1
(30)
(29)
l2M
10 This is the same as replacing bodies of a distributed mass by
the component masses placed at their respective centroids for the
calculation of the centroid of the total body system.
(i)
X̄k¡1
=
8 (ik¡1 )
fx̂k¡1jk¡1 , ¹k¡1
: ik¡1 2 M k¡1 g
MMSE-CMM
>
(ik¡1 )
>
>
>
>
>
ff(ik¡1 ) (xk¡1 j z k¡1 ), ¹k¡1
: ik¡1 2 M k¡1 g
>
(ik¡1 )
>
>
<
MAP-CMM
(ik¡n ,:::,ik¡1 )
(ik¡n ,:::,ik¡1 )
>
fx̂k¡1jk¡1
, ¹k¡n,:::,k¡1
: (ik¡n , : : : , ik¡1 ) 2 M n¡1 g
>
>
>
>
>
>
GPBn
>
>
: (j)
(j)
fx̂k¡1jk¡1 , ¹k¡1 , j 2 Mg
LI & JILKOV: SURVEY OF MANEUVERING TARGET TRACKING. PART V: MULTIPLE-MODEL METHODS
:
GPB2
(32)
1273
Fig. 4. Structure of single-scan recursive MM estimation algorithms.
Fig. 5. GPB1, GPB2, and IMM reinitializations. Line on left of round node indicates filtering operation; a node outputs a weighted
sum of its inputs to provide reinitialization. (a) GPB1. (b) IMM. (c) GPB2.
For GPB1 and MMSE-AMM (first generation),
(i)
(i)
X̄k¡1
has only one element x̄k¡1jk¡1
, given by
8 (i)
(i)
k¡1
x̄k¡1jk¡1 = x̂k¡1jk¡1
= E[xk¡1 j z k¡1 , m(i)
]
>
>
<
(i)
= x̂k¡1jk¡1 = E[xk¡1 j z k¡1 ]
x̄k¡1jk¡1
>
>
P
:
(j)
(j)
Pfmk¡1
j z k¡1 g
= j2M x̂k¡1jk¡1
AMM
GPB1:
c) IMM: A significantly more cost-effective
reinitialization than those of GPB1’s and GPB2’s is
(i)
= E[xk¡1 j z k¡1 , mk(i) ]
x̄k¡1jk¡1
(j)
, mk(i) , z k¡1 ] j z k¡1 , mk(i) g
= EfE[xk¡1 j mk¡1
X (j)
(j)
x̂k¡1jk¡1 Pfmk¡1
j z k¡1 , mk(i) g:
(33)
=
j2M
1274
This leads to the IMM algorithm [56, 55, 58]. Like
the GPB1 algorithm, the IMM algorithm also runs
each of the M filters only once at each recursion.
(i)
Compared with the GPB1 reinitialization x̄k¡1jk¡1
=
E[xk¡1 j z k¡1 ], the extra conditioning on mk(i) in the
(i)
IMM reinitialization x̄k¡1jk¡1
= E[xk¡1 j z k¡1 , mk(i) ]
is both legitimate and effective [190, 192]. It is
legitimate because mk(i) is assumed true anyway
(i)
when calculating x̂kjk
in the conditional filtering; it
is effective because mk(i) carries valuable information
about mk¡1 , since the mode sequence is dependent,
which in turn affects xk¡1 .
The reinitialization in the GPB1, GPB2,
and IMM algorithms is illustrated in Fig. 5 and
explained as follows [192]. The GPB1 algorithm
reinitializes each filter with the “best possible”
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
Fig. 6. Structure of IMM estimation algorithm (with three models).
common single quasi-sufficient statistic–the
previous overall estimate; the elemental filters
interact with one another only through the use
of this common input at each recursion, which
carries information from all elemental filters. In
the IMM algorithm, each filter i at time k has its
(i)
own, individualized reinitialization x̄k¡1jk¡1
, which
forms the “best possible” quasi-sufficient statistic of
all old information and the knowledge/assumption
that model m(i) is in effect at k. Such individualized
reinitialization clearly is intuitively more appealing
than GPB1’s common reinitialization. The superiority
of IMM to GPB1 stems from this smart extra
conditioning or individualized reinitialization, known
as mixing, as evidenced by numerous applications
detailed later in Section VE. Note that GPB1 uses
a single merging operation for both output and
conditional filtering, whereas IMM and GPB2 both
use two separate ones.
Note that the IMM reinitialization (33) differs
from (29) of the GPB2 algorithm only in the time
at which the model is assumed to be true. In this
sense, the IMM algorithm does the mixing at a better
time (before conditional filtering) than the GPB2
algorithm (after conditional filtering) [58]. This is
clearly seen by comparing Fig. 5(b) and Fig. 5(c). For
the case where each model-based system dynamics
(j,i)
is linear, state-prediction mixing,11 with x̂kjk¡1
=
(j)
E[xk j mk¡1
, z k¡1 , mk(i) ],
(i)
x̂kjk¡1
= E[xk j z k¡1 , mk(i) ]
(j)
= EfE[xk j mk¡1
, z k¡1 , mk(i) ] j z k¡1 , mk(i) g
X (j,i)
(j)
x̂kjk¡1 Pfmk¡1
j z k¡1 , mk(i) g
=
j2M
11 The
benefit of the extra conditioning is more evident for
state-prediction mixing than for reinitialization.
Fig. 7. Semi-Markov process.
is equivalent to the IMM reinitialization (33). If the
model-based system dynamics is nonlinear, it is more
accurate in general than reinitialization (33) but is
computationally less efficient because it requires M
predictions in each conditional filtering operation,
rather than a single one if (33) is used.
The architecture of the IMM algorithm is
illustrated in Fig. 6 with three models. A complete
recursion of the IMM algorithm with Kalman filters
as its elemental filters is summarized in Table II
for the Markov jump-linear system (4)—(5) with
white Gaussian process and measurement noises. A
straightforward implementation may have numerical
problems for some applications, especially those
with a wide model separation. A numerically robust
version of the IMM algorithm (as well as the AMM
algorithm) was presented in [216].
d) IMM with semi-Markov models: As explained
in Section III.I of Part I [209], a semi-Markov process
model has a greater modeling power and suits better
to more real-world problems than a Markov model.
The evolution of a semi-Markov process can be
visualized as follows (Fig. 7). From any mode m(i) ,
the next mode m(j) to take place is chosen at random
according to the transition probability ¼ij , and the
time between m(i) and m(j) (i.e., the sojourn time
¿ (i) in mode m(i) ) is chosen at random according
to sojourn time pdf fij (¿ ). A class of semi-Markov
LI & JILKOV: SURVEY OF MANEUVERING TARGET TRACKING. PART V: MULTIPLE-MODEL METHODS
1275
TABLE II
One Cycle of IMM Estimator
1. Model-conditioned reinitialization (for i = 1, 2, : : : , M):
P
¢
Predicted mode probability: ¹(i) = Pfmk(i) j z k¡1 g =
¼ ¹(j)
j ji k¡1
kjk¡1
jji
¢
(j)
(j)
Mixing weight:
¹k¡1 = Pfmk¡1 j mk(i) , z k¡1 g = ¼ji ¹k¡1 =¹(i)
kjk¡1
Mixing estimate:
(i)
x̄k¡1jk¡1
= E[xk¡1 j mk(i) , z k¡1 ] =
Mixing covariance:
¢
(i)
P̄k¡1jk¡1
=
P
j
(j)
P
jji
¹
x̂(j)
j k¡1jk¡1 k¡1
(j)
(j)
jji
(i)
(i)
[Pk¡1jk¡1 + (x̄k¡1jk¡1
¡ x̂k¡1jk¡1 )(x̄k¡1jk¡1
¡ x̂k¡1jk¡1 )0 ]¹k¡1
2. Model-conditioned filtering (for i = 1, 2, : : : , M):
Predicted state:
(i)
(i) (i)
(i)
(i)
x̂kjk¡1
= Fk¡1
x̄k¡1jk¡1 + Gk¡1
w̄k¡1
Predicted covariance:
(i)
(i)
(i)
(i) 0
(i)
(i)
(i) 0
P̄k¡1jk¡1
Pkjk¡1
= Fk¡1
(Fk¡1
) + Gk¡1
Qk¡1
(Gk¡1
)
Measurement residual:
(i)
z̃k(i) = zk ¡ Hk(i) x̂kjk¡1
¡ v̄k(i)
Residual covariance:
(i)
Sk(i) = Hk(i) Pkjk¡1
(Hk(i) )0 + Rk(i)
Filter gain:
(i)
Kk(i) = Pkjk¡1
(Hk(i) )0 (Sk(i) )¡1
Updated state:
(i)
(i)
x̂kjk
= x̂kjk¡1
+ Kk(i) z̃k(i)
Updated covariance:
(i)
(i)
Pkjk
= Pkjk¡1
¡ Kk(i) Sk(i) (Kk(i) )0
3. Mode probability update (for i = 1, 2, : : : , M):
¢
assume
Model likelihood:
= p[z̃k(i) j mk(i) , z k¡1 ] = N (z̃k(i) ; 0, Sk(i) )
L(i)
k
Mode probability:
=
¹(i)
k
¹(i)
L(i)
kjk¡1 k
P
j
4. Estimate fusion:
Overall estimate:
Overall covariance:
x̂kjk =
Pkjk =
(j)
(j)
¹kjk¡1 Lk
P
x̂(i) ¹(i)
i kjk k
[P (i) + (x̂kjk
i kjk
P
models, highly relevant to MM tracking, is the
sojourn-time dependent Markov (STDM) process
[258, 69]. Here the process representing the modal
state is characterized by the sojourn-time dependent
transition probabilities, defined by
¼ij (¿ ) = Pfsk = m(j) j sk¡1 = ¢ ¢ ¢ = sk¡¿ = m(i) 6
= sk¡¿ ¡1 g
(i)
(i)
, ¿k¡1
= ¿g
= Pfmk(j) j mk¡1
(i)
where ¿k¡1
is the time already stayed in mode m(i) at
time k ¡ 1.
An extension of the IMM configuration to the case
of an STDM process was proposed in [69]. It operates
in the same manner as the standard IMM algorithm
except that for the recursive cycle (k ¡ 1 ! k) each
transition probability ¼ij is replaced by average
(expected value) of ¼ij (¿ ) over all possible ¿ values:
¢
(i)
¼ˆ ij,k¡1 = Pfmk(j) j mk¡1
, z k¡1 g
=
k¡1
X
¿ =1
(i)
k¡1
¼ij (¿ )p(i)
]:
k¡1 (¿ ) = E[¼ij (¿ ) j mk¡1 , z
(i)
(i) k
An “exact” recursion for p(i)
k (¿ ) = Pf¿k = ¿ j mk , z g
was derived in [259]. Later [282] showed that this
recursion is actually approximate, and the IMM
1276
(i)
(i) 0 (i)
¡ x̂kjk
)(x̂kjk ¡ x̂kjk
) ]¹k
(i)
reinitialization x̄k¡1jk¡1
= E[xk¡1 j z k¡1 , mk(i) ] loses its
magic in the case of an STDM process and turns out
to be similar to GPB2’s reinitialization (29):
E[xk¡1 j z k¡1 , mk(i) ]
(j)
, mk(i) , z k¡1 ] j z k¡1 , mk(i) g
= EfE[xk¡1 j mk¡1
X (j,i)
(j)
j z k¡1 , mk(i) g:
x̂k¡1jk¡1 Pfmk¡1
=
j
Use of the standard IMM reinitialization
(33) here essentially amounts to ignoring the
¢
(j,i)
sojourn-time dependence because x̂k¡1jk¡1
=
¢
(j)
(j)
, mk(i) , z k¡1 ] = E[xk¡1 j mk¡1
, z k¡1 ] =
E[xk¡1 j mk¡1
(j)
x̂k¡1jk¡1
holds only if the Markov chain is not
sojourn-time dependent.
From a practical point of view, the STDM model
appears rather complicated to design mainly because
the required knowledge of the sojourn-time dependent
transition probabilities is hard to come by. The
problem can be simplified slightly by considering a
narrower class–homogeneous semi-Markov processes
[337, 241] specified by an embedded Markov chain
with given initial mode probabilities, transition
probabilities, and sojourn-time pmfs of each mode
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
m(i) , defined by
¹̄(i)
= m(i) j sk¡1 = ¢ ¢ ¢ = sk¡¿ = m(i) 6
= sk¡¿ ¡1 g
k (¿ ) = Pfsk 6
(i)
(i)
= Pfsk 6
= m(i) j mk¡1
, ¿k¡1
= ¿ g:
That is, ¹̄(i)
k (¿ ) is the probability that a jump occurs
at time k given that the last jump wasP
at time k ¡ ¿
(i)
to mode m(i) . It is time invariant and 1
¿ =1 ¹̄k (¿ ) = 1
due to homogeneity of the process. In this case, the
STDM transition probabilities can be determined by
¼ij (¿ ) = ¹̄(i)
k (¿ )¼ij [337, 241], which can serve as a
useful guideline to build an STDM process model.
2) Other Merging-Based Algorithms:
a) Bayesian filter bank: Generally speaking,
the mode transition may be base-state dependent
and thus neither Markov nor semi-Markov (but
the base state and mode together are Markov).
Reference [296] proposed a general Bayesian
density-based scheme for hybrid estimation with
nonlinear measurements and such non-Markovian
mode jumps. The scheme is linear in the number of
models and a computational implementation based
on the Gaussian-sum approximation techniques was
proposed. For the particular case of homogeneous
MJLS this scheme can be used for point estimation
and can be implemented via standard techniques of
merging close components and/or pruning unlikely
components. A more detailed description will be
given in a subsequent part of this survey. A systematic
treatment of mixture component reduction techniques
in a more general setting can be found in [299], [300],
and [384].
b) Change of measure: By change of measure,
a measure-theoretic technique based on the
Radon-Nikodym theorem, [326, 325, 327] developed
a Gaussian wavelet estimator (GWE)12 based on
hypothesis merging to limit the growth of the
Gaussian mixture. Its merging strategy resembles
the GPB2 strategy very much.13 However, it differs
fundamentally from the IMM (and GPB) algorithms
in computing the mode probabilities. It accounts for
the effect of the measurement residuals z̃k(i) on the state
estimate update directly in the state space:
(i)
Pfmk¡1
, mk(j) j z k g
¢¤
£ ¡ (i,j) 2
(i,j)
k(P (i,j) )¡1 ¡ kx̂kjk¡1
k2(P (i,j) )¡1
/ exp 12 kx̂kjk
kjk
kjk¡1
where kxk2A = x0 Ax. This formula gives more weight
to the models that have a larger normalized change
in the magnitude of the updated mean, unlike the
conventional, intuitively appealing IMM and GPB1
formulas, which give more weight to those with a
12 A Gaussian wavelet is simply a Gaussian mixture, where the
mother wavelet is simply the Gaussian distribution function.
13 In [326], [325], and [327], dynamic modes and measurement
models are denoted by different indices, resulting in triple indices.
smaller normalized measurement residual: Pfmk(i) j z k g
/ exp(¡ 12 kz̃k(i) k2(S (i) )¡1 ). Reference [328] included
k
a comparative study between GWE and the IMM
algorithm for scenarios with different data rates. While
for high rates the differences were tiny, for low rates
the GWE showed significant improvement.
More generally, an exact estimator with all
components of the mixture was presented in [109]
by change of measure, along with simulation results
illustrating its improvement over the IMM algorithm.
Recently, [108] presented simulation results of an
approximate implementation with a fixed complexity
by “exact pruning” (without explanation), showing
superior performance to the IMM algorithm for a
passive tracking example.
c) Enhancement by mode observations: Tracking
performance can be improved by using additional
observations of target features, such as target attitude
and image, provided by, e.g., an imaging sensor.
These observations are related more directly to the
target motion mode than the kinematic measurements.
Many of them are actually used in target recognition
and feature-aided tracking. As such, incorporation
of such observations in tracking is closely related
to joint tracking and recognition. Issues in modeling
and utilization of such information for target tracking
have been studied extensively (see, e.g., [163], [188],
[8], [329], [330], [93]). Many of them are beyond the
scope of this paper and their coverage is planned in a
subsequent part. We present here only a brief review
of recent results using mode observations within the
MM context.
Loosely speaking, a modal sensor can be modeled
as a classifier over a set M [327] (possibly augmented
by a “no-decision” event [363, 364, 112]). Let yk
and y k be the mode observations at and through
time k, respectively, along with the kinematic
measurements z k . It follows from straightforward
Bayesian calculus [110, 112] that for the MJLS
(1)—(2), the joint posterior density p(xk , sk j z k , y k )
of the hybrid state is again a mixture density with
an exponentially increasing number of components,
similar to the case with kinematic measurements only.
This fact was established earlier (see [109], [326])
in terms of unnormalized14 joint posterior densities
q(xk , sk j z k , y k ) by change of measure in discrete time
[107]. Main efforts thereafter have been focused on
developing tractable approximate estimators with fixed
computation/memory.15
14 Using unnormalized density is advantageous in providing linearity
of the fundamental Bayesian recursion [296].
15 Fortunately, it turns out that for the special case of
mode-observation-only tracking, optimal estimates for Pfmk(i) j y k g
and f(xk j y k ) can be obtained recursively with fixed (nonincreasing)
computational requirements. The corresponding algorithms were
proposed in [364] for Pfmk(i) j y k g and in [100] for E[xk j y k ].
Further results along this line can be found in [176] and [177].
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Based on the optimal Bayesian mixture-density
representation, [111], [110], and [112] proposed
an extension of the IMM algorithm, referred to
as image-enhanced IMM (IE-IMM), derived by
IMM-like merging. It consists of all the familiar
IMM steps plus an additional step to update the mode
probabilities with the mode (image) observations by16
1
Pfmk(i) j z k , y k g = p(sk j mk(i) , z k , y k¡1 )Pfmk(i) j z k , y k¡1 g
c
(34)
where the likelihood p(sk j mk(i) , z k , y k¡1 ) is provided by
the model of the modal sensor and Pfmk(i) j z k , y k¡1 g
comes from the standard IMM part after the update
with the kinematic measurement zk . The IE-IMM
design for tracking included a constant-velocity (CV)
and two CT models with known turn rates. Simulation
results demonstrated that mode observations indeed
enhance IMM’s performance significantly in the case
of a high quality modal sensor, but the enhancement
diminishes as the quality of the mode observations
becomes poorer.
As mentioned above, a recursion for the
unnormalized joint posterior density q(xk , sk j z k , y k )
can be obtained by change of measure. An exact
estimator in this class was presented in [109]. Based
on the recursion for the unnormalized density,
[326], [325], and [327] proposed the approximate
implementation, GWE described above. Again, the
entire effect of the mode observations is on the mode
probability, also given by (34). The comparative
simulation results for IMM, IE-IMM, and GWE
algorithms presented in [327] indicated that generally
GWE provides an improvement over IE-IMM and
IMM, which in some cases (e.g., good imager, poor
kinematic data) may become considerable (25% over
IE-IMM and 50% over IMM) to justify its increase
in computation. An exact hybrid filter based on
change of measure that accounts for intermittent mode
measurements was presented in [3], along with an
approximate, GPB-type implementation and EM-based
estimation of the transition probability matrix. More
details of GWE and image-enhanced MM estimation
can be found in [323], [189], and [324]. Feature-aided
IMM was also studied in [376] and [385].
d) Multirate IMM: References [138] and [139]
proposed to use the IMM strategy to combine a
bank of filters, each with a different data rate that
is consistent with its assumed target dynamics (e.g.,
a CA filter would benefit a higher data rate more
than a CV filter). A filter with a lower data rate
updates less frequently, and thus compared with the
corresponding full-rate IMM algorithm using similar
16 This
makes perfect sense since the mode observations are
modeled as a classifier, which carries no information about target’s
kinematic state directly.
1278
models, this multirate IMM algorithm provides certain
computational savings at a cost of somewhat larger
peak errors at maneuver onset. The multirate data
bank is obtained from the original data by a discrete
wavelet transform.
e) Weighted-model based single filter: To reduce
the computational complexity of MM algorithms,
[237] proposed an MM-based single filter algorithm,
where the filter is based on an average model,
which is a probabilistically weighted sum of the
models used. The weights are updated over time
by a recursive formula of hypothesis probabilities
in the multihypothesis version of the Shiryayev
sequential probability ratio test (i.e., the quickest
change-point detector under some conditions)
[238] and approximate model likelihoods assuming
Gaussianity.
3) Hypothesis Reduction by Pruning: The
basic idea of hypothesis reduction by pruning has
been explained in Section IVC under the title “hard
decision,” although that subsection is for output
processing. More specifically, for hypothesis reduction
at time k a “good” subset Bk of model sequences is
identified and maintained while discarding/pruning
less likely ones in the set Mk of all possible model
sequences. The number of model sequences in Bk may
or may not change over time.
a) B-best: This approach intends to maintain
only a number Bk of the best (i.e., most probable
or likely) model sequences at time k. This idea has
an exact and straightforward batch implementation:
select Bk sequences with the largest probabilities
(or likelihoods). But recursive implementations
are more realistic. Consider a recursion at time k
with Bk¡1 “best” model histories (with associated
(ik¡1 )
and mode-sequence
base-state estimates x̂k¡1jk¡1
k¡1
probabilities ¹(ik¡1 ) ) and M models (elemental filters).
Each elemental filter performs Bk¡1 conditional
(ik¡1 )
as input
filtering operations, using one x̂k¡1jk¡1
in each operation, yielding as many as Bk¡1 M
(ik¡1 ,j)
updated estimates fx̂kjk
, (ik¡1 , j) 2 Bk¡1 £ Mg and
the corresponding mode-sequence probabilities
¹k(ik¡1 ,j) . Only Bk of these Bk¡1 M model sequences
(and the associated base-state estimates) with the
largest ¹k(ik¡1 ,j) are retained17 (and renormalized)
for the next recursion (and output at k). The above
mode-sequence probabilities ¹k(ik¡1 ,j) can be replaced
by model-sequence likelihoods. As explained in
Section IVC, this recursive implementation actually
does not guarantee that the Bk model sequences are
actually most probable among all possible sequences,
because the most probable ones may have less
17 As
(ik¡1 ,j)
such, updated estimates x̂kjk
not among the Bk best
sequences actually need not be computed, but the corresponding
model-sequence likelihoods are needed to obtain ¹k k¡1 .
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
(i
,j)
OCTOBER 2005
Fig. 8. Viterbi algorithm. (a) Hypotheses with link likelihoods. (b) Most likely paths (k = 3). (c) Most likely paths (k = 4).
probable partial histories and thus been discarded at
an earlier recursion. Another drawback is that some
or many of the B most probable model sequences
may be quite similar and had better be merged to
save processing resources. This B-best idea is the one
underlying many hypothesis reduction techniques,
such as the one in the multiple hypothesis tracking
(MHT) algorithms (see, e.g., [293], [280], [34]). For
maneuvering target tracking, it was first proposed in
[338], [129], [338], [336], and [337], and extended
to Markov jump systems with unknown transition
probabilities in [335] and to MM smoothing in [241].
Bk is usually fixed over time, as in these publications,
but the approach works for time-varying Bk . For
instance, Bk can be determined automatically by
requiring all survived sequences to have a probability
above a threshold, as proposed in [215], [92], and
[63], or probably better to have their ratios of
probability to the largest one above a threshold [215].
b) Viterbi algorithm: The Viterbi algorithm (see,
e.g., [115]), also known as the forward dynamic
programming, aims at finding the best path (model
sequence) over a time horizon in a recursive manner.
Fig. 8 illustrates this procedure for a hypothesis
tree of three modes similar to the one in Fig. 3(a).
The number next to a link (i.e., mode transition) in
Fig. 8(a) is its log-likelihood of being the true one;
the number next to a node (mode) in Figs. 8(b) and
8(c) is the log-likelihood of reaching the node by the
most likely path (sequence of transitions), which is
assumed to be the sum of the log-likelihoods of the
links on the path; only the most likely paths reaching
each node are indicated in Figs. 8(b) and 8(c). Clearly,
the most likely paths (the thicker lines) through time
k = 3 and k = 4 are (1, 2, 2) and (2, 3, 3, 1), respectively.
Note that at k = 3 the path (2, 3, 3) is not most likely,
but it must be the most likely one arriving at node
3. This idea has a straightforward implementation
for hypothesis reduction [11]. Consider a recursion
at time k with M model histories (with associated
(ik¡1 )
and ¹k¡1
) and M models. Each elemental
x̂k¡1jk¡1
(ik¡1 )
filter j performs M conditional filtering operations,
(ik¡1 )
using one x̂k¡1jk¡1
as input in each operation, yielding
k¡1
(i
M updated estimates x̂kjk
,j)
and the corresponding
mode-sequence probabilities ¹k(ik¡1 ,j) . Only the
sequence with the largest ¹k(ik¡1 ,j) for each j is retained
for the next recursion (and output at k). In this way,
the best model history for each model at any time is
identified. It always has M survived model histories
at any time. Each of them is the best model history
for an elemental filter at the time. These M histories
k
include the best one m̂MAP
but are not necessarily the
M best ones because, for example, the second best
model history for a model may be better than the best
one for another model. These suboptimal histories are
needed to guarantee the inclusion of the best sequence
for the future.
The above recursive procedure intends to find the
k
most probable sequence m̂MAP
, namely, one with the
largest probability
¹k(ik ) = Pfm(ik k ) j z k g
1
k¡1
g
= f(zk j m(ik k ) , z k¡1 )Pfmk(ik ) j m(ik¡1
k¡1 ) , z
c
k¡1
g
¢ Pfm(ik¡1
k¡1 ) j z
ln ¹k(ik )
=
ln ¹(ik¡1
k¡1 )
(35)
+ ¢ik¡1 ,k
¢ik¡1 ,k = ln f(zk j m(ik k ) , z k¡1 )
k¡1
g ¡ ln c:
+ ln Pfmk(ik ) j m(ik¡1
k¡1 ) , z
If the mode transition log-likelihood ¢ik¡1 ,k
were independent of the mode history m(ik¡2
k¡2 ) =
(i
)
k¡2
), the above Viterbi algorithm
(m1(1) , m2(2) , : : : , mk¡2
k
would yield the most probable sequence m̂MAP
.
However, for a hybrid system ¢ik¡1 ,k actually depends
on the mode history m(ik¡2
k¡2 ) , which is analogous to
the case with Fig. 8(a) in which the number next
to a link depends on the path reaching its left node.
Consequently, the above procedure is only suboptimal
even if the mode sequence is a Markov chain such
(ik¡1 )
k¡1
that Pfmk(ik ) j m(ik¡1
g = Pfmk(ik ) j mk¡1
g = ¼ik¡1 ,ik
k¡1 ) , z
k¡1
k
¤
because in this case ln ¹(ik ) = ln ¹(ik¡1 ) + ¢ik¡1 ,k could
¤
¤
< ln ¹k¡1
for
be larger than ln ¹k(ik ) even if ln ¹k¡1
(ik¡1 )
(ik¡1 )
¤
k¡2
some mode history m(ik¡2
k¡2 different from m(ik¡2 ) . This
)
¤
dependence of ¢ik¡1 ,k on m(ik¡2
k¡2 ) is a major (but subtle)
difference between the hidden Markov models
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1279
(HMM) of a hybrid system (or HMM for target
tracking18 ) and the more standard HMM found in
such applications as speech processing (see [289]
and [290] for a tutorial), where f(zk j m(ik k ) , z k¡1 )
reduces to f(zk j mk(ik ) ) because direct (continuousor discrete-valued) observations of the mode are
available. However, this dependence may be removed
in the framework of the EM algorithm [287],
discussed later in Section VB5.
4) Merging versus Pruning: Experience
indicates that base-state estimators for maneuvering
target tracking based on merging are usually superior
to those based on pruning. We believe that this
performance difference stems mainly from their
difference in sensitivity to the mode-model mismatch,
similar to our discussion in Section VA that contrasts
MMSE estimation and some versions of MAP
estimation. Simply put, the resultant sequence by
merging is not limited to the assumed set of model
sequences and in this sense merging is less sensitive
than pruning to the mismatch between the assumed
models and the true model. In contrast, pruning
appears more intuitively appealing than merging for
target tracking in the presence of clutter, where a
measurement is better treated either from a target
or clutter, rather than possibly from something in
between.
Another major difference between merging
and pruning is that the sum of the probabilities of
the merged model sequences is always one, while
without renormalization that of a fixed number of the
survived model sequences after pruning is constantly
decreasing, usually dramatically, as time elapses. For
example, for a problem with ten possible models at
each time, if after pruning ten model sequences are
retained, their sum of probabilities at time k could
be as low as 10=10k = 10¡(k¡1) (assuming a uniform
distribution) or possibly even lower because many
good sequences could have been deleted earlier
in a recursive implementation. Even if the most
likely sequences have a probability a million times
that of the average probability of a sequence, this
sum has an upper bound 10 £ 106 =10k = 10¡(k¡7) ,
8k ¸ 7, which still drops dramatically as time goes.
There is hardly any reason to expect that the mean
or global maximizer of a mixture density of 10k
components can be approximated well by one with
the 10 components.
It should be clear from the above discussion that
a test scenario in which the true model (sequence)
is one of those assumed by the MM algorithm
favors implicitly the pruning or selection based
techniques, which is however not very realistic.
In other words, merging outperforms pruning and
selection particularly when the true model differs
18 See
1280
[284] for a tutorial on HMM for tracking.
significantly from those assumed, as evidenced by
the comparative study of seven MM algorithms,
including AMM, GPB1, GPB2, IMM, B-best, Viterbi,
and reweighted IMM (RIMM), for a few simple
maneuvering target tracking scenarios, reported in
[283].
a) IMM versus B-best: Surprisingly few
implementations of pruning strategies for tracking
a single maneuvering target (without clutter) have
been reported in the literature. One reason is probably
that early simulation results by the authors of
the B-best pruning indicated that generally the
performance of the B-best pruning is inferior to
GPB-type merging of the same computational
complexity [336]. Nevertheless, [302] reported an
implementation of a B-best strategy based tracker
for homing missile guidance that features rapid and
large variation in target acceleration. For this quite
realistic scenario the B-best strategy implemented
with 11 models (quantized acceleration levels in
[¡9 g, 9 g]) showed a fairly satisfactory accuracy, but
at the very high cost of keeping 253 model sequences.
A comparison between a two-model (CV and Singer)
IMM algorithm and two B-best strategy based MMSE
and MAP filters, respectively, over a simplistic
one-dimensional scenario was reported in the short
note [356]. It claimed that the B-best strategy
provided better accuracy than the IMM algorithm,
but no indication was given of how many filters were
used. A more thorough and realistic comparison
between two trackers based on the IMM and B-best
strategies, respectively, was presented in [146] and
[147] for civilian (2D air traffic control (ATC)
tracking) and military (3D antiaircraft gun tracker)
scenarios. The results showed that at a comparable
computational complexity the two trackers were
highly competitive in terms of estimation accuracy but
had complementary strengths and trade-off patterns:
the IMM algorithm led to smaller peak errors at
maneuver onset, while the B-best algorithm resulted
in smaller steady-state errors during nonmaneuvering
and maneuvering motions. Although these results
make sense intuitively, they provide a rather indirect
comparison of the two techniques as hypothesis
reduction strategies because different underlying
designs were used for the filters. While the IMM
strategy used two kinematic models with a Markov
model for mode transitions, the B-best strategy
used three kinematic models and a sequence of
independent binary random variables for the mode
evolution. It would be more representative to
compare the two strategies with more similar
designs and parameters.
b) IMM versus Viterbi: Both the Viterbi and IMM
algorithms were implemented and compared in [11]
for a model-set design of 12 models, quantizing a
2D acceleration vector of a magnitude up to 4g. The
simulation results showed a comparable accuracy
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
of the Viterbi and IMM algorithms with generally
smaller peak errors of the latter during maneuver
onset. The Viterbi algorithm was slightly more
accurate in cases with small modal separation. A
closely related algorithm, along with simulation
comparison results, was presented in [287] using the
EM technique, discussed later in Section VB5.
c) Combined pruning and merging: It is intuitively
appealing to combine pruning and merging strategies,
or more generally, hard decision with soft decision.
Obviously, numerous ways of combination exist. A
simple, integrated idea is to prune all mode histories
ending at model m(j) at time k if its probability
(j)
k
¹(j)
k = Pfmk j z g
X
(j)
(j)
k¡1
Pfmk(j) j z k , m(ik¡1
j zk g
=
k¡1 ) , mk gPfm(ik¡1 ) , mk
ik 2M k
=
X
¹k(ik¡1 ,j)
This is often hard or intractable if the likelihood
function fµ (Z) is unavailable (e.g., without a closed
form). In many cases, however, fµ (Y, Z) for some
“complete” data (Y, Z) is available and has a simple
form, where Y is some “nuisance” random parameter,
known as missing data, hidden data, unobservable
data, or left-out data. Estimation based on fµ (Z) is
in this sense an “incomplete-data” problem. It is
intuitively appealing to replace fµ (Z) with f̂µ (Z j µ̂)
¢
= E[fµ (Y, Z) j Z, µ̂] given the best available estimate
µ̂ of µ, where the average is over Y for the given Z
and µ̂ in E[fµ (Y, Z) j Z, µ̂], which is, for example,
sometimes equal to fµ (Ŷ(Z, µ̂), Z) with Ŷ(Z, µ̂) =
E[Y j Z, µ̂]. The basic idea of the EM algorithm is to
solve the problem of arg maxµ fµ (Z) by the iteration
µ̂j+1 = arg maxµ f̂µ (Z j µ̂j ) with a better and better
estimate µ̂ of µ in the hope that
µ̂j+1 = arg max E[fµ (Y, Z) j Z, µ̂j ]
ik¡1 2M k¡1
µ
is below a threshold, or prune those with the same
(i)
, mk(j) ) if its probability
pair (mk¡1
X
(j)
(i)
k
¹(i,j)
¹k(ik¡2 ,i,j)
k¡1,k = Pfmk¡1 , mk j z g =
j!1
¡! arg max E[fµ (Y, Z) j Z, µ] = arg max fµ (Z) = µ̂ML :
µ
ik¡2 2M k¡2
is below a threshold. This is actually an integration of
(i,j)
merging and pruning since ¹(j)
k and ¹k¡1,k correspond
to merging (see Section VB1) and can be obtained
(approximately) by the IMM (or GPB1) and GPB2
algorithms, respectively, although the corresponding
(i,j)
(i)
x̂kjk
and x̂kjk
are not needed directly here.
5) Iteration-Based Algorithms: MMSE
estimation and MAP estimation are actually problems
of integration and maximization of a posterior density,
respectively. ML estimation amounts to finding the
global maximizer of the likelihood function. The
above hypothesis reduction strategies take advantage
of the structure (i.e., mixture density) of the base-state
distribution. Alternatively, these problems can be
solved numerically without explicit reliance on the
mixture-density structure (see, e.g., [175]).
Although a large number of numerical integration
algorithms are available, to our knowledge, few
have been proposed for finding an MMSE estimator
specifically in the MM context. Effort along this
line appears worthwhile. The situation is better for
MAP-based MM estimation. A class of iterative search
based algorithms have been proposed, almost all of
which rely on the so-called EM algorithm.
a) EM algorithm: The EM algorithm [87] is
an iterative procedure of finding a maximizer of
a likelihood function, particularly suitable for the
so-called incomplete-data problems (see, e.g., [249]).
Consider the problem of estimating a parameter µ
using data Z by the ML method, given by
µ̂ML = arg max fµ (Z):
µ
(36)
µ
More specifically, starting with some initial estimate
µ̂0 of µ, each iteration of the EM algorithm consists
of two conceptual steps (they are often combined
actually):
E-step (expectation): Q(µ j µ̂j ) = E[ln fµ (Y, Z) j Z, µ̂j ]
M-step (maximization): µ̂j+1 = arg maxµ Q(µ j µ̂j ).
The iteration stops when kµ̂j+1 ¡ µ̂j k is below some
threshold and then µ̂j+1 is taken to be µ̂ML . Clearly,
Q(µ j µ̂j )jµ=µ̂ ¸ Q(µ j µ̂j )jµ=µ̂j . It follows from Jensen’s
j+1
inequality that this iteration enjoys the monotone
property fµ (Z)jµ=µ̂ ¸ fµ (Z)jµ=µ̂j , which guarantees
j+1
global convergence19 of the likelihood values
fµ (Z)jµ=µ̂j for a bounded fµ (Z) under mild regularity
conditions and likewise for global convergence of µ̂j
under more stringent conditions (see, e.g., [249]).
The key in the application of the EM algorithm
is to identify the missing data Y and come up with
Baum’s auxiliary function Q(µ j µ̂j ) that is equivalent
to E[ln fµ (Y, Z) j Z, µ̂j ] and has an easy-to-find
maximum. The EM algorithm is attractive mainly for
its simplicity, wide applicability, low computation and
storage per iteration, and global convergence. Its main
shortcoming is the lack of guarantee to converge to
global maxima. A main application domain of the
EM algorithm is estimation problems with a mixture
density [292, 249], to which most target tracking and
MM estimation problems belong. The best known
applications so far of the EM algorithm in target
19 That
is, convergence from any starting point to a stationary point,
including saddle points as well as (local and global) maxima.
LI & JILKOV: SURVEY OF MANEUVERING TARGET TRACKING. PART V: MULTIPLE-MODEL METHODS
1281
tracking are those with the probabilistic multiple
hypothesis tracking (PMHT) method [318, 319,
320, 119, 359], where the above likelihood function
fµ (Z) is replaced by the posterior density f(µ j Z) or
equivalently the joint density f(µ, Z).
b) EM-based MM estimation: For estimation
of a hybrid system with base-state sequence xk ,
mode sequence mk , and data z k , the complete data
is fµ, Y, Zg = fxk , mk , z k g (the set is unordered here).
Then, two natural choices are (µ, Y, Z) = (xk , mk , z k )
and (µ, Y, Z) = (mk , xk , z k ), depending on what is to
be estimated. The first choice aims at estimating the
base-state sequence by treating the mode sequence as
missing/hidden data, leading to the EM formulation
k
for the base-state sequence MAP estimation x̂MAP
=
k
k
arg maxxk f(x j z ) with
E:
M:
k
k
Q(xk j x̂[j]
) = E[ln p(xk , mk , z k ) j z k , x̂[j]
]
(37)
k
k
x̂[j+1]
= arg maxxk Q(xk j x̂[j]
)
where subscript [j] stands for iteration j. The second
choice is for mode-sequence estimation, which treats
the base-state sequence as hidden data, leading to the
k
EM formulation m̂MAP
= arg maxmk f(mk j z k ) with
E:
M:
k
k
Q(mk j m̂[j]
) = E[ln p(xk , mk , z k ) j z k , m̂[j]
]
k
k
m̂[j+1]
= arg maxmk Q(mk j m̂[j]
)
:
(38)
For a Markov jump system, we have the following
key decomposition
p(xk , mk , z k ) = f(zk j xk , mk )f(xk j mk , xk¡1 )
¢ p(mk j mk¡1 )p(xk¡1 , mk¡1 , z k¡1 ):
The results in the remainder of this subsection
are valid only for an MJLS with white, mutually
independent, Gaussian process and measurement
noises.
k
As stated in [228] and [229], maxxk Q(xk j x̂[j]
)
is equivalent to minimizing a sum of weighted
squares of base-state prediction errors, measurement
prediction errors, and initial state estimation error of a
linear Gaussian system. This system is an “average”
Gaussian MJLS over possible modes at each time
in that it depends on the mode only through its
(i) k
k
(i) (i)
probability ¹̄(i)
· = P(m· , z j x̂[j] ) = ®· ¯· =c· , · · k,
(i)
where c· is the normalization factor, ®(i)
· and ¯·
are computed via HMM-type forward and backward
recursions, respectively, as given in [228] and [229].
From the equivalence of the optimal weighted
least-squares, MAP, and Kalman smoothing for
such systems it thus follows: in each batch iteration
(using z k ) of the EM algorithm for MAP estimation
k
of the base-state sequence x̂MAP
= arg maxxk f(xk j z k ),
(i)
k
the required f¹̄· , · · kg based on x̂[j]
can be
computed (in the E-step) by HMM-type forward and
backward recursions for the above “average” linear
1282
k
can be obtained
Gaussian system; and then x̂[j+1]
(in the M-step) by a fixed-interval Kalman smoother
[228, 229].
k
As derived in [228] and [229], maxmk Q(mk j m̂[j]
)
is equivalent to maxl ±k (l), where ±k (l) is the maximum
score of a model sequence mk with m(l) in effect
at time k (i.e., mk = m(l) ). Importantly, ±k (l) has
a recursive form ±· (l) = maxi [±·¡1 (i) + ln ¼il ] +
gl (z· , ȳ· , y·2 ), where ¼il is the mode transition
probability (6), gl (¢) is a known function of model
0
k
]0 , ȳ· = E[y· j z k , m̂[j]
], and y·2 =
m(l) , y· = [x·0 , x·¡1
0
k
k
E[y· y· j z , m̂[j] ]. As such, within each batch iteration
(using z k ) of the EM algorithm for MAP estimation
k
of the mode sequence m̂MAP
= arg maxmk p(mk j z k ),
the required fȳ· , y·2 , · · kg can be obtained (in
the E-step) by an efficient fixed-interval Kalman
smoother for a system with base state y· equivalent
k
to the original linear system based on m̂[j]
; and then
k
m̂[j+1] can be obtained (in the M-step) by solving
the optimal path problem with the above score
±· (l) efficiently via dynamic programming (Viterbi
algorithm) [228, 229]. Along similar lines, a simpler
EM-based ML estimation algorithm of the constant
mode sequence (i.e., for the first generation AMM
algorithm) was given in [254] and [73] in the context
of the so-called mixture of experts (see Section VIII).
c) EM-based MM estimation for tracking: Clearly,
the above EM-based hybrid estimation techniques
can be applied to maneuvering target tracking using
multiple models explicitly or implicitly. Indeed, a
number of such algorithms have been developed in
[285], [286], [287], [231], [230], [358], [359], [27],
and [28]. Here the key question is: What amounts
to a maneuver model? Similar to the decision-based
methods surveyed in Part IV [208], two answers have
been proposed: 1) the unknown input (acceleration)
uk [285, 286, 287, 231, 230] and 2) the statistics
(mean and covariance) of process noise wk [358, 359,
27, 28].
In [285], [286], [287], [231], and [230], the target
maneuver is described by a linear time-invariant
system in white Gaussian noise
xk = Fxk¡1 + Guk + wk ,
zk = Hxk + vk ,
wk » N (0, Qk )
vk » N (0, Rk )
with an unknown input (acceleration) uk modeled as a
homogeneous Markov chain having M possible levels
u(1) , : : : , u(M) and initial and transition probabilities
p(u1 ), p(uk j uk¡1 ). The base-state sequence xk is
treated as missing data in [285], [286], and [287] for
k
mode-sequence estimation, where Q(mk j m̂[j]
) of (38)
reduces to
Q(uk j ûk[j] )
=
k
X
·=1
[j]
[j]
fln p(u· j u·¡1 ) ¡ 12 kGu·¡1 ¡ x̂·jk
+ F x̂·¡1jk
k2Q¡1 g:
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
·
OCTOBER 2005
The E-step clearly boils down to the computation of
[j] ¢
x̂·jk
= E[x· j z k , ûk[j] ], obtainable by a fixed-interval
Kalman smoother given ûk[j] , and the M-step can be
implemented exactly by the Viterbi algorithm given
[j]
x̂·jk
since the score Q(u· j ûk[j] ) ¡ Q(u·¡1 j ûk[j] ) at time
· depends only on the transition (u·¡1 , u· ) [285,
286, 287]. Note that the transition score ¢ik¡1 ,k of
(35) used in [11] depends on the entire history
mk¡1 only through xk because f(zk j mk , z k¡1 ) =
E[f(zk j xk , mk ) j mk , z k¡1 ]. It would be independent
of mk¡1 if f̂(zk j mk , z k¡1 ) = E[f(zk j xk , mk ) j m̂k , z k¡1 ]
were used, as in the EM formulation, where average is
over xk only.
As is typical for the PMHT approach, [231] and
[230] treat discrete uncertainties (uk here) as missing
data20 for base-state sequence estimation via the EM
formulation (37), leading to
k
Q(xk j x̂[j]
)
1X
2
=¡
fkz· ¡ Hx· k2R¡1 + kx· ¡ Fx·¡1 ¡ Gû[j]
· kQ·¡1 g
·
2
k
·=1
+ kx0 ¡ x̄0 k2P ¡1
0
PM
(i) (i)
(i)
(i)
k k
where û[j]
· =
i=1 u ¹· and ¹· = Pfu· = u j z , x̂[j] g
is computed (E-step) via a forward-backward HMM
k
k
smoother given x̂[l]
. The maximizer of Q(xk j x̂[j]
)
k
given û[j] is found (M-step) by a fixed-interval
Kalman smoother [231, 230].
The approach of [358] and [359] differs from
that of [231] and [230] in that the maneuver models
differ in their process noise wk , governed by a
hidden Markov chain with different covariance
levels Q(i) , rather than the unknown input levels.
This approach was combined with the so-called
turbo-PMHT in [298] for maneuvering target tracking
in clutter. Further, by choosing the complete-data
problem as (µ, Y, Z) = ((xk , mk ), rk , z k ), where rk
stands for the sequence of data association events,
another version was proposed in [298], in which the
forward-backward smoothing for M-step is replaced
by an IMM smoother to handle the uncertainty in
rk . The EM formulation here is, however, directed
to data association, not motion-mode uncertainty.
Comparative results of all the above EM-based
trackers were also reported in [298]. Similarly, [27]
and [28] also model various maneuvers by process
noise having M covariance levels (two levels were
implemented). However, the base state is treated as
a “nuisance.” Conceptually, the E-step amounts to
fixed-interval Kalman smoothing, but the M-step
is greatly simplified by assuming jumps among
the levels are independent over time: it is trivially
20 In
fact this work accounted also for clutter measurements in a
PMHT framework.
implemented for each · = 1, 2, : : : , k independently
without a need for dynamic programming. Still
another popular formulation of maneuvers is in terms
of turn rate. To our knowledge there is no EM-based
such formulation in the literature so far.
d) Remarks: The optimal MMSE-CMM estimator
has an exponentially (geometrically) increasing
computational complexity on the order of M k , while
the above EM-based MAP-CMM algorithms have
a linearly increasing complexity on the order of
(M 2 + n3x )k in each iteration for the batch from initial
time to time k, where nx is the dimension of the base
state. The price paid by these EM-based algorithms
to achieve this linear complexity is the following.
In general there is no hope to obtain the exact MAP
estimate in finite iterations and no guarantee to
converge to the exact MAP estimate even with infinite
iterations. Like almost all iterative algorithms for
optimization problems, there is no guarantee for the
global convergence of an EM algorithm to the global
maximum–it may well converge to a local maximum
or saddle point or possibly even to a minimum in rare
cases (see [249] for a simple example). A widely
used strategy here is to try different initial points
to enhance the chance of converging to the global
maximum at the cost of substantially increased
computation. While practitioners would not insist
on having an exact optimal estimate, giving up the
requirement of being close to the global maximum is
a major relaxation of the MAP estimation goal. There
is no reason to believe that with such a relaxation
the corresponding MMSE-based algorithm with a
comparable complexity could not be developed.
Nevertheless, the EM-based approach has certain
undeniable merits: it is systematic, theoretically
elegant, and very powerful. This does not imply
that it is free of serious drawbacks for practical,
real-time applications in maneuvering target tracking.
(See [359] for a comprehensive discussion.) What
is probably worst is that the EM algorithm requires
batch processing of data, just like other MAP
estimators. This is acceptable for some applications,
such as trajectory determination, but not for most
maneuvering target tracking problems, where real-time
processing is necessary. Although no recursive form
in general, the EM algorithm may serve as a basis
for developing approximate, recursive algorithms, as
described next.
e) EM-based recursive MM estimation for tracking:
Two such EM-based recursive algorithms for
maneuvering target tracking have been proposed
recently in [285], [286], [287], [157], and [158].
The recursive algorithm proposed in [285], [286],
and [287] is an approximation of the batch solution
for mode-sequence MAP estimation presented
therein. It reduces the batch processing to the
Viterbi algorithm combined with a one-step Kalman
filter by 1) modifying Baum’s auxiliary function
LI & JILKOV: SURVEY OF MANEUVERING TARGET TRACKING. PART V: MULTIPLE-MODEL METHODS
1283
Q(uk j ûk[j] ) = E[ln p(xk , uk , z k ) j z k , ûk[j] ] to an
approximate sequential version q(uk j ûk¡1 ) =
E[ln p(xk , uk , z k ) j z k , uk¡1 ]juk¡1 =ûk¡1 suitable for
recursion, 2) replacing the smoothed state estimates
by the filtered state estimates, and 3) ignoring the
dependence of xk on uk¡1 given E[xk j uk¡1 , z k ],
cov(xk j uk¡1 , z k ), and uk . The resulting recursive
algorithm has M 2 Kalman filtering cycles at each
recursion and is essentially the same as that of [11].
At time k for each transition (link) (uk¡1 , uk ) of
the Viterbi trellis, measurement residual z̃k and its
covariance Sk are obtained by a one-step Kalman
filter, and then the best path for each level of uk is
determined by the Viterbi algorithm based on the
transition costs
¢ik¡1 ,k = ¢k¡1,k = ln p(uk j uk¡1 ) ¡ 12 kz̃k k2S̃ ¡1
k
where S̃k = Sk (Hk Qk Hk )¡1 Sk . As pointed out in
[285], [286], and [287], it differs in effect from
that of [11] only in that S̃k is in place of Sk . No
interpretation for this replacement was given,
although it appears beneficial judging from simulation
results presented therein with an assumed model set
f0, §0:15, §0:3g m/s2 in each of the two coordinates
(thus there are 25 models). A simple scenario with
a true input level jumping from 0 to 0:15 m/s2 and
then back to 0 again was considered, which does not
seem representative of the reality. Contradicting the
results shown in [11], much poorer results from the
IMM algorithm were also given in [287], possibly
due to the particular design/implementation. More
surprisingly, [287] claimed that the computational time
of the IMM algorithm, which has M complexity, is 6
times that of the recursive algorithms of [285], [286],
[287] and of [11], which both have M 2 complexity.
The above recursive EM-based algorithm is based
on mode-sequence estimation. More natural is a
recursive EM-based algorithm for base-sequence
estimation directly. Such an algorithm was proposed
in [157] and [158], named “reweighted IMM
(RIMM)” algorithm by its authors. It is identical to
the IMM algorithm except that the mixing and output
formulas are replaced by a new weighted sum in
which the weights account for not only the probability
of each model being true but also the accuracy of the
estimate from each elemental filter:
(j)
(j)
x̂kjk¡1
= Pkjk¡1
X (i,j)
ijj
(i,j)
(Pkjk¡1 )¡1 x̂kjk¡1
¹k¡1
(39)
i2M
(j)
)¡1 =
(Pkjk¡1
X (i,j)
ijj
(Pkjk¡1 )¡1 ¹k¡1
i2M
x̂kjk = Pkjk
X (i)
(i) (i)
(Pkjk )¡1 x̂kjk
¹k
i2M
¡1
Pkjk =
X (i)
(Pkjk )¡1 ¹(i)
k
i2M
1284
(40)
ijj
where the mixing weights ¹k¡1 and mode probability
¹(i)
k are computed as in the IMM algorithm (see
Table II) and
(i,j)
(j)
(i)
(i)
x̂kjk¡1
= E[xk j z k¡1 , mk¡1
, mk(j) ] = Fj x̂k¡1jk¡1
+ Gj uk¡1
(i,j)
Pkjk¡1
=
¼ij
(i)
Fk(j) Pk¡1jk¡1
(Fk(j) )0
(j)
¹kjk¡1
+ Gk(j) Qk(j) (Gk(j) )0 :
Both “reweighted” sum formulas are combinations
of a probabilistic weighted sum for MMSE-based
MM estimation and the “parallel resistors” formula
for fusion of (probabilistically correct) estimates
with uncorrelated errors (see [19, Sec. 8.3.3]). Note
that at each recursion, the RIMM algorithm requires
M 2 predictions but M updates, while the IMM
algorithm requires M predictions and M updates
(see Section VB1). Similar to the GPB2 case, let
(i)
(i)
X̄k¡1 = fx̂k¡1jk¡1
, Pk¡1jk¡1
, ¹(i)
k¡1 , i 2 Mg. The reweighted
output formula follows from minimizing a sum of
quadratic errors of fitting (xk¡1 , xk ) to measurement
(i)
zk , dynamics, and the estimates x̂k¡1jk¡1
, i2M
with a given probabilistic weight ¹(j)
k , j 2 M, that
is, x̂kjk = arg minxk q(xk j X̄k¡1 ) with q(xk j X̄k¡1 ) =
minxk¡1 q(xk¡1 , xk j X̄k¡1 ), where
q(xk¡1 , xk j X̄k¡1 )
=
X
(i)
kxk¡1 ¡ x̂k¡1jk¡1
k2
¹(i)
(P (i) )¡1 k¡1
i2M
+
X
fkzk ¡ Hk(j) xk k2
k¡1
(j)
+ kxk ¡ Fk¡1
xk¡1 k2
(j) ¡1
)
k
(R
j2M
(j) (j) (j)0 ¡1
Q G )
k
k
k
(G
g¹(j)
:
k
As shown in [157] and [158], q(xk j X̄k¡1 ) can be
written either in the GPB2 form of M 2 pairs of
quadratics
q(xk j X̄k¡1 ) =
X
i,j2M
fkzk ¡ Hk(j) xk k2
(j) ¡1
)
k
(R
(j) (i)
+ kxk ¡ Fk¡1
x̂k¡1jk¡1 k2
ijj ¡1
(i,j)
=¹
)
kjk¡1 k¡1
(P
g¹(i)
k
or in the IMM form of M pairs of quadratics:
q(xk j X̄k¡1 )
=
X
j2M
fkzk ¡ Hk(j) xk k2
(j) ¡1
)
k
(R
(j)
+ kxk ¡ x̂kjk¡1
k2
(j)
)¡1
kjk¡1
(P
g¹(i)
k
(j)
(j)
with mixing estimate x̂kjk¡1
and covariance Pkjk¡1
given by (39). Equation (40) thus follows easily
from minimizing q(xk j X̄k¡1 ) in this IMM form.
As explained in Section VB1, these two forms
are equivalent for output processing, but not for
conditional filtering. Each pair of the quadratics
above is minimized by a Kalman conditional filter.
As argued in [157] and [158], ¡q(xk¡1 , xk j X̄k¡1 )
can be interpreted/justified as an approximation
K
of Baum’s auxiliary function Q(xk j x̂[l]
)=
K
k+1 K
K K
E[ln p(x , m , z ) j z , x̂[l] ], 8k · K for a data
batch z K = (z1 , : : : , zk , zk+1 , : : : , zK ) for an alternating
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OCTOBER 2005
expectation conditional maximization (AECM)
algorithm [255, 249]. The approximations arise from
truncation of xk to (xk¡1 , xk ) and replacement of
the smoothed estimates from multiple iterations by
filtered estimates from a single forward path. The
AECM algorithm is an extension of the so-called
expectation conditional maximization algorithm
[256], which replaces a complex M-step of the EM
algorithm with several computationally simpler
maximization steps conditioned on some constraints
on the estimatee, called CM steps. In the AECM
algorithm the specification of the complete data could
be different for each CM step. This provides more
flexibility needed for formulating sequential problems
than most other EM-based algorithms. Reference
[158] included simulation results suggesting that a
two-model RIMM design had a more favorable speed
error compared with the standard two-model and
three-model IMM designs of [18]. More comparative
studies are needed to draw a more general conclusion.
f) Other iterative algorithms: An iterative
algorithm for JMAP estimation of the base state and
mode sequences was proposed in [229] based on the
block component optimization [125]:
M1:
k
k
x̂[j+1]
= arg maxxk f(xk , z k j m̂[j]
)
k
k
= arg maxmk p(mk , z k j x̂[j+1]
)
M2: m̂[j+1]
which belongs to the general class of the so-called
coordinate ascent (or descent) methods [232, 30]. For
a Gauss-Markov jump-linear system, the M1 and M2
steps can be accomplished by a Kalman smoother and
the Viterbi algorithm, respectively [229]. In fact, we
believe it is better to swap the above M1 and M2 steps
as follows
k
k
M1: m̂[j+1]
= arg maxmk p(mk , z k j x̂[j]
)
M2:
k
k
= arg maxxk f(xk , z k j m̂[j+1]
)
x̂[j+1]
because m̂k is usually not sensitive to x̂k (think,
e.g., the case with mk = uk above) but x̂k depends
on m̂k significantly. Another coordinate ascent
algorithm was proposed in [90] to obtain the MAP
estimate of the mode sequence mk by treating each
m· as one coordinate through iterations m̂·[j] =
[j]
[j¡1]
, m·+1
, : : : , mk[j¡1] )
arg maxm· p(m· j z k , m1[j] , : : : , m·¡1
for · = 1, : : : , k. Likewise for the MAP estimation of
the base-state sequence xk . It was demonstrated in
[90] by simulation results from maneuvering target
tracking that these two MAP algorithms outperform
the corresponding EM-based MAP algorithms of
[229] discussed above.
C. Multiple-Model Smoothing
In maneuvering target tracking a number of
problems exist that allow offline processing. One
example is trajectory reconstruction (see, e.g., [273]).
Also, if an estimation delay can be tolerated the
tracking performance may be improved dramatically
by smoothing [236, 95, 96, 97, 173, 174, 77, 78].
Besides, smoothing can also be used as an integral
part of a nonlinear filtering algorithm to improve
performance without time delay.
Smoothing is estimation (or more precisely
“retrodiction” [95, 96, 97]) of a process at or through
time n using data z k through time k (k > n). In the
Bayesian setting, the complete solution amounts to
finding the distribution function f(yn j z k ) or f(y n j z k ).
For hybrid systems, y could be the base state x, mode
m, or hybrid state » = (x, m). Then, finding f(yn j z k )
and f(y n j z k ) are state smoothing and state sequence
(or trajectory) smoothing, respectively. Formal
solutions to many of these smoothing problems in the
sense of MMSE and MAP point estimation have been
presented in Section VA. Here we focus on base-state
smoothing for point (not density) estimation, that
is, x̂njk with n < k, since almost all existing results
are limited to this case. As given in Section VA, the
MMSE
MMSE-optimal smoother x̂njk
= E[xn j z k ] is a
k
(i )
weighted sum of x̂njk
= E[xn j z k , m(ik k ) ] given mode
sequence m(ik k ) with weights ¹k(ik ) . The probabilistic
weights ¹k(ik ) = Pfm(ik k ) j z k g can be obtained by Bayes’
rule. For a jump-linear system with white Gaussian
(ik )
noise, each conditional smoothed estimate x̂njk
is
given by the well-known Kalman smoother (see,
e.g., [291, 251, 250, 7]). As in the case of MMSE
filtering, unfortunately, this optimal solution has
an exponentially increasing complexity and is thus
infeasible for real-time applications. So a number of
suboptimal solutions with polynomial or even linear
complexity have been proposed [241, 59, 132, 133,
76, 174, 78]. Some of the issues associated with
smoothing for target tracking were discussed in [95],
[96], and [97].
For smoothing, particularly recursive smoothing,
three common classes have been traditionally
considered: fixed interval (x̂njk , n = 1, 2, : : : , k with a
fixed k), fixed point (x̂pjk , k = 1, 2, : : : with a fixed p),
and fixed lag (x̂k¡Ljk , k = 1, 2, : : : with a fixed L).
MM smoothing has been largely limited to the cases
of fixed interval and fixed lag except that
of [154].
Fixed-Interval Smoothing: Reference [59]
presented general results for time-reversion of
discrete-time Markov jump systems (MJSs) and, in
particular, models in reverse time that are equivalent
to the original MJLS. As an application, it presented
an optimal solution for fixed-interval smoothing of
an MJS based on fusion of posterior distributions
obtained by two optimal MM estimators, one running
forward for the original system and the other running
backward using the equivalent reverse-time model.
The approach is quite general, not limited to an
MJLS or point estimation. The main difficulty is
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to obtain the equivalent reverse-time model and the
optimal forward and backward MM estimators. For an
approximate implementation the IMM algorithm was
suggested to replace the optimal MM estimators. This
implementation was demonstrated (by simulation via
trajectory smoothing of the state of an MJLS) to have
very good estimation accuracy and mode identification
at a low computational cost. The approximate
fixed-interval smoother of [132] is conceptually
similar in that it also consists of fusing forward and
backward IMM estimators, but with two significant
differences. First, fusion is based on a simpler but
more restrictive rule–the “parallel-resistor” formula
(see, e.g., [116] or [19, Sec. 8.3.3]). As a result,
the backward IMM estimator has to be initialized
without any prior knowledge of the state.21 Second, it
bypassed the task of finding equivalent reverse-time
model and derived the required backward IMM
algorithm directly from the original MJLS with
white Gaussian noise. The simulation results for
maneuvering target tracking demonstrated a dramatic
improvement of the smoothed estimates in comparison
with the forward/backward IMM estimators alone. In
both smoothers, fusion is done between every pair of
forward and backward conditional filters, resulting
in M 2 fusion operations per time step, although
fusion between the overall estimates of the two IMM
estimators would reduce it to just one fusion operation
per time step, but with some performance loss. These
IMM smoothers are both MMSE based, although the
general approach of [59] is not limited to the MMSE
criterion.
As pointed out in Section VA, the components of
kjk
an MMSE sequence estimate x̂MMSE = E[xk j z k ] are
kjk
MMSE
= E[xn j z k ]: x̂MMSE
MMSE smoothed estimates x̂njk
MMSE
MMSE
= (x̂1jk
, : : : , x̂kjk
), but the components of a MAP
kjk
sequence estimate x̂MAP = arg maxxk f(xk j z k ) are not
MAP
MAP smoothed estimates x̂njk
= arg maxxn f(xn j z k ):
kjk
MAP
MAP
x̂MAP = (x̂1jk , : : : , x̂kjk )MAP 6
= (x̂1jk
, : : : , x̂kjk
)
because the peak location of the joint pdf f(x, y) is
not (x¤ , y ¤ ), where x¤ and y ¤ are the peak locations of
the marginal pdfs f(x) and f(y), respectively. Thus,
the EM-based algorithms, discussed in Section VB5,
for MAP sequence estimation do not provide the
fixed-interval MAP smoothed estimates in one shot.
However, they can be modified easily for MAP
smoothing of a state sequence. More important, a
MAP sequence estimate appears more meaningful
and useful in practice than a sequence of such MAP
smoothed estimates.
Fixed-Lag Smoothing: The fixed-lag smoothing
algorithm of [241] differs from the MM filtering
21 This fusion rule can be replaced by a more general one (e.g.,
those of [225]) so that the backward estimator can be initialized as
desired.
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algorithm of [338], [129], [338], [336], and
[337] based on the B-best pruning strategy (see
Section VB3) only in that conditional filtering
is replaced by conditional smoothing, achieved
by fixed-lag Kalman smoothers [7]. Two MM
algorithms were developed in [131] and [133] for
one-step fixed-lag smoothing based on two different
representations of f(xk¡1 j z k ) via the total probability
theorem
A:
B:
f(xk¡1 j z k ) =
k
f(xk¡1 j z ) =
M
X
(i)
f(xk¡1 j mk¡1
, z k )¹(i)
k¡1jk
M
X
f(xk¡1 j mk(i) , z k )¹(i)
k :
i=1
i=1
Algorithm A involves M 2 one-step smoothers after
(i)
(i)
approximating f(xk¡1 j mk¡1
, z k ) and f(xk¡1 j mk¡1
,
(j) k
mk , z ) by single Gaussians via moment
matching,22 while algorithm B involves only M
one-step smoothers based on the standard IMM
approximation of f(xk¡1 j mk(i) , z k ) by a single
Gaussian. Algorithm B was extended to L-step
fixed-point smoothing in [154]. The central idea of the
L-step fixed-interval smoother (over a sliding window)
0
0
of [76] is to form a grand state (xk0 , xk¡1
, : : : , xk¡L
)0
by state stacking (augmentation) and run the IMM
estimator for this augmented system to produce
0
0
the smoothed estimate E[(xk0 , xk¡1
, : : : , xk¡L
)0 j z k ]
“automatically.” A key enabling approximation
is that f(xk , xk¡1 , : : : , xk¡L j mk(j) , z k ) is Gaussian,
which is much stronger than the approximation B
above. Another implication is that no mode jumps
within the interval (k ¡ L, : : : , k ¡ 1] are accounted
for explicitly. Additional strong approximations
were made to evaluate the retrodicted probabilities
(i)
Pfmk¡n
j z k g, n = 1, : : : , L since the IMM estimator
gives only Pfmk(i) j z k g. Another approximate way
of evaluating these mode probabilities was given
in [278]. Simulation results over the scenario of
[132] demonstrated that even with only a small lag
this smoother can outperform the IMM filtering
algorithm very significantly at the cost of an increase
in computation about (L + 1) times. Further successful
tracking applications of this state-stacking based
IMM smoother were reported in [77] for a single
maneuvering target in clutter (smoother coupled with
the probabilistic data association (PDA) filter), and
in [78] for multiple maneuvering targets in clutter
(smoother coupled with the joint-PDA filter). It was
found during our investigation for [154] that this
state-stacking based IMM smoother and the one-step
IMM smoother of [133] had almost equal accuracy for
22 These
M 2 one-step smoothers were lumped into M one-step
smoothers by a hardly justifiable heuristic in [131].
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OCTOBER 2005
the scenario simulated, while the latter is much less
computationally demanding.
For real-time applications, the above smoothing
results can only provide time-delayed estimates.
Nevertheless, such delayed estimation can be used
to improve filtering results (i.e., without a delay) of
an MM algorithm, which is inherently nonlinear, by
refining its past estimates and rerunning the MM
algorithm using the refined estimates. This approach
was taken consciously and promoted in [154] for
performance enhancement.
D. Convergence of CMM Estimation Algorithms
For a stochastic jump-linear system, a hybrid
estimation algorithm that converges exponentially
exists if several conditions, including those on
observability, given in [145] are satisfied. Here the
exponential convergence [145] refers to an algorithm
which correctly identifies the system mode in finite
time and has a base-state estimate sequence with
a unique mean and convergent covariance, and an
estimation-error mean converging exponentially
to a bounded set with a guaranteed rate. Earlier,
[281] considered the problem of mode-sequence
identification. It defines
M
¹̃(j)
k =
1X L
(¦ )ij ¹̃(i)
k¡1 exp[¡Asj ]
c
i=1
Asj = lim kZk(s) ¡ Zk(j) k2 =(L¾ 2 )
L!1
(j)0
(j)0
(j)0 0
where Zk(j) = [z(k¡1)L+1
, z(k¡1)L+2
, : : : , zkL
] is the
measurements over the block (interval) of L discrete
times, [(k ¡ 1)L + 1, (k ¡ 1)L + 2, : : : , kL] if m(j) is the
correct model over the block; s stands for the true
model; (¦ L )ij stands for the (i, j)th element of the Lth
power of the transition probability matrix ¦; and c
is the sum of the numerators over i = 1, : : : , M. For a
hybrid system, it was shown in [281] that the weights
¹̃(j)
k will converge and the true model s has the largest
steady-state value limk!1 ¹̃(s)
k under the condition that
all mode transitions are possible but infrequent and
the true model has the best fit to the data. Note that
¹̃(j)
k is an approximation of the following de facto
mode probability in the Gaussian case
M
¹(j)
k =
1X L
(j) 2
(s)
2
(¦ )ij ¹(i)
k¡1 exp[¡kZk ¡ Zk k =(L¾ )]
c
i=1
assuming no mode transition within the block, by
replacing the exponential factor with its stead-state
value as the block size increases. As such, the above
(j)
results for ¹̃(j)
k hold approximately true for ¹k , which
is meaningful if ¹(j)
k is used as a fitness measure for an
MM estimator. In fact, such a block MM estimator was
proposed in [281] for mode sequence MAP estimation.
By a theoretical analysis of a generic CV-CA
IMM algorithm, [88] concluded that even if the true
system has a CA plant, the error of the CV filter
remains bounded and the steady-state RMS error can
be estimated from the acceleration estimates of the CA
filter.
E. Tracking Applications
Almost all tracking applications of the second
generation MM algorithms so far are those of
the IMM algorithm [12, 14, 15]. Many of these
applications have been documented in the survey
of [246]. Since then numerous new successful
applications have been reported. They further
demonstrate the good performance of the IMM
algorithm for various tracking problems. While
addressing older results briefly, many already
discussed in [246], we will pay more attention to
more recent ones.
1) Surveillance for Air Traffic Control: The first
real application of the IMM algorithm is probably
the jump-diffusion prototype tracker developed by
Blom’s team as the track maintenance part of a
multisensor multitarget tracking system [60] (see
also [61] and [55]) for Eurocontrol, the European
organization for air navigation. This sophisticated
tracker included four models for horizontal motion
(straight constant/changing speed motion, and
left/right turns) and two for vertical motion (level
motion and changing altitude). Such a model set
represents well typical en-route motions of a civilian
aircraft–horizontal CV motions most of the time
with occasional changes in speed, or horizontal
constant turns, or small vertical climb/descent
maneuvers. It also allows decoupled tracking of
the altitude and horizontal motions. An EKF was
employed to handle the nonlinearity involved in
some of the horizontal models (with 2D position,
ground speed, course and transversal acceleration
as the state components) (see [209, Sec. V.B.2]).
The more efficient second-order-dependence model
(3) was employed to govern the mode transitions.
The comprehensive performance evaluation with
both simulation and real ATC data presented in
[60] demonstrated accurate state estimation, fast
response to mode changes, and high credibility
of the tracker. Overall the tracker outperformed
by far the existing ones based on a single model
(®-¯-° or EKF based). This IMM-based prototype
tracker has been implemented and installed in the
Air Traffic Management Surveillance Tracker and
Server (ARTAS) of Eurocontrol, which is operational
in many European countries and is the basis for the
Surveillance Data Processing and Distribution (SDPD)
system in Europe [136], [54].
Another comprehensive study of the capabilities of
the IMM algorithm for advanced ATC systems within
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the Hadamard project of the French civil aviation
administration was reported in [340], including
detailed design and evaluation of six different MM
configurations. It was found that the best trade-off
between performance and computation was achieved
by a two-model IMM configuration of a CV model
and a CT model with an unknown turn rate ([209,
Sec. VB]) as a state component and fictitious process
noise for the longitudinal acceleration. In contrast
to [60], no explicit model for the longitudinal
acceleration was included because it is rare and small
in civil aircraft motion. This two-model CV-CT
IMM configuration was shown to meet the rather
stringent requirements of the project very well. A
principal conclusion of this study was that “stringent
speed estimation can be obtained only with MM
algorithms.”
A detailed design and evaluation of an IMM
algorithm with a two-model (CV-CT) configuration
was given in [199] (see also [18] and [21]), along
with many guidelines and insights on parameter
selection and tuning of the algorithm. In particular, it
was demonstrated that the CT model is well suited to
the ATC applications and best results are obtained if it
is included as a maneuver model. An implementation
of a two-model (CV-CT) IMM algorithm was reported
in [161] for the Micro En Route Automated Radar
Tracking System (¹EARTS) that again showed the
superiority of the IMM estimator to single-model
(adaptive ®-¯ or EKF) based filters. The series
of papers [367, 345, 171] presented the work for
a large-scale (about 1000 targets) multisensor
multitarget tracking system for ATC surveillance,
which was developed into the software package
MATSurv (for “Multisensor Air Traffic Surveillance”)
[366]. It combines the IMM algorithm for state
estimation with assignment algorithms for radar data
association in a dense environment. This combination
is supported by the prior comparative study of
IMMPDA versus IMM-assignment on real ATC
surveillance data reported in [172]. More specifically,
[367] implemented an IMM configuration with
two second-order linear models. Reference [345]
reported a significant performance enhancement by
utilizing a nonlinear CT model in the (CV-CT) IMM
configuration, with an improvement of 10—50% in the
horizontal prediction errors on real data obtained from
five ATC radars. Additionally, the IMM configuration
facilitated data association better than the Kalman
filter. More results on real and simulated data were
presented in [171], along with a discussion of
parallelized algorithms with superlinear speedup for
multitarget tracking using the IMM estimator. Other
references addressing the design of IMM tracking
filters for ATC surveillance include [342] and [127].
2) Defense Applications: Compared with the
ATC applications, which involve relatively benign
maneuvers, tracking hostile or noncooperative targets,
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such as evasive manned aircraft, can be significantly
more difficult and challenging. For example, these
targets possess very strong maneuverability, often
not well known to trackers; their motion behavior
is quite unpredictable without sufficient knowledge
of their types, missions, tactics, etc.; they may
apply countermeasures to degrade the quality of
measurements and hamper tracking efforts. The
good news is that data quality and rates usually
are significantly higher than in the ATC case. As
evidenced by the vast majority of studies, the MM
approach appears to be the most powerful framework
available, capable of meeting the challenges of
maneuvering target tracking in a feasible way.
a) Benchmark tracking problems: Several
benchmark problems were initiated in 1994 for a
unified performance evaluation and fair comparison of
tracking algorithms using a phased-array radar, which
will be discussed in greater detail in a subsequent
part. Comparative studies and designs of a variety
of tracking algorithms were reported for the first
benchmark [40, 48] and second benchmark [47,
45, 49]. They suggested that among all solutions
proposed only the IMM algorithm was able to handle
satisfactorily the wide range of maneuver scenarios,
varying from moderate 2—3 g turns of cargo aircraft
to intense series of severe 5—7 g turns of fighter
aircraft [44, 86 355, 168, 35 (using 3 models), and
155 (using 2 models)]. Some of these performance
studies were verified by real experiments. The IMM
design of [355] used CV, constant-thrust, and 3D CT
models. The constant-thrust model was implemented
adaptively within the standard CA filter by correcting
the predicted acceleration vector (before and after
mixing) so as to make it parallel to the predicted
velocity vector. In a similar manner the predicted
acceleration vector of the 3D CT filter was made
perpendicular to the predicted velocity and the
speed was kept “nearly” constant by means of the
kinematic-constraint technique of [4]. The main
approaches to the second benchmark problem [50] all
use the IMM algorithm as a base-state estimator. They
include the one in [355] that combines IMM with
the integrated PDA filter, the IMMPDA solutions of
[167], and the IMM-MHT solution of [36]. Reference
[167] used three coordinate-uncoupled models: a CV
model with low process noise for benign motion,
a CV model with high process noise for ongoing
maneuver, and a CA model with high process noise
for maneuver onset/termination. Reference [36]
employed a horizontal CT model with polar velocity
(see [209, Sec. V.B.2]), a CV model, and a CA model,
where the CA and CT models allow altitude changes
(see also [35] and [68]). Generally speaking, the
IMMPDA solution reduced more radar time while the
IMM-MHT reduced more radar energy. More recently
[298] presented comparative results for the second
benchmark problem between maneuvering (i.e., MM)
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OCTOBER 2005
PMHTs (see Section VB5) showing that the PMHT
performed reasonably well, almost as well as the
above two solutions. The use of the IMM algorithm in
the subsequent benchmark problems as a base tracking
filter appears beyond question. Additional information
for adaptive sampling and waveform selection in
phased-array radars using the IMM algorithm can be
found in [313], [194], [213], and [316].
b) Ballistic targets: In recent years the IMM
algorithm proved very useful for another practical
problem of vital importance–tracking of tactical
ballistic missiles (TBMs) in all flight phases:
boost (including postboost), coast (free flight), and
reentry (possibly maneuverable). The motion of
a TBM is much more constrained than that of a
manned maneuvering aircraft and can be modeled
relatively well during any particular flight phase
(mode) (see [205]). In contrast to the hard decision
based AMM approach of [104] and [103] (see
Section IVE), various IMM-based algorithms, which
make soft decisions, have been proposed to avoid the
deficiencies associated with a hard decision [322, 257,
39, 141, 142, 80]. The IMM algorithm was employed
in a prototype system for TBM in [322]. Four models
were used: specialized boost, coast, and reentry
models, and an auxiliary “general-purpose” CA model
intended to provide a “back-up” estimate to the other
filters (through the mixing mechanism) in cases when
the specialized models are inadequate (e.g., due to
unexpected maneuvers, such as trajectory corrections,
retargeting). A critical issue in the MM tracking of
ballistic targets is to design the transition probabilities
since they are time varying, because the possible
transitions are strongly dependent on the current
mode of flight. This dependence was accounted for in
[322] by switching among five transition probability
matrices depending on the estimated target altitude
and flight phase. For example, the boost and coast
models/filters are dropped when reentry phase is
established. Another major issue, mixing of different
target states/covariances, was avoided by mixing only
the common position and velocity components. The
implementations of [39], [257], and [141] considered
only the boost and coast phases. Reference [39]
used a three-model configuration with CV for coast
phase, CA for a “generic” filter, and a detailed flight
dynamics based ten-state model for boost phase [205].
Specific for this implementation is the unconventional
ad hoc state/covariance mixing, allowed only between
the boost and CA models, and between the coast
and CA models. This seems to make sense if the
CA filter is accurate enough in all conditions to
provide a backup in case the boost or coast filter
does not perform well due to a mode change. For a
similar implementation (2-model boost-coast IMM)
[257] proposed and analyzed different time-varying
distributions of the boost-to-coast transition based
on a predicted burnout time and the uncertainty of
this prediction. The study showed that the 2-model
IMM algorithm is able to “detect” the burnout and
provide highly accurate estimates shortly thereafter.
No rocket staging, however, was allowed in this study.
It seems that a CA or other (e.g., correlated) generic
filter for acceleration would help cope with possible
staging. For modeling the boost-to-coast transition in a
two-model IMM configuration, [141, 142] proposed
a sigmoidal function for the transition probability,
depending on the estimated altitude and a prior i
altitude at which the booster cutoff is likely to occur.
Performance comparison of this version of an IMM
algorithm having constant transition probabilities with
the EKF and ®-¯-° trackers over simulated and real
data demonstrated its superior capabilities provided
the parameter guess does not mismatch the truth by
far. Another implementation of IMM for tracking of a
TBM in the entire flight was given in [80].
c) GPB1 applications: The early paper [267]
formulated the GPB1 algorithm for a semi-Markov
mode sequence, but one with exponential distributed
sojourn times was simulated, which is actually
Markov. Reference [260] applied this algorithm
to passive tracking of a submarine with vertical
maneuvers. By quantizing the unknown input into
several known levels, the GPB1 algorithm reduces
to a single Kalman filter with a probabilistically
weighted input for a linear target motion. Submarine
tracking was studied further in [262], [263], [261],
and [264] by GPB1 tracking in range, velocity,
and depth using passive time delay measurements.
References [123] and [266] presented detailed
GPB1 designs for realistic scenarios of 3D manned
maneuvering aircraft tracking. A Singer model with
several known quantized mean levels of acceleration
was employed for an MM description of the target
dynamics and the target maneuver was modeled
by (semi-) Markov transitions between the levels.
Versions in rectangular and spherical coordinates
were developed and investigated, showing a good
accuracy and filter stability over a wide range of target
acceleration, unreachable by a conventional single
(e.g., EKF) filter. Other related implementations and
applications of the GPB1 algorithm can be found in
[341], [265], and [71]. Reference [160] proposed a
six-model GPB1-type MM tracker for a maneuvering
reentry vehicle (MaRV) that quantizes the acceleration
vector to represent the possible maneuvers, left/right
turns, climb/dive, and deceleration within the MaRV
model [75, 205].
3) Tracking in Presence of Correlated Noise,
Glint, or Multipath:
a) Correlated noise: Radar tracking at high
sampling rate leads to significant temporal correlation
of the measurement errors, which degrades the
performance of those trackers relying on whiteness
of the measurement noise. Techniques for handling
correlated measurements within the IMM framework
LI & JILKOV: SURVEY OF MANEUVERING TARGET TRACKING. PART V: MULTIPLE-MODEL METHODS
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were proposed in [128] and [361]. The approach of
[128] is based on modeling the errors as a first-order
Markov process with known coefficients. After
decorrelation using the standard measurement
differencing method, the IMM algorithm was applied
straightforwardly (see also [246] for a discussion).
Reference [361] proposed a more general technique
that performs the decorrelation in an adaptive manner
within the IMM framework without the assumption of
known correlation parameters. It was demonstrated
that this adaptive version and the one with known
correlation parameters have similar performance.
b) Glint: When tracking large targets at
short distance, the resulting radar measurement
errors (known as glint noise) is characterized by
non-Gaussian distributions with a heavy tail due to
the interference caused by reflections from different
elements of the target. The presence of glint can
seriously degrade tracking performance if white
Gaussian measurement errors are assumed by the
tracker. Modifications of the Kalman filter capable of
accounting for glint can be found in [134], [239], and
[240] (see also [246]). A good approximate model for
the distribution of a glint is a mixture of a Gaussian
with a moderate variance and a Laplacian distribution
with a large variance and a small weight [360]. This
model has been generally accepted now. It was used
in [360] to develop a tracking filter implementing
the Masreliez filter for non-Gaussian noise [239]
for the approximate spherical target-measurement
model of [123]. In the context of maneuvering
target tracking, this approach was extended in [362],
where a two-model IMM configuration with two
modified23 Masreliez filters (instead of Kalman filters)
was proposed. A different approach was suggested
in [84] and [85]. Two measurement models were
included in the IMM design–one matched to the
system with Gaussian observation noise and the
other to the system with Laplacian noise of a large
variance. Conditional filtering was implemented in
[84] by EKFs for both models.24 Further, to handle
the maneuvering target case, the IMM design was
expanded in [85] with a (Singer) maneuver model.
Under the assumption that the model sets describing
the target motion and measurement noise, respectively,
are independent,25 a “layered” version of the IMM
algorithm was developed, which has computational
23 Since the Masreliez filter requires in general performing a
convolution operation, an efficient approximation based on normal
expansion of the predicted measurement distribution was developed.
24 The use of an LMMSE-based EKF, rather than the MMSE filter,
for Laplacian noise was motivated by its great computational
advantage at an acceptable loss of accuracy as compared with the
exact nonlinear MMSE filter (derived therein).
25 Target motion and glint seem coupled due to the target attitude
changes during maneuver on/off that could cause glint to
appear/disappear or change.
1290
advantages. A somewhat similar approach was
followed in [373]. The performance evaluation
over two scenarios from the benchmark problem of
[49] showed a significant advantage of this layered
IMM over the algorithm of [362] in terms of noise
reduction, faster response to mode changes, and better
mode identification. A possible enhancement here is
to replace the EKFs in [84] by better filters, such as
those based on measurement conversion (see [206])
or the approximate best linear unbiased filters for
polar/spherical measurements of [372]. For a 2D
homing missile scenario, [317] implemented an IMM
algorithm with two decoupled models for range and
bearing in Gaussian and Laplacian noise, respectively,
as in [84]. In this setting the measurement equations
are linear. Another study of target tracking in glint
was presented in [331]. It employed mixture reduction
techniques [300] for MM estimation.
c) Multipath: The multipath propagation effects
arise in radar or sonar tracking especially when the
target is in a close vicinity of a reflecting surface.
For example, due to the combination of the return
from a low elevation target and sea-surface reflected
returns, the measurement error can be huge relative
to the “normal” ones. The effect is very complex
and may be devastating to a tracker that assumes
normal and uncorrelated measurement errors. An
elegant solution to this problem based on the IMM
method was proposed in [17]. As shown therein
analytically, this is essentially a hybrid estimation
problem due to the jumpwise behavior of the
multipath error process arising on top of the standard
measurement error process. It was demonstrated that
the multipath effect can be successfully alleviated by
an IMM mechanism with a “no multipath” model
and a first-order autocorrelated multipath model,
without the need for a detailed physical model.
Somewhat related, unknown noise can be identified
by an IMM algorithm, as proposed in [200] and
[20].
VI. THE THIRD GENERATION:
VARIABLE-STRUCTURE MM ESTIMATION
A. Theoretical Foundation of VSMM Estimation
The first two generations have a fixed structure
(FS) in the sense that they use a fixed set of models
throughout the time as reflected by Assumption A2.
The third generation abandons A2, resulting in having
a variable structure, hence the name variable-structure
MM (VSMM) estimation. Although not necessary,
many VSMM algorithms rely on Assumption A10
that the mode sequence is Markov or semi-Markov,
mostly because the second-generation algorithms serve
as building blocks of the VSMM algorithms.
State Dependency of Mode Set: A key concept in
VSMM estimation is the state dependency of a mode
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OCTOBER 2005
set [191, 201, 192]. Simply put, given the current
mode (and base state), the set of possible modes at the
next time is a subset of the mode space, determined
by the mode transition law. Consider tracking a
car with three models: straight (m(1) = 1), left turn
(m(2) = 2), and right turn (m(3) = 3). Initially it goes
straight on a street at k = 1; it arrives at a four-way
intersection at k = 10, where it could go straight or
take a left or right turn; at k = 11, if it took a left turn
at k = 10 the car could either go straight or continue
the left turn (making it a U turn); then it goes straight
until it enters into an open space at k = 20, where any
motion-mode could be converted to any other. The
state-dependent mode sets through k = 20 are
i = 1, 2, 3:
The sequence of possible mode sets through k = 20 is
S20 = fS1 , : : : , S10 , S11 , S12 , : : : , S20 g
= ff1g, : : : , f1, 2, 3g, f1, 2g, f1g, : : : , f1, 2, 3gg
S
where Sk = i S(i)
k is the union of state-dependent
mode sets at k. Note that the set of possible modes at
time k depends on mode sk¡1 in effect at k ¡ 1 and the
base state xk¡1 and xk . It was shown in [191], [201],
and [192] that an MM estimator cannot be optimal if
at some time k it uses a model set Mk different from
the mode space Sk . As such, the use of a fixed model
set, say, M = f1, 2, 3g, is clearly not preferable for this
example.
The second generation abandons the constant
mode assumption of the first generation; instead
it imposes a Markov-type property on the mode
sequence. Somewhat similarly, the third generation
abandons the constant mode-space assumption of
the first two generations and explores the state
dependency of the mode set.
Clearly, the state dependency of the mode set
cannot be described by the mode set itself. That
is why a graph-theoretic formulation of the MM
estimation was proposed in [198], [201], and [192],
where a mode and a possible transition from one
mode to another are represented by a node and a
directed edge, respectively, resulting in a directed
graph (digraph) as a representation of a mode set and
its associated state dependency. This formulation has
certain advantages, as elaborated in [201] and [192],
and is the basis of a class of VSMM algorithms
[215, 195, 346].
Optimal VSMM Estimation: As presented in [198],
[191], [201], and [192], the MMSE-optimal VSMM
estimator is given by
x̂kjk = E[E[xk j Sk , z k ] j z k ] =
X
Sk
k
(S )
PfSk j z k g
x̂kjk
X (Sk )
(Sk )
(Sk ) 0
[Pkjk + (x̂kjk ¡ x̂kjk
)(x̂kjk ¡ x̂kjk
) ]PfSk j z k g
Sk
(42)
(Sk )
x̂kjk
(Sk )
Pkjk
and
are the optimal estimate and its
where
MSE matrix respectively at time k assuming that the
true mode-set sequence is Sk , given by
k
(S )
= E[E[xk j sk , Sk , z k ] j Sk , z k ]
x̂kjk
=
X
(s )
Pfsk j Sk , z k g
x̂kjk
X
(S )
(s )
(S )
(s ) 0
(s )
[(x̂kjk
¡ x̂kjk
)(x̂kjk
¡ x̂kjk
) + Pkjk
]Pfsk j Sk , z k g
sk 2Sk
(Sk )
Pkjk =
sk 2Sk
k
k
k
k
k
k
k
(1)
(1)
(1)
S(1)
1 = ¢ ¢ ¢ = S9 = f1g, S10 = f1, 2, 3g, S11 = f1g
(3)
(i)
S(2)
11 = f1, 2g, S11 = f1g, : : : , S20 = f1, 2, 3g,
Pkjk =
(41)
(s )
where x̂kjk
is the optimal estimate at time k assuming
(sk )
the true mode sequence is sk , and Pkjk
is its MSE
matrix. The summations in (41)—(42) are over all
mode-set sequences such that every possible mode
sequence is in one and only one Sk . Note, however,
that a mode-set sequence may contain more than one
possible mode sequence.
The optimal VSMM estimator in the MAP sense is
given by x̂kjk = arg maxxk f(xk j z k ), where
X
f(xk j Sk , z k )PfSk j z k g
f(xk j z k ) =
Sk
=
XX
Sk sk 2Sk
f(xk j sk , z k )Pfsk j Sk , z k gPfSk j z k g
is a mixture density, each component f(xk j Sk , z k ) of
which is itself a mixture density.
RAMS Approach: The optimal VSMM estimator
is computationally infeasible. For most applications,
its higher level with multiple model-set sequences
should be replaced, due to limited computational
resources, by a single (hopefully “best”) model-set
sequence, obtained in practice through model-set
adaptation in a recursive manner. This is the recursive
adaptive model-set (RAMS) approach [196, 198, 191,
201, 195]. In general, each recursion of a RAMS
algorithm has two tasks.
1) Model-set adaptation determines at each time
the model set to use for the MM estimation, utilizing
posterior information contained in the data as well
as prior knowledge. This is unique for VSMM
estimation. Different RAMS algorithms differ from
each other primarily with respect to how the model-set
adapts.
2) Model-set sequence conditioned estimation
intends to provide best possible estimates given a
model-set sequence. It consists of a) initialization–
assign initial probabilities to new models and initialize
the filters based on them–which is absent in the first
two generations, and b) cooperation strategies and
conditional filtering, similar to those of the first two
generations.
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Model-Set Adaptation: Model-set adaptation can
be decomposed as model-set expansion and model-set
reduction [196, 195]. This decomposition has
several significant advantages over naive model-set
switching in terms of tractability, performance, and
generality [196, 224, 195]. Model-set expansion
is often more important than model-set reduction:
Although inclusion of an impossible model could be
as bad as missing a possible model, the performance
of an MM algorithm will suffer greatly if a highly
likely model is missed, but only slightly if a highly
unlikely model is included. As a result, a delay in
including the correct model will always result in
significant performance deterioration, while a delay in
terminating an incorrect model usually does not incur
great performance loss if the correct model is in the
set. Unfortunately, model-set expansion is in general
much more difficult than model-set reduction.
Both expansion and reduction of a model-set
require two functional tasks: model-set “candidation”,
which determines candidate sets for expansion or
reduction, and model-set decision, which selects and
retains the best candidate set(s). Model-set candidation
for expansion amounts to activation or generation
of a set of new models, which is the main task of
each model-set adaptation algorithm, discussed in
Section VID. This candidation is much easier for
model-set reduction than expansion.
model-set M1 to M2 , which is equal to the product
of model-set marginal likelihood ratios under mild
conditions [196]
B. Model-Set Decision Given Candidate Sets
in a sequential setting, where their union M =
M1 [ M2 is used before a decision is made. Note
that M1 and M2 may include common models. This
problem is solved optimally in [196], [195], and
[214] by the following model-set sequential likelihood
ratio test (MS-SLRT) for some thresholds A and B:
choose M1 when ¤k ¸ B; choose M2 when ¤k · A;
otherwise use M, go to the next time cycle, ask for
one more measurement, and continue to test. This test
is optimal in the sense of making quickest decisions
with guaranteed conditional decision error bounds
Model-set decision may be formulated as a
statistical decision problem, in particular, a problem
of testing statistical hypotheses in a sequential setting,
which is natural since observations are available
sequentially, and beneficial in terms of decision delay
and threshold determination. As hypotheses are always
assumed fixed in the standard theory of hypothesis
testing, in this subsection, the true mode s in effect
and the model sets are assumed constant during the
time period over which the test is performed.
Model-Set Likelihood and Probability: Since
the task is to decide on the right model set, the
probabilities and/or likelihoods of the model sets
involved are naturally of major interest. The marginal
likelihood of a model-set M at time k is the sum of
the predicted probabilities Pfmk(i) j s 2 M, z k¡1 g times
the marginal likelihoods f[z̃k j s = m(i) , z k¡1 ] of all the
models m(i) in M [196, 224, 195]:
¢
k¡1
]
LM
k = f[z̃k j s 2 M, z
X
=
f[z̃k j s = m(i) , z k¡1 ]Pfs = m(i) j s 2 M, z k¡1 g
m(i) 2M
where z̃k is the measurement residual. The joint
¢
likelihood of the model-set M is defined as LkM =
f[z̃ k j s 2 M]. Let ¤k be the joint likelihood ratio of
1292
¢
¤k =
LkM1
LkM2
=
Y LM1
·
k0 ···k
2
LM
·
where k0 is the test starting time. The (posterior)
probability that the true mode is in M is defined by
¢
k
¹M
k = Pfs 2 M j s 2 M, z g
X
X (i)
=
Pfmk(i) j s 2 M, z k g =
¹k
m(i) 2M
m(i) 2M
which is the sum of the probabilities of all modes
in M, where M is the union of all model sets under
consideration, including M as a subset. The mode
probability ¹(i)
k is available from an MM estimator
using M.
Several hypothesis tests were proposed in [196],
[195], and [214] and applied in [224], [219], [195],
[204], [203], [222], and [346] for model-set decision
given candidate sets based on model-set likelihoods or
probabilities.
Model-Set Decision Given Two Model Sets:
Assume s 2 M. Consider the problem of choosing
between two model sets M1 and M2 , that is, testing
H1 : s 2 M1
versus H2 : s 2 M2
PfChoose M2 j s 2 M1 g · ®,
PfChoose M1 j s 2 M2 g · ¯,
0 < ®, ¯ < 1
for any given ® and ¯. Replacing the likelihood ratio
M2
1
¤k in the above by the probability ratio P k = ¹M
k =¹k
yields the model-set sequential probability ratio test
(MS-SPRT) [196, 195, 214]. It is optimal in the sense
of making quickest decisions with guaranteed joint
decision error bounds
PfChoose M2 , s 2 M1 g · ®,
PfChoose M1 , s 2 M2 g · ¯,
0 < ®, ¯ < 1:
The thresholds A and B are given approximately by
A = ¯=(1 ¡ ®), B = (1 ¡ ¯)=®. Clearly, MS-SLRT
and MS-SPRT can be used to answer such important
questions as “Which model set is better to use, M1
or M2 ?” and “Is it better to delete a subset M1 from
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
the current model set M?” They are also basis for
solutions of problems involving more than two model
sets.
Model-Set Decision Given More Than Two Model
Sets: Consider the problem of whether it is better to
add one of the model sets M1 , : : : , MN to the current
set M. This can be formulated as testing
H0 : s 2 M versus H1 : s 2 M1 ¢ ¢ ¢ versus HN : s 2 MN
in a sequential setting, where M is used before a
decision is made. The following multiple model-set
sequential likelihood ratio test (MMS-SLRT) was
proposed in [196], [195], and [214] as a solution to
this problem.
S1. Perform N MS-SLRTs simultaneously for
N pairs of hypotheses (H0 : s 2 M versus H1 : s 2
M1 ), : : : , (H0 : s 2 M versus HN : s 2 MN ). These tests
are one-sided in the sense that H0 is never rejected,
which is implemented by using thresholds B =
(1 ¡ ¯)=® (or B = 1=®) and A = 0. This step ends
when only one of the hypotheses H1 , H2 , : : : , HN
remains. Specifically, reject all Mi for which
¢
¤k = LkM =LkMi ¸ B and continue to the next time
cycle to test for the remaining pairs with one more
measurement until only one of the hypotheses
H1 , H2 , : : : , HN , say Hj , is not rejected.
S2. Perform an MS-SLRT to test H0 : s 2 M
versus Hj : s 2 Mj , where Hj is the winning hypothesis
in S1.
With a slight modification, this test can be used to
answer other important questions, such as “Is it better
to delete one of the model sets M1 , : : : , MN from the
current set M?”.
Probably the most versatile test developed so far
for model-set decision is the mode-set probability
sequential ranking test (MSP-SRT) [196, 195, 214]. It
is based on ranking of mode-set probability: At each
time k, rank all Nk of the model sets M1 , M2 , : : : , MN
that have survived (i.e., not yet rejected or accepted)
by time k as M(1) , M(2) , : : : , M(Nk ) such that their
M(i) ¢
mode-set probabilities ¹k
decreasing order:
M(1)
¹k
M(2)
¸ ¹k
= Pfs 2 M(i) j z k g are in a
M(Nk )
¸ ¢ ¢ ¢ ¸ ¹k
:
Then, a sequential decision is made by comparing a
mode-set probability ratio P k with a pair of thresholds
M
A and B, where P k = ¹k (i) =¹M
k if the current model set
M is involved, such as to answer the question “is it
better to add/delete some (unknown number) of the
model sets M1 , M2 , : : : , MN to/from M,” otherwise,
M
M
P k = ¹k (i) =¹k (1) , such as for the problems of choosing
one, L (known), or some (unknown) out of the model
sets M1 , M2 , : : : , MN .
Alternative solutions were also presented in [196]
and [195], where the MS-SLRTs and model-set
likelihood ratio ¤k in MMS-SLRT are replaced
by MS-SPRTs and mode-set probability ratio P k ,
respectively, and P k in MSP-SRT are replaced by ¤k .
In addition, a so-called multiple-level test was also
presented in [196]. See [196], [195], and [214] for
more details, along with simulation results of testing
model sets with some simple models typically used in
maneuvering target tracking.
These tests are general, intuitively appealing,
computationally efficient, and easy to implement
because they use only model-set likelihoods or
probabilities, which are available in MMSE-based
MM algorithms if the model sets are already
used. Note that an adaptation of the model set is
accomplished whenever a model set other than the
current one is accepted. As such, model-set adaptation
requires a series of hypothesis tests.
C.
MM Estimation Given Model-Set Sequence
In this subsection, it is assumed that the sequence
of model sets has been determined by, say, model-set
adaptation. For simplicity, Mk and Mk are also
used to denote the events fsk 2 Mk g and fsk 2 Mk g,
respectively, and a perfect match between modes and
models is assumed.
Initialization of New Models and Filters: A model
is a new one if it is in Mk but not in Mk¡1 . Two
important questions arise naturally: 1) How to assign
initial probabilities to the new models? 2) How to
obtain initial estimates and error covariances for the
filters based on the new models? Answers to these
questions are essential for the implementation of any
MM algorithm of a truly variable structure. Many
heuristics and ad hoc treatments have appeared in the
literature. In fact, the key to the optimal initialization
of new models and the corresponding elemental filters
is the state dependency of the mode set, explained
in Section VIA. As applied to model and filter
initialization here with a single state estimate and
mode probability, the optimal assignment of the initial
probability to a new model accounts only for the
probabilities of those models that may switch to it;
and the optimal initial state estimate for a filter based
on a (new or old) model is determined only from
the estimates (and the probabilities) of the filters
based on those models that may switch to the new
model.
After writing down formulas for the optimal
initialization, it can be recognized that they are similar
to those in the model-conditioned reinitialization
(mixing) step of the IMM estimator (see Table II).
This recognition leads to the VS-IMM recursion
(Table III), presented in [193] and [195]. It gives a
generic recursion for VSMM estimation based on a
time-varying model set. It was shown in [193] that the
VS-IMM recursion is optimal in the MMSE sense for
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1293
TABLE III
VS-IMM Recursion
1. Model-set conditioned (re)initialization [8m(i) 2Mk ]:
¢
P
predicted mode probability:
= Pfmk(i) jMk ,Mk¡1 ,z k¡1 g=
¹(i)
kjk¡1
mixing weight:
¹k¡1 = Pfmk¡1 j mk(i) ,Mk¡1 ,zk¡1 g=¼ji ¹(j)
=¹(i)
k¡1 kjk¡1
mixing estimate:
(i)
= E[xk¡1 j mk(i) ,Mk¡1 ,z k¡1 ]=
x̄k¡1
mixing covariance:
(i)
P̄k¡1
=
¢
jji
(j)
¢
P
(j)
m(j) 2Mk¡1
m(j) 2Mk¡1
P
(j)
k¡1
¼ji ¹
jji
(j)
¹
x̂
m(j) 2Mk¡1 k¡1jk¡1 k¡1
(j)
(j)
jji
(i)
(i)
[Pk¡1jk¡1 + (x̄k¡1
¡ x̂k¡1jk¡1 )(x̄k¡1
¡ x̂k¡1jk¡1 )0 ]¹k¡1
2. Model-conditioned filtering [8m(i) 2Mk ]:
¢
predicted state:
(i)
(i) (i)
x̂kjk¡1
= E[xk j mk(i) ,Mk¡1 ,z k¡1 ]=Fk¡1
+G(i) w̄(i)
x̄
k¡1
k¡1 k¡1
predicted covariance:
(i)
(i)
(i)
(i) 0
(i)
(i)
(i) 0
P̄k¡1
Pkjk¡1
= Fk¡1
(Fk¡1
) + Gk¡1
Qk¡1
(Gk¡1
)
measurement residual:
(i)
z̃k(i) = zk ¡ E[zk j mk(i) ,Mk¡1 ,zk¡1 ]=zk ¡Hk(i) x̂kjk¡1
¡v̄ (i)
k
residual covariance:
(i)
Sk(i) = Hk(i) Pkjk¡1
(Hk(i) )0 + Rk(i)
filter gain:
(i)
Kk(i) = Pkjk¡1
(Hk(i) )0 (Sk(i) )¡1
updated state:
(i)
(i)
x̂kjk
= E[xk j mk(i) ,Mk¡1 ,zk ] =x̂kjk¡1
+K (i) z̃ (i)
k k
updated covariance:
(i)
(i)
Pkjk
= Pkjk¡1
¡ Kk(i) Sk(i) (Kk(i) )0
¢
¢
3. Mode probability update [8m(i) 2Mk ]:
¢
model likelihood:
= p[z̃ (i) j mk(i) ,Mk¡1 ,z k¡1 ]
L(i)
k
mode probability:
¢
¹(i)
= Pfmk(i) jMk ,Mk¡1 ,zk ]=
k
4. Fusion:
overall estimate:
overall covariance:
¢
x̂kjk = E[xk jMk ,Mk¡1 ,zk ] =
Pkjk =
P
m(i) 2Mk
E[xk j mk(i) , z k ] = E[xk j mk(i) , Mk¡1 , z k ]
8 m(i) 2 Mk
and the linear-Gaussian assumption of the Kalman
filter given the system mode.
The VS-IMM recursion automatically initializes
all new models and filters “optimally”: All new
models are assigned the optimal initial probabilities
and the filters based on these models are initialized
with the “optimal” initial conditions (estimates and
error covariances). This VS-IMM recursion is almost
identical to one cycle of the IMM algorithm (compare
Tables III and II). It is a natural extension of the
IMM algorithm given a time-varying model set. It is
extremely useful for VSMM estimation because of its
cost-effectiveness, efficiency, and applicability. Other
than the model sets Mk and Mk¡1 and the transition
law between their models, it requires exactly the same
thing as the IMM algorithm does. This recursion is
used in most VSMM algorithms developed so far.
Another nice feature of the VS-IMM recursion is that
as shown in [193], it uses the transition probabilities
1294
=
P
N (z̃ (i) ;0,S (i) )
k
k
¹(i)
L(i)
kjk¡1 k
m(j) 2Mk
¹(j)
L(j)
kjk¡1 k
P
x̂(i) ¹(i)
m(i) 2Mk kjk k
(i)
(i)
(i) 0 (i)
[Pkjk
+ (x̂kjk ¡ x̂kjk
)(x̂kjk ¡ x̂kjk
) ]¹k
a Markov jump-linear system under the following two
fundamental assumptions of the RAMS approach (of
zero depth):
Pfmk(i) j Mk , z k g = Pfmk(i) j Mk , Mk¡1 , z k g,
assume
¼ij with respect to the total set M, rather than Mk
or Mk¡1 . Were this not true, each possible model set
would require a distinct design of the corresponding
set of transition probabilities.
Fusion of Two MM Estimates: A question
important for MM estimation is the following: Given
two separate MM estimates based on two model sets,
respectively, how to obtain the estimate based on all
models in these two sets? For example, a model-set
adaptation algorithm may decide to add a set M2 of
models to the current model set M1 after the estimates
based on model set M1 have been obtained. The
solution to this problem is the following optimal
fusion rule, presented in [193] and [195]. Consider
two optimal MM estimators based on a common
model-set history Mk¡1 but two distinct model sets
M1 and M2 at time k, respectively:
(i)
(i) (i) (i)
fx̂kjk
, Pkjk
, Lk , ¹kjk¡1 gm(i) 2M1
(i)
(i) (i) (i)
fx̂kjk
, Pkjk
, Lk , ¹kjk¡1 gm(i) 2M2
(i)
where L(i)
k and ¹kjk¡1 are model likelihood and
predicted model probability respectively of model
m(i) in set M1 or M2 . It was shown in [193] that
the optimal MM estimator based on model set M =
M1 [ M2 at time k and a common history Mk¡1 is
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given by
x̂kjk =
X
(i) (i)
¹k
x̂kjk
X
(i)
(i)
(i)
[Pkjk
+ (x̂kjk
¡ x̂kjk )(x̂kjk
¡ x̂kjk )0 ]¹(i)
k
m(i) 2M
Pkjk =
m(i) 2M
where
¢
(i)
¹(i)
k = Pfmk
k¡1
j M, M
k
,z g = P
(i)
L(i)
k ¹kjk¡1
(i) (i)
m(i) 2M Lk ¹kjk¡1
:
Note that for a common model m(i) of the two sets,
(i)
(i) (i) (i)
its x̂kjk
, Pkjk
, Lk , ¹kjk¡1 are identical for the two MM
estimators (i.e., they do not depend on which model
set is used at k). If M1 and M2 have a common
model m(j) , then ¹(i)
k above can be obtained from all
(ijM1 ) ¢
= Pfmk(i) j sk 2 M1 , z k g and
model probabilities ¹k
(ijM2 ) ¢
¹k
= Pfmk(i) j sk 2 M2 , z k g of the two MM estimators
(i)
directly without knowledge of L(i)
k and ¹kjk¡1 :
¹(i)
k =
1 (ijM1 )
,
¹
® k
(jjM1 )
¹(i)
k =
¹k
(jjM )
®¹k 2
8 m(i) 2 M1
(ijM2 )
¹k
,
8 m(i) 2 M2
where
®=
X
(ijM1 )
¹k
m(i) 2M1
(jjM1 )
+
¹k
X
(jjM )
¹k 2 m(i) 2(M2 ¡M1 )
(ijM2 )
¹k
:
Most VSMM algorithms developed so far, including
those of [224], [219], [215], [195], [207], [212], and
[346], use this optimal fusion rule.
D. VSMM Algorithms
We believe that MM estimation will eventually
develop into one of a “kit of tools,” represented by
various FS and VS algorithms. Development of good
model-set adaptation algorithms is perhaps the most
important task in VSMM estimation. As stated before,
model-set adaptation consists of model-set candidation
and model-set decision given candidate sets. Very
general and satisfactory results have been obtained for
model-set decision, but no such results are available
or even in sight for model-set candidation, which as
a result, becomes the main task for each individual
model-set adaptation algorithm. In other words,
different VSMM algorithms may have the same
procedure to select the best set from the candidate
sets but they differ from one another primarily in
model-set candidation, namely, how the candidate sets
are determined.
An adaptive structure is a variable structure in
which the structure varies via adaptation in real time.
Many adaptive structures are possible. They can be
classified into two broad families, active model-set and
model-set generation [195], depending on whether the
total model-set (i.e., the set of all possible models) can
be specified as a finite set in advance or not.
Active Model-Sets: In the active model-set
family [195], the total model-set is finite and can
be determined in advance before any measurement
is received. At any given time it uses an active or
working subset of the total model-set determined
adaptively, hence the name. Its underlying idea is
somewhat similar to that of the active-set method
for constrained optimization problems: At each time
some models may be terminated and others may be
activated.
As outlined in [198], [201], and [192], model-set
switching is one of the simplest classes of active
model-set structures in which the active set is
determined by switching among a number of
predetermined subsets of the total model-set. These
subsets are the candidate sets for model-set adaptation.
The switching can be soft as well as hard, similar
to soft and hard decision for output processing
(see Section IVC). The soft switching assumes
that each predetermined subset at any time has
a certain probability of having a member model
matching the true mode [224]. A hard switching is
one based on a set of “hard” rules. The key task with
this class of structures is the design of the model
subsets, determination of the candidate subsets, and
decision procedure for switching. Such a structure,
called model-group switching (MGS) algorithm, was
presented in [224] and [218] with a comprehensive
design example given in [219] and [217], where each
“group” represents a certain cluster of closely related
system modes, hence the name. This MGS algorithm
uses a two-stage hard switching: In addition to the
current group, a candidate group is activated first if
deemed appropriate by a hard rule, the union of the
groups is run, with the help of the optimal fusion rule
of two MM estimates in Section VIC, until a decision
is made between the two groups by the sequential
tests of Section VIB. It runs only one group most
of the time and thus provides a substantial saving
in computation over an FS algorithm using the total
model-set, as demonstrated in [219], [217], and [195].
Different groups may have common models, which
facilitate group switching and initialization of newly
activated filters. Several designs of the model subset
switching (see, e.g., [226], [149], [150], [219], [217],
[170]) have been reported for maneuvering target
tracking, along with illustrations of their superior cost
effectiveness to the FS-IMM algorithm.
Another simple class of active model-set structures
is called likely-model set (LMS) structure, outlined
in [198], [201], and [192]. Simply put, its active
set is formed by deleting the models in the total
set that are unlikely to match the true mode at the
given time. In order to follow the true mode that
may jump, it must have a mechanism of expanding
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the active model set, i.e., determination of the
candidate sets. There are various possible ways of
expansion. One of the most natural ideas is based
on the concept of the state dependency of the mode
set (see Section VIA) that given the current mode the
set of possible modes at the next time is a subset of
the mode space, determined by the mode transition
law (i.e., the adjacency relations of the modes). A
simple implementation is the following, resulting in
the so-called LMS algorithm [215, 195]. Identify
each model in the model set Mk¡1 in effect at time
k ¡ 1 to be unlikely (e.g., if its probability is below
a threshold t1 ), principal (if its probability exceeds
t2 ), or significant (if its probability is in between
t1 and t2 ). Then, the model set Mk can be obtained
as follows: 1) discard the unlikely ones; 2) keep
the significant ones; and 3) activate the models to
which the principal ones may switch directly. Clearly,
the model activation relies on the graph-theoretic
representation of the model set [198, 201, 195],
as briefly mentioned in Section VIA. The unlikely
models in Mk¡1 are those whose ratios of probability
to the largest mode probability are below a certain
threshold and can be eliminated following the
sequential ranking test of Section VIB. Alternatively,
a simpler but less accurate way is to delete all the
models in Mk¡1 except the B models of the largest
probabilities, where B is a constant, determined from
computational considerations. As demonstrated in
[215] and [195] for a maneuvering target tracking
example, this LMS algorithm is somewhat more
cost effective than the MGS algorithm of [224] and
substantially outperforms the FS-IMM algorithm. A
simplified version of the LMS idea was proposed in
[346], where the model set used at any time is the
state-dependent set of models that can be switched
from a principal model, called minimal submodel set
in [346], including the principal model itself–the
significant models are not necessarily kept and the
unlikely models are not necessarily deleted. The
principal model is identified as either the one with
the largest probability and likelihood or the one that
is closest to the estimated true mode at the time.
Adaptation of the model set then amounts to switching
among the state-dependent model sets, determined by
the sequential tests of Section VIB. It is substantially
more cost effective than the FS-IMM algorithm, as
demonstrated in [346].
Still another simple class is those with a
hierarchical architecture. The active set in this
hierarchical model-set structure consists of hierarchical
levels of models [73, 297]. The makeup (i.e.,
model subset) of a lower level as a candidate set is
determined under the guidance of the higher level(s).
An MM estimator typically operates at each level but
interactions among levels are generally beneficial
[297]. If some models that form one or more levels
are generated (instead of activated) in real time, the
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corresponding hierarchical structure may be deemed
to belong to the model-set generation family. Not
all hierarchical MM algorithms have an adaptive
structure. For example, those proposed in [118], [348],
[85], and [202] are hierarchical MM algorithms of
a fixed structure since the model set used is time
invariant, albeit of a hierarchical structure.
Model-Set Generation: In the model-set
generation family [195], new models are generated in
real time and thus it is impossible to specify the total
model-set as a finite set in advance.
A natural idea for model-set generation is to
augment the working set Mk of models by one (or
more) that match an estimate m̂k of the true mode
at time k, leading to the so-called estimated-mode
augmentation. The augmented model m̂k can be an
estimate of the mode under any optimality criterion in
principle, such as 1) the expected mode (conditional
mean)
MMSE
m̂kjk
= E[sk j sk 2 Mk , z k ]
X
=
m(i) Pfsk = m(i) j sk 2 Mk , z k g
m(i) 2Mk
resulting in the expected-mode augmentation [207,
ML
212], 2) the estimate m̂kjk
= arg maxm f(z k j sk = m)
that is the model with the largest likelihood, resulting
in the ML model augmentation [297], and 3) MAP
estimate m̂kMAP = arg maxm Pfsk = m j z k g, which is
the model with the largest posterior probability. A
promising alternative is to augment the model set also
by the predicted modes, such as
MMSE
m̂k+1jk
= E[sk+1 j sk 2 (Mk [ m̂kjk ), z k ]
ML
= arg max f(z k j sk+1 = m)
m̂k+1jk
m
MAP
m̂k+1jk
= arg max Pfsk+1 = m j z k g
m
to anticipate the next mode transition, leading to what
can be called predicted-mode augmentation. As shown
in [207] and [212] theoretically, such an augmentation
improves accuracy of MM estimation, which is
supported by the demonstrations given in [186], [207],
[212], and [297] for a variety of applications. Since
the estimated mode is added constantly to the working
set Mk of models, the optimal fusion rule of two MM
estimators of Section VIC is instrumental here. More
generally, the model set Mk can be augmented by two
or more models using average modes over a (likely)
subset of Mk [207, 212] or models with the largest
likelihoods or probabilities. Augmentations based
on MMSE, ML, and MAP estimates have distinct
characteristics. While the expected mode is in the
convex hull formed by models in Mk , this is not
necessarily so for the MAP and ML mode estimates.
MMSE-based augmentation is limited to the case
where all models are in the same vector space (and
thus their sum is meaningful) and depends on the
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current model set Mk used, but allows clearly an
, z k ], Mk[j] = M[j¡1]
[
iteration m̂k[j] = E[sk j sk 2 M[j¡1]
k
k
[j]
m̂kjk
to improve the mode estimation. This is not
the case for ML and MAP estimates. Note that the
mode estimates need not be obtained by filters based
on models in Mk . For example, an adaptive IMM
algorithm for radar tracking of a maneuvering target
was proposed in [186] that uses an acceleration model
determined by a separate Kalman filter on top of a
fixed set of models (i.e., augmented by this model).
Another natural and simple class of adaptive
structure is the so-called adaptive grid structure, where
a model is represented by a grid point. As outlined
in [198], [201], and [192], it quantizes the mode
space unevenly and adaptively. It usually starts from a
coarse grid and adjusts the grid in real time based on
data as well as prior information. The grid adjustment
usually includes a local grid refinement over one or
more highly likely subsets of the mode space. The
possible locally-refined grids form the candidate
model sets here, which are not given explicitly though.
This structure belongs to the model-set generation
family unless all models in all the grid levels can be
determined in advance. The problem here is closely
related to model-set design, where theoretical results
of Section VIIA provide guidelines. Many practical
schemes for adaptation of the grid are possible. The
algorithms/designs of [185], [120], [244], [271],
[105], [150], [344], [306], [114], and [288] are
examples of this structure, where illustrations of their
superior cost effectiveness to the FS-IMM algorithm
were also given.
More specifically, adaptation of an initial coarse
grid to a subsequent fine grid was proposed in [120]
for an AMM algorithm combined with the PDA filter
[19]. Also for an AMM algorithm, [244] presented
a filter bank that moves over a predefined fixed grid
according to a decision logic, including five versions
based on measurement residuals, expected mode,
variation in mode estimates, mode probabilities, and
error covariance, respectively. This moving-bank
method was adopted in several applications [126,
301, 343]. References [198] and [201] suggested to
employ the expected mode m̂kMMSE as the center of an
adaptive grid for an example of nonstationary noise
identification. A set of target acceleration models was
proposed in [270] and [271] where the acceleration
of the center model is determined by an additional
Kalman filter in a two-stage filtering setting (see [208,
Sec. 6.3]). It appears that the performance can be
enhanced if the (conditional and/or fused) estimates
from the MM estimator is utilized in the two-stage
filter as well. Reference [89] proposed to replace
the above two-stage filter with a fuzzy Kalman filter
characterized by a fuzzified process noise covariance.
Reference [105] proposed a moving set of CT models
centered around one with a turn rate determined by
the magnitude of the acceleration divided by the
speed of the target. In [149] and [150], a set Mk
of three CT models–left, center, and right–were
made adaptive by online adjustment of their assumed
turn rates !kL , !kC , and !kR centered around !kC based
on their posterior mode probabilities ¹Lk , ¹Ck , ¹Rk ,
C
; that is,
with the expected turn rate taken to be !k+1
C
k
L
L
C
!k+1 := !ˆ kjk = E[!k j z , Mk ] = !k ¹k + !k ¹Ck + !kR ¹Rk .
As pointed out in [207] and [212], an alternative
C
is to use the predicted turn rate !k+1
:= !ˆ k+1jk =
k
L L
C C
E[!k+1 j z , Mk ] = !k ¹k+1jk + !k ¹k+1jk + !kR ¹Rk+1jk ,
where ¹k+1jk are predicted mode probabilities. As
C
presented in [288], choosing !k+1
:= !ˆ kjk or !ˆ k+1jk
L
R
and the corresponding !k+1 and !k+1 by a marginal
model-set likelihood ratio test yields improved
performance. An enhancement of the motion and
sizing of the moving bank was proposed in [344]
based on a probabilistic discretization of the mode
space locally centered around m̂kMMSE using the
probability of the normalized measurement residual
¡1
0
z̃(i) of each elemental filter as a measure
squared z̃(i)
S(i)
of the model-mode mismatch. The “filter spawning”
technique proposed in [114] for fault detection and
estimation first detects a mode change (by a MAP
test), decides on the direction of the new mode, and
then locally refined models (grid points) are spawned
along that direction with the help of the current
mode estimate. Performance superiority of these
adaptive-grid structures to the corresponding fixed
structures were demonstrated in all the publications
above. Two perturbation model based adaptive grid
schemes were proposed in [307], [304], [305] and
[306] for ship and aircraft tracking. The first scheme
uses a fixed grid but each elemental filter includes the
deviation (perturbation) of its assumed mode value
from the true mode as a state variable and estimates
it, resulting in de facto an adaptive-grid scheme. The
second scheme differs from the first one in that the
fixed grid is replaced by an adaptive grid, where each
grid point is updated by the corresponding estimated
perturbation at each recursion. To avoid the mask
of the model differences by their corresponding
elemental filters, it was proposed in [233] to adjust
assumed model parameters in real time to keep
the inter-residual distance measure ²0ij Sij ²ij below a
threshold, where ²ij = z̃(i) ¡ z̃(j) and Sij > 0 is a weight
matrix. Another implementation using UKF instead
of EKF is [382]. Reference [274] reported simulation
results of automatic adjusting model parameters by an
if-then rule for process noise covariance and heuristic
estimation of turn rate based on kinematics. Reference
[185] proposed a simplex-directed mathematical
programming scheme for an AMM algorithm, where
the grid is formed by the vertices of a simplex
and updated by certain rules (e.g., replacement of
worst vertices by their mirror images and scaling
up or down of the simplex) based on mode (vertex)
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probabilities. Other programming schemes are of
course possible here, as discussed next.
Closely related to the adaptive-grid structures,
still another class of variable structures relies on
optimization techniques. Here, the problem is
formulated as that of finding the optimal model
set given data. Although applicable to model-set
adaptation, it is actually more suitable for model-set
design, to be discussed in Section VIIA. Such an
algorithm was proposed in [162] based on the
genetic algorithm (GA) [137]. This algorithm uses
a population of n strings (chromosomes) of real or
binary codes, M = fm(1) , m(2) , : : : , m(n) g, where each
string m(i) (not necessarily an index) represents a
possible model. Starting from a random sample
uniformly distributed over the mode space as the
initial population M0 , it runs an AMM algorithm
to obtain posterior model probabilities in each
generation. The posterior model probability serves as
the objective function, known as the fitness function
in GA. The next generation is produced by the genetic
operations of selection, crossover, and mutation. This
process is repeated as desired. Selection is a process
by which individual strings are tentatively selected as
candidate parents of the next generation based on their
fitness values. It is an implementation of the “survival
of the fittest.” Crossover (or recombination) produces
the new generation of strings with hopefully improved
fitness by randomly selecting mating pairs from the
tentative parent pool and crossing over of these pairs
(e.g., crossover of ABDCE and abcde to generate
ABcde and abCDE). It guarantees the diversity and
improvement of each generation and is generally
considered the most important and representative GA
operation. Mutation is the occasional (with a small
probability) random alteration of (the single digit of)
the value of a string. Other less popular/fundamental
GA operations [124], such as inversion, were not used
in [162]. Although numerous possibilities exist, no
concrete information was provided therein as how the
operations of selection, crossover, and mutation were
implemented, except that the so-called biased roulette
wheel selection was hinted where a string is selected
at random with a probability proportional to its
fitness. It was demonstrated in [162] via three simple
examples that this GA-based algorithm converges to
the true model in dozens of generations, each using
only one measurement update and having a population
of size 10 (i.e., 10 models), although a more typical
size is 50 in most GA applications. The MAP estimate
and the associated error covariance were chosen to
reinitialize all elemental filters; however, how the
prior probabilities of the models in the new generation
are assigned is not clear. GA was also suggested in
[72] to update the parameters of the entire model
set in the context of “mixture of experts.” Other
generally applicable optimization techniques (see, e.g.,
the survey of [175]), such as the popular simulated
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annealing and tabu search, are potentially applicable
here as well. The simplex-directed scheme of [185]
and the recursive quadratic programming of [72], [73]
also belong to this class in some sense.
It is possible to include one or more adaptive
models or elemental filters, in addition to fixed
models, within an adaptive structure. The above
estimated-mode augmentation is an example. This
makes sense intuitively since the fixed models can
obtain rapidly a rough initial estimate for the adaptive
models, which fine-tune themselves automatically
to yield accurate estimates. This leads to a two-level
adaptive structure, meaning that both the models (or
its elemental filters) and the model sets are adaptive. It
belongs to the model-set generation family in general.
These adaptive structures are easily implementable
and more cost-effective than the state-of-the-art
second-generation algorithm. They are particularly
suitable for different classes of problems and thus are
complementary to each other. Their combinations are
certainly possible and may be advantageous for certain
problems.
E. Tracking Applications
A challenging application tackled very actively in
the most recent years is tracking of ground targets, in
particular, in a road network. This is usually aided by
reports of a ground motion target indicator (GMTI).
This problem is characterized mainly by the presence
of a large number of constraints on the target motion,
depending on the target type as well as the terrain
conditions, available in the form of topography
information, such as road maps. The existence of
these constraints requires the use of a model set
that is too large for conventional FSMM estimation
algorithms. That is why the only application here of
the first two generations of MM algorithms known
to us is that of an FS-IMM algorithm to tracking dim
ground targets in heavy clutter without any road and
terrain constraints by a ground based infrared search
and track (IRST) sensor, reported in [38]. The actual
mode set for any given target and transitions between
modes are naturally time varying and state dependent,
making the VSMM method ideally suited to this
problem. To our knowledge the VS-IMM solution
is the only effective one to this problem, used by
all contractors in the Affordable Moving Surface
Target Engagement (AMSTE) study sponsored by
U.S. Defense Advanced Research Projects Agency.
A formulation of the problem and a comprehensive
VS-IMM solution were first published in [169]
and [170]. The specific design implemented the
VS-IMM recursive algorithm (see Table III) with
an individual model-set adaptation for each target
based on its current and predicted state and the
known topography of the surveillance region. Main
situations that require addition or deletion of a model
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at each revisit time are: on road/off road motion,
motion in junctions/intersections, road entry/exit
conditions, road obscuration. Furthermore, [166]
incorporated an additional “stopped-target” model
into the total model set to cope with the possible
“move-stop-move” evasion strategy of targets since
the GMTI is incapable of detecting slow motion
or stationary targets. Quite significant advantages
of the VS-IMM algorithm over the FS-IMM
algorithm were demonstrated by the simulations.
Similar VS-IMM approaches and results were also
presented in [310] and [275]. References [79] and
[53] proposed to combine a VS-IMM estimator
with a joint belief-probability data association
approach to track, identify, and group multiple moving
targets. A variable structure was used to capture the
behavior of highly maneuverable targets through
move-stop-move cycle, incorporating features such
as motion constraints on road networks and high
maneuver terrain. References [9] and [295] employed
particle filters into the framework of [170] to cope
with the non-Gaussianity of the posterior densities
when constraints are applied. The VS-IMM method
was employed in [113] and [379] to handle another
interesting ground target problem–monitoring
the motion of aircraft and vehicles in an airport
area based on surface movement radar data. Such
surveillance is an essential part of airport movement
guidance and control systems. Embedding map
information was made by incorporating an elegant
kinematic constraint technique [314] (e.g., by
using the curvature of a taxiway) in the CT model
with polar velocity (see [209, Sec. V.B.2]). Its
comparison with the EKF and FS-IMM algorithm
using synthesized and real data demonstrated again
that the VS-IMM algorithm is highly beneficial
in terms of mode identification, accuracy, and
computational savings. Other, most recent references
that study VSMM tracking of ground targets subject
to constraints include [377], [375], [383], and [378].
VII. MM ALGORITHM DESIGN ISSUES
A. Model-Set Design
Model-set design and choice/decision are closely
related but different. Model-set choice/decision deals
with the problem of deciding which set is the best
given a family of candidate model sets, discussed in
Section VIB. Model-set design does not have a given
family of candidate sets in general. It determines the
model set to be used for a given problem. Clearly,
model-set choice can be viewed as an integral part
of model-set design. Model-set design is the most
important issue in the application of MM estimation.
The performance of an MM algorithm for a given
problem depends largely on the set of models used
and the primary difficulty in the application of the
Fig. 9. Minimum distribution-mismatch design of model set
M=fm(1) ,:::,m(6) g.
MM method is the design of the model set. Numerous
publications have appeared in which ad hoc designs
were presented, as surveyed in Sections VE and VIIC.
There are two types of model-set design: online
and offline. Offline design is for the total model set M
used by the first two generations or the initial model
set in VSMM estimation. In an FSMM algorithm,
the model set used cannot vary and is determined
a priori by model-set design. In a VSMM algorithm,
the model set in effect at any time is determined
by model-set adaptation, discussed in Section VIB,
which may be viewed as an online (real-time) design
process. This section focuses on offline model-set
design.
General Design Methods: Model-set offline design
was formulated in [222] and [197] mathematically
as a problem of approximating the true mode as
a random variable s with a certain distribution
by a discrete random variable m (random model)
with a certain pmf. The range of the variable m
is the model set and the pmf is the initial model
probabilities needed for MM estimation. Three
general design methods were proposed in [221],
[222], and [197] based on this formulation: minimum
distribution-mismatch, minimum modal-distance, and
moment matching.
The minimum distribution-mismatch design
minimizes the mismatch (or distance) kFs ¡ Fm k =
maxx jFs (x) ¡ Fm (x)j between the cumulative
distribution functions (cdf) Fs (x) and Fm (x) of the
mode s and model m. Given the number of models
M = jMj, it was shown in [221], [222], and [197]
that this method yields in the scalar case the optimal
model set M = fm(1) , : : : , m(M) g such that Fm (m(i) ) =
(2i ¡ 1)=2M = Fs (x)jx=m(i) , meaning that m(i) can be
determined as follows: Divide the range of the cdf
Fs (x) by 2M equal intervals and the value of x such
that Fs (x) = (2i ¡ 1)=2M is then the optimal location
of m(i) (see Fig. 9). This optimal model set has a
uniform pmf pm (m(i) ) = 1=M. A procedure that uses
a minimal number of models given any tolerance on
the distribution mismatch was also given in [222] and
[223] for the case where m is a vector.
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The minimum modal-distance design minimizes
the distance ks ¡ mk between s and m in the
mode/model space, rather than in the space of
distribution functions. The problem of (scalar or
vector) quantization and data compression [121] is
in a sense a special case of the model-set design
problem in this formulation. Significant theoretical
results for this design were presented in [221], [222],
and [197], including the following conditions for a
model set to be optimal and properties of an optimal
model set. Assume that S = fS1 , : : : , SN g is a partition
of the mode space where Si is covered by model
m(i) exclusively in the sense fs 2 Si g = fm = m(i) g.
Then, the following conditions hold for the optimality
in the sense of minimizing distance ks ¡ mk2 =
E[(s ¡ m)0 (s ¡ m)] (and some more general metrics):
1) given any partition S = fS1 , : : : , SN g of mode space,
a model set M = fm(1) , : : : , m(M) g is optimal if each
model m(i) is the centroid (mean) of the corresponding
partition member Si : m(i) = arg minm E[(s ¡ m)0 (s ¡ m) j
s 2 Si ]; 2) given any model set M = fm(1) , : : : , m(M) g,
a partition is optimal if and only if points in any
partition member Si are closer to m(i) than to any
other m(j) 2 M; that is, a point s must be assigned
to its nearest neighbor m(i) in M. Simply put, the
optimal model set is within the class in which models
are located at the centroid (mean) of members of a
nearest-neighbor partition of the mode space. This
result suggests iterative procedures to find an optimal
model set. For example, we may start with an initial
partition of mode space; find a candidate of the model
set as the centroid of each partition member; use the
nearest-neighbor rule to update the partition; and
repeat this process. This centroid model set that covers
each Si by its centroid m(i) = E[s j s 2 Si ] exclusively
has several nice and intuitive properties [221]. For
example, the (random) model and mode have the same
mean: E[m] = E[s]; the modeling error is orthogonal
to the model: E[m(s ¡ m)0 ] = 0; the cross power of the
mode and model is equal to the power of the model
E[m0 s] = E[s0 m] = E[m0 m]. Many of these results
were inspired by those in vector quantization and data
compression.
The above designs require knowledge of the
distribution of the true mode, which is hard to come
by in practice. The moment-matching design matches
the moments of the model to those of the mode,
which is much more easily available. Let s̄ and
Cs be the mean and covariance matrix of mode s.
It was shown in [222], [197], and [221] that we
can always use a model set with as few as but not
fewer than rank(Cs ) + 1 models to match s̄ and Cs
exactly. A set of concrete moment-matching designs
was presented in [221], [222], and [197], including
those with minimum number of models, those with
symmetric pmfs, and those with equal inter-model
distance (called diamond-set designs in [221],
[222], and [197] because the model locations form a
1300
diamond geometrically). In each of these designs, the
probability mass and the location of every model are
determined. The simplest possible diamond-set design
(with one at the center and six on the first layer) was
implemented in [211] and [207] for an example of
maneuvering target tracking using MM algorithms.
Examples of some of these design methods can be
found in [365], [211], [207], [212], [221], [222], and
[223].
As pointed out in Section VID, model-set design
can be formulated as that of finding the optimal model
set given data, based on optimization techniques.
Such an algorithm was proposed in [162] based on
the GA [137]. Similar to the algorithm described in
Section VID, this algorithm uses a population of N
strings (vectors), where each string is M-dimensional,
representing a set of M models. Starting from N
random samples uniformly distributed over the
mode space as the initial population, it runs N
AMM algorithms in parallel for K measurement
updates to obtain the probabilities of all models in
each generation. The maximum model probability
maxi fPfs = m(i) j s 2 Mj g, i = 1, : : : , Mg in each
string Mj serves as the fitness of the string. The next
generation is produced by the genetic operations of
selection, crossover, and mutation, as described in
Section VID. This process is repeated as desired. This
algorithm was demonstrated in [162] to converge
to the true model in 40 generations, each over a
batch of K = 50 measurement updates. Note that this
GA-based method is applicable only to the case with
a known, fixed number of models. As pointed out in
Section VID, other generally applicable optimization
techniques, such as simulated annealing and tabu
search, are potentially applicable here and some
would allow a variable number of models. A key issue
here is the choice of objective (fitness) function for
optimization. Many objective functions are possible,
particularly those discussed in [222], [204], [203],
and [223]. The use of model probability calculated
within each model set as the fitness function in [162]
does not appear desirable since the probability of a
model is relative only to others within the set and thus
is meaningful for comparison only within a model
set but not across different sets. For example, m(1)
in the set M1 = fm(1) , m(2) , m(4) g may have a larger
probability than m(3) in the set M2 = fm(1) , m(2) , m(3) g
even if m(3) is closer to the true model: Pfm =
m(1) j m 2 M1 g > Pfm = m(3) j m 2 M2 g > Pfm =
m(1) j m 2 M2 g. A simple way out is to calculate
model probabilities over the union of the model sets,
obtainable from the model likelihoods in the N AMM
estimators, rather than within each model set.
Guidelines for Model-Set Design: Clearly, many
criteria/measures for model-set design are possible
and their choice is important. An array of such
criteria and measures were proposed and discussed
in [222], [204], and [203] for different purposes of
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OCTOBER 2005
MM estimation. These include: kx ¡ x̂M k for base-state
estimation, where x̂M is an estimate of the base state
x using model set M and ku ¡ vk2 = E[(u ¡ v)0 (u ¡ v)
j z];26 ks ¡ ŝk for mode estimation, where ŝ is the
estimate of the mode s; rates (probabilities) of
correct, incorrect, and no identification for mode
identification; k» ¡ »ˆM k for hybrid-state estimation,
where »ˆM is an estimate of the hybrid state » using
model set M; and more general information-theoretic
measures based on the Kullback-Leibler information
and mutual information.
One of the most natural and simplest measures is
kx ¡ x̂M k for base-state estimation, formally introduced
in [190], [191], and [201] for model-set design. More
theoretical results for model-set design are available
based on this measure than on other measures. Let
x̂M = E[x j z, S = M] be the optimal MM estimators
assuming model sets M is the optimal model set, where
z is the data. Given an arbitrary model set M and let
D = (M ¡ S) [ (S ¡ M) be its symmetric difference
from the optimal set S. Note first that it follows
from (17) that kx ¡ x̂B k · kx ¡ x̂A k if and only if
kx̂S ¡ x̂B k · kx̂S ¡ x̂A k, where x̂S is the optimal MM
estimator using the mode space S. It was shown in
[191], [201], and [192] that kx̂S ¡ x̂M k = j1 ¡ cj kx̂S ¡
x̂D k, where c = Pfs = m(i) js 2 Mg=Pfs = m(i) js 2 Sg for
any model m(i) common to M and S, which implies
that use of too many models is as bad as use of
too few models. Moreover, consider the problem of
adding an arbitrary model set A to another arbitrary
model set M without overlap (i.e., M \ A = Ø). Let
M0 = M [ A. It was shown in [191], [201], and [192]
that set M0 is better than set M in the sense that x̂M0
has a smaller MSE than x̂M if and only if
p
b2 cos2 µ + 1 ¡ b2 ¡ b cos µ
r<
(43)
1¡b
where b = Pfs = m(i) j s 2 M0 g=Pfs = m(i) j s 2 Mg for
any model m(i) in M, r = kx̂S ¡ x̂A k=kx̂S ¡ x̂M k, and
cos µ = (x̂S ¡ x̂M )0 (x̂S ¡ x̂A )=(kx̂S ¡ x̂M kkx̂S ¡ x̂A k). Note
that (43) describes a ball of radius 1=(1 ¡ b) centered
at (¡b=(1 ¡ b), 0, 0, : : : , 0) if x̂M and x̂S are placed at
(1, 0, 0, : : : , 0) and (0, 0, 0, : : : , 0), respectively. A simple
example was given in [222] and [223] that illustrates
how this result, which requires knowledge of the
optimal MM estimator x̂S , can be used in practice.
The above result holds even if M0 is not a subset of
the mode space S. If M0 ½ S (e.g., when M0 is a set
of discrete points of a continuous mode space S as a
parameter space), as shown in [211], assuming x̂A and
x̂M have uncorrelated estimation errors, set M0 is better
than set M (i.e., adding A to M is better) in the sense
that x̂M0 has a smaller MSE than x̂M if and only if the
posterior probability of the model set A is below a
26 With
this definition of the norm, kx ¡ x̂M k2 is actually the
conditional MSE of x̂M .
threshold:
Pfs 2 A j s 2 (M [ A), zg <
2kx ¡ x̂M k2
:
kx ¡ x̂M k2 + kx ¡ x̂A k2
This condition always holds if kx ¡ x̂A k < kx ¡ x̂M k,
which should be the case if x̂A =: x̂ŝ = E[x j z, s = ŝ]
(i.e., augment M by A = fŝg). This result provides
a theoretical support for the estimated-mode
augmentation (see Section VID) for VSMM
estimation, as presented in [211], [207], and [212].
Even if x̂A is worse than x̂M , x̂M0 can still be better
than x̂M if and only if Pfs 2 A j s 2 (M [ A), zg satisfies
0
the above inequality when E[x̃M
x̃A ] = 0, or if and only
0
0
0
if E[x̃M
= 0 [211].
x̃A ] < E[x̃M
x̃M ] when E[x̃M
x̃A ] 6
In order to apply the MM method to problems
with uncertain parameter s over space S, two
important questions are: 1) which quantity is best
selected as the estimatee (i.e., the quantity to be
estimated) and 2) how to quantize the parameter space
S. The following general guideline was presented
in [194] for estimatee selection: If the ultimate
goal is to estimate a parameter s, which is related
to another parameter p nonlinearly, then a model
set fs1 , : : : , sM g in the space of s is superior to a
model set fp1 , : : : , pM g in the space of p even if p
has a better physical interpretation. For the second
question, a procedure to determine the choice of the
quantization points M = fm(1) , : : : , m(M) g was presented
in [311], given the number
of quantization points M.
R
It minimizes J(M) = S kx ¡ x̂M k2W f(s)ds, where W is
a weighting matrix,
specified by the designer, and the
R
pdf f(s) = 1= S ds. The resultant choice is optimal in
the sense of having the minimum average weighted
MSE for the true mode over the set S. In the Gaussian
case, this vector minimization problem can be solved
numerically in a straightforward fashion. An example
was given in [311] that demonstrates its superiority to
several heuristic choices, including the simple, popular
uniform quantization scheme.
Caution must be exercised in model-set design.
For example, there should be enough separation
between models so that they are “distinguishable,”
“observable,” or “identifiable.” This separation should
well exhibit itself in the measurement residuals
[145], especially between the filters based on the
true model and mismatched models, respectively.
Otherwise, the MM estimator will not be selective
in terms of choosing the correct model because
the residuals have a dominant effect on estimation
results. A necessary condition for the effective
performance of MM estimation was presented in [70]
for a stochastic linear time-invariant system with an
uncertain parameter. For a single-input single-output
system with an uncertain input bias, the dc gain Gdc
of the system transfer function from the input to
the output (measurement) must be non-zero. This
makes sense intuitively. The steady-state output
(i)
(measurement residual) z̃ss
is proportional to the dc
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1301
gain times the bias difference (as the step input),
which is the actual input bias b minus the input bias
(i)
b(i) assumed in filter i: z̃ss
= (I ¡ B)Gdc (b ¡ b(i) ), with
B depending on the steady-state Kalman filter gain for
filter i, where I is the identity matrix. For the case of
uncertain system matrix parameters but known input
(i)
(i)
(i)
u, z̃ss
= (I ¡ B)[Gdc ¡ Gdc
]u, where Gdc
is the input
gain for filter i. For a multiple-input multiple-output
system, the necessary condition becomes that each
column of the dc gain matrix (difference) must have at
least one non-zero element. Other relevant results can
be found in, e.g., [23]. Such results are beneficial for
performance enhancement of MM estimation, such as
those presented in [243].
Model Efficacy: Each model has a certain
effective coverage region of the true mode within
the model set in use. Knowledge about such relative
efficacy is quite useful in model-set design. A
concept of relative efficacy of a model in terms of its
coverage, along with its quantitative measures, was
introduced in [222]. More specifically, a “window”
function wi (x) was introduced to quantify the efficacy
of model m(i) in covering the true mode s = x relative
to other models in the set, where wi (x) is a function
of x. The larger wi (x) is, the more effective the model
m(i) is (relative to other models in the set) given s = x.
Two versions of wi (x) were defined in [222]. Consider
a model set M = fm(1) , : : : , m(M) g. A probability-based
efficacy of model m(i) in M is wi (x) = Pfm = m(i) j
s = x, m 2 Mg; that is, wi (x) is the probability that the
random model m will take on the value m(i) given that
it has to take on a value in M and the fact that the true
mode s is equal to x. Its effect on model
R probability
is clear through Pfm = m(i) j m 2 Mg = wi (x)fs (x)dx,
where fs (x) is the pdf of the true mode. Alternatively,
the testing-based model efficacy, defined by wi (x) =
PfHi not rejected j s = xg=L, is the probability that Hi
is not rejected by an (optimal) test given s = x divided
by the number L of hypotheses that are not rejected at
the end of the test for the hypothesis testing problem
H1 : m = m(1)
versus ¢ ¢ ¢ versus 2HM : m = m(M)
using all available data. This definition is theoretically
equivalent to but implementationally advantageous
than the definition wi (x) = Pfaccept Hi j s = xg. A
simple example of these two model efficacies can
be found in [222]. More related, theoretically elegant
results can be found in [386].
B. Determination of Transition Probabilities
Theoretically, post-first-generation MM estimators
assume that the transition probability matrix
(TPM) governing the mode jumps is completely
known. In target tracking, however, it is practically
unknown, since it depends critically on the unknown
control inputs, or worse, the mode sequence is not
really Markov. The determination (design, tuning,
1302
adaptation) of the TPM amounts to identifying a
Markov transition law that “best” fits the unknown
truth, similar to tuning of the process noise
covariance Q in the Kalman filtering. Fortunately, the
performance of MM estimation is not very sensitive to
the choice of the TPM provided it is not too far off;
but to a certain degree this choice provides a trade-off
between the peak estimation errors at the onset and
termination of a maneuver and the steady-state errors
during CV motion (see, e.g., [199] and [21]).
Offline Design: Traditionally, the TPM has been
considered in tracking as a design parameter chosen
a priori. Numerous designs and tuning results have
been reported in the literature. Most of them are ad
hoc, but some are more or less systematic, including
those proposed and studied in [43], [18], [68], [39],
[257], and [62]. A simple design of TPM ¦ =
(¼ij )M£M , appeared as early as in [267], [260], and
[123] and used by many, is ¼ii = q, ¼ij = (1 ¡ q)=
(M ¡ 1), i, j = 1, : : : , M directly in discrete time
for some large q (e.g., q = 0:97 [11]). Such a
design directly in discrete time is questionable for
a discretized system with a nonuniform sampling
(revisit) interval since the TPM depends on the
sampling intervals as well as target behavior (in
continuous time). A more systematic design is
based on modeling the Markov jump process in the
continuous time [43, 68, 39]. It follows from the
forward and backward Kolmogorov equations [279]
that ¦(T) = e¤T , where ¤ = (¸ij )M£M is the transition
density matrix of the process, defined
P similarly as for
¦, with ¸ii < 0, ¸ij > 0, i 6
= j and M
j=1 ¸ij = 0. The
diagonal elements ¸ii of ¤ and the mean sojourn time
¿¯i of mode m(i) are related by ¸ii = ¡1=¿¯i since the
sojourn time ¿i of a state (mode) m(i) of a Markov
jump process has an exponential distribution with
parameter ¡¸ii . Its direct discrete-time counterpart
is ¼ii = 1 ¡ 1=¿¯i , where ¿¯i is expressed in discrete
time [18, 19]. From ¸ii = ¡1=¿¯i and ¦(T) = e¤T it
follows approximately that ¼ii = e¡T=¿¯i , which is
more widely used, such as in [60] for the design
of an IMM-based ATC surveillance system and in
[155] for TPM adaptation in a two-model IMM
tracker with adaptive sampling for the first benchmark
problem [48]. Another design, used in [161], is
¼ii = 1 ¡ T=¿¯i , which is in fact the above direct
discrete-time design but with ¿¯i in continuous time
and is equal to the linear approximation of the
approximate continuous-time model ¼ii = e¡T=¿¯i . This
was modified to ¼ii = maxfqi , 1 ¡ T=¿¯i g in [199] and
[19], where qi is the minimum probability for mode
m(i) to stay on, as opposed to jumping to any other
mode. The choice of qi = ¹(i)
1 was suggested in [39],
where ¹(i)
is
the
steady-state
probability of mode m(i) ,
1
independent of the initial mode. As presented in [68],
¦ can be designed as follows if ¹(i)
1 is known for
each mode. First determine (numerically) ¤ from the
relationship limT!1 (e¤T )0 = ¦(1)0 = [¹1 , ¹1 , : : : , ¹1 ],
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
(2)
(M) 0
where ¹1 = [¹(1)
1 , ¹1 , : : : , ¹1 ] , and then get ¦ =
¦(T) = e¤T for the given sampling interval T. This
method was used in [68] and [39] for different IMM
configurations with a nonuniform T for air defence
system applications. Compared with those based on
mean sojourn time ¿¯i above, this method has pros and
cons: It obtains off-diagonal elements ¼ij as well as
diagonal elements ¼ii , but it relies on knowledge of
¯
¹(i)
1 , which is often harder to come by than ¿i , and an
asymptotic relationship, based on which results are
usually less accurate. This reliance on the asymptotics
0
can be replaced by ¦(T0 ) = e¤T if ¦(T0 ) is known
0
for some T . Note that a simple use of ¼ij and ¼ii
from the above two
P methods would usually violate
the requirement M
j=1 ¼ij = 1. It would be better to
combine them by solving (approximately, if needed)
0
limT!1 (e¤T )0 = (¹1 , ¹1 , : : : , ¹1 ), or ¦(T0 ) = e¤T
0
if ¦(T ) is known, for ¤ with ¸ii = ¡1= ¿¯i , 8i, if
possible.
Online Adaptation: The offline design, being
completely a priori in nature, does not provide
estimates of the TPM using online data. In some
cases, prior information about the TPM may be
inadequate or lacking. The “unreasonable” need to
provide TPM a priori even in the case of insufficient
information has been cited by some as one of the
main reservations about using Markov-chain-based
MM estimation algorithms (see, e.g., [114]). A
number of algorithms have been proposed recently
in [151], [153], [152], and [91] for online estimation
of the TPM. These algorithms are naturally and
easily incorporable into a typical MM (e.g., IMM)
estimation algorithm, resulting in TPM-adaptive MM
estimation. More specifically, [151], [153], and [152]
developed a Bayesian framework and proposed several
suboptimal algorithms for recursive MMSE estimation
ˆ =
of the TPM starting from an initial estimate ¦
0
(1)
(M) 0
ˆ 0 ] . Among them, the most cost-effective
ˆ 0 ,:::,¼
[¼
one–the so-called quasi-Bayesian estimator–assumes
a Dirichlet prior distribution of the TPM and its
recursion is given by
¼ˆ k(ij) =
1
®(ij) ,
k+1 k
(ij)
®(ij)
k = ®k¡1 + PM
ˆ k(i) = [¼ˆ k(i1) , : : : , ¼ˆ k(iM) ]0
¼
1
(ij) (ij)
j=1 ®k¡1 gk
gk(ij) = 1 +
1
ˆ L
¹0k ¦
k¡1 k
(ij) (ij)
®k¡1
gk
(j)
ˆ (i)0
¹(i)
k [Lk ¡ ¼k¡1 Lk ]
(M) 0
(1)
(M) 0
where Lk = [L(1)
k , : : : , Lk ] and ¹k = [¹k , : : : , ¹k ]
are vectors of model likelihoods and probabilities,
respectively. This algorithm was shown in [151],
[153], and [152] to have reasonable performance at
an almost negligible computational expense. Also
adopting the Dirichlet prior, [91] derived recursive
hybrid estimation schemes with an unknown TPM
by obtaining posterior marginal pdfs of the base
and modal states analytically. Note that these online
adaptation algorithms are more generally applicable
than the offline designs.
It is intuitively appealing to combine offline
design with online adaptation to take advantage of
both prior knowledge and online information: the
a priori designed TPM is refined by the online TPM
adaptation using online data from the current scenario;
the adaptation may be slow if the prior TPM is
(nearly) “noninformative” (e.g., uniformly distributed)
but could be speeded up by a good initial TPM.
C.
Various MM Designs and Performance Studies
Successful application of any particular MM
algorithm to a real-world maneuvering target tracking
problem largely amounts to design of the model
set, ad hoc adjustment of the algorithm, tuning of
parameters, and choice of the best trade-off variant by
performance evaluation and comparison. The tracking
literature is abundant in various studies on model-set
design, parameter choice/tuning, and performance
evaluation/comparison. Many of them are generic
enough to be applied to a wide range of problems and
situations, although a “universal” approach best for all
applications is impossible. Here we give a brief review
of these more generic designs. More problem-specific
implementations were addressed in Sections IVE, VE,
and VIE.
Most MM algorithm designs follow two basic
ideas concerning how maneuvers are modeled:
parametric and structural. Parametric designs select
one or more parameters to represent the effect of
the target maneuvers; each model is characterized
by a quantized value of the parameters. Structural
designs use models of different structures to describe
different maneuvers. All designs include one or more
CV models. The reader is referred to [209] for model
details.
Parametric Designs: Typical parameters to be
quantized are input (acceleration), process noise
level, and turn rate. Most common examples are:
exponentially correlated acceleration (ECA) model
(i.e., the Singer model) with constant mean levels
[260, 123, 266], second-order linear kinematic model
with multiple acceleration levels (see, e.g., [11]) or
with multiple process noise levels (see, e.g., [367] and
[298]). In particular, [260] designed a linear target
motion model, with unknown input uk quantized into
several known levels u(i) . The corresponding GPB1
algorithm reduces to a single Kalman filter with
input ūk , which is a probabilistically weighted sum
of the known input levels u(i) (see [208, Sec. 5.3.8
of Part IV]). This technique has been used by many
others. It is also applicable in principle when the
input level is over a continuum if the transition pdf
between any pair (uk¡1 , uk ) is known. This, however,
involves integration in general to obtain the average
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ūk . Assuming the transition pdf between (uk¡1 , uk ) is
Gaussian and uk¡1 is Gaussian distributed, the average
ūk can be obtained in a close form, and such a form
was obtained in [187] for a constant-position model
with u as velocity bias. References [11] and [10]
investigated a 2D design consisting of one CV and
12 CA models with acceleration values distributed
symmetrically over a 2D region centered at zero and
bounded by 40 m/s2 . The tracking performance was
found to depend on j¢ajT2 =¾, where j¢aj is the
quantization step, T the sampling time, and ¾ 2 the
measurement noise variance. This design has later
been used in a number of theoretical and comparative
studies (see, e.g., [226], [219], [215], and [207]). A
shortcoming associated with quantization of the input
acceleration is that many models/filters are needed to
cover even a moderate range. A much smaller number
of models are needed if instead the process noise level
is quantized [367, 298].
Structural Designs: Structural and hybrid
(structural/parametric) designs can be much more
efficient than quantization only. Typically, CV, CA,
and/or CT models are used here. Most common
examples are: CV-CA [16], [18], CV-CT (see,
e.g., [101], [21]), and CV-CA-CT [36]. More
specifically, [16] presented two IMM designs with
two models (CV-CA) and three models (CV-CA-CA),
respectively, denoted as IMM2 and IMM3, and
demonstrated that they beat the input-estimation
method [64] significantly in accuracy and dramatically
in computation. Reference [140] suggested the
inclusion of a CA model with higher process
noise to cover the transitions between CV and CA
motions and provide a faster response. Reference
[52] introduced an innovative model, exponentially
increasing acceleration (EIA), particularly suitable
for fast maneuver detection and demonstrated that
an IMM algorithm with a CV-EIA-ECA model
set improved the accuracy achieved by the IMM2
during maneuvers. Explicit modeling of CT motion
was proposed in [101] and [102], where two model
sets were designed, one CA-CT-CT (left/right turns
with known turn rates) and the other CV-2DCT
(with estimated turn rate). Using explicit CT models
proved to be very beneficial for precision tracking
during turns. Further significant enhancements were
presented in the series of papers [351], [352], and
[353]. While utilizing the EIA for speedy maneuver
responses, the proposed IMM design includes a
3D CT model, implemented with the kinematic
constraint [4], to provide constant speed prediction.
The resulting sophisticated tracker using three
models (CV-EIA-3DCT) with kinematic constraint
outperformed considerably the CV-2DCT of [101] in
tracking nonhorizontal planar maneuvers [351]. These
and many other comprehensive studies of various
IMM designs, proposed by Blair and Watson [46,
51, 43, 44, 354], preceded and led to the solution of
1304
the second benchmark problem proposed in [355].
Solutions to the benchmark problems were discussed
in Section VE and more discussions will be given in a
subsequent part.
Although using only three filters the above designs
are computationally involved since nonlinear and/or
correlated models are used. If computational load is
of a great concern, an alternative is the interacting
multiple bias model scheme [42, 350, 349] (see also
[246]). The main idea is to model the maneuver
acceleration as an isolated system bias and employ
the two-stage (bias free + bias only) reduced-order
estimator [117] rather than the complete Kalman filter
for the augmented state (including the bias state). (See
[208] for a survey and discussion of issues associated
with the two-stage filtering.) Its two-model (CV and
bias) version, referred to as interacting acceleration
compensation algorithm, was demonstrated in [350] to
achieve about 50% reduction in computation relative
to the CV-CA configuration with similar performance
if the data rate is high enough.
A comprehensive study reported in [68], [37],
[39], and [36] evaluated configurations with CA, CV,
Singer models and several new models, including a
horizontal CT model with polar velocity [122] and
a 3D version with two additional states (velocity
elevation angle and its rate) [66]. Comprehensive
simulations over a great variety of scenarios showed
that the horizontal CT model combined with
decoupled altitude filtering performed slightly better
than the complete 3D model overall. The former was
included into the IMM-MHT solution to the second
benchmark problem (see Section VE). Reference [32]
includes an evaluation of an IMM design with normal
and tangential accelerations within its proposed 2D
curvilinear model (see [209]).
Hybrid Designs: Parametric and structural designs
can of course be integrated, leading to hybrid designs,
which involve both models with different structures
and quantized parameters within structures. Examples
of hybrid designs include: CV and CA models with
multiple process noise levels (see, e.g., [18] and [21])
and CV and CT models with multiple turn rates (see,
e.g., [101], [199], and [18]).
Additional references concerning design,
performance evaluation/comparison, and/or
other aspects related to IMM configurations for
maneuvering target tracking include [67], [159], [88],
[41], [165], [277], [135], [303], [143], [31], [144],
[294], and [380].
VIII. NON-PROBABILISTIC/STATISTICAL
TECHNIQUES
MM approach is a general methodology, not
limited to the probabilistic/statistical setting of the
previous sections. Many nonstatistical methods have
been proposed (see, e.g., [272]). In this section,
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
we discuss only those that have been proposed for
maneuvering target tracking. They are based on
alternative means of modeling and handling the
target motion-mode uncertainty, including evidential
reasoning [369], neural networks [72, 73, 6], fuzzy
logic [247, 89, 374], genetic algorithms [29, 162], and
deterministic algorithms [281, 5, 135].
Much of the pros and cons of the probabilistic/
statistical methods of MM estimation stems from
their reliance on the total probability/expectation
theorem and Bayes’ rule. These theorems require
partitioning of the probability space (i.e., one and
only one member is true) even when information is
not sufficient for doing so. In other words, they force
us to overstate the degree of certainty when evidence
or knowledge is actually incomplete or subjective, and
thus the claim of probabilistic exactitude or optimality
is actually more or less artificial. This is probably
the main weakness of the probabilistic formalism. Its
main strengths include: it is rigorous, systematic, and
particularly suitable for sequential processing, among
other things. Nonstatistical approaches offer a variety
of alternatives with distinctive flexibilities that are
valuable for handling many real-world problems. For
example, the Dempster-Shafer reasoning overcomes
the partitioning limitation of the Bayesian methods
by allowing representation of neither exclusive nor
exhaustive hypotheses of posterior evidence. As
such, in the context of MM estimation it may offer
a framework potentially better to handle possible
model-truth mismatch. Like other nonprobabilistic
approaches, however, its standard version does not
provide a way of fusing old knowledge and new
information as Bayes’ rule does in the probabilistic
setting. As a result, not surprisingly we are not aware
of any effective nonstatistical methods for conditional
filtering, which requires sequential processing (fusion)
of old evidence and new data with perfect knowledge
of the underlying model.
Nonstatistical techniques can be applied to all
the other components of MM estimation: model-set
determination, cooperation strategy, and output
processing. The use of genetic algorithms for
model-set determination (adaptation and design)
has been discussed in Sections VID and VIIA.
Application of nonstatistical methods to output
processing is most natural (see, e.g., [72], [73], [247],
and [374]) since it amounts to fusion of evidence
obtained at the same time. By the similarity between
output processing and (hard and soft) decision based
cooperation strategies, these methods can also be used
for cooperation strategies, although we are unaware of
such use in the literature.
Reference [369] proposed an approach that
integrates maneuver detection with two-model
MM estimation based on the Dempster-Shafer
evidential reasoning (see, e.g., [309] and [39]).
Consider sequential testing of no-maneuver (H0 )
against maneuver (H1 ) hypotheses. The belief Bel(Hi )
and plausibility Pls(Hi ) of Hi are obtained by an
extension of the Dempster-Shafer theory. They can be
interpreted loosely as lower and upper probabilities,
respectively, in that probability is an interval (not
a single number) PfHi g = [Bel(Hi ), Pls(Hi )].27 The
interval length Pls(Hi ) ¡ Bel(Hi ) reflects residual
ignorance in our knowledge. If Bel(H0 ) > Pls(H1 ),
or equivalently (for two-model case), Bel(H0 ) > 0:5,
the target is deemed not maneuvering, its estimate
is based on the nonmaneuver model alone, and a
maneuver onset detector is turned on, which declares
a maneuver onset if Bel(H0 , H1 ) > Pls(H0 , H0 ). If
Bel(H1 ) > Pls(H0 ) or equivalently Bel(H1 ) > 0:5,
the target is deemed maneuvering, its estimate is
based on the maneuver model alone, and a maneuver
termination detector is turned on, which declares a
maneuver termination if Bel(H1 , H0 ) > Pls(H1 , H1 ). In
other cases (i.e., when the intervals PfH0 g and PfH1 g
have an overlap), including after a maneuver onset
or termination is declared, the target state estimate is
the weighted sum of estimates x̂(0) and x̂(1) from both
models using their normalized belief values as weights
x̂ = [x̂(0) Bel(H0 ) + x̂(1) Bel(H1 )]=[Bel(H0 ) + Bel(H1 )]:
Given new measurements, the belief is updated
based on Dempster’s rule of combination. By the
random-set theory, however, this rule holds only if the
bodies of evidence being combined are independent,
which is actually not the case here. Earlier, this way
of integrating hard decision and soft decision was
proposed in [332] in a probabilistic setting. The
approach of [369] was developed within the AMM
framework since x̂(0) and x̂(1) are obtained by two
elemental filters working independently, but it can
certainly be extended to the later generations.
References [72] and [73] considered
MM estimation by means of the so-called
“mixture-of-experts” system, gained popularity in
the neural network literature recently. Here each
conditional filter (rather than neural network) is
viewed as an expert. The overall estimate is a
weighted sum (mixture) of the output of all experts.
The weights are computed by the softmax operation
P 0 (i)
zk0 a(i)
¹(i)
= i ezk a , which is a differentiable
kjk¡1 = e
version of the “winner takes all” strategy, meaning
that the filter (expert) with the best performance
will have a weight close to unity. Here zk is the
measurement. The internal weight vector a(i) is
updated (for the use at the next time) by a steepest
descent search with a step size (learning rate) ´:
(i)
(i)
a(i) := a(i) + ´(¹(i)
k ¡ ¹kjk¡1 )zk , where ¹k follows from
¹(i)
kjk¡1 and likelihood by Bayes’ rule as in the IMM
27 For
instance, we may think the probability of raining tomorrow is
in between 50% and 70% (not in California, of course).
LI & JILKOV: SURVEY OF MANEUVERING TARGET TRACKING. PART V: MULTIPLE-MODEL METHODS
1305
algorithm (see Table II). A hierarchical version was
also explored in [73]. The approach of [72] and [73]
is essentially an AMM algorithm, but the parameters
of the best model are made adaptive by quadratic
programming, resulting in an algorithm with a certain
VSMM flavor. The simulation results presented
seem to suggest that this estimator is slightly better
than the probabilistic AMM algorithm in terms of
response time to mode jumps as well as best filter
selection when the truth is not in the model set. Its
performance, however, is sensitive to the design
parameter ´ and its competitiveness relative to the
IMM algorithm is not known.
References [247] and [374] proposed to use fuzzy
weights as an alternative to the probabilistic weights
in the IMM algorithm. In [247], the space of true
mode s (e.g., acceleration a [247] or turn rate ! [374])
is quantized to obtain a mode set fm(1) , : : : , m(M) g.
A corresponding set of (Gaussian or triangular)
membership functions ¹(i) (s) 2 [0, 1], centered at m(i) ,
is designed as a measure of the “validity” of each
model m(i) . A mode estimate ŝk = âkjk is obtained
by an independent CA filter. ThePmodel weights are
(i)
(j)
then computed as ¹(i)
j ¹ (ŝk ) and an
k = ¹ (ŝk )=
IMM algorithm is run using these weights as if they
were the posterior probability weights Pfmk(i) j z k g.
Such total reliance on the not-so-accurate estimate
ŝk = âkjk for weight update is undesirable. The
inferior performance of this fuzzy variant relative
to the standard IMM algorithm, as indicated by
the simulation results, does not justify the extra
design effort required here. Reference [374] differs
from [247] by using ad hoc fuzzy if-then rules to
obtain ¹(i) (s) based on normalized measurement
residual squared. Note that these fuzzy weights do
not fit particularly well into the IMM configuration;
they can be used equally well/poorly for any other
merging-based MM estimation algorithm. As
mentioned in Section VID, [89] proposed to use a
fuzzified process noise covariance in the Kalman filter
to provide an acceleration estimate for grid adaptation
in VSMM estimation; simulation results of automatic
adjusting model parameters by an if-then rule for
process noise covariance were reported in [274]. In
general, such heuristic blending of a probabilistic
formalism with fuzzy techniques is not appealing.
As a side effect of their flexibility, many
nonstatistical techniques are more susceptive to
misuse than statistical methods. For example, a main
weakness of these nonstatistical methods stems from
their lack of solid, systematic weight update, while the
statistical methods have a built-in solid mechanism for
sequential update of the weights thanks to Bayes’ rule.
The above fuzzy variants rely on ad hoc heuristics
for weight update, while the “mixture of experts” of
[72] and [73] had recourse to an optimization (search)
algorithm with the help of Bayes’ rule.
1306
A two-model MM algorithm was derived in [5]
using a deterministic approach. It can be classified
as a second generation algorithm with a cooperation
strategy that resembles a mixture of the IMM and
GPB1 strategies. As pointed out in [135], however, an
IMM algorithm with uniform transition probabilities
¼ij = 0:5, 8i, j has the same state estimate formula yet
a superior formula for the error covariance.
As pointed out before at the beginning of
Section IV and in IVD, although fundamental for
Bayesian MM estimation, Assumptions A1, A10 ,
A2b, and A2c of Sections IV and V are not needed
for classical (non-Bayesian) estimation, such as ML
estimation. This is more so for nonstatistical methods
of this section, although these methods can still be
classified into the three generations as above.
IX. CONCLUDING REMARKS
The MM estimation approach provides the
state-of-the-art solutions to many maneuvering target
tracking problems. There are basically two directions
to improve the existing solutions.
The first one is to design a better set of models.
Numerous publications have appeared in which
various ad hoc designs were presented. This will
certainly continue. It is extremely challenging to
obtain effective, systematic, and generally applicable
results for model-set design. Relevant theoretical
results are scarce and this deserves more attention.
The other direction is to develop and design
better algorithms. The MM approach started with the
first-generation AMM algorithms in which elemental
filters work independently. Its advantage over
many non-MM approaches stems from its superior
processing of results from elemental filters for output
a posteriori. The first generation has significant
applications for nontracking problems, but limited
value for maneuvering target tracking, because of
its inability to account for information contained in
one elemental filter for better performance of another
filter.
Represented by the IMM algorithm, the
second-generation (CMM) algorithms explore effective
cooperation strategies among elemental filters while
inheriting the first generation’s superior rule for
output processing. The IMM algorithm has been
so successful in solving a number of maneuvering
target tracking problems in the real world that it
has become a standard tool for maneuvering target
tracking. Significant advances have been made in
recent years and further developments are sure to
come. However, their fundamental limitations are clear
and not minor: They believe at any given time one of
their elemental filters is perfect and none of them may
provide incorrect, misleading, confusing, or any other
harmful information. In short, they trust themselves
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 4
OCTOBER 2005
so much that they refuse to adapt themselves to the
outside world, although their estimates are adaptive to
a certain degree. They cannot be expected to perform
well if they are exposed to an environment to which
none of the existing elemental filters fit well, such as
one that is unknown or new to them.
The third generation is potentially much
more advanced in the sense of having an open
architecture–a variable structure–than its ancestors,
which have a closed architecture. Not only does it
inherit the second generation’s effective cooperation
strategies and the first generation’s superior output
processing, but it also adapts to the outside world
by producing new elemental filters if the existing
ones are not good enough and by eliminating those
elemental filters that are harmful. The decisions on
terminating harmful ones are relatively easier–general
and rather successful rules have been obtained. The
task of producing good new filters systematically
in a general setting is much more challenging. A
breakthrough here would be a new milestone in
MM estimation. Similar to model-set design, ad hoc
designs for producing new filters are almost always
obtainable that outperform the first two generations,
given a particular problem. Many products in this area
can be expected in the future. The main drawback of
most algorithms in this generation is their complexity.
The first two generations do have certain
“intelligence” at different levels in that they learn the
environment during the course of estimation, along
with their capability of self-assessment, but they stop
short of drastically adjusting themselves for better
performance. The third generation is more intelligent
as characterized by its self-adjustment to the outside
world, in particular its ability to (re)produce elemental
filters, for the best performance possible.
The operation of a non-MM algorithm amounts to
deciding on the best single individual first, letting him
perform, and then sending out his estimation results.
For the MM estimation, the first generation can be
thought of as a fixed group of individuals working
independently. Its superiority to non-MM algorithms
stems from the fact that its output is generated after
all individuals have performed, which allows, for
example, use of the best performance a posteriori and
optimal combination of individual results. The price
paid is that all these individuals have to perform. The
elemental filters in the second generation in effect
form a cooperative team with a fixed membership.
It outperforms the first generation because of its
team work via cooperation. The third generation can
be likened to an adaptive, cooperative team with a
possibly variable membership. It may recruit new
members and fire bad or incompetent members or put
them on probation. This additional flexibility enables
the third generation to handle a wider spectrum of
intricate and challenging problems in uncertain,
complex, and changing situations.
All three generations have their reasons to exist
because they have their best domains of application.
Clearly, a non-MM algorithm would be optimal if
the best possible individual for the task at the time
could always be chosen. This is possible only in the
absence of uncertainty about the task. If the task
were fixed in time but unknown over a set and the
group were formed by the best possible individuals
for every task in the set, the first generation would be
optimal. The second generation would potentially be
optimal if the task might be changing over time within
a set and the team were formed by the best possible
individuals for every task in the set. If either the best
possible individual for each of the tasks is not part
of the team or some team members do not match any
of the tasks, it would be possible for a variable team
(third generation) to outperform the champions of the
first two generations.
ACKNOWLEDGMENTS
Help from a number of people for writing this part
of the survey is appreciated. The authors would like
to thank particularly Yaakov Bar-Shalom, Henk Blom,
and Dave Sworder.
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X. Rong Li (S’90–M’92–SM’95–F’04) received the B.S. and M.S. degrees
from Zhejiang University, Hangzhou, Zhejiang, PRC, in 1982 and 1984,
respectively, and the M.S. and Ph.D. degrees from the University of Connecticut,
USA, in 1990 and 1992, respectively. He joined the Department of Electrical
Engineering, University of New Orleans in 1994, where he is now university
research professor, department chair, and director of Information and Systems
Technology Research Center. During 1986—1987 he did research on electric
power at the University of Calgary, AB, Canada. He was an assistant professor
at the University of Hartford, West Hartford, CT, from 1992 to 1994. He has
authored or coauthored four books: Estimation and Tracking (with Yaakov
Bar-Shalom, Norwood, MA: Artech House, 1993), Multitarget-Multisensor
Tracking (with Yaakov Bar-Shalom, Storrs, CT: YBS Publishing, 1995),
Probability, Random Signals, and Statistics (Boca Raton, FL: CRC Press, 1999),
and Estimation with Applications to Tracking and Navigation (with Yaakov
Bar-Shalom and T. Kirubarajan, New York: Wiley, 2001); seven book chapters;
and more than 200 journal and conference proceedings papers. His current
research interests include signal and data processing, target tracking, information
fusion, stochastic systems, statistical inference, and electric power.
Dr. Li has served the International Society of Information Fusion as president
(2003), vice president (1998—2002), a member of Board of Directors (since
1998), general chair for 2002 International Conference on Information Fusion,
and steering chair or general vice-chair for 1998, 1999, and 2000 International
Conferences on Information Fusion; served IEEE Transactions on Aerospace
and Electronic Systems as associate editor from 1995 to 1996 and as editor from
1996 to 2003; served Communications in Information and Systems as editor since
2001; received a CAREER award and an RIA award from the U.S. National
Science Foundation. He received 1996 Early Career Award for Excellence in
Research from the University of New Orleans and has given numerous seminars
and short courses in North America, Europe, Asia, and Australia. He won several
outstanding paper awards, is listed in Marquis’ Who’s Who in America and Who’s
Who in Science and Engineering, and consulted for several companies.
Vesselin P. Jilkov (M’01) received his B.S. and M.S. degree in mathematics
from the University of Sofia, Bulgaria in 1982, the Ph.D. degree in the technical
sciences in 1988, and the academic rank senior research fellow of the Bulgarian
Academy of Sciences in 1997.
He was a research scientist with the R&D Institute of Special Electronics,
Sofia, (1982—1988) where he was engaged in research and development of radar
tracking systems. From 1989 to 1999 he was a research scientist with the Central
Laboratory for Parallel Processing–Bulgarian Academy of Sciences, Sofia,
where he worked as a key researcher in numerous academic and industry projects
(Bulgarian and international) in the areas of Kalman filtering, target tracking,
multisensor data fusion, and parallel processing. Since 1999 Dr. Jilkov has been
with the Department of Electrical Engineering, University of New Orleans, where
he is currently an assistant professor, and is engaged in teaching and conducting
research in the areas of hybrid estimation and target tracking. His current research
interests include stochastic systems, nonlinear filtering, applied estimation, target
tracking, information fusion.
Dr. Jilkov is author/coauthor of over 55 journal articles and conference papers.
He is a member of ISIF (International Society of Information Fusion).
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