BIOMETRICS
55, 678-687
September 1999
Simulated Likelihood Methods for Complex
Double-Platform Line Transect Surveys
Tore Schweder,l>*Hans J. Skaug,2 Mette L a n g a a ~ and
, ~ Xeni K. Dimakos
'Department of Economics, University of Oslo, Box 1095 Blindern, 0317 Oslo, Norway
21nstitute of Marine Research, Box 1870 Nordnes, 5817 Bergen, Norway
3Norwegian Computing Center, Box 114 Blindern, 0314 Oslo, Norway
4Department of Mathematics, University of Oslo, Box 1053 Blindern, 0316 Oslo, Norway
* email: tore. schweder@econ.uio.n o
SUMMARY.The conventional line transect approach of estimating effective search width from the perpendicular distance distribution is inappropriate in certain types of surveys, e.g., when an unknown fraction
of the animals on the track line is detected, the animals can be observed only at discrete points in time,
there are errors in positional measurements, and covariate heterogeneity exists in detectability. For such
situations a hazard probability framework for independent observer surveys is developed. The likelihood of
the data, including observed positions of both initial and subsequent observations of animals, is established
under the assumption of no measurement errors. To account for measurement errors and possibly other
complexities, this likelihood is modified by a function estimated frgm extensive simulations. This general
method of simulated likelihood is explained and the methodology applied to data from a double-platform
survey of minke whales in the northeastern Atlantic in 1995.
KEY WORDS: Abundance estimation; Covariate heterogeneity; g ( 0 ) ; Hazard probability; Measurement errors; Minke whales.
1. Introduction
Consider data gathered in a double-platform line transect survey of animals that are available for observation at discrete
points in time. Examples are whales when they surface and
birds when they sing. The likelihood function of such complex
data is available in an explicit form under simplified hazard
probability models. When errors exist in the recorded positions of the detected animals and uncertainty in the identification of sightings made by both teams, the likelihood function
is too complex to be evaluated directly. This paper presents a
hazard probability model for line transect experiments with
discrete availability and the method of simulated likelihood
to obtain inference in complex stochastic models. We illustrate the methods on data from an extensive double-platform
survey of minke whales in the northeastern Atlantic in 1995.
The hazard probability model for line transect surveys with
discrete availability was proposed by Schweder (1974) in the
context of whale surveys. He considered a temporal-spatial
stochastic point process of surfacings and an observer that
moved at constant speed along the transect line. The sighting
efficiency of the observer was characterized by a hazard probability of sighting a surfacing at a given specified position,
given that the observer previously was unaware of the whale.
Schweder and Host (1992) used the hazard probability approach to model data from line transect surveys. However, the
likelihood function was not easily derived for the complex data
that had been gathered. Schweder and H0st (1992) proposed
estimating a set of descriptive parameters in a simple statistical model and mapping these into estimates of the parameters
of interest by a function estimated from extensive simulations.
The resulting estimates were called maximum simulated likelihood estimates, and we will use the term simulated likelihood
to denote an approximate likelihood of observed data that is
partially estimated from simulated data.
As opposed to more standard line transect surveys (see
Buckland et al., 1993), minke whale surveys in the northeastern Atlantic are characterized by the following:
0
0
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Discrete availability: The animals are available for observation only when they surface. Surfacings last only a few
seconds, whereas the average dive lasts more than a minute.
The surfacings are treated a s discrete points in time.
Reduced detection on the track line: Because of diving and
the general difficulty in sighting a minke whale, the probability that a whale on the track line is detected is less
than 1.
Measurement errors: The radial distance to a sighted surfacing is visually estimated, and the sighting angle is read
from an angle board fixed to the platform. Both radial distance and sighting angle might be biased, and random errors are unavoidable.
Covariate heterogeneity: Sighting efficiency is influenced by
covariates such as sea state, weather, observer team, and
vessel.
Simulated Likelihood Methods
The methodology presented in this paper is devoted especially to situations in which these four points are of importance. We start by outlining methodology related to line
transect surveys in Section 2. In Section 3 the simulated likelihood method is presented, and in Section 4 the methods are
illustrated by application to the minke whale survey data.
The application was presented to the Scientific Committee
of the International Whaling Commission in Schweder et al.
(1997) and was discussed in International Whaling Commission (1997).
2. Hazard Probability Modeling
Hazard probability modeling is useful for line transect surveys
with the following characteristics:
0
0
0
The animals in the population are available for detection
only at short discrete time points. Each event of availability
is called a szgnal, and tbe time period between two signals
is called the znterszgnal tame.
Animal movement is ignorable relative to the speed at which
the observer moves. This holds for both responsive and nonresponsive animal movement.
There are two independent observers, A and B, surveying
the area simultaneously.
The same animal may give several different signals within
the detectable range. The observers record the estimated
relative position of the animal (radial distance and sighting
angle) for every detected signal.
Consider an animal located at perpendicular distance z
from the transect line. Let the probability of detecting the
animal at distance z, i.e., detecting at least one signal, be
denoted by g ( z ) . The function g is referred to as the detection function. The essential parameter when estimating abundance from line transect data is the effective strip half-width
(see Buckland, 1993, p. 38), defined by
1
00
w=
g(x)dx.
(1)
An estimate of w is obtained by substituting an estimate of
g into this formula.
The detection function can be defined in cases of both continuous and discrete availability. In the discrete case, g reflects
not only the observer’s ability to detect animals but also how
often the animals are available for detection. The two main
components of hazard probability models are (i) the hazard
probability function, which governs the observer’s ability to
detect animals, and (ii) the process that governs the availability of the animals, which we call the signal process. Our aim
is to make inference about the hazard probability function.
To gain information about the signal process, additional data
are needed, e.g., dive time data for whales (see Schweder et
al., 1997).
2.1 Data from Independent Observer Surveys
Each of the two observers is exposed to a sequence of signals
from the animals, which are within observable distance. The
sequence of detected signals from a single animal is called a
track. For each signal in the track, time and position (relative
to the observer) are recorded. The tracks of observer A are
called the A tracks, and those of observer B are called the
679
B tracks. If an A track corresponds to a B track, in the sense
that they refer to the same animal, the two tracks are said
to be duplicate tracks. Duplicate tracks need not be identical
because one observer might have detected the animal before
the other, or a subsequent signal might have been missed.
Corresponding signals in duplicate tracks are called duplicate
signals. The identification of duplicate tracks and duplicate
signals is a nontrivial task in the presence of errors in recorded
times and positions. In this section we assume that the
duplicate identification is done without errors. Sections 3 and
4 discuss how measurement error and erroneous duplicate
identification can be accounted for.
The first step in constructing a likelihood function is to
identify the independent data units. We define the combined
observer, AUB. An animal is detected by AUB if it is detected
by either A or B or by both. The animals detected by A U B
are the independent data units. If the animal is detected only
by A, the AUB track is simply the A track and similarly if the
animal is only detected by B. If the animal is detected by both
A and B, the combined track consists of the union of signals
in the two tracks. Thus, data have a two-level hierarchical
structure (signals within animals).
Because we aim only to estimate the hazard probability
function and not the signal process, we construct a conditional
likelihood function in which we condition on all aspects of
data that do not depend on the hazard probability function.
The information in a A U B track is split into three parts:
(a) The position of the initial (first detected) signal. Denote
the relative position of the initial signal by (r,6’), where
r is the radial distance from the observer to the animal
and 6’ is the sighting angle, i.e., the angle between the
sighting line and the transect line.
(b) Was the initial signal detected by only one of the
observers or by both? Let u E {1,2} denote this number.
(c) When the initial signal was detected by only one of the
observers, there is a Bernoulli trial for each subsequent
signal, with outcome “success” if the signal is also
detected by the other observer and outcome “failure” if
not. Let ( r i ,6’i) denote the position of the ith subsequent
signal and define d i by d i = 1 if the i t h Bernoulli trial is
a success and d i = 0 otherwise.
Once an observer has detected a signal from an animal,
the probability that he detects subsequent signals is no
longer governed by the hazard probability function. Thus, the
number, I , of Bernoulli trials in (c) is defined a s the first value
of i for which di = 1. If all d i = 0, we define I a s the number
of the last signal given before the animal is passed, according
to some definition. If no subsequent signals are detected, we
let I = 0, and if the initial signal is detected by both A and
B ( u = a), we let I = 0.
2.2 Likelihood of Independent Observer Data
The likelihood of a A U B track has three components,
corresponding to each of the three data components described
previously. Denote the probability density ( r ,0) in (a) by
fl(r,6’). Further, let f2(u1 r , 6’) be the conditional probability
function of u in (b), given that A U B made an initial
observation at (r,6’). Also, let f S ( d i I ri, 6,) be the conditional
probability function of d i in (c), given that a subsequent
Baometrics, September 1999
680
signal was given at (ri,Qi). The likelihood of a track from the
combined observer is
signal, given that the animal eventually is detected, is qz/E q i ,
where
1>2
where the last term is taken to be unity if I = 0.
For simplicity we refer to the last two terms in (2) as the
Bernoulli part of the likelihood, while f l ( r , 8) is referred to as
the positional part. The Bernoulli and the positional parts
contribute with different kinds of information. As seen in
the following, the Bernoulli part depends only on the hazard
probability function, whereas the positional part also depends
on the signal process.
2.3 Hazard Probability Model
The hazard probability function Q(r,8) is defined as the
conditional probability of detecting an animal that signalizes
at ( r , 8 ) , given that the observer is unaware of the animal
previous to the signal. For simplicity we assume that the
hazard probability functions of A and B are identical. Then
the hazard probability function of the combined observer
AUBis
Under the assumption that animals stay at fixed positions
and that the intersignal times follow a Poisson point process,
Schweder et al. (1996) derived a simple analytical formula for
f l . In Section 2.4 we derive an alternative method which is
not based on the Poisson assumption and which automatically
accounts for measurement errors in distance and sighting
angle.
The functions f 2 and f3 in ( 2 ) depend on Q and Q A ~ as
B
follows:
Let ( ~ ~ and
8 ) (r',8') denote the measured and true
coordinates of the initially detected signal, respectively.
Denote the conditional density of ( r ,Q), given ( T ' , Q'), by
h(r,Q 1 r',8'). Given that an animal is detected and given
its true signal series, W, the density of ( r ,Q)is
In a population of J animals making signal series
density of the position of an initial sighting is
W1,. . . , W J ,the
where p , = C, qt(W,) is the probability of detecting the j t h
animal. This formula provides a way of calculating f l ( r ,8) by
simulation. If we assume that animals do not move, a signal
series W is generated as follows. First, the perpendicular
distance, z, is drawn uniformly from the interval [0,X I , when:
X is chosen so large that the probability of detecting ail
animal at distance z > X is negligible. Second, a sequence of
intersignal times is drawn randomly from a pool of available
sequences, and within the chosen sequence a start point
is drawn at random. This determines the positions of the
signals that constitute W . By repeating this procedure for
a large number of animals, an approximation to f l ( r , 8 ) is
obtained from (5). As a by-product we obtain the following
approximation to the effective strip half-width
.7
f ~ ( dI T, 8 ) = Q(r,Q)d (1 - Q ( r ,8 ) > 1 - d .
As a parametric model for Q , we use Q(T,8) = QO. Q1 ( r ) .
Q 2 ( Q ) , where
where J is the number of simulated animals.
3. Measurement Errors and Simulated Likelihood
It is an assumption of Section 2 that the position of a
Q I ( T ) = 1ogit-l {-x,(T
- qy.))/logit-l(Xrv,)
(4) signaling animal is measured without error. Usually, distances
and sighting angles are measured visually in line transect
Q ~ ( o=) logit-1 {-x,(e - q o ) )/logit-l(XoqQ).
surveys, and thus measurement error (systematic or random)
Here logit-l(z)
= exp(z)/{exp(z)
1) and
= is an important aspect of the data. The direct application of
( q e v e l r A,, v,, Xe, ~ 0 is) the basic model parameter vector. In
the likelihood (2), which is based on the assumption of no
the application presented in Section 4, no angle effect exists, measurement error, will lead to biased estimates of model
such that Q 2 ( Q ) = 1. The parameter
can be estimated by parameters and thus to biased estimates of effective strip
maximizing the log likelihood, l y j o ~-( f l .
half-width. The solution offered by this paper is the method
2.4 Posztzonal Lzkelzhood
of simulated likelihood. This method requires that a model
In this section a simulation technique for evaluating fl ( r ,8 ) for the measurement error is available. The estimation of the
is developed. The two main features of the approach are that measurement error model is outside the scope of the present
sequences of intersignal times are used to represent the signal paper and is not discussed.
In essence, the method of simulated likelihood provides a
process, and the automatic correction for measurement error
bias correction of the likelihood (a),obtained by transforming
in (r,8).
Let W = { ( r i , 8 i ) , z 2 1) be the sequence of true the argument, i.e., the parameters in the model. Determining
positions where a specific animal gives its signals. Here, z = 1 the bias-removing transformation involves the simulation of
corresponds to the last signal before the animal is passed, data with measurement errors added, thus the name simulated
and the index z runs backwards in time. The conditional likelihood. The method is also used to account for duplicate
probability that the zth signal becomes the initially detected identification errors, which result from inaccurate timing and
Qo = logit-'
(3)
( ~ 1 ~ ~ ~ 1 )
+
+
+
Simulated Likelihood Methods
measurements of positions, by simulating data with a realistic
proportion of duplicate identification errors.
The method of simulated likelihood is quite general and
can be used to remove bias due to factors not included in the
likelihood. The method is presented in the following and is
illustrated by a simple example.
3.1 Simulated Likelihood
The basic idea behind the method of simulated likelihood is
to find an approximate statistical model for the data that
allows an explicit likelihood function to be constructed in a
set of auxiliary parameters. This likelihood is regarded as an
approximation to the likelihood in the proper model but in
parameters that at the outset might have an unknown relation
to the parameters of the proper model. The simplified model
is called the shallow model and its parameters the shallow
parameters. The proper model is called the deep model.
The deep model is based on substantive theory describing a
complex stochastic or nonstochastic data-generating process.
It has its strength in interpretation of parameters and in
predictive power but is often difficult to estimate. The shallow
model is a statistical model describing the structure of the
data. The strength of the shallow model lies in its explicit
likelihood. Denote by $ the deep parameter and by $s the
shallow parameter. Bridging the deep and the shallow model
consists of finding a transformation from the deep parameter
II, into the shallow parameter +s. The transformation is called
a bridge and is denoted by F . As an estimate of the bridge, we
use a smooth function that can be obtained from regression
analysis of extensive data obtained by simulating the deep
model at various values of $. At each of a number of points
in $ space, simulated data are obtained. These are then
assumed to come from the shallow model, and a maximum
likelihood estimate, +s, of gS is calculated. The set ($,?lS)
over the design set for $ is then used as data for a regression
is obtained
analysis, and a smooth function F : $ -+
as the prediction function. The bias corrected log-likelihood
function, called the simulated log likelihood, is
Gs
M$)= l s h d , o w ( F ( $ H .
The value of $ that maximizes l s ( $ ) is called the maximum
simulated likelihood estimate and is denoted by
We
emphasize that the original data enter 1, only through &hallow,
whereas the simulated data enter only through F .
The validity of this approach might need to be discussed.
First, when the amount of simulated data becomes large and
F will converge to the true bridge F . Then the approximate
log likelihood is defined as l a ( $ ) = &f,allo,v(F($)). The
estimates based on l a ( $ ) are consistent under very general
conditions, as might be inferred from, e.g., Gourieoux,
Monfort, and Renault (1995). Because of the nature of
this paper, we will not elaborate on this point further.
It might be difficult t o find a simple regression model
that provides a global approximation to the bridge. In our
application it was practical to use linear regression to obtain
a local approximation. However, some iteration was required
to find a design set for $ in the appropriate neighborhood
of the resulting estimate of the deep parameter. If the bridge
is markedly nonlinear, an estimate based on linear regression
depends substantially on the design set for $ that is used to
obtain input data, (II,,qs),to the regression analysis. Thus,
681
only after some iteration, a design set is found that produces
a locally linear estimate of the bridge that in turn provides a
maximum simulated likelihood estimate that lies in the design
set.
The difference between the deep log likelihood and the
simulated log likelihood consists of the difference due to
model approximation, l&ep - l a , and the difference due to
simulation and bridge estimation, la - I,. The adequac?
of the approximation of the deep likelihood estimate, $,
by the simulated likelihood estimate, 4, depends on the
information content of the simulations carried out to estimate
the approximate bridge, the assumptions made regarding the
structure of the approximate bridge and the appropriateness
of the shallow statistical model. Standard diagnostic methods
are available to check that the shallow model fits the observed
data, and statistical theory is helpful in identifying a model
that gives an efficient summary of the observed data.
Simulated likelihood was first used in the context of line
transect sampling by Schweder and Host (1992). However, the
approach has been used earlier in other contexts and in other
ways and with other terminology than ours (see Diggle and
Gratton, 1984, and the collection of van Dijk, Monfort, and
Brown, 1995). Our bridge function is parallel to the binding
function of Gourieroux et al. (1995), who call the approximate
likelihood the pseudolikelihood and the simulated likelihood
the simulated pseudolikelihood but who estimate the binding
function in a different manner from our bridge function.
In our line transect case with discrete availability, the
shallow model is taken as the hazard probability model but
disregarding measurement errors in the positional recordings
and assuming perfect duplicate identification. An example is
presented to illustrate how the bridge is estimated in a simple
example.
3.2 A Simple Example
In a conventional line transect situation, assume that the
observed perpendicular distance, Y , is a grouped version of
a half-normal variate. With $ being the scale parameter
of interest and @ being the cumulative normal distribution
function, the deep model is
4.
f$ (y) = 2
[.(-->+
2y
1
-@
(3g)
]
I
Y = 1,2, . . .
Inference about $ should be based on the deep log likelihood
Ideep($)
= ~ v y n y l o g ( f + ( y ) ) where
,
ny is the observed
frequency of Y = y.
The deep, the approximate, and the simulated log
likelihoods are compared on the basis of n = 100 simulated
data points generated by simulation of Y = r a n d ( 10.75Zl),
where Z is drawn from the standard normal distribution, i.e.,
$ = 0.75. The simulation resulted in the values y = 0, 1 and
2 appearing with frequencies 51, 44 and 5, respectively.
As our shallow model we assume that the distribution of Y
is half-normal with parameter $s, with shallow log likelihood
lshallow($'s) = -1/(2q!~;)CY,~ - nlog(+s) and the shallow
maximum likelihood estimator qs = [(l/n)Cr=l yi2]1/2. We
use a linear function in $ as our regression model for the
bridge between the deep and the shallow model. On the basis
Biornetrics, September 1999
682
of the simulated observations of {Yt 1
for $ in [0.5,1.0],
the linear bridge is estimated by regressing the shallow
maximum likelihood estimate
on the true deep parameter
$. In this simple example we can also calculate the true
bridge, F ( $ ) = [E(Y2 I
yielding the approximate log
likelihood Za(F(+)).
For the observed data, the (deep) maximum likelihood
estimate is $ 0.75. The approximate maximum likelihood
estimate is +a = argmaxl,(F(+))
= 0.75, and the
maximum simulated likelihood estimate is = 0.74, obtained
from transforming the estimate gS = 0.80 of the shallow
parameter by the estimated bridge. The deep, approximate,
and simulated log-likelihood function fell virtually on top of
one another and are shown together with the shallow log
likelihood in Figure 1.
Rounding or heaping is not uncommon in line transect
data. Buckland et al. (1993) discuss the method of smearing
out heaped observations before applying standard estimation
techniques (Butterworth, 1982). The approximate or the
simulated likelihood approach illustrated previously might
outperform that method because it is conceptually cleaner
and is likely to make better use of the data.
Gs
4
4. Application
The methodology of Sections 2 and 3 is illustrated by
application to the data from the Norwegian double-platform
minke whale survey in 1995 (see Schweder et al., 1997). A
full description of the procedures for collecting data and the
screening process of the covariates is found in the same paper.
4.1 Data and Data Processang
The survey covered 824,000 square nautical miles in the
northern North Sea, the Norwegian Sea, the Greenland Sea,
and the Barents Sea. The surveyed area consisted of 18
survey blocks. An IWC management area consists of one
or several survey blocks. Four management areas, EB, ES,
EC, and EN, define the northeastern Atlantic management
area. An additional management area, CM, was partly
covered by the Norwegian survey (see Figure 2). A total
of 11 vessels traversed the areas on primary search mode,
i.e., on predefined transect lines under acceptable weather
and sighting conditions. On each vessel were two separated
observer platforms, A and B, usually in the crows nest and
on the wheel-house roof. The individuals were grouped in
fixed teams of three, with two on duty and the third resting
on a rotational basis. In the notation of Section 2, each
platform equipped with one team constituted an observer
unit. The observational protocol instructed the individual
making an initial sighting to record the positional data of tht:
surfacing and to follow the whale and record all subsequent
sightings until the whale had been passed. The positional
data consisted of an estimate of the radial distance ( r )
from the vessel to the whale and the sighting angle (0)
between the transect line and the animal, found from an angle
board. This positional observation, together with the time
( t )of every sighted surfacing, was recorded on tape. To gain
information about the measurement error, distance and angle
measurement experiments were conducted in addition to the
ordinary survey activities.
An automated routine is used to identify duplicate tracks
and duplicate surfacings in the parallel data from the two
observer units: the A platform and the B platform. This paper
presents recalculated results for the automatic routine DIR2
and repeats the results for the not fully automatic routine
DIRl from Schweder et al. (1997). Both routines are described
in the reference paper. From the identified tracks and from
identified duplicate surfacings, positional data and Bernoulli
trials are constructed. The shallow likelihood is obtained from
the Bernoulli trials data. The duplicate identification routine
is also applied to simulated data as part of the simulatcd
likelihood analysis.
4.2 Covariates
The sightings are identified by vessel, platform, and team and
characterized by sighting conditions in addition to position
and time. The sighting conditions are measured by the
weather and the sea state (the Beaufort scale). The covariates
are made subject to initial screening to simplify the scales and
reduce the number of parameters in the linear model relating
I
,
I
I
I
05
06
07
08
0.9
10
Figure 1. Log likelihoods of the simplified example. The
plot shows the deep log likelihood (solid), the approximate log
likelihood (dotted), the simulated likelihood (dashed-dotted),
and the shallow log likelihood (dashed) as functions of 4.
-20
0
20
40
60
Figure 2. North Atlantic small management areas. The
dotted line represents the ice edge.
Simulated Likelihood Methods
qr(qlevel
683
= -1)
Figure 3. Estimated bridge for the Bernoulli log likelihood. The upper two panels show the
shallow qlevel plotted as a function of the deep q e v e 1 , keeping the remaining components of @
fixed, with deep qr = 950 to the left and qr = 700 to the right. The figure shows the bridge
evaluated at five points, with simulation uncertainty indicated by a box plot. The solid line
represents the linear estimate of the bridge. The dashed line is the identity function. Similarly,
the lower panel shows the shallow qr plotted as a function of the deep qr for the deep qlevel = -1.
the covariates to the linear predictors qlevel and qr in ( 3 ) and
(4).
The screening procedure results in a team specific covariate
(RTEAM) in qr and (LTEAM) in qlevel, each with three levels,
reflecting the sighting efficiency of the teams. The categories
in the radial and level component are named {SHORT, AVERAGE, LONG} and {HIGH, MEDIUM, LOW}, respectively. The
weather and sea state are used to construct a combined sighting condition covariate (WB) with three levels {WBI, WBII,
WBIII}. The levels of WB are ordered in the sense that WE1
indicates blue sky and less than 20% cloud coverage and low
Beaufort; WBII partly cloudy, rain, fog or drizzle, and low
Beaufort; and WBIII high Beaufort. The covariate WB, as well
as the platform covariate (PLAT), enters the radial component. Sum-to-zero constraints are used to identify the parameters of the categorical covariates in the linear predictors
LTEAM
Wevel
qr
N
RTEAM+WB+PLAT,
presented in standard GLM formulation (see McCullagh and
Nelder, 1989). The observed angles are nearly uniformly distributed, and thus the angle component of the hazard probability is excluded by assuming Qa(Q)zz 1, leaving the basic
parameter - = ( q e v e l , AT, qr). Suppressing parameters de-
+
termined by the sum-to-zero constraints, the parameters to
be estimated are the slope parameter X r and the regression
parameters
P=
(P',P SHORT ,P LONG IPWE1 PWB", P
1
,plevel,
PLOW>PMEDIUM
)3
where
P'
and
/?level
are intercept terms.
4.3 Deep Mode2
The deep model describes the complex data-generating
mechanism. From the distance and angle measurement experiments and from identified duplicate sightings, a simple model
of errors in positional recordings and time measurements
is estimated (see Schweder, 1997). Measurement errors are
assumed independent and identically distributed for all
observations and observers.
In addition to the measurement error model and the
duplicate identification rule, information on truncation of
the data, probability of missing positional data, probability
of losing track of a whale, probability of holes in tracks,
and spatial distribution of whales according to an estimated
Neyman-Scott point process are parts of the deep model. This
is taken into account in the simulated likelihood analysis (see
Schweder et al., 1997).
Biometrics, September 1999
684
Level intercept
Radial intercept (p’)
@level)
I
\
1
.sxI
.xu
SHORT RTEAM
ZW
53a
LONG RTEAM
,103
(PSHoRT)
am
(PLoNG)
LOW LTEAM
00
02
(pLow)
0a
04
MEDIUM LTEAM
(PMEDIUM)
Figure 4. Trace plots along selected DIRZ parameter estimates of Table 1 for the simulated
log likelihood of the Bernoulli data 15 (solid line), the positional log likelihood 11 (dotted),
and the total log likelihood Z T ~ T (dashed). The plots are translated so that the maximum
log-likelihood value equals zero for all three log likelihoods.
4.4 Shallow Model and Bridge
The shallow model is a simplification of the deep model,
in which measurement errors are neglected, assuming that
the duplicate identification is correct. The shallow model
likelihood is given in (2).
The positional likelihood is judged to be rather insensitive
to duplicate identification errors, and because measurement
errors in the positional recordings are accounted for when
f1 ( r ,6) is calculated as in Section 2.4, no bridge is needed
for this part of the likelihood. For the Bernoulli part of
the likelihood, a bridge, F , from the deep parameter
to
its shallow counterpart
is estimated. This is d o n e by
simulating the deep model over a grid of values of +, and
gs,
+
A
applying multivariate linear regression to the data (+,
_ +s),
_
where - is the maximum likelihood estimate obtained by
applying the shallow Bernoulli likelihood to the simulated
survey data.
6s
Simulated Likelihood Methods
The deep model is computationally demanding, and the
number of simulations have to be kept within limits. Because
the linear predictors q e v e l and 7, are judged to be most affected by the simplifications made in the shallow model, the
effort is spent on estimating a bridge for these two parameters, and the identity bridge is assumed for A,. Aspects of the
estimated bridge are shown in Figure 3. The model is simulated over a 5 x 5 grid in the intercept term of 7leve1 and
7,. Since the predicted values of q e v e l and 17, basically vary
within the region of support for the bridge as the covariates
vary, no extra bridging is needed for estimating the regression
coefficients determining these predicted values.
4.5 The Maximum Simulated Likelihood Estimate
For a fixed set of slopes A,, the positional part of the log
likelihood, when calculated as in Section 2.4, is a continuous
function. The simulated Bernoulli part of the log likelihood is
also continuous. To find the maximum simulated likelihood
estimate, the total simulated log likelihood ITOT(+) - =
Ir(2) l ~ ( F ( 2 is) )maximized. Here l r ( + ) is the positional
part of the log likelihood, and l ~ ( k ( +is i the Bernoulli
part adjusted with the estimated bridgeTThe optimization is
performed using the variable metric method (see, e.g., Gill,
Murray, and Wright, 1981). Figure 4 shows trace plots of
the log likelihood, and Table 1 gives the resulting parameter
estimates. The results for DIRl is taken from Schweder et al.
(1997). Because of minor differences in the calculation of the
positional likelihood, the recalculated estimates are slightly
different from those given for DIR2 in Schweder et al. (1997).
+
4.6 Abundance Calculatzons
The parameter estimates of Table 1 influence the abundance
estimate through the strip half-width defined by (1)and are
estimated as explained in Section 2.4. The strip half-width
quantifies the sighting efficiency as the combined effect of the
observer efficiency characterized by the hazard probability
function and animal behavior characterized by the signal
process. The expected number of sighted whales over a
transect leg of length L, with constant strip half-width w, is
2wLD, where D is the spatial density of whales. For a given
estimate of the slope A, and the regression parameter p, the
strip half-width is calculated for each setting of the covazates.
The effective strip half-width varies along the transect line
as the covariates change. The average strip half-width for a
L
survey block is w =
w(t)dlfL, where w(t) is the strip halfwidth at position 1 and L is the total survey length over all
the vessels that have contributed in the block.
Abundance estimates are found for each survey block.
Whereas a sighting from the combined platform A U B is
regarded as the sampling unit in the parameter estimation,
so
685
each platform is regarded as a separate unit in the abundance
estimation. This is done to exclude errors imposed by the
duplicate identificationlroutine. For a given survey block, the
abundance estimate is N = n Af a , where n is the total number
of sighted whales in the block, a = 2Lw is the estimated
effective search area, and A is the total area of the survey
block. Because the two platforms are exposed to the same
whales, the number of sighted whales and the estimated strip
half-widths of the platforms are correlated. This correlation
is accounted for in the variance calculation of the abundance
estimates (see Schweder et al., 1997). The variance in the
counts, n, is estimated by fitting Neyman-Scott point process
models describing the clustering of whales over the ocean.
The block estimates are aggregated to form estimates for
the management areas, as shown in Table 2. The abundance
estimate for a management area is the sum of the block
estimates of the corresponding blocks. Sum estimates are
given for the total survey area (SUM) and for the northeastern
Atlantic management areas (SUME), which excludes the area
CM.
Standard deviations for the abundance estimates of Table
2 are found by using a parametric bootstrapping technique
described in Schweder et al. (1997). The standard deviations
and coefficients of variance presented in Table 2 are based
on the duplicated identification rule DIR1. These are also
reasonable estimates for the rule DIR2.
5 . Discussion
The overall purpose of this paper is to establish an
approximation to the likelihood function of independent
observer data when discrete availability of animals and
measurement errors in positions are intrinsic features of
the data. This is achieved in two steps. In Section 2 the
likelihood function is derived explicitly under the assumption
of no measurement errors, and in Section 3 the simulated
likelihood approach is used to account for measurement errors
and consequential duplicate identification errors. Many of
the ideas presented in this paper originate from discussions
within the Scientific Committee of the International Whaling
Commission (IWC). However, the methodology should be
applicable in a broader context.
The simulated likelihood method has the potential of
providing inference in situations in which the basic model
is too complicated to allow the likelihood t o be determined.
The method should therefore be of help in complex situations
in which the researcher traditionally has been faced with
the dilemma of either oversimplifying his model to allow
an explicit likelihood or using ad hoc methods to obtain
inferential results. When using simulated likelihood, the work
could be more evenly shared between the subject matter
Table 1
Estimates for the scale and covariate parameters (DIR1 results from Schweder et al., 1997)
DIR2est.
DIRl est.
DIRl SD
A,
pr
pSHORT
pLONG
pWB1
pWBII
pPLATB
plevel
PLOW
PMEDIUM
0.0065
0.0065
0.0005
851
828
44
-247
-223
31
309
290
26
226
247
51
28
3
34
-126
-99
31
-1.09
-1.00
0.19
-0.80
-0.81
0.11
0.37
0.38
0.08
Biometries, September 1999
686
Table 2
Abundance estimates, N, aggregated over management areas
Area
EB
ES
EC
EN
CM
SUM
SUME
N
NDIR~
SDDIR~
CVDIR~
56,233
25,706
2445
28,536
6295
56,330
25,969
2462
27,364
6174
7651
2908
562
5626
2203
0.136
0.112
0.228
0.206
0.337
119,215
112,920
118,299
112,125
12,212
11,639
0.103
0.104
scientist and the statistician. The deep model would normally
be a responsibility of the subject matter scientist, whereas the
shallow model would be joint work with the statistician. The
model for the bridge would normally be worked out by the
statistician. Without the requirement that the deep model
should allow an explicit likelihood, one has more freedom
when developing the deep model. On the other hand, the
bridge allows the shallow model to be specified without being
overburdened with subject matter theory.
The degree of complexity of the deep model is a matter of
choice. In the whale abundance application, a rather complex
model was found necessary to account for factors such a s measurement errors and duplicate identification errors. The simulated likelihood approach further allows complicating factors,
such as responsive behavior of the animals, random movement, random heterogeneity in signal strength, and correlated
behavior in close animals, to be incorporated in the analysis.
This would require explicit modeling of these various phenomena, to make the data simulated from the deep model t o be
generated according to the mare complex specification. The
gain obtained by adding complexities is the ability to study
the interacting effects of these phenomena on the estimates
of abundance and to remove any further possible bias.
If the deep model can be facilitated, more explicit statistical models are available. In the present double-platform line
transect context with discrete signals, signals can be assumed
to follow a Poisson point process if the clustering is not too
severe. Further, assuming a convenient parametric model for
the hazard probability function, Skaug and Schweder (1998)
obtained the likelihood function on the basis of the observed
perpendicular distances from the track line of the initial sighting. The likelihood was obtained in a model without measurement errors and without covariate heterogeneity, but these
restrictions might turn out to be unnecessary. The perpendicular distance method is far less computationally demanding
than the simulated likelihood method.
The data from the Norwegian survey in 1995, together with
data from previous surveys, are lodged with the secretariat of
the IWC. Scientists accredited to the commission have full
access to all data used in this paper and in previous papers
presented to the IWC. The software for carrying out the simulated likelihood analysis is also lodged with the commission.
The software have been validated (see International Whaling
Commission, 1997) but is complex and extensive and does not
have a commercial interface.
RESUME
L’approche conventionnelle utilisant la distribution des distances au transect pour estimer la largeur efficace de recherche
dans l’kchantillonnage par transect , est inapproprike dans certain types d’enqugtes: par exemple quand une fraction inconnue des animaux est dktect6e sur le sentier, lorsque les animaux ne peuvent &re observes qu’8 certain moments, lorsqu’il
y a des erreurs dans les mesures de localisation ou qu’il existe
des sources d’hktkrogknkitk dans la dktection. Pour de telles
situations, nous avons dkveloppk un cadre probabiliste p o u ~
des observateurs indkpendants. La vraisemblance des donnkes.
incluant la position observke des animaux pour le premiei
contact et les suivants, est ktablie sous l’hypothkse d’absence
d’erreur de mesure. Pour prendre en compte ces erreurs dc
mesure et la possibilitk d’autres phknomknes complexes, cettr
vraisemblance est modifike par une fonction estimke 8 partir
d’un grand nombre de simulations. Cette mkthode gknkrale
de vraisemblance simul6e est expliquke et la mkthodologie est
appliquke h des donnkes issues d’une enquste sur des baleines
dans la partie NE de I’Atlantique en 1995.
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