IEEE TRANS. ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, TO APPEAR, 2013
1
Clustering Dynamic Textures with the
Hierarchical EM Algorithm for Modeling Video
Adeel Mumtaz, Emanuele Coviello, Gert. R. G. Lanckriet, Antoni B. Chan
F
Algorithm 2 Kalman filter and Kalman smoothing filter
A PPENDIX A
S ENSITIVITY
ANALYSIS
SMOOTHING FILTER
OF
THE
K ALMAN
In this section, we derive a novel efficient algorithm for performing
sensitivity analysis of the Kalman smoothing filter, which is used
to compute the E-step expectations. We begin with a summary
of the Kalman smoothing filter, followed by the derivation of
sensitivity analysis of the Kalman filter and Kalman smoothing filter.
Finally, we derive an efficient algorithm for computing the expected
log-likelihood using these results.
A.1
1: Input: DT parameters Θ = {A, Q, C, R, µ, S, ȳ}, video y1:τ .
2: Initialize: x̃1|0 = µ, Ṽ1|0 = S.
3: for t = {1, · · · , τ } do
4:
Kalman filter – forward recursion:
Ṽt|t−1 = AṼt−1|t−1 AT + Q,
T
Kt = Ṽt|t−1 C (C Ṽt|t−1 C
x̃t|t−1 = Ex|y1:t−1 [xt ],
(42)
+ R)
(50)
Ft = A − AKt C.
−1
,
Jt−1 = Ṽt−1|t−1 AT Ṽt|t−1
Ht−1 = A−1 − Jt−1 ,
Ṽt−1|τ = Ṽt−1|t−1 + Jt−1 (Ṽt|τ −
Ṽt−1,t−2|τ =
T
Ṽt−1|t−1 Jt−2
T
Ṽt|t−1 )Jt−1
,
x̃t|t−1 = Ft−1 x̃t−1|t−2 + Gt−1 (yt−1 − ȳ),
x̃t−1|τ = Ht−1 x̃t|t−1 + Jt−1 x̃t|τ ,
+ Jt−1 (Ṽt,t−1|τ −
where {Ft , Gt , Jt , Ht } are defined in (52), (53).
Finally, note that the conditional covariances, Ṽt|τ and Ṽt,t−1|τ
in (54) and (55), and matrices {Ft , Gt , Jt , Ht }, are not functions
of the observed sequence y1:τ . Hence, we have
(r)
(r)
V̂t = Ey|Θb [Ṽt|τ ] = Ṽt|τ ,
(r)
(r)
(46)
V̂t,t−1 = Ey|Θb [Ṽt,t−1|τ ] = Ṽt,t−1|τ .
• A. Mumtaz and A. B. Chan are with the Department of Computer Science,
City University of Hong Kong.
E-mail: adeelmumtaz@gmail.com, abchan@cityu.edu.hk.
• E. Coviello and G. R. G. Lanckriet are with the Department of Electrical
and Computer Engineering, University of California, San Diego.
E-mail: emanuetre@gmail.com, gert@ece.ucsd.edu.
(53)
(54)
(55)
(56)
9: end for
10: Output: Kalman filter matrices {Ṽt|t−1 , Ṽt|τ , Ṽt,t−1|τ , Gt , Ft , Ht },
and state estimators {x̃t|t−1 , x̃t|t , x̃t|τ }.
A.2 Sensitivity Analysis of the Kalman smoothing
filter
We consider the
associated Kalman
(r)
(44)
(45)
(52)
T
AṼt−1|t−1 )Jt−2
,
x̃t−1|τ = x̃t−1|t−1 + Jt−1 (x̃t|τ − Ax̃t−1|t−1 ).
(43)
Both filters are summarized in Algorithm 2. The Kalman filter
consists of a set of forward recursive equations (Alg. 2, line 4),
while the Kalman smoothing filter contains an additional backward
recursion (Alg. 2, line 8).
For sensitivity analysis, it will be convenient to rewrite the state
estimators in (50) and (56) as functions only of x̃t|t−1 and x̃t|τ ,
(51)
5: end for
6: Initialize: Ṽτ,τ −1|τ = (I − Kτ C)AṼτ −1|τ −1 .
7: for t = {τ, · · · , 2} do
8:
Kalman smoothing filter – backward recursion:
x̃t|τ = Ext |y1:τ [xt ],
Ṽt|τ = covxt |y1:τ (xt ),
(48)
(49)
x̃t|t = x̃t|t−1 + Kt (yt − C x̃t|t−1 − ȳ),
while the Kalman smoothing filter estimates the state conditioned
on the fully observed sequence y1:τ ,
Ṽt,t−1|τ = covxt−1,t |y1:τ (xt , xt−1 ).
,
x̃t|t−1 = Ax̃t−1|t−1 ,
Gt = AKt ,
Ṽt|t−1 = covx|y1:t−1 (xt ),
−1
Ṽt|t = (I − Kt C)Ṽt|t−1 ,
Kalman smoothing filter
The Kalman filter [36, 26] computes the mean and covariance of the
state xt of an LDS, conditioned on the partially observed sequence
y1:t−1 = {y1 , . . . , yt−1 },
(47)
T
(r)
(r)
two LDS, Θb and Θr , and their
(b)
(b)
(b)
(b)
(b)
filters {Ft , Gt , Ht , x̃t|t−1 , x̃t|τ } and
(r)
(r)
{Ft , Gt , Ht , x̃t|t−1 , x̃t|τ }. The goal is to compute the mean
and covariance of the Kalman smoothing filter for Θr , when the
source distribution is Θb ,
h
i
(r)
(r)
x̂t = Ey|Θb x̃t|τ , κ̂t = covy|Θb (yt , x̃t|τ ),
(57)
(r)
(r)
(r)
χ̂t = covy|Θb (x̃t|τ ), χ̂t,t−1 = covy|Θb (x̃t|τ , x̃t−1|τ ).
To achieve this, we first analyze the forward recursion, followed by
the backward recursion.
A.2.1
Forward recursion
For the forward recursion, the Kalman filters for Θb and Θr are
recursively defined by (44),
" (b) # " (b) (b)
#
(b)
(b)
x̃t|t−1
Ft−1 x̃t−1|t−2 + Gt−1 (yt−1 − ȳb )
=
,
(r)
(r) (r)
(r)
(b)
x̃t|t−1
Ft−1 x̃t−1|t−2 + Gt−1 (yt−1 − ȳr )
(b)
where {yt } are the observations from source Θb . Substituting (2)
(b)
(b)
(b)
of the base model, i.e., yt−1 = Cb xt−1 + wt−1 + ȳb , and including
IEEE TRANS. ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, TO APPEAR, 2013
Algorithm 3 Sensitivity Analysis of Kalman filter
Algorithm 4 Sensitivity Analysis of Kalman smoothing filter
1: Input: DT parameters Θb and Θr , Kalman filter matrices
(b)
(b)
(r)
(r)
{Gt , Ft , Gt  , F
τ.
t }, length 

Sb 0 0
µb



0 0.
2: Initialize: x̂1 = µb , V̂1 = 0
0
0 0
µr
3: for t = {2, · · · , τ + 1} do
4:
Form block matrices:


Ab
0
0
(b)
G(b) C
0 
At−1 =  t−1 b Ft−1
,
(r)
(r)
0
Ft−1
Gt−1 Cb


(59)


 
0
0
I
(b) 

B = 0 , Ct−1 = Gt−1  , Dt−1 =  0  .
(r)
(r)
0
Gt−1
Gt−1
5:
2
1: Input: DT parameters Θb and Θr , Kalman smoothing filter matrices
(b)
(b)
(r)
(r)
(r)
(r)
{Gt , Ft } and {Gt , Ft , Ht , Jt }, Kalman filter sensitivity
analysis {x̂t , V̂t }, length τ .
[3]
−1
−1 [3,3] −T
2: Initialize: x̂τ = A−1
r x̂τ +1 , χ̂τ = Ar V̂τ +1 Ar , Lτ = Ar ,
Mτ = 0.
3: for t = {τ, · · · , 1} do
4:
Compute cross-covariance:
(r) [3,2]
(r)
[1,1]
(r)
ρ̂t = Lt Ft V̂t
+ (Lt Gt Cb + Mt )V̂t
CbT + Lt Gt Rb .
(67)
5:
6:
if t > 1 then
Compute sensitivity:
(r)
ω̂t = Lt Ft
x̂t−1 =
Update means and covariances:
x̂t = At−1 x̂t−1 + Dt−1 (ȳb − ȳr ),
h
(r)
χ̂t−1 = Ht−1
(60)
T
T
V̂t = At−1 V̂t−1 AT
t−1 + BQb B + Ct−1 Rb Ct−1 .
the recursion of the associated state-space
given in (1), we have

 (b)  
(b)
(b)
Ab xt−1 + vt
xt
(b)
(b)
(b)

x̃(b)  F (b) x̃(b)

 t|t−1 = t−1 t−1|t−2 + Gt−1 (Cb xt−1 + wt−1 )
(r)
(b)
(b)
(r) (r)
(r)
x̃t|t−1
Ft−1 x̃t−1|t−2 + Gt−1 (Cb xt−1 + wt−1 + ȳb − ȳr )
(b)
(b)
+ Dt−1 (ȳb − ȳr ),
(r)
Mt−1 =
(r)
(r)
(r)
= Ht−1 x̂t + Jt−1 x̂t ,
(62)
with initial condition
x̂τ = E
=
(r)
y|Θb [x̃τ +1|τ ]
E
=
[3]
A−1
r x̂τ +1 .
+ Mt )Ab .
(73)
(r)
(r)
(r)
(r)
(r)
(66)
(r)
(r)
A.3
Expected Log-Likelihood
log p(y|Θr ) =
τ
X
log p(yt |y1:t−1 , Θr )
t=1
τ
X
(r)
log N (yt |Cr x̃t|t−1 + ȳr , Σt )
t=1
τ
X
−1
tr
2
h
(74)
T
Σ−1
t (yt − ȳr − Cr x̃t|t−1 )(yt − ȳr − Cr x̃t|t−1 )
(r)
(r)
i
(64)
Substituting (45) into the definition of χ̂t,t−1 yields
m
2
log(2π),
(75)
(r)
noting that Ṽt|t−1 and Σt are not a functions of the observations
y1:t−1 ,
` = Ey|Θb [log p(y|Θr )]
τ
h
i
X
T
T
T
−1
=
tr Σ−1
t (Ût − λ̂t Cr − Cr λ̂t + Cr Λ̂t Cr )
2
(r)
covy|Θb (A−1
r x̃τ +1|τ )
[3,3] −T
= A−1
Ar .
r V̂
log |Σt | −
(r)
(r)
(r)
−T
A−1
r covy|Θb (x̃τ +1|τ )Ar
1
2
where Σt = Cr Ṽt|t−1 CrT + Rr . Taking the expectation of (75), and
where ω̂t = covy|Θb (x̃t|t−1 , x̃t|τ ), and the initial condition is
=
(72)
Finally, the cross-covariances, ω̂t = covy|Θb (x̃t|t−1 , x̃t|τ ) and
(r)
κ̂Tt = ρ̂t = covy|Θb (x̃t|τ , yt ), are calculated efficiently using the
recursion in (67) and (68), where {Lt , Mt } are recursively given by
(72) and (73). The derivation is quite involved and appears in the
Appendix B. The algorithm for sensitivity analysis of the Kalman
smoothing filter is summarized in Algorithm 4.
−
= covy|Θb (Ht−1 x̃t|t−1 + Jt−1 x̃t|τ )
" (r) T #
h
i [3,3]
V̂t
ω̂tT (Ht−1 ) ,
(r)
(r)
= Ht−1
Jt−1
(r)
ω̂t
χ̂t
(Jt−1 )T
=
,
t=1
(r)
χ̂τ =
(r)
(r)
(r)
=
χ̂t−1 = covy|Θb (x̃t−1|τ )
(r)
covy|Θb (x̃τ |τ )
(71)
= ω̂t (Ht−1 )T + χ̂t (Jt−1 )T .
(63)
Taking the covariance of (45), we obtain the recursion,
(r)
(r)
(r)
(r)
Jt−1 (Lt Gt Cb
(r)
=
A−1
r
, (70)
= covy|Θb (x̃t|τ , Ht−1 x̃t|t−1 + Jt−1 x̃t|τ )
(r)
= Ht−1 Ey|Θb [x̃t|t−1 ] + Jt−1 Ey|Θb [x̃t|τ ]
(r)
y|Θb [x̃τ |τ ]
#
The expected log-likelihood Ey|Θb [log p(y|Θr )] can be calculated
efficiently using the results from the sensitivity analysis for Kalman
filters. First, the observation log-likelihood of the DT is expressed
in “innovation” form
x̂t−1 = Ey|Θb [x̃t−1|τ ]
[3]
ω̂tT
χ̂t
(r)
(Ht−1 )T
(r) T
(Jt−1 )
8:
end if
9: end for
10: Output: {x̂t , χ̂t , χ̂t,t−1 , κ̂t = ρ̂T
t }.
(58)
Taking the expectation of (45) yields a recursion for x̂t ,
(r)
[3,3]
V̂t
ω̂t
"
(r)
(r)
Backward recursion
(r)
(69)
Lt−1 = Ht−1 + Jt−1 Lt Ft
(r)
(r)
(68)
χ̂t,t−1 = covy|Θb (x̃t|τ , x̃t−1|τ )
where xt = [(xt )T , (x̃t|t−1 )T , (x̃t|t−1 )T ]T , and the block matrices
{At−1 , B, Ct−1 , Dt−1 } are defined in (59).
Finally, taking the expectation of (58), with respect to
{x1:τ , y1:τ } ∼ Θb , yields the recursive equations for x̂t in
(60). Similarly, taking the covariance of (58) yields a recursive
equation for V̂t in (61), where we have used the fact that
(b)
(r)
(b)
(b)
(b)
(b)
(b)
⊥ wt . The
⊥ {xt−1 , x̃t−1|t−2 , x̃t−1|t−2 }, and vt ⊥
{vt , wt−1 } ⊥
recursive equations for the sensitivity analysis of the Kalman filter
are summarized in Algorithm 3.
A.2.2
,
Update matrices:
which can be rewritten succinctly as
+
(r)
Jt−1
i
(r)
7:
(b)
Ct−1 wt−1
[2,3]
χ̂t,t−1 = ω̂t (Ht−1 )T + χ̂t (Jt−1 )T .
(b)
xt−1
xt = At−1 xt−1 +
+
(r)
+ (Lt Gt Cb + Mt )V̂t
(r)
Jt−1 x̂t ,
(61)
6: end for
7: Output: x̂t , V̂t .
(b)
Bvt
[3,3]
V̂t
(r) [3]
Ht−1 x̂t
(65)
t=1
−
1
m
log |Σt | −
log(2π),
2
2
(76)
IEEE TRANS. ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, TO APPEAR, 2013
3
or in general, for s > t,
where
(r)
(r)
(r)
(r)
(r)
= covy|Θb (x̃t|t−1 ) + Ey|Θb [x̃t|t−1 ]Ey|Θb [x̃t|t−1 ]T
[3,3]
= V̂t
[3]
[3]
+ x̂t (x̂t )T ,
(r)
(r)
x̃s|s−1 = Fs−1,t+1 (Ft x̃t|t−1 + Gt yt )
|
{z
}
(r)
Λ̂t = Ey|Θb [x̃t|t−1 (x̃t|t−1 )T ]
αs
(77)
s−1
X
+
(r)
Fs−1,j+1 Gj yj = αs + βs ,
(83)
j=t+1
and
{z
|
(r)
λ̂t = Ey|Θb [(yt − ȳr )(x̃t|t−1 )T ]
(r)
[2,3]
(r)
[1]
[3]
+ (Cb x̂t + ȳb − ȳr )(x̂t )T .
A PPENDIX B
E FFICIENT CALCULATION
OF
(78)
CROSS -
THE
(
I
,s < t
, and the quantities
Fs Fs−1 · · · Ft , s ≥ t
αs and βs as above. Note that βt+1 = 0.
(r)
We now substitute x̃s|s−1 = αs + βs into (79). First, substituting
the αs terms into (79),
where we define Fs,t =
= covy|Θb (yt − ȳr , x̃t|t−1 ) + Ey|Θb [yt − ȳr ]Ey|Θb [x̃t|t−1 ]T
= Cb V̂t
(r)
ρ̂t = κTt = covy|Θb (x̃t|τ , yt ).
=
τ
+1
X
as a function of only
and observations yt:τ . Filling in the
backward recursion of the Kalman smoothing filter in (45),
(r)
(r)
=
+
(r)
(r)
(r)
(r)
(r) (r)
Jt (Ht+1 x̃t+2|t+1
τ
+1
X
Lt =
(r)
+
(r)
(r)
(r)
(r) (r)
= Ht x̃t+1|t +
(r)
s−2
Y
(
(r)
(r)
(r)
= Ht
+(
Ji )A−1
r x̃τ +1|τ
(r)
(r)
i=t
τ
+1
X
=
!
(r)
Ft+1
Jt+1,s−2 Ĥs−1 Fs−1,t+2
(r)
(r)
+ Jt Lt+1 Ft+1 ,
(87)
Lτ = Jτ,τ −1 Ĥτ Fτ,τ +1 = Ĥτ = A−1
r .
(r)
Jt,s−2 Ĥs−1 x̃s|s−1 ,
(79)
β̂t =
(r)
Ht
A−1
r
where we define Ĥt =
(
I
,t > s
. Next,
(r) (r)
(r)
Jt Jt+1 · · · Js
,t ≤ s
,t < τ
, and Jt,s
,t = τ
we
rewrite
(r)
x̃s|s−1 ,
the
(88)
Next, we substitute the βs terms into (79),
s=t+1
(
(86)
where in (85) we have separated the first term of the summation
(r)
(s = t + 1) , and in (86) we have used Jt,s−2 = Jt Jt+1,s−2 and
(r)
Fs−1,t+1 = Fs−1,t+2 Ft+1 , when s ≥ t. The initial condition for
the Lt backward recursion is
(r)
Ji )Hs−1 x̃s|s−1
s=t+2 i=t
τY
−1
τ
+1
X
s=t+2
(r)
(r)
τ
X
Jt,s−2 Ĥs−1 Fs−1,t+1
(r)
Jt
= Ĥt +
(r)
+ Jτ −2 (Hτ −1 x̃τ |τ −1 + Jτ −1 A−1
r x̃τ +1|τ ) · · · ))
(r)
τ
+1
X
s=t+2
= Ht x̃t+1|t + Jt (Ht+1 x̃t+2|t+1 + Jt+1 (· · ·
(r)
τ
+1
X
= Ĥt +
(r)
Jt+1 (· · ·
+ Jτ −2 (Hτ −1 x̃τ |τ −1 + Jτ −1 x̃τ |τ ) · · · ))
(r) (r)
Jt,s−2 Ĥs−1 Fs−1,t+1
s=t+1
(r)
(r)
(85)
}
s=t+2
(r)
(r)
Jt,s−2 Ĥs−1 Fs−1,t+1
{z
= Ht x̃t+1|t + Jt (Ht+1 x̃t+2|t+1 + Jt+1 x̃t+2|τ )
(r) (r)
Ht x̃t+1|t
(84)
Lt
= Jt,t−1 Ĥt Ft,t+1 +
(r) (r)
x̃t|τ = Ht x̃t+1|t + Jt x̃t+1|τ
(r) (r)
(r)
where Lt can be computed recursively,
(r)
x̃t|t−1
(r) (r)
(r)
Jt,s−2 Ĥs−1 Fs−1,t+1 (Ft x̃t|t−1 + Gt yt ),
s=t+1
(r)
First, we derive an expression of the Kalman smoothing filter x̃t|τ
(r)
Jt,s−2 Ĥs−1 αs
s=t+1
|
(r)
(r)
τ
+1
X
α̂t =
COVARIANCE TERMS
In this section, we derive efficient expressions for calculating the
cross-covariance terms,
ω̂t = covy|Θb (x̃t|τ , x̃t|t−1 ),
}
βs
τ
+1
X
Jt,s−2 Ĥs−1 βs
s=t+1
=
=
filter terms
where s > t, as a function of
and
{yt , · · · , ys−1 }. Note that we drop the constant ȳr term since it
will not affect the covariance operator later. Substituting the forward
recursions in (44), we have
Jt,s−2 Ĥs−1
s=t+1
Kalman
(r)
x̃t|t−1
τ
+1
X
=
τ
+1
X
s−1
X
(r)
Fs−1,j+1 Gj yj
j=t+1
s−1
X
(r)
Jt,s−2 Ĥs−1 Fs−1,j+1 Gj yj
s=t+2 j=t+1
=
τ
X
τ
+1
X
(r)
Jt,s−2 Ĥs−1 Fs−1,j+1 Gj yj ,
(89)
j=t+1 s=j+1
(r)
(r)
(r)
x̃t+1|t = Ft x̃t|t−1 + Gt yt ,
(r)
x̃t+2|t+1
(r)
x̃t+3|t+2
=
(r)
Ft+1 x̃t+1|t
=
(r)
(r)
(r)
Ft+1 (Ft x̃t|t−1 + Gt yt )
(r)
(r)
Ft+2 x̃t+2|t+1 + Gt+2 yt+2
=
(r)
(r)
+
(80)
(r)
Gt+1 yt+1
(r)
(r)
+ Gt+1 yt+1 ,
(r)
x̃t|τ =
(r)
= Ft+2 (Ft+1 (Ft x̃t|t−1 + Gt yt )
(r)
(81)
(r)
+ Gt+1 yt+1 ) + Gt+2 yt+2 ,
(r)
where in (89) we have collected the Gj yj terms by switching the
double summation. Finally, using {α̂t , β̂t }, we rewrite the Kalman
(r)
smoothing filter of (79) as a function of x̃t|t−1 and yt+1:τ ,
(82)
τ
+1
X
Jt,s−2 Ĥs−1 (αs + βs ) = α̂t + β̂t
s=t+1
(r) (r)
(r)
= Lt (Ft x̃t|t−1 + Gt yt ) + β̂t .
(90)
IEEE TRANS. ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, TO APPEAR, 2013
Note that β̂t is a function of {yt+1 , · · · , yτ }.
4
Looking at the covariance with β̂t ,
covy|Θb (β̂t , yt )
B.1
Calculating ω̂t =
(r)
(r)
covy|Θb (x̃t|τ , x̃t|t−1 )
= covy|Θb (
ω̂t =
(r)
Jt,s−2 Ĥs−1 Fs−1,j+1 Gj yj , yt )
j=t+1 s=j+1
Next, we will derive an expression for ω̂t
(r)
(r)
covy|Θb (x̃t|τ , x̃t|t−1 )
τ
+1
X
τ
X
= covy|Θb (α̂t +
(r)
β̂t , x̃t|t−1 ).
(91)
τ
+1
X
τ
X
=
(r)
Jt,s−2 Ĥs−1 Fs−1,j+1 Gj covy|Θb (yj , yt )
j=t+1 s=j+1
Looking at the covariance with the β̂t term,
τ
+1
X
τ
X
=
(r)
[1,1]
Jt,s−2 Ĥs−1 Fs−1,j+1 Gj Cb Abj−t V̂t
CbT
j=t+1 s=j+1
(r)
[1,1]
covy|Θb (β̂t , x̃t|t−1 )
= covy|Θb (
= Mt V̂t
τ
+1
X
τ
X
(r)
(r)
Jt,s−2 Ĥs−1 Fs−1,j+1 Gj yj , x̃t|t−1 )
=
τ
+1
X
(r)
ρ̂t = covy|Θb (x̃t|τ , yt )
(r)
(r) (r)
(r)
=
τ
+1
X
(r)
= covy|Θb (Lt (Ft x̃t|t−1 + Gt yt ) + β̂t , yt )
Jt,s−2 Ĥs−1 Fs−1,j+1 Gj covy|Θb (yj , x̃t|t−1 )
j=t+1 s=j+1
τ
X
(98)
Finally, using (90), the cross-covariance is
j=t+1 s=j+1
τ
X
CbT .
(r)
(r)
(r)
= Lt Ft covy|Θb (x̃t|t−1 , yt ) + Lt Gt covy|Θb (yt )
(r)
[2,3]
V̂t
Jt,s−2 Ĥs−1 Fs−1,j+1 Gj Cb Aj−t
b
,
+ covy|Θb (β̂t , yt )
(92)
[3,2]
(r)
j=t+1 s=j+1
{z
|
= Lt Ft V̂t
}
Mt
[1,1]
(r)
CbT + Lt Gt (Cb V̂t
CbT + Rb )
[1,1]
+ Mt V̂t CbT
(r)
[1,1]
(r) [3,2]
CbT
+ (Lt Gt Cb + Mt )V̂t
= Lt Ft V̂t
where in the last line we have used (107). Mt can be computed
with a backward recursion,
(r)
+ Lt Gt Rb ,
τ
X
Mt =
τ
+1
X
"
=
B.3
τ
+1
X
τ
X
Jt+1,s−2 Ĥs−1
i
Ab
·Fs−1,j+1 Gj Cb Aj−t−1
b
"
=
(r)
Jt
τ
+1
X
Useful properties
In this section, we derive some properties used in the previous
section. Note that in the sequel we remove the mean terms, ȳb and
ȳr , which do not affect the covariance operator. First, we derive the
covariance between two observations, for k > 0,
j=t+1 s=j+1
(r)
#
(r)
Jt+1,s−2 Ĥs−1 Fs−1,t+2 Gt+1 Cb
(b)
+
τ
+1
X
(b)
covy|Θb (yt+k , yt )
(93)
s=t+2
τ
X
(100)
where (99) follows by using (107), (98), and covy|Θb (yt ) =
Cb covx|Θb (xt )CbT + Rb .
(r)
Jt,s−2 Ĥs−1 Fs−1,j+1 Gj Cb Aj−t
b
j=t+1 s=j+1
(r)
Jt
(b)
(r)
j−(t+1)
(b)
(b)
(r)
= Jt (Lt+1 Gt+1 Cb + Mt+1 )Ab ,
(r)
(r)
ω̂t = covy|Θb (x̃t|τ , x̃t|t−1 )
=
(r) (r)
covy|Θb (Lt (Ft x̃t|t−1
+
=
(r)
(r)
Lt Ft covy|Θb (x̃t|t−1 )
+
(r)
+ β̂t , x̃t|t−1 )
(r)
(r)
Lt Gt covy|Θb (yt , x̃t|t−1 )
=
=
+
+
Calculating ρ̂t =
[1,1]
Cb Akb V̂t
(r)
(b)
(b)
(96)
k
X
(102)
CbT ,
(103)
(b)
(r)
(b)
(b)
=
=
Cb Akb covy1:t−1 |Θb (
=
(b)
(r)
Cb Akb covy|Θb (x̃t|t−1 , x̃t|t−1 )
(b)
(r)
Abk−l vt+l ) + wt+k , x̃t|t−1 )
l=1
(b)
(r)
covxt ,y1:t−1 |Θb (Cb Akb xt , x̃t|t−1 )
(95)
(105)
(b)
(r)
xt |y1:t−1 [xt ], x̃t|t−1 )
(106)
E
(r)
=
(104)
[2,3]
Cb Akb V̂t ,
(107)
(b)
(r)
where in (105) we use vt+l ⊥
⊥ x̂t|t−1 for l ≥ 1 and wt+k ⊥
⊥ x̃t|t−1
for k ≥ 0, and in (106), covx,y (x, y) = covy (Ex|y [x], y).
We now derive an expression for ρ̂t ,
(r)
(b)
(101)
(b)
(b)
where in (102) we have rewritten xt+k as a function of xt ,
P
(b)
(b)
(b)
i.e., xt+k = Akb xt + kl=1 Ak−l
vt+l , in (101) we have used
b
(b)
(b)
(b)
(b)
(b)
(b)
xt ⊥
⊥ wt+k and xt+k ⊥
⊥ wt for k > 0, and in (103), xt ⊥
⊥ vt+l
= cov(Cb (Akb xt +
(r)
covy|Θb (x̃t|τ , yt )
ρ̂t = covy|Θb (x̃t|τ , yt ) = covy|Θb (α̂t + β̂t , yt ).
(b)
Ak−l
vt+l , xt )CbT
b
l=1
(b)
k
T
Cb Ab cov(xt )Cb =
(b)
where in (95) we have used (107) and (92).
B.2
=
k
X
covy|Θb (yt+k , x̃t|t−1 ) = cov(Cb xt+k + wt+k , x̃t|t−1 )
(r)
(r)
[2,3]
[2,3]
Lt Gt Cb V̂t
+ Mt V̂t
(r)
[2,3]
(Lt Gt Cb + Mt )V̂t ,
(b)
for l ≥ 1. Next, we derive the covariance between the one-step
ahead state estimator and an observation, for k ≥ 0,
(r)
Gt yt )
+ covy|Θb (β̂t , x̃t|t−1 )
(r) [3,3]
Lt Ft V̂t
(r) [3,3]
Lt Ft V̂t
(b)
= Cb cov(Akb xt +
(94)
where in (93) we have separated the first term of the summation
(j = t + 1), and in (94) we have used the definition of Lt+1 and
Mt+1 . The initial condition is Mτ = 0. Finally using (90), the
cross-covariance is
(b)
(b)
= Cb cov(xt+k , xt )CbT + covΘb (wt+k , wt )
Ab
j=t+2 s=j+1
(r)
(b)
(b)
= cov(Cb xt+k + wt+k , Cb xt + wt )
!
Jt+1,s−2 Ĥs−1 Fs−1,j+1 Gj Cb Ab
(99)
(97)