Filtering and smoothing of state vector
for di use state space models
S. J. Koopman
Department of Econometrics,
Free University Amsterdam,
de Boelelaan 1105, 1081 HV Amsterdam, Netherlands.
J. Durbin
Department of Statistics,
London School of Economics and Political Science,
Houghton Street, London WC2A 2AE, UK.
Abstract
This paper presents exact recursions for calculating the mean and mean square error
matrix of the state vector given the observations for the multivariate linear Gaussian state
space model in the case where the initial state vector is (partially) diuse.
Keywords: Diuse initialisation Kalman lter Smoothing State space.
1 Introduction
In this paper we develop e cient recursions for ltering and smoothing of the state vector for
the multivariate linear Gaussian state space time series model given by
yt = Zt t + "t
t+1 = Tt t + Rt t "t N(0 Ht)
(1)
t N(0 Qt) t = 1 : : : n
where yt is the N 1 vector of observations and t is the m 1 unobserved state vector,
in a situation where the initial state vector 1 is, at least partially, diuse. The matrices
Zt Tt Rt Ht and Qt are assumed to be known. The N 1 disturbance vector "t and the r 1
disturbance vector t are assumed to be serially independent and independent of each other at
all time points. Similar results to those in this paper can also be derived for the more general
model of de Jong (1991).
1
The initial state vector 1 is specied as
1 = a + A + R0 0 0 N(0 Q0)
(2)
where is a q 1 vector of unknown quantities, the m 1 vector a is known, the m q matrix
A and the m (m ; q) matrix R0 are selection matrices, that is, they consist of columns of the
identity matrix Im they are dened so that when taken together, their columns constitute all
the columns of Im so A R0 = 0 and = A 1. The matrix Q0 is assumed to be positive denite
and known. In most cases vector a will be treated as a zero vector unless some elements of the
initial state vector are known constants. The vector is random and we assume that
0
0
N(0 Iq )
(3)
where > 0 is given. Therefore, the initial conditions for the state vector become E( 1) = a
and Var( 1) = P where
P = P + P 1
(4)
and where we let ! 1 where appropriate. Here P = AA and P = R0 Q0R0 given the
matrix structure of A, it follows that P is an m m diagonal matrix with q elements equal to
one and the other elements equal to zero. Without loss of generality, when a diagonal element of
P is non-zero we take the corresponding element of a to be zero. A vector with distribution
N(0 Iq ) as ! 1 is said to be diuse.
The Kalman lter is a forward recursion for calculating at+1 = E( t+1jYt) with Yt =
fy1 : : : ytg together with the mean square error matrix Pt+1 = Var( t+1 jYt) for t = 1 : : : n.
The disturbance smoother is a backward recursion for calculating ^"t = E ("t jYn) and ^t =
E (t jYn), together with their corresponding mean square error matrices while the state smoother
is a backward recursion for calculating ^ t = E ( t jYn) together with Vt = Var ( t jYn) for
t = n : : : 1. Well-known Kalman lter and smoothing recursions are available for model (1)
and (2) with given > 0. The objective in this paper is to derive limiting versions of these
recursions when ! 1.
A simple technique for obtaining an approximation to the exact limiting results is to replace
in (4) by a large value such as 107 and use the standard recursions see Harvey and Phillips
(1979). However, the numerical solution obtained in this way will contain numerical inaccuracies
and is theoretically unsatisfactory. Two main approaches have appeared in the literature on
the exact treatment of a diuse initial vector for ltering and smoothing:
0
1
0
1
1
An exact solution as ! 1 was rst considered in detail by Ansley and Kohn (1985).
They provided intricate equations for state ltering and smoothing. Ansley and Kohn
(1990) presented a more simple treatment for mainly univariate models. The solution
consists of modications to the Kalman lter which remain until the inuence of has
disappeared from the updating equations. The transition to the usual Kalman lter at
2
some time point t = d is automatic there is no choice of doing a transition or not. The
usual state smoothing recursions can be applied for t = n : : : d + 1. The modications
for state smoothing are only required for the initial period t = d : : : 1.
An alternative and simpler approach for diuse likelihood evaluation and state ltering
and smoothing was given by de Jong (1991). This solution requires augmenting the vector
recursions of the Kalman lter and smoother to matrix recursions and adding an extra
matrix recursion. Regression calculations applied to the auxiliary part of the Kalman
lter and smoother provide the solution. A transition to the usual Kalman lter after
some time point t = d is optional but it is normally advisable to collapse at the earliest
time possible. However, as is the case for the Ansley and Kohn solution, modications
for initial state smoothing are required when a collapse has taken place. Details of these
modications are given by Chu-Chun-Lin and de Jong (1993).
Other contributions on exact treatments of diuse initialisation have appeared in the literature but these do not specically deal with the problem of both ltering and smoothing of
the state vector. Bell and Hillmer (1991) consider likelihood evaluation of ARIMA models
Snyder and Saligari (1996) present a strategy for initialising the Kalman lter using a square
root formulation but their technique does not provide a solution for initial state smoothing
(t = d : : : 1) Koopman (1997) presents a more transparent treatment of diuse ltering and
likelihood evaluation using the Ansley and Kohn approach in which the collapse to the Kalman
lter is automatic. In this paper we provide a solution to diuse state smoothing using the
approach of Koopman (1997). The paper is organised as follows. Sections 2 and 3 provide exact
initial recursions for diuse state ltering and smoothing, respectively. Section 4 presents useful
matrix formulations of the ltering and smoothing recursions. Proofs are given in Appendices.
2 State ltering
2.1 Non-diuse state ltering
The Kalman lter for model (1) for given > 0 is
at+1 = Tt at + Ktvt Pt+1 = Tt PtLt + Rt Qt Rt 0
0
t = 1 : : : n
(5)
where
vt = yt ; Ztat Kt = Tt PtZt Ft 1
0
Ft = ZtPtZt + Ht
Lt = Tt ; Kt Zt
0
;
(6)
with a1 = a and P1 = P . The proof can be found in Anderson and Moore (1979, Chapter 3)
and Durbin and Koopman (2001, x4.2.1).
3
2.2 Diuse state ltering
The mean square error matrix Pt is decomposed in a similar way to matrix P in (4), that is
Pt = P t + P t + O( 1) t = 1 : : : n
(7)
where P t and P t do not depend on . It is shown by Ansley and Kohn (1985) and Koopman
(1997) that the inuence of the term P t will disappear after a limited number of updates d
of the exact initial Kalman lter. Therefore, the state ltering equations (5) apply without
change for t = d + 1 : : : n when ! 1. Note that when all state elements follow stationary
processes or are known, P = 0 and d = 0.
Our approach relies on the expansion of the inverse of matrix
Ft = F t + F t + O( 1 ) t = 1 : : : n
which appears in the denition of Kt in (6). The expression of Ft is analogous to the decomposition of (7). The solution that we shall give for both ltering and smoothing apply to all
univariate series. For multivariate series we give explicit solutions for the special cases F t
nonsingular and F t = 0. These two cases apply to nearly all time series occurring in practice.
Although an explicit solution can be given as in Koopman (1997), we have found that for the
rare cases where F t is a nonzero singular matrix, it is more e cient to convert the multivariate
series to a univariate series in the way described in Koopman and Durbin (2000) and to apply
the recursions of this paper to the resulting univariate series.
In Appendix A we show that for t = 1 : : : d the exact initial state ltering equations when
F t is nonsingular are given by
at+1 = Tt at + K tvt (8)
P t+1 = Tt P tL t
P t+1 = Tt P tL t ; K tF tK t + Rt QtRt where
vt = yt ; Zt at F t = Zt P tZt F t = ZtP tZt + Ht
K t = Tt P tZt F 1t L t = Tt ; K tZt K t = (TtP t Zt ; K tF t)F 1t
with the initialisations a1 = a, P 1 = P and P 1 = P . The recursions (8) are referred to as
the exact initial Kalman lter following Koopman (1997).
In the case F t = 0, we have
at+1 = Tt at + K t vt
(9)
P t+1 = Tt P tTt P t+1 = Tt P tL t + RtQt Rt where
vt = yt ; Ztat F t = ZtP t Zt + Ht (10)
K t = TtP t Zt F t1 L t = Tt ; K tZt
for t = 1 : : : d.
;
1
1
1
1
;
1
1
1
1
1
1
0
1
1
1
0
0
1
1
1
0
0
1
0
1
1
0
1
;
1
1
1
1
1
0
1
1
0
0
0
0
;
0
1
;
4
1
1
3 State smoothing
3.1 Non-diuse state smoothing
For given , the conditional mean of the state t given all the observations Yn, that is ^ t =
E ( t jYn), together with its mean square error matrix Vt can be evaluated recursively by wellknown smoothing recursions provided by Anderson and Moore (1979). A computationally more
e cient set of recursions is developed by de Jong (1988), de Jong (1989) and Kohn and Ansley
(1989). For model (1) and for given > 0 these are
rt 1 = Zt Ft 1vt + Lt rt Nt 1 = Zt Ft 1Zt + Lt Nt Lt (11)
^ t = at + Ptrt 1 Vt = Pt ; PtNt 1 Pt t = n n ; 1 : : : 1
with rn = 0 and Nn = 0. Storage space is required for the quantities vt , Ft , Kt , at and Pt,
which are calculated by the Kalman lter, for t = 1 : : : n.
0
;
0
;
0
;
0
;
;
;
3.2 Diuse state smoothing
The state smoothing recursions (11) apply to the period t = n n ; 1 : : : d + 1 for which
contribution to Pt of terms depending on vanishes as ! 1. We show in Appendix B that
for the initial time period t = d d ; 1 : : : 1, the exact initial state smoothing equations when
F t is nonsingular are given by
1
^ t = at + P trt(0)1 + P trt(1)1
(12)
Vt = P t ; P tNt(0)1 P t ; < P tNt(1)1 P t > ;P t Nt(2)1 P t
where for any square matrix A, < A >= A + A , together with the backwards recursions
1
;
;
;
1
;
1
1
;
0
rt(0)1 = L t rt(0) Nt(0)1 = L t Nt(0) L t 0
;
1
0
;
1
1
rt(1)1 = Zt (F 1tvt ; K t rt(0) ) + L trt(1) Nt(1)1 = Zt F 1tZt + L tNt(1) L t ; < L tNt(0) K t Zt > (13)
Nt(2)1 = Zt F# tZt + L tNt(2) L t; < L t Nt(1) K t Zt >
0
;
0
;
0
0
1
1
;
0
1
;
0
1
1
0
;
1
0
0
1
1
1
where
F# t = K t Nt(0) K t ; F 1tF t F 1t
0
;
1
;
1
and with initialisations rd(0) = rd , rd(1) = 0 and Nd(0) = Nd, Nd(1) = Nd(2) = 0.
In the case F t = 0, we have
1
rt(0)1 = Zt F t1vt + L t rt(0) Nt(0)1 = Zt F t1Zt + L t Nt(0) L t rt(1)1 =
Nt(1)1 =
Nt(2)1 =
Storage space is required for the quantities vt, F t , F t, K t, K
calculated by the exact initial Kalman lter.
0
0
;
;
;
;
;
0
5
1
(14)
0
;
0
;
Tt rt(1) Tt Nt(1) L t Tt Nt(2) Tt :
t , at , P t and P
0
0
1
1
t
which are
4 Matrix formulations
A convenient representation of the exact initial state ltering equations for the case F
gular is given by
1
at+1 = Tt at + K tvt
Pt+1 = Tt Pt Lt + (Rt Qt Rt 0) y
1
y
y0
t = 1 : : : d
0
t nonsin-
(15)
with the initialisation a1 = a and P1 = P = (P P ). Elements of a corresponding to diuse
elements of 1 can be put equal to zero. The matrices Pt and Lt are dened as
y
y
1
y
"
Lt = L
Pt = (P t P t) y
#
t ;K t Zt
:
0
L t
1
y
1
y
(16)
1
It is noteworthy that, apart from redenitions of matrices, the initial recursions (15) have the
same structure as the standard recursions (5).
A similar representation to (15) for the case F t = 0 can be obtained by replacing K t in
(15) by K t from (10) and the matrix Lt by
1
1
y
"
#
Lt = L t 0 :
0 Tt
Dene
"
#
Zt = Zt 0 0 Zt
vt =
y
y
vt
0
!
Ft =
"
y
F
0
;
#
;
1
1
1
(0) !
r
F 1t
;F 1tF t F
t
"
;
1
;
1
1
t
#
(0)
(1)
rt = t(1) Nt = Nt(1) Nt(2) :
rt
Nt Nt
A convenient representation of the exact initial state smoothing equations for the case F
nonsingular is given by
y
y
1
rt 1 = Zt Ft vt + Lt rt ^ t = at + Pt rt 1 y
y0
y
y
y0
y
;
y
y
;
Nt 1 = Zt Ft Zt + Lt Nt Lt Vt = P t ; Pt Nt 1 Pt t = d : : : 1
y
y0
y
y
y0
y
y
;
y
y
y0
t
(17)
;
with rd = (rd 0) and Nd = diag (Nd 0), where the partioned matrices Lt and Pt are as in (16).
As with the lter (15), these recursions have the same structure as the standard recursions (11)
apart from redenition of matrices. These results are obtained after some minor manipulation
of the equations in xx2.2 and 3.2.
When F t = 0, the smoothing equations (17) apply with Ft and Lt replaced by
y
0
0
y
y
y
1
"
#
"
1
Ft = F t 0 0 0
;
y
#
Lt = L t 0 0 Tt
respectively.
6
y
Acknowledgements
The rst author was working at the Center for Economic Research of Tilburg University, The
Netherlands as a Research Fellow of the Royal Netherlands Academy of Arts and Sciences during
the main part of this project. The nancial support of the Academy is gratefully acknowledged.
Appendix A Diuse ltering
In this Appendix we derive the diuse ltering results. The decomposition (7) leads to the
similar decompositions for Ft = ZtPt Zt + Ht and Mt = Tt Pt Zt , that is
0
Ft = F
0
t + F t + O( 1 )
Mt = M
;
1
t + M t + O( 1 )
(18)
;
1
where
F
M
1
= ZtP tZt = Tt P tZt t
1
F t = Zt P tZt + Ht M t = Tt P tZt 0
0
1
t
0
(19)
0
1
for t = 1 : : : d. The derivation of the exact initial Kalman lter for the case F t nonsingular
is based on the expansion for Ft 1 = F t + F t + O( 1)] 1 as a power series in 1, that is
1
;
;
1
;
;
; 3
Ft 1 = Ft(0) + 1Ft(1) + 2 Ft(2) + O ;
;
;
(20)
;
for large . Since Ip = Ft Ft 1 we have
;
Ip = (F
1
t+F t+
;
1F
at +
;
2F
(0) + 1 F (1) + 2 F (2) + : : : ):
b t + : : : )(Ft
;
;
t
t
On equating coe cients of j for j = 0 ;1 ;2 : : : we obtain
F tFt(0) = 0
F tFt(0) + F tFt(1) = Ip
Fa tFt(0) + F tFt(1) + F tFt(2) = 0 etc.
1
1
1
(21)
We need to solve equations (21) for Ft(0) , Ft(1) and Ft(2) further terms will not be required.
For F t nonsingular, we have from (21),
1
Ft(0) = 0
Ft(1) = F 1t
Ft(2) = ;F 1t F tF 1t:
;
;
1
1
;
1
(22)
The matrix Kt = Mt Ft 1 depends on the inverse matrix Ft 1 so it also can be expressed as a
power series in 1 . We have
;
;
;
Kt = M t + M t + O( 1)]( 1 Ft(1) + 2Ft(2) + : : : )
= K t + 1K t + O( 2 )
;
1
;
1
;
7
;
;
where
K
1
K
M tFt(1)
M tF 1t
M t Ft(1) + M tFt(2)
M t F 1t ; M tF 1tF tF
(M t ; K tF t )F 1t:
=
=
=
=
=
t
t
1
;
1
1
1
;
;
1
1
1
(23)
1
;
1
t
;
1
1
The updating equation (5) for at+1 can now be expressed as
at+1 = Ttat + Kt vt
; = Ttat + K t + 1K t + O 2 ]vt ;
;
1
which becomes as ! 1,
at+1 = Tt at + K tvt :
(24)
1
The updating equation (5) for Pt+1 = P t+1 + P t+1 is
1
Pt+1 = Tt Pt Lt + Rt Qt Rt
= Tt P t + P t + O( 1 )] Tt ; K
0
0
;
1
t+
1K
;
1
t+O
; 2 Zt + Rt Qt Rt :
0
;
0
Consequently, the updates for P t+1 and P t+1 as ! 1 are given by
1
P
P
1
= Tt P t(Tt ; K tZt) t+1 = Tt P t (Tt ; K t Zt ) ; Tt P t Zt K t + Rt Qt Rt :
t+1
0
1
1
0
0
1
1
0
0
(25)
The matrix Pt+1 also depends on terms in 1 , 2, etc. but these terms will not be multiplied
by or higher powers of within the Kalman lter recursions. Thus the updating equations
for Pt+1 do not need to take account of these terms when ! 1.
For the case F t = 0, we have P tZt = 0, M t = 0 and
;
1
;
1
1
Ft 1 = F t + O( 1)] 1 = F t1 + 1 Fa t + O( 2):
;
;
;
;
;
;
It follows that
Kt = M t + 1Ma t + O( 2)]F t1 + 1Fa t + O( 2 )]
= K t + 1 Ka t + O( 2)
;
;
;
;
;
;
;
where
K t = M t F t1
Ka t = M t Fa t + Ma t F t1:
;
;
8
(26)
We will show below that no explicit denitions for Fa t , Ma t and Ka t are required. The ltering
equations are given by
at+1 = Tt at + K t + 1Ka t + O( 2)]vt Pt+1 = Tt P t + P t + O( 1)]fTt ; K t + 1Ka t + O( 2 )]Ztg + Rt Qt Rt :
;
;
;
1
;
;
0
0
When ! 1, we obtain
at+1 = Tt at + K tvt P t+1 = Tt P tTt P t+1 = Tt P t(Tt ; K tZt ) + Rt Qt Rt 0
1
1
0
(27)
0
since P tZt = 0.
0
1
Appendix B Diuse smoothing
B1 Derivation of smoothed mean of state
To obtain the limiting recursions for the smoothing equation ^ t = at + Ptrt 1 given in (11) for
t = d : : : 1, we consider the recursion rt 1 = Zt Ft 1vt + Lt rt . Since rt 1 depends linearly on
Ft 1 and Kt which can both be expressed as power series in 1 we write
;
;
0
0
;
;
;
;
rt 1 = rt(0)1 + 1rt(1)1 + O( 2)
;
;
t = d : : : 1:
;
;
;
(28)
Substituting the relevant expansions into the recursion for rt 1 in (11) we have for the case
F t nonsingular,
;
1
rt(0)1 + 1rt(1)1 + : : : = Zt 1Ft(1) + 2Ft(2) + : : : vt
;
;
0
;
;
;
+ Tt ; K
t+
;
1
1K
; 2 (0)
(1)
1
Zt rt + rt + : : : t+O 0
;
;
leading to recursions for rt(0) and rt(1) ,
rt(0)1 = L trt(0) rt(1)1 = Zt F 1tvt ; K t rt(0) + L trt(1) 0
;
1
0
;
;
0
0
1
1
for t = d : : : 1 with rd(0) = rd and rd(1) = 0. Note that L
The smoothed state vector is
1
t
(29)
= Tt ; K tZt .
1
^ t = at + Ptrt
1
; (0)
(1)
(0)
(1)
1
1
= at + P t rt 1 + rt 1 + P t rt 1 + rt 1 + O 1
; = at + P trt(0)1 + P trt(0)1 + P trt(1)1 + O 1 :
;
;
;
;
1
;
;
;
;
1
1
;
9
;
;
;
(30)
It is obvious that for this expression to make sense we must have P trt(0)1 = 0 for all t. Thus
we need to show that Var( tjYn) is nite as ! 1. Analogously to the arguments in de Jong
(1991) we can express t as a linear function of 0 1 : : : t 1 . But Var(jYd) is nite by
denition of d so Vt = Var( t jYn) must be nite as ! 1 since d < n. Also, Qj = Var(j ) is
nite so Var(j jYn) is nite for j = 0 : : : t ; 1. It follows that Var( t jYn) is nite for all t as
! 1 so from (30) P trt(0)1 = 0. Letting ! 1 we obtain the expression for ^ t in (12).
For the case F t = 0, the recursion for rt becomes
1
;
;
1
;
1
;
rt(0)1 + 1 rt(1)1 + : : : = Zt F t1 + 1Fa t + : : : vt
; 2 (0)
(1)
1
1
+ Tt ; K t + Ka t O Zt rt + rt + : : : ;
0
;
;
;
;
;
0
;
;
leading to recursions for rt(0) and rt(1) ,
rt(0)1 = Zt F t1vt + L t rt(0) (1)
(0)
rt 1 = Zt Fa t vt ; Ka t rt + L t rt(1) 0
;
;
0
0
0
(31)
0
;
for t = d : : : 1 with matrix L t given by (10). Since rt(1)1 is premultiplied by matrix P
(12) and since P tZt = 0, the recursion for rt(1) reduces to the one given in (14).
1
;
t
in
0
1
B2 Derivation of smoothed variance of state
To obtain exact nite expressions for Vt and Nt 1 in (11) when F t is nonsingular and ! 1,
for t = d : : : 1, we need to take three-term expansions instead of the two-term expressions
previously employed:
;
1
Nt = Nt(0) + 1Nt(1) + 2Nt(2) + O( 3):
(32)
Ignoring residual terms and on substituting in the expression Nt 1 = Zt Ft 1Zt + Lt Nt Lt , we
obtain the recursion for Nt 1 as
Nt(0)
1 + 1 Nt(1)1 + 2Nt(2)1 + : : :
= Zt 1Ft(1) + 2 Ft(2) + : : : Zt
; + Tt ; K t + 1K t + 2K# t + O 3 Zt Nt(0) + 1 Nt(1) + 2Nt(2) + : : :
; Tt ; K t + 1K t + 2K# t + O 3 Zt for the case F t is nonsingular. We will argue below that no explicit denition for K# t is
needed. This leads to the set of recursions
Nt(0)1 = L tNt(0) L t
Nt(1)1 = Zt F 1tZt + L tNt(1) L t; < Zt K tNt(0) L t >
(33)
Nt(2)1 = ;Zt F tF 1tF t Zt + L tNt(2) L t; < Zt K tNt(1) L t >
(0)
(0)
; < Zt K# tNt L t > +Zt K t Nt K tZt ;
;
;
0
;
0
;
;
;
;
;
0
;
;
;
;
;
1
;
;
;
;
0
;
;
;
1
1
0
;
1
1
0
;
;
0
1
1
0
0
1
0
0
;
0
;
0
1
1
1
1
0
0
1
0
10
0
1
with Nd(0) = Nd and Nd(1) = Nd(2) = 0. Note that < A >= A + A for any square matrix A.
Substituting the power series in 1, 2 , etc. and the expression Pt = P t + P t into the
relation Vt = Pt ; Pt Nt 1 Pt we obtain
0
;
;
1
;
Vt = P t + P t ; (P t + P t) Nt(0)1 + 1Nt(1)1 + 2Nt(2)1 + : : : (P t + P t)
= ;2 P tNt(0)1 P t
+ P t ; P tNt(0)1 P t ; P tNt(0)1 P t ; P tNt(1)1 P t
+P t ; P t Nt(0)1 P t ; P tNt(1)1 P t ; P tNt(1)1 P t ; P tNt(2)1 P t
; +O 1 :
1
;
1
1
;
;
1
1
;
1
;
;
1
;
;
1
;
1
1
;
1
1
;
;
1
;
(34)
1
;
It was argued in the previous section that Vt = Var( tjYn) is nite for t = 1 : : : n. Thus the
two matrix terms associated with and 2 in (34) must be zero. Letting ! 1, the smoothed
state variance matrix is given by
Vt = P t ; P tNt(0)1 P t; < P tNt(1)1 P
;
1
;
> ;P tNt(2)1 P t:
t
1
;
(35)
1
Since we have argued that
P tNt(0)1 P
1
1
;
t
= 0 where Nt(0)1 = L tNt(0) L t
0
;
it follows that P tL t Nt(0) = 0, when F
as if the recursion for Nt(2)1 in (33) is
1
1
t
1
1
1
is nonsingular. Therefore, we can proceed in eect
;
Nt(2)1 = ;Zt F tF 1tF t Zt + L t Nt(2) L t ; < Zt K tNt(1) L
0
;
;
1
0
0
1
1
0
0
1
t
> +Zt K t Nt(0) K tZt :
0
0
It is convenient that K# t drops out from our calculations for Nt(2) .
For the case F t = 0, the recursion for Nt becomes
1
Nt(0)1 + 1 Nt(1)1 + 2Nt(2)1 + : : :
;
= Zt F t1 + 1Fa t + : : : Zt
+ Tt ; K t + 1Ka t + 2Kb t + O(
Tt ; K t + 1 Ka t + 2Kb t + O(
;
;
;
0
;
;
;
;
;
;
;
3 ) Z t
;
;
;
0
3 ) Z t
Nt(0) + 1Nt(1) + 2Nt(2) + : : :
;
;
leading to the recursions
Nt(0)1 = Zt F t1Zt + L t Nt(0) L t Nt(1)1 = Zt Fa t Zt + L tNt(1) L t ; < Zt Ka t Nt(0) L t >
Nt(2)1 = Zt Fb t Zt + L t Nt(2) L t ; < Zt Ka tNt(1) L t >
; < Zt Kb tNt(0) L t > +Zt Ka t Nt(0) Ka t Zt
0
;
;
0
0
0
;
0
0
0
;
0
0
0
0
0
0
11
0
(36)
for t = d : : : 1 with matrix L t given by (10). Similar to the arguments below (35), we have
P tNt(0)1 P
= 0 and P tNt(0)1 P t = P tL t Nt(0) L t P t = 0
since P tZt = 0. Further, matrix Nt(1)1 is pre- or post-multiplied by matrix P t in (12) and
since P tZt = 0 and P tL t Nt(0) = 0, the recursion for Nt(1) reduces to the one given in (14).
Similar reductions are obtained for the recursion for Nt(2) because it is pre- and post-multiplied
by matrix P t in (12). We do not need to give explicit denitions for the matrices Fa t , Fb t,
Ka t and Kb t due to these reductions.
1
;
1
t
0
1
;
1
1
1
0
1
1
;
0
0
1
1
1
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