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8.4 The Kalman Recursions 273
Kalman Prediction:
For the state-space model (8.1.1)–(8.1.2), the one-step predictors ˆXt : Pt−1(Xt)
and their error covariance matrices t E ′
Xt − ˆXt Xt − ˆXt are uniquely
determined by the initial conditions
ˆX1 P(X1|Y0), 1 E
X1 − ˆX1 X1 − ˆX1
and the recursions, for t 1,...,
where
and −1
t
ˆXt+1 Ft ˆXt + t −1
t
Yt − Gt ˆXt
t+1 FttF ′
t + Qt − t −1
t GttG ′
t + Rt,
t FttG ′
t ,
t ′
t
is any generalized inverse of t.
′
, (8.4.1)
, (8.4.2)
Proof We shall make use of the innovations It defined by I0 Y0 and
+ Wt, t 1, 2,....
It Yt − Pt−1Yt Yt − Gt ˆXt Gt
Xt − ˆXt
The sequence {It} is orthogonal by Remark 1. Using Remarks 3 and 4 and the relation
Pt(·) Pt−1(·) + P(·|It) (8.4.3)
(see Problem 8.12), we find that
where
ˆXt+1 Pt−1(Xt+1) + P(Xt+1|It) Pt−1(FtXt + Vt) + t −1
t It
Ft ˆXt + t −1
t It, (8.4.4)
t E(It I ′
t
) GttG ′
t
t E(Xt+1I ′
t ) E
FttG ′
t .
+ Rt,
FtXt
+ Vt
Xt − ˆXt
′
G ′
t
To verify (8.4.2), we observe from the definition of t+1 that
.
t+1 E Xt+1X ′
t+1 − E
ˆXt+1
ˆX ′
t+1
+ W′
t