Springer - Read

274 Chapter 8 State-Space Models
With (8.1.2) and (8.4.4) this gives
t+1 FtE(XtX ′ ′
t )F t + Qt − FtE
FttF ′
t + Qt − t −1
t ′
t .
ˆXt
ˆX
′
t F ′ −1
t − tt ′
t
h-step Prediction of {Yt} Using the Kalman Recursions
The Kalman prediction equations lead to a very simple algorithm for recursive calculation
of the best linear mean square predictors PtYt+h, h 1, 2,.... From (8.4.4),
(8.1.1), (8.1.2), and Remark 3 in Section 8.1, we find that
and
PtXt+1 FtPt−1Xt + t −1
t (Yt − Pt−1Yt), (8.4.5)
PtXt+h Ft+h−1PtXt+h−1
.
(Ft+h−1Ft+h−2 ···Ft+1) PtXt+1, h 2, 3,..., (8.4.6)
PtYt+h Gt+hPtXt+h, h 1, 2,.... (8.4.7)
From the relation
we find that (h)
t
Xt+h − PtXt+h Ft+h−1(Xt+h−1 − PtXt+h−1) + Vt+h−1, h 2, 3,...,
(h)
t
: E[(Xt+h − PtXt+h)(Xt+h − PtXt+h) ′ ] satisfies the recursions
(h−1)
Ft+h−1
t
F ′
t+h−1 + Qt+h−1, h 2, 3,..., (8.4.8)
with (1)
t t+1. Then from (8.1.1) and (8.4.7), (h)
t : E[(Yt+h − PtYt+h)(Yt+h −
PtYt+h) ′ ]isgivenby
(h)
t
(h)
Gt+h
t G′
t+h + Rt+h, h 1, 2,.... (8.4.9)
Example 8.4.1 Consider the random walk plus noise model of Example 8.2.1 defined by
Yt Xt + Wt, {Wt} ∼WN 0,σ 2
w ,
where the local level Xt follows the random walk
Xt+1 Xt + Vt, {Vt} ∼WN 0,σ 2
v .
Applying the Kalman prediction equations with Y0 : 1, R σ 2 w , and Q σ 2 v ,we
obtain
ˆYt+1 PtYt+1 ˆXt + t
Yt − ˆYt
t
(1 − at) ˆYt + atYt