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66 Chapter 2 Stationary Processes
where Xn (Xn,...,X1) ′ and
⎡
⎢
⎣
1 φ φ 2
··· φ n−1
φ 1 φ ··· φ n−2
. . . . .
φ n−1 φ n−2 φ n−3
⎤⎡
⎤
a1
⎥⎢
⎥
⎥⎢
a2 ⎥
⎥⎢
⎥
⎥⎢
⎥
⎦⎣
. ⎦
··· 1
⎡
φ
⎢ φ
⎢
⎣
2
.
φ n
⎤
⎥
⎥.
⎦
(2.5.11)
A solution of (2.5.11) is clearly
an (φ, 0,...,0) ′ ,
and hence the best linear predictor of Xn+1 in terms of {X1,...,Xn} is
PnXn+1 a ′
n Xn φXn,
with mean squared error
an
E(Xn+1 − PnXn+1) 2 γ(0) − a ′
n γn(1)
σ 2
1 − φ 2 − φγ(1) σ 2 .
A simpler approach to this problem is to guess, by inspection of the equation defining
Xn+1, that the best predictor is φXn. Then to verify this conjecture, it suffices to check
(2.5.10) for each of the predictor variables 1,Xn,...,X1. The prediction error of the
predictor φXn is clearly Xn+1 − φXn Zn+1. But E(Zn+1Y) 0 for Y 1 and for
Y Xj,j 1,...,n. Hence, by (2.5.10), φXn is the required best linear predictor
in terms of 1,X1,...,Xn.
Prediction of Second-Order Random Variables
Suppose now that Y and Wn, ..., W1 are any random variables with finite second
moments and that the means µ EY, µi EWi and covariances Cov(Y, Y ),
Cov(Y, Wi), and Cov(Wi,Wj) are all known. It is convenient to introduce the random
vector W (Wn,...,W1) ′ , the corresponding vector of means µW (µn,...,µ1) ′ ,
the vector of covariances
γ Cov(Y, W) (Cov(Y, Wn), Cov(Y, Wn−1),...,Cov(Y, W1)) ′ ,
and the covariance matrix
Ɣ Cov(W, W) Cov(Wn+1−i,Wn+1−j) n
i,j1 .
Then by the same arguments used in the calculation of PnXn+h, the best linear predictor
of Y in terms of {1,Wn,...,W1} is found to be
P(Y|W) µY + a ′ (W − µW ), (2.5.12)
where a (a1,...,an) ′ is any solution of
Ɣa γ. (2.5.13)