By Randall L. Eubank
ISBN-10: 0824723651
ISBN-13: 9780824723651
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Example text
Some further simplifications occur if we focus on large values of t. 25), produces S(t|t) = W0 H2 1− W0 . 27) R(t) Consequently, R(t) = H 2 F 2 W0 H2 1− W0 R(t − 1) 2 2 = F W0 + H Q0 + W0 − + Q0 2 F 2 W0 R(t − 1) + W0 . If we now let C1 = F 2 W0 + H 2 Q0 + W0 and C2 = 2 , it then follows that F 2 W0 R(t) = C1 − C2 R(t − 1) C2 = C1 − C1 − C2 R(t − 2) C2 = C1 − C2 C1 − C2 C1 − C1 − C2 R(t − 3) which reveals a continued fraction representation for R(t) as described, for example, in Khinchin (1997).
N at which observations are taken. 4 we see that the below diagonal blocks of ΣXε have a relatively simple representation as σXε (t, j) = F t−j S(j|j − 1)H T , j ≤ t − 1. Expressions for the above diagonal entries are more complicated except in the case of univariate state and response variables. To proceed further let us now specialize to the instance where p = q = 1 so that H, F, W0 and Q0 are all scalar © 2006 by Taylor & Francis Group, LLC The Fundamental Covariance Structure 47 valued. 25) with initializing values provided by S(0|0) = 0, 2 S(1|0) = F S(0|0) + Q0 = Q0 , 2 R(1) = H Q0 + W0 , Q0 H 2 M (1) = F 1 − R(1) and S(1|1) = Q0 − 2 Q2 0H .
K). The common component in all these factors is the innovation vectors whose computation is linked directly to the Cholesky factorization of Var(y). Consequently, the Cholesky decomposition is the unifying theme for all that follows and is the perspective we will adopt for viewing developments throughout the text. 30). Then, in Chapter 3, we show how this structure can be exploited to obtain a computationally efficient, modified Cholesky factorization of Var(y) as well as Var−1 (y). , the smoothing step) Kalman filter recursions.
A Kalman Filter Primer by Randall L. Eubank
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