By Randall L. Eubank
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For this purpose it will be worthwhile to first set out a few important facts that will arise again and again in our study of state-space models. Specifically, we can see that the following properties are immediate consequences of the state-space model assumptions and the definition of the innovations: (F1) ε(t) is uncorrelated with ε(s), s = t and e(s) for s > t. (F2) e(t) is uncorrelated with x(s) for all s. (F3) u(t) is uncorrelated with x(s), y(s), ε(s) for s ≤ t. 3) we obtain t−1 ε(t) = y(t) − Cov(y(t), ε(j))R −1 (j)ε(j) j=1 = H(t)x(t) + e(t) t−1 − Cov(H(t)x(t) + e(t), ε(j))R −1 (j)ε(j) j=1 t−1 = H(t)[x(t) − Cov(x(t), ε(j))R −1 (j)ε(j)] j=1 +e(t) = H(t)[x(t) − x(t|t − 1)] + e(t).
N, will not be available unless we have already evaluated S(t|t−1), t = 1, . , n. Consequently, if we want to compute the S(t|t − 1) and R(t) in tandem with evaluation of ΣXε we need a slightly more subtle strategy. Now, in general, for the tth row block the above diagonal blocks appear like σXε (t, j) = S(t|t − 1)M © 2006 by Taylor & Francis Group, LLC T (t) · · · M T (j − 1)H T (j) A Kalman Filter Primer 42 for j = t+ 1, . , n. So, computations above the diagonal can be carried out by storing and updating matrices of the form A(t, j) = S(t|t − 1)M T (t) · · · M T (j − 1).
3. This two-stage approach can be perfectly satisfactory and we will see this reflected in some of the forward and backward recursions for computing signal and state vector estimators in Chapters 4 and 5. However, there are also cases where it would be more convenient to have the entire matrix ΣXε in hand after a single forward pass. One way to evaluate the entirety of ΣXε in a single recursion is to work forward from the upper left hand corner of the matrix in an L-shaped pattern. 2. Then the above diagonal blocks for the (t + 1)st column block are evaluated.