By Randall L. Eubank
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Additional info for A Kalman Filter Primer
4) for t = 2, . 5) for t = 3, . , n and j = 1, . , t − 2. Proof. 16) we have for t > j that L(t, j) = Cov(y(t), ε(j))R−1 (j). But, y(t) = H(t)x(t) + e(t) and e(t) is uncorrelated with ε(1), . , ε(t − 1). 4. Efficient order n2 (for p, q small relative to n) recursions for L can now be obtained in several ways. One © 2006 by Taylor & Francis Group, LLC A Kalman Filter Primer 56 approach is to build L row by row. 1 to see that the first column for L is I H(2)F (1)S(1|0)H T (1)R−1 (1) H(3)F (2)F (1)S(1|0)H T (1)R−1 (1) H(4)F (3)F (2)F (1)S(1|0)H T (1)R−1 (1) H(5)F (4) · · · F (1)S(1|0)H T (1)R−1 (1) .
F (n−2) F (n − 2) · · · F (1)S(1|0)H T (1) ×F (n−1) F (n − 1) · · · F (1)S(1|0)H T (1) © 2006 by Taylor & Francis Group, LLC A Kalman Filter Primer 32 and S(2|1)H T (2) ×F (2) F (2)S(2|1)H T (2) ×F (3) . . ×F (n−2) F (n − 2) · · · F (2)S(2|1)H T (2) ×F (n−1) F (n − 1) · · · F (2)S(2|1)H T (2) By extrapolating from what we have observed in these special cases we can determine that the diagonal and below diagonal blocks of ΣXε can be computed on a row-by-row basis by simply “updating” entries from previous rows through pre-multiplication by an appropriate F (·) matrix.
In this regard, we develop a forward recursion that produces the matrix row by row starting from its upper left block entry. 3 then provides a parallel result pertaining to L−1 . 2 Recursions for L For the developments in this and subsequent sections it will be convenient to introduce a final piece of notation for the so-called Kalman gain matrices. These matrices arise naturally in formulae for both L and L−1 and, not surprisingly, appear in various signal and state vector prediction formulae that we will encounter in the next chapter.