By Randall L. Eubank

ISBN-10: 0824723651

ISBN-13: 9780824723651

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**Additional resources for A Kalman Filter Primer**

**Sample text**

Finally, to complete our treatment of the Kalman filter, a more general state– space model is introduced in Chapter 8 and we describe how the results from the previous chapters extend to this case. 1 Introduction In this chapter we will lay the foundation for Chapters 3–5. In that regard, our goal is to obtain a complete characterization of the covariance relationship between the innovations and the state vectors. 3. 3 as well as other results in subsequent chapters. 2 Some tools of the trade To begin let us recall some of the basic formulations from Chapter 1.

6) and (F3). 10). Then, R(t) = H(t)S(t|t − 1)H for t = 1, . , n. 14) A Kalman Filter Primer 28 Proof. First note that R(1) = Var(ε(1)) = Var(y(1)) = Var[H(1)x(1) + e(1)] = H(1)Var(x(1))H T (1) + W (1) due to (F2). The result for t = 2, . 9) and (F2). 3 we can recursively update S(t|t − 1), S(t|t) and R(t) via the following algorithm. 1 This algorithm computes S(t|t−1), R(t) and S(t|t) for t = 1, . n. 1 that for each t the amount of work involved in evaluating S(t|t − 1), S(t|t) and R(t) depends only on the dimensions of the matrices and not on n.

Column (n − 1) © 2006 by Taylor & Francis Group, LLC A Kalman Filter Primer 34 is seen to have the form S(1|0)M T (1) · · · M T (n − 2)H T (n − 1) S(2|1)M T (2) · · · M T (n − 2)H T (n − 1) . . T S(n − 3|n − 4)M (n − 3)M T (n − 2)H T (n − 1) T S(n − 2|n − 3)M (n − 2)H T (n − 1) S(n − 1|n − 2)H T (n − 1) F (n − 1)S(n − 1|n − 2)H T (n − 1) while the blocks in column n are S(1|0)M T (1) · · · M T (n − 1)H T (n) S(2|1)M T (2) · · · M T (n − 1)H T (n) . . T S(n − 3|n − 4)M (n − 3) · · · M T (n − 1)H T (n) T S(n − 2|n − 3)M (n − 2)M T (n − 1)H T (n) S(n − 1|n − 2)M T (n − 1)H T (n) S(n|n − 1)H T (n) The pattern that appears here is similar to what we saw in working out the forward recursion for the diagonal and below diagonal elements of ΣXε in the sense that each row can be updated (or “downdated”) as we move (backward) to the row above through the use of a common pre-multiplier.

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