Probability Statistics

A Kalman Filter Primer (Statistics: A Series of Textbooks by Randall L. Eubank

By Randall L. Eubank

Process kingdom estimation within the presence of noise is necessary for regulate platforms, sign processing, and plenty of different functions in a number of fields. constructed many years in the past, the Kalman clear out is still a tremendous, robust device for estimating the variables in a method within the presence of noise. even if, while inundated with idea and immense notations, studying simply how the Kalman filter out works could be a daunting activity. With its mathematically rigorous, “no frills” method of the fundamental discrete-time Kalman clear out, A Kalman clear out Primer builds a radical knowing of the internal workings and simple techniques of Kalman clear out recursions from first rules. rather than the common Bayesian standpoint, the writer develops the subject through least-squares and classical matrix equipment utilizing the Cholesky decomposition to distill the essence of the Kalman filter out and display the motivations in the back of the alternative of the initializing nation vector. He offers pseudo-code algorithms for a number of the recursions, permitting code improvement to enforce the filter out in perform. The e-book completely experiences the advance of contemporary smoothing algorithms and techniques for settling on preliminary states, in addition to a finished improvement of the “diffuse” Kalman filter out. utilizing a tiered presentation that builds on uncomplicated discussions to extra advanced and thorough remedies, A Kalman clear out Primer is the right creation to speedy and successfully utilizing the Kalman clear out in perform.

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Extra resources for A Kalman Filter Primer (Statistics: A Series of Textbooks and Monographs)

Example text

9) along with (F2) to complete the proof. 2 . 10). Then, for t = 1, S(t|t − 1) = F (t − 1)S(t − 1|t − 1)F T (t − 1) +Q(t − 1). 12) Proof. 1 has the consequence that for t = 2, . , n x(t) − x(t|t − 1) = F (t − 1)x(t − 1) + u(t − 1) −F (t − 1)x(t − 1|t − 1) = F (t − 1)[x(t − 1) − x(t − 1|t − 1)] +u(t − 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).

4. 9) and (F2). 2) to see that x(t) = F (t − 1) · · · F (j)x(j) + Z(t) with Z(t) depending only on u(t − 1), . , u(j). 15). 18) is somewhat more difficult to establish. 18) for j = t + 1. By exactly the same process we used for j = t + 1 we find that Cov(x(t), ε(t + 2)) has the form Cov(x(t), x(t + 1) − x(t + 1|t + 1))F T (t + 1)H T (t + 2). 15) we can express x(t + 1) − x(t + 1|t + 1) as t x(t + 1) − Cov(x(t + 1), ε(j))R −1 (j)ε(j) j=1 −Cov(x(t + 1), ε(t + 1))R −1 (t + 1)ε(t + 1) = F (t)[x(t) − x(t|t)] −S(t + 1|t)H T (t + 1)R −1 (t + 1)ε(t + 1) + u(t).

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 .

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