Approximate Kalman Filtering by Guan Rong Chen

By Guan Rong Chen

Kalman filtering set of rules provides optimum (linear, independent and minimal error-variance) estimates of the unknown nation vectors of a linear dynamic-observation process, less than the average stipulations akin to ideal information details; whole noise records; precise linear modelling; excellent will-conditioned matrices in computation and strictly centralized filtering. In perform, even if, a number of of the aforementioned stipulations is probably not chuffed, in order that the normal Kalman filtering set of rules can't be at once used, and consequently ''approximate Kalman filtering'' turns into priceless. within the final decade, loads of cognizance has been keen on enhancing and/or extending the traditional Kalman filtering strategy to deal with such abnormal circumstances. This booklet is a suite of numerous survey articles summarizing contemporary contributions to the sector, alongside the road of approximate Kalman filtering with emphasis on its functional features

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1 sTS~ls)/a2 48 V. Gomez and A. Maravall If S in (8) is singular, de Jong leaves the diffuse log-likelihood undefined. In order to define the limiting expressions of Theorem 3 when S is singular, we have to consider model (9) with an R matrix that is not of full column rank. Let K be a selector matrix formed by zeros and ones such that KSKT has a rank equal to rank(R) and replace model (9) by v = RKT6l+e, (10) where £ t ~ N(c, cr2C), with C nonsingular and 6_x is the vector formed by choosing those components in 8 corresponding to the selected columns RK .

T - Then, t we can write v 7 = Ri6 + ej vi[ = Ru6 + eu (11a) (116) The next theorem shows how to implement the collapsing of the EKF or DKF to the KF. Theorem 9. Under Assumption A, let J with \J\ = 1 be a matrix like those used by AK to define their likelihood, with corresponding submatrices Ji and J% such that j J\R / 0 and J2R = 0, and let p(v//|v/, 5/) be t i e density ofvn -- £ { v /E{VH\VI,6J}, where E{vn\vj,6j} >/} is the conditional expectation ofvn given vj in model (11a) and (116), considering 6_ fixed (C = 0), and 6, replaced by its maximum likelihood estimator 8_j in model (11a).

Proof: For simplicity, consider that Ri is of full column rank. If not, we would use generalized inverses, but the proof would not be affected. From model (11a) and (116), we have, considering 6. fixed, £ { v / / | v / } = Rn6 + E 2 iE]- 1 1 (v / - RiS) , where E21 = Cov(e_jj,er) and E n = Var{e_j). Then, E{vji\vj,6_j] = RJJRJ vj and VJI — E{VJJ\VJ,6_J} = v// — RnRJ1vj. Define the matrix J = (Jj T , Jj)T with Ji = (1,0) and J2 = {—R[iRJl,I). Then, J is a matrix of the type used by AK to define their likelihood and v// — £ { v / / | v / , 6_j) = J 2 v.

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