Simultaneous Estimation of Filtered and Smoothed State Probability Density Functions by Multiple Distribution Estimation

Abstract

This paper shows that for the multiple distribution estimation filter (MDEF)[1] [2], the one-step-behind (OSB) smoothed state probability density function (PDF) used for the estimation of the filtered state PDF is the key factor for the high filtering accuracy of the MDEF. The MDEF calculates the OSB smoothed state PDF prior to the calculation of the filtered state PDF at the current epoch and the filtered state PDF is estimated by the marginalization of the conditional state PDF with respect to this smoothed state PDF. Since the OSB smoothed state PDFs are obtained at every time step prior to the observation update, the MDEF can be regarded as providing the simultaneous estimation of the filtered and OSB smoothed state PDFs. In this paper, we numerically evaluate the estimation accuracy for the OSB smoothed state estimates by the MDEF using the benchmark filtering problems [3]–[5] and compare it with those for the particle and the Gaussian smoothers employed to the MDEF. We confirmed that the smoothing accuracy for the OSB smoothed state estimates was more accurate than that for the Gaussian smoother and almost comparable to that for the particle smoother, while the calculation cost was significantly lowered than that for the particle smoother.