Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models

Abstract A new algorithm for the prediction, filtering, and smoothing of non-Gaussian nonlinear state space models is shown. The algorithm is based on a Monte Carlo method in which successive prediction, filtering (and subsequently smoothing), conditional probability density functions are approximated by many of their realizations. The particular contribution of this algorithm is that it can be applied to a broad class of nonlinear non-Gaussian higher dimensional state space models on the provision that the dimensions of the system noise and the observation noise are relatively low. Several numerical examples are shown.

[1]  H. Sorenson,et al.  Nonlinear Bayesian estimation using Gaussian sum approximations , 1972 .

[2]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[3]  H. Kunzi,et al.  Lectu re Notes in Economics and Mathematical Systems , 1975 .

[4]  C. Masreliez.  Approximate non-Gaussian filtering with linear state and observation relations , 1975 .

[5]  M. Netto,et al.  On the optimal and suboptimal nonlinear filtering problem for discrete-time systems , 1977 .

[6]  Howell Tong,et al.  Some Comments on the Canadian Lynx Data , 1977 .

[7]  Andrew P. Sage,et al.  Estimation theory with applications to communications and control , 1979 .

[8]  B. Anderson,et al.  Optimal Filtering , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[9]  G. Kitagawa,et al.  A Smoothness Priors–State Space Modeling of Time Series with Trend and Seasonality , 1984 .

[10]  M. West,et al.  Dynamic Generalized Linear Models and Bayesian Forecasting , 1985 .

[11]  J. E. Miller,et al.  A non-Gaussian state space model and application to prediction of records , 1986 .

[12]  G. Kitagawa.  Non-Gaussian State—Space Modeling of Nonstationary Time Series , 1987 .

[13]  G. Kitagawa,et al.  Akaike Information Criterion Statistics , 1988 .

[14]  G. Kitagawa,et al.  Akaike Information Criterion Statistics , 1988 .

[15]  Richard J. Meinhold,et al.  Robustification of Kalman Filter Models , 1989 .

[16]  Genshiro Kitagawa,et al.  Non-Gaussian seasonal adjustment , 1989 .

[17]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter , 1990 .

[18]  William H. Press,et al.  Numerical Recipes: FORTRAN , 1988 .

[19]  L. Fahrmeir,et al.  On kalman filtering, posterior mode estimation and fisher scoring in dynamic exponential family regression , 1991 .

[20]  G. Kitagawa.  A nonlinear smoothing method for time series analysis , 1991 .

[21]  L. Fahrmeir.  Posterior Mode Estimation by Extended Kalman Filtering for Multivariate Dynamic Generalized Linear Models , 1992 .

[22]  Nicholas G. Polson,et al.  A Monte Carlo Approach to Nonnormal and Nonlinear State-Space Modeling , 1992 .

[23]  Sylvia Schnatter.  Integration-based Kalman-filtering for a dynamic generalized linear trend model , 1992 .

[24]  P. Hodges,et al.  A COMPUTATIONAL METHOD FOR ESTIMATING DENSITIES OF NON‐GAUSSIAN NONSTATIONARY UNIVARIATE TIME SERIES , 1993 .

[25]  Hisashi Tanizaki,et al.  Nonlinear filters , 1993 .

[26]  W. Press,et al.  Numerical Recipes in Fortran: The Art of Scientific Computing.@@@Numerical Recipes in C: The Art of Scientific Computing. , 1994 .

[27]  S. Frühwirth-Schnatter.  Data Augmentation and Dynamic Linear Models , 1994 .

[28]  G. Kitagawa.  The two-filter formula for smoothing and an implementation of the Gaussian-sum smoother , 1994 .