General State Space Modeling

Non-Gaussian state space modeling of time series and the more general, general state space modeling are treated in this chapter. They were introduced because of the need to model time series with abrupt discontinuities, and time series with outliers and to model time series whose state and or observation processes were nonlinear. The general state space model and its recursive formulas for prediction, filtering and smoothing are treated in Section 6.2 after an introduction in Section 6.1. Three alternative computational approaches for realizing the general state space modeling, numerical integration method, a Gaussian mixture-two filter formula method and the most recently developed Monte Carlo “particle-path tracing” method are discussed respectively in Sections 6.3, 6.4 and 6.5. In addition, starting from the general state space model, an alternative derivation of the Kalman filter is shown in Section 6.6. Applications of the non-Gaussian methods appear in Chapters 8 through 16. In those chapters where appropriate, the non-Gaussian and general state space methods performance are contrasted with the linear Gaussian state space analysis methods.