Self-Organizing Time Series Model

Abstract

The generalised state-space model (GSSM) that we deal with in this study is defined by a set of two equations,

$$system\;\bmod el\quad {x_t} = f({x_{t - 1}},{v_t})$$

(20.1.1)

$$observation\;\bmod el\quad {y_t} \sim r( \cdot |{x_t},{\theta _{obs}})$$

(20.1.2)

where x _t is an n _x × 1 vector of unobserved sate variables, and y _t is an n _y dimensional vector observation. ${\mathbb{R}^{{n_x}}} \times {\mathbb{R}^{{n_v}}} \to {\mathbb{R}^{{n_x}}}$ is a given function. {v _t } is an independent and identically distributed (i.i.d.) random process with v _t ~ q(v|θ _sys ). r is the conditional distribution of y _t given x _t ∙ q(∙|∙) and r(∙|∙) are, in general, non-Gaussian densities specified by the unknown parameter vectors, θ _sys and θ _obs respectively. In this study, we set θ = [θ′ _sys ,θ′ _obs ]′. The initial state x ₀ is distributed according to the density p ₀(x).