Nonlinear Filters

1Introduction

1.1 State of a Dynamic System

In many branches of science and engineering, deriving a probabilistic model for sequential data plays a key role. System theory provides guidelines for studying the underlying dynamics of sequential data (time series). In describing a dynamic system, the notion of state is a key concept [1]:

Definition 1.1 State of a dynamic system is the smallest collection of variables that must be specified at a time instant in order to be able to predict the behavior of the system for any time instant . To be more precise, the state is the minimal record of the past history, which is required to predict the future behavior.

According to the principle of causality, any dynamic system may be described from the state perspective. Deploying a state‐transition model allows for determining the future state of a system, , at any time instant , given its initial state, , at time instant as well as the inputs to the system, , for . The output of the system, ...