Nonlinear Filters

Theory and Applications

Peyman Setoodeh, Saeid Habibi, Simon Haykin

• Publisher: Wiley
• Year: 2022
• Print ISBN:
  9781118835814
• Edition: 1
• eBook ISBN:
  9781119078159
• Topic: Technology & Engineering
• Subtopic: Electrical Engineering & Telecommunications
• Index: Technology & Engineering

NONLINEAR FILTERS

Discover the utility of using deep learning and (deep) reinforcement learning in deriving filtering algorithms with this insightful and powerful new resource

Nonlinear Filters: Theory and Applications delivers an insightful view on state and parameter estimation by merging ideas from control theory, statistical signal processing, and machine learning. Taking an algorithmic approach, the book covers both classic and machine learning-based filtering algorithms.

Readers of Nonlinear Filters will greatly benefit from the wide spectrum of presented topics including stability, robustness, computability, and algorithmic sufficiency. Readers will also enjoy:

• Organization that allows the book to act as a stand-alone, self-contained reference
• A thorough exploration of the notion of observability, nonlinear observers, and the theory of optimal nonlinear filtering that bridges the gap between different science and engineering disciplines
• A profound account of Bayesian filters including Kalman filter and its variants as well as particle filter
• A rigorous derivation of the smooth variable structure filter as a predictor-corrector estimator formulated based on a stability theorem, used to confine the estimated states within a neighborhood of their true values
• A concise tutorial on deep learning and reinforcement learning
• A detailed presentation of the expectation maximization algorithm and its machine learning-based variants, used for joint state and parameter estimation
• Guidelines for constructing nonparametric Bayesian models from parametric ones

Perfect for researchers, professors, and graduate students in engineering, computer science, applied mathematics, and artificial intelligence, Nonlinear Filters: Theory and Applications will also earn a place in the libraries of those studying or practicing in fields involving pandemic diseases, cybersecurity, information fusion, augmented reality, autonomous driving, urban traffic network, navigation and tracking, robotics, power systems, hybrid technologies, and finance.

1
Introduction

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,

, is a function of the state, which can be computed using a measurement model. In this regard, state‐space models are powerful tools for analysis and control of dynamic systems.

1.2 State Estimation

Observability is a key concept in system theory, which refers to the ability to reconstruct the hidden or latent state variables that cannot be directly measured, from the measured variables in the minimum possible length of time [1]. In building state‐space models, two key questions deserve special attention [2]:

1. (i) Is it possible to identify the governing dynamics from data?
2. (ii) Is it possible to perform inference from observables to the latent state variables?

At time instant

, the inference problem to be solved is to find the estimate of

in the presence of noise, which is denoted by

. Depending of the value of

, estimation algorithms are categorized into three groups [3]:

1. (i) Prediction:
   
   ,
2. (ii) Filtering:
   
   ,
3. (iii) Smoothing:
   
   .

Regarding the mentioned two challenging questions, in order to improve performance, sophisticated representations can be deployed for the system under study. However, the corresponding inference algorithms may become computationally demanding. Hence, for designing efficient data‐driven inference algorithms, the following points must be taken into account [2]:

1. (i) The underlying assumptions for building a state‐space model must allow for reliable system identification and plausible long‐term prediction of the system behavior.
2. (ii) The inference mechanism must be able to capture rich dependencies.
3. (iii) The algorithm must be able to inherit the merit of learning machines to be trainable on raw data such as sensory inputs in a control system.
4. (iv) The algorithm must be scalable to big data regarding the optimization of model parameters based on the stochastic gradient descent method.

Regarding the important role of computation in inference problems, Section 1.3 provides a brief account of the foundations of computing.

1.3 Construals of Computing

According to [4], a comprehensive theory of computing must meet three criteria:

1. (i) Empirical criterion: Doing justice to practice by keeping the analysis grounded in real‐world examples.
2. (ii) Conceptual criterion: Being understandable in terms of what it says, where it comes from, and what it costs.
3. (iii) Cognitive criterion: Providing an intelligible foundation for the computational theory of mind that underlies both artificial intelligence and cognitive science.

Following this line of th...

Table of contents

• Cover
• Table of Contents
• Title Page
• Copyright
• Dedication
• List of Figures
• List of Table
• Preface
• Acknowledgments
• Acronyms
• 1 Introduction
• 2 Observability
• 3 Observers
• 4 Bayesian Paradigm and Optimal Nonlinear Filtering
• 5 Kalman Filter
• 6 Particle Filter
• 7 Smooth Variable‐Structure Filter
• 8 Deep Learning
• 9 Deep Learning‐Based Filters
• 10 Expectation Maximization
• 11 Reinforcement Learning‐Based Filter
• 12 Nonparametric Bayesian Models
• References
• Index
• Wiley End User License Agreement