Principles of Nonlinear Filtering Theory

This text presents a comprehensive and unified treatment of nonlinear filtering theory, with a strong emphasis on its mathematical underpinnings. It is tailored to meet the needs of a diverse readership, including mathematically inclined engineers and scientists at both graduate and post-graduate levels. What sets this book apart from other treatments of the topic is twofold. Firstly, it offers a complete treatment of filtering theory, providing readers with a thorough understanding of the subject. Secondly, it introduces updated methodologies and applications that are crucial in today’s landscape. These include finite-dimensional filters, the Yau-Yau algorithm, direct methods, and the integration of deep learning with filtering problems. The book will be an invaluable resource for researchers and practitioners for years to come.

With a rich historical backdrop dating back to Gauss and Wiener, the exposition delves into the fundamental principles underpinning the estimation of stochastic processes amidst noisy observations—a critical tool in various applied domains such as aircraft navigation, solar mapping, and orbit determination, to name just a few. Substantive exercises and examples given in each chapter provide the reader with opportunities to appreciate applications and ample ways to test their understanding of the topics covered. An especially nice feature for those studying the subject independent of a traditional course setting is the inclusion of solutions to exercises at the end of the book.

The book is structured into three cohesive parts, each designed to build the reader's understanding of nonlinear filtering theory.  In the first part, foundational concepts from probability theory, stochastic processes, stochastic differential equations, and optimization are introduced, providing readers with the necessary mathematical background. The second part delves into theoretical aspects of filtering theory, covering topics such as the stochastic partial differential equation governing the posterior density function of the state, and the estimation algebra theory of systems with finite-dimensional filters. Moving forward, the third part of the book explores numerical algorithms for solving filtering problems, including the Yau-Yau algorithm, direct methods, classical filtering algorithms like the particle filter, and the intersection of filtering theory with deep learning.

Stephen Shing-Toung Yau (Life Fellow, IEEE) received the Ph.D. degree in mathematics from the State University of New York, Stony Brook, NY, USA, in 1976. He was a Member of the Institute of Advanced Study, Princeton, NJ, USA, from 1976 to 1977 and 1981 to 1982. He was a Benjamin Pierce Assistant Professor with Harvard University, Cambridge, MA, USA, from 1977 to 1980. He then joined the Department of Mathematics, Statistics and Computer Science (MSCS), University of Illinois at Chicago (UIC), Chicago, IL, USA, and served for more than 30 years. From 2005 to 2011, he was a Joint Professor with the Department of Electrical and Computer Engineering, MSCS, UIC. After retiring in 2011, he joined the Department of Mathematical Sciences at Tsinghua University in Beijing, China, where he served for over 10 years. In 2022, he became a research fellow at the Beijing Institute of Mathematical Sciences and Applications (BIMSA) in Beijing, China, to begin his new research.  His research interests include nonlinear filtering, bioinformatics, complex algebraic geometry, Cauchy–Riemann geometry, and singularities theory. Dr. Yau has been the Managing Editor and Founder of Journal of Algebraic Geometry since 1991 and the Editor-in-Chief and Founder of Communications in Information and Systems since 2000. He was the General Chairman of the 1995 IEEE International Conference on Control and Information. He received the Sloan Fellowship in 1980, the Guggenheim Fellowship in 2000, and the American Mathematical Society Fellow Award in 2013. In 2005, he was entitled the UIC Distinguished Professor. In 2019, He won the Chern Prize of Lifetime Achievement in Mathematics.

Xiuqiong Chen received the B.S. degree in the School of Mathematical Sciences, Beihang University, Beijing, China, in 2014, and the Ph.D. degree in applied mathematics from the Department of Mathematical Sciences, Tsinghua University, Beijing, China in 2019. After her graduation, she was a Postdoctoral Scholar with Yau Mathematical Sciences Center, Tsinghua University, Beijing, China, from 2019 to 2021. She joined in Renmin University of China, Beijing, China, since 2021. She is currently an Assistant Professor with School of Mathematics, Renmin University of China. Her research interests include nonlinear filtering and deep learning.

Xiaopei Jiao received his Bachelor's degree from Shanghai Jiao Tong University in 2017 and completed his Ph.D. from the Department of Mathematics at Tsinghua University in 2022. From 2022 to 2024, he worked as a postdoctoral researcher at the Beijing Institute of Mathematical Science and Application. He is currently employed at the University of Twente in the Netherlands as postdoctoral researcher. His research focuses on nonlinear filtering, Lie estimation algebra, physics-informed deep learning, and bioinformatics.

Jiayi Kang received the B.S. degree from the college of mathematics, Sichuan University, Chengdu, China, in 2019 and Ph.D. degree from Department of Mathematical Sciences at Tsinghua University, China in 2024. He is currently an assistant professor at the Beijing Institute of Mathematical Sciences and Applications in Beijing, China.

Zeju Sun received the B.S. degree from Department of Mathematical Sciences, Tsinghua University, Beijing, China, in 2020. He is currently pursuing the Ph.D. degree in mathematics with the department of Mathematical Sciences, Tsinghua University, Beijing, China.

Yangtianze Tao received the B.S degree in College of Mathematics, Sichuan University, Sichuan, China in 2019. Now he is pursuing Ph.D. degree with Department of Mathematical Sciences, Tsinghua University, Beijing, China, under the supervision of Prof. Stephen Yau in the field of applied mathematics. His research interests include deep learning, machine learning and nonlinear filtering.