Offered in the fall of odd numbered years.
Brief description:
This course is structured to introduce the senior undergraduates and junior graduates in applied mathematics the basic concepts of stochastic processes, parameter estimation and filtering. The course is divided into three parts. The first two parts deal with fundamentals while the third part deals with parameter and state estimation which are major topics in applied stochastic process. The topics are designed so that the material will be covered in 36 lecture hours. This course will form the basis for more advanced courses in Applied Stochastic Processes, Advanced topics in Stochastic Processes and Random Dynamical Systems.
Part I sets the stage for the following two parts. This is concerned with a review of random variables, expectations, conditional probabilities, conditional expectations, convergence of a sequence of random variables and limit theorems. The later sections of part I focus on minimum mean square error estimation and the orthogonality principle, where we will introduce the first glimpse of the building blocks of estimation theory.
Part II introduces the notion of random process, briefly covers several examples and classes of random processes. A substantial material is covered on countable-state Markov models. Classification and convergence of both discrete-time and continuous-time Markov chains are presented in detail. Markov processes, orthogonal increment processes, and Brownian motion are covered in greater depth in the later sections.
Part III covers the main results of linear stochastic systems and estimation theory. First few sections will cover Expectation Maximization (EM) algorithm, which is a technique for finding maximum likelihood parameter estimates for a broad range of problems. Then innovation sequences and linear stochastic equations are introduced. Kalman Filter is one of the most celebrated and popular data assimilation algorithms in the field of information processing. This section provides a simple and intuitive derivation of discrete time Kalman filter in multi-dimension. State and parameter estimation will be dealt in detail in this part of the course with applications to biological, financial and space systems.
Prerequisites:
One of AMATH 250, 251, 350, MATH 211/ECE 205, MATH 218, 228, and STAT 230 or 240.