Abstract

Linear systems have been used extensively in engineering to model and control the behavior of dynamical systems. In this note, we present the Expectation Maximization (EM) algorithm for estimating the parameters of linear systems (Shumway and Stoffer, 1982). We also point out the relationship between linear dynamical systems, factor analysis, and hidden Markov models. Introduction The goal of this note is to introduce the EM algorithm for estimating the parameters of linear dynamical systems (LDS). Such linear systems can be used both for supervised and unsupervised modeling of time series. We first describe the model and then briefly point out its relation to factor analysis and other data modeling techniques. The Model Linear time-invariant dynamical systems, also known as linear Gaussian state-space models, can be described by the following two equations: x t+1 = Ax t +w t (1) y t = Cx t + v t : (2) Time is indexed by the discrete index t. The output y t is a linear function of ...

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