Parameter estimation for linear dynamical systems
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 Sto er, 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 rst describe the model and then brie y 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: xt+1 = Axt +wt (1) yt = Cxt+ vt: (2) Time is indexed by the discrete index t. The output yt is a linear function of the state, xt, and the state at one time step depends linearly on the previous state. Both state and output noise, wt and vt, are zero-mean normally distributed random variables with covariance matrices Q and R, respectively. Only the output of the system is observed, the state and all the noise variables are hidden. Rather than regarding the state as a deterministic value corrupted by random noise, we combine the state variable and the state noise variable into a single Gaussian random