Introduction to Pattern Recognition: A Matlab Approach

1.6 The Expectation-Maximization Algorithm 13

1.6 THE EXPECTATION-MAXIMIZATION ALGORITHM

Let us assume that we are given a set of N points, x

i

∈ R

l

, i = 1, 2,...,N , whose statistical properties

are described by a pdf that is expanded as in Eq. (1.6). Adopting a val ue for J, the task is to use these

data points to estimate the parameters that enter in the expansion—that is, the probability parameters

P

j

, j = 1,2, ...,J and the parameters associated with each one of the terms p(x|j), j = 1, 2,...,J.For

example, if we assume each one of the summand pdfs to be a Gaussian distribution with σ

2

j

I covariance

matrix:

p(x|j) =

1

(2π)

l/2

σ

l

j

exp



−

(x −m

j

)

T

(x −m

j