Page 81 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

2. Probability and Random Process 69

Figure 2-1. Clustering Example with Four Clusters
Estimating the best values for mean vectors and covariance matrices such
that the probability P D is maximized is described below.
Maximization is obtained by differentiating P D = P D=p(x 1) p(x 2)…p(x m)
with respect to the unknown parameters m 1, m 2, m 3, … m n, cov 1, cov 2,
cov 3…cov n and equate to zero. Solving the above set of obtained equations
gives the best estimate of the unknown parameters, which are listed below

m 1= [(p(c 1/x 1)*(x 1) + p(c 1/x 2)*(x 2)+ p(c 1/x 3)*(x 3)+…… p(c 1/x m)*x m)]
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p(c 1/x 1)+ p(c 1/x 2)+ p(c 1/x 3)+…… p(c 1/x m)

where p (c 1/x 1) is computed as [p(x 1/c 1) * p (c 1)] / [p(x 1)]

Note that p(x 1/c 1) is the probability of ‘x 1’ computed using Gaussian
density function of ‘x’ with mean ‘m 1’ and covariance ‘cov 1’. Also p(x 1) =
p(c 1) p(x 1/c 1) + p (c 2) p(x 1/c 2) + p(c 3 ) p(x 1/c 3) + . . . p(c n) p(x 1/c n)