Page 315 - Computational Statistics Handbook with MATLAB

304 Computational Statistics Handbook with MATLAB

We continue through the data set, one point at a time, adding new terms as
necessary. Our density estimate is then given by

N
ˆ ˆ ˆ ˆ
(
,
f AM x() = ∑ p i φ x;µ i Σ i . ) (8.49)
i = 1
This allows for a variable number of terms N, where usually N << n . The
adaptive mixtures technique is captured in the procedure given here, and a
function called csadpmix is provided with the Computational Statistics
Toolbox. Its use in the univariate case is illustrated in Example 8.12.

ADAPTIVE MIXTURES PROCEDURE:

1. Initialize the adaptive mixtures procedure using the first data point
1 ()
x :

ˆ 1 () 1 () ˆ 1 () ˆ 1() ,
µ 1 = x , p 1 = 1 , and Σ 1 = I
where I denotes the identity matrix. In the univariate case, the
variance of the initial term is one.
( n + 1)
2. For a new data point x , calculate the squared Mahalanobis
distance as in Equation 8.44.
, then create a
3. If the minimum squared distance is greater than t C
new term using Equations 8.46 through 8.48. Increase the number
of terms N by one.
4. If the minimum squared distance is less than the create threshold
, then update the existing terms using Equations 8.41
t C
through 8.43.
5. Continue steps 2 through 4 using all data points.

In practice, the adaptive mixtures method is used to get initial values for
the parameters, as well as an estimate of the number of terms needed to
model the density. One would then use these as a starting point and apply the
iterative EM algorithm to refine the estimates.

Example 8.12
In this example, we illustrate the MATLAB code that implements the univari-
ate adaptive mixtures density estimation procedure. The source code for
these functions are given in Appendix D. We generate random variables
using the same three term mixture model that was discussed in Example
8.9.Recall that the model is given by

© 2002 by Chapman & Hall/CRC