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306 Computational Statistics Handbook with MATLAB

plot for the three term mixture model in Example 8.12. Note that the adaptive
mixture approach yields more than three terms. This is a problem with mix-
ture models in general. Different models (i.e., number of terms and estimated
component parameters) can produce essentially the same function estimate
ˆ
or curve for f x() . This is illustrated in Figures 8.16 and 8.17, where we see
that similar curves are obtained from two different models for the same data
set. These results are straight from the adaptive mixtures density estimation
approach. In other words, we did not use this estimate as an initial starting
point for the EM approach. If we had applied the iterative EM to these esti-
mated models, then the curves should be the same.
The other issue that must be considered when using the adaptive mixtures
approach is that the resulting model or estimated probability density func-
tion depends on the order in which the data are presented to the algorithm.
This is also illustrated in Figures 8.16 and 8.17, where the second estimated
model is obtained after re-ordering the data. These issues were addressed by
Solka [1995].

8.5 Generating Random Variables

In the introduction, we discussed several uses of probability density esti-
mates, and it is our hope that the reader will discover many more. One of the
applications of density estimation is in the area of modeling and simulation.
Recall that a key aspect of modeling and simulation is the collection of data
generated according to some underlying random process and the desire to
generate more random variables from the same process for simulation pur-
poses. One option is to use one of the density estimation techniques dis-
cussed in this chapter and randomly sample from that distribution. In this
section, we provide the methodology for generating random variables from
finite or adaptive mixtures density estimates.
We have already seen an example of this procedure in Example 8.11 and
Example 8.12. The procedure is to first choose the class membership of gen-
erated observations based on uniform (0,1) random variables. The number of
random variables generated from each component density is given by the
corresponding proportion of these uniform variables that are in the required
range. The steps are outlined here.

PROCEDURE - GENERATING RANDOM VARIABLES (FINITE MIXTURE)

(
1. We are given a finite mixture model (p i , g x;θ ) ) with c compo-
i
i
nents, and we want to generate n random variables from that
distribution.

© 2002 by Chapman & Hall/CRC