Page 305 - Computational Statistics Handbook with MATLAB

294 Computational Statistics Handbook with MATLAB

c
ˆ ˆ ˆ ˆ
,
(
f FM x() = ∑ p i φ x;µ i Σ i , ) (8.33)
i = 1
ˆ
where x is a d-dimensional vector, µ i is a d-dimensional vector of means, and
ˆ is a d ×
Σ i d covariance matrix. There are still c-1 mixing coefficients to esti-
mate. However, there are now c × d values that have to be estimated for the
⁄
means and cd c +(( 1)) 2 values for the component covariance matrices.
The dF representation has been extended [Solka, Poston, Wegman, 1995] to
show the structure of a multivariate finite mixture, when the data are 2-D or
3-D. In the 2-D case, we represent each term by an ellipse centered at the
ˆ
mean of the component density µ i , with the eccentricity of the ellipse show-
ing the covariance structure of the term. For example, a term with a covari-
ance that is close to the identity matrix will be shown as a circle. We label the
center of each ellipse with text identifying the mixing coefficient. An example
is illustrated in Figure 8.14.
A dF plot for a trivariate finite mixture can be fashioned by using color to
represent the values of the mixing coefficients. In this case, we use the three
dimensions in our plot to represent the means for each term. Instead of
ellipses, we move to ellipsoids, with eccentricity determined by the covari-
ance as before. See Figure 8.15 for an example of a trivariate dF plot. The dF
plots are particularly useful when working with the adaptive mixtures den-
sity estimation method that will be discussed shortly. We provide a function
called csdfplot that will implement the dF plots for univariate, bivariate
and trivariate data.

Example 8.10
In this example, we show how to implement the function called csdfplot
and illustrate its use with bivariate and trivariate models. The bivariate case
is the following three component model:

p 1 = 0.5 p 2 = 0.3 p 3 = 0.2 ,

– 1 1 5
µ 1 = µ 2 = µ 3 = ,
– 1 1 6

10 0.5 0 10.5
Σ 1 = Σ 2 = Σ 3 = .
01 00.5 0.5 1

% First create the model.
% The function expects a vector of weights;
% a matrix of means, where each column of the matrix

© 2002 by Chapman & Hall/CRC