Page 326 - Computational Statistics Handbook with MATLAB

Chapter 8: Probability Density Estimation 315

8.11. In this chapter, we showed how you could construct a kernel density
estimate by placing a weighted kernel at each data point, evaluating
the kernels over the domain, and then averaging the n curves. In that
implementation, we are looping over all of the data points. An alter-
native implementation is to loop over all points in the domain where
you want to get the value of the estimate, evaluate a weighted kernel
at each point, and take the average. The following code shows you
how to do this. Implement this using the Buffalo snowfall data.
Verify that this is a valid density by estimating the area under the
curve.

load snowfall
x = 0:140;
n = length(snowfall);
h = 1.06*sqrt(var(snowfall))*n^(-1/5);
fhat = zeros(size(x));
% Loop over all values of x in the domain
% to get the kernel evaluated at that point.
for i = 1:length(x)
xloc = x(i)*ones(1,n);
% Take each value of x and evaluate it at
% n weighted kernels -
% each one centered at a data point, then add them up.
arg = ((xloc-snowfall)/h).^2;
fhat(i) = (sum(exp(-.5*(arg)))/(n*h*sqrt(2*pi)));
end
8.12. Write a MATLAB function that will construct a kernel density esti-
mate for the multivariate case.
8.13. Write a MATLAB function that will provide the finite mixture den-
sity estimate at a point in d dimensions.
8.14. Implement the univariate adaptive mixtures density estimation pro-
cedure on the Buffalo snowfall data. Once you have your initial
model, use the EM algorithm to refine the estimate.
8.15. In Example 8.13, we generate a random sample from the kernel
estimate of the Old Faithful geyser data. Repeat this example to
obtain a new random sample of geyser data from the estimated
model and construct a new density estimate from the second sample.
Find the integrated squared error between the two density estimates.
Does the error between the curves indicate that the second random
sample generates a similar density curve?
8.16. Say we have a kernel density estimate where the kernel used is a
normal density. If we put this in the context of finite mixtures, then
2
what are the values for the component parameters (p i , µ i , σ i ) in the
corresponding finite mixture?

© 2002 by Chapman & Hall/CRC