Introduction - On Statistical Pattern Recognition in Independent Component Analysis Mixture Modelling - page 7

Fig. 1.3 Types of statistical classifiers. The types of methods proposed in this work are
highlighted

1.1.1 Density Estimation

The probability density function is a fundamental concept in statistics. The
probability density function f for a random quantity X gives a natural description
of the distribution of X and also allows probabilities associated with X to be found
from the relation

Þ ¼ Z

b

Pa\X\b ð f ð x Þ dx for all a\b ð 1 : 3 Þ

a

Suppose that we have a set of observed data points assumed to be samples from
an unknown probability density function. Density estimation is the construction of
an estimate of the density function from the observed data. One approach to
density estimation is parametric, which assumes that the data are drawn from one
of a known parametric family of distributions (e.g., the normal distribution with
mean l and variance r 2 ). The density f underlying the data could then be esti-
mated by finding estimates of l and r 2 from the data and substituting these
estimates into the formula for the normal density. The non-parametric approach
imposes less rigid assumptions about the distribution of the observed data.
Although it is assumed that the distribution has a probability density f , the data is