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110 4 Statistical Classification

reproducibility are less accessible to formal analysis, and in practice they have to
be assessed in a purely experimental way.

4.3.1 The Parzen Window Method

As explained previously, in order to estimate p(x) we will choose a sufficiently
small region R centred at x. Lel us select R as a d-dimensional hypercube whose
edge h(n) varies with n. By varying h(n) we are able to select an appropriate
hypercube size depending on how many training patterns we have available. The
volume of R is:

Let us define the following counting function:

1 if (~~11112, k=l, ..., d;
v(x) =
0 otherwise.
Therefore, if a point falls inside the unit hypercube centred at the origin, it is
counted; otherwise, it is not counted. Using this function we can express compactly
the number k(n) of points xi falling inside any hypercube centred at x, as:

Tn this formula function p is scaled by the hypercube edge length h(n), and the
counting criterion depends on the difference vector x - xi, between a feature vector
x and a training set feature vector xi.
With this k(n) we can now express equation (4-32) as:

From formula (4-36) we see that if there is a large agglomeration of points in the
immediate neighbourhood of x one obtains a high value of p(x, n) . If the number
of such points is small, the value of j(x, n) is also small.
Figure 4.27 illustrates the Parzen window method applied to a one-dimensional
distribution and using a rectangular window. The pdf estimates at the regularly
spaced marks are given by the height of the solid bars, proportional to the number
of points (circles) that fall inside the window associated with a specific position.