Since the distribution of the weight vectors of the SOM approximates that of the training data, the SOM can be utilized in estimating the probability density function (PDF) of a data set. The estimation can be done using e.g. the reduced kernel density estimation (RKDE) [10]:

where *M* is the number of neurons in the map, *K* is a
kernel function and the weights and the smoothing
parameters are estimated from the data. The batch version
of the algorithm proceeds as follows:

- Train the kernel centers using the SOM.
- Calculate the weights , where is the number of samples in the Voronoi region of the reference vector .
- Calculate a single smoothing parameter
*h*, or perhaps different for each kernel. This can be done e.g. by computing the distance between and its nearest model vector, setting , and optimizing as the Breiman-Meisel-Purcell estimator.

Tue May 27 12:40:37 EET DST 1997