Consistance d'un estimateur de minimum de variance \'etendue

TL;DR

This paper generalizes the K-means variance criterion using neighborhood structures, proving the strong consistency of the estimator under certain conditions, with applications to Kohonen maps.

Contribution

It introduces a generalized variance criterion incorporating neighborhood structures and proves the estimator's strong consistency under specified assumptions.

Findings

Proposed a neighborhood-based extension of the K-means variance criterion.

Proved the strong consistency of the extended variance estimator.

Applicable to models like Kohonen maps for quality quantification.

Abstract

We consider a generalization of the criterion minimized by the K-means algorithm, where a neighborhood structure is used in the calculus of the variance. Such tool is used, for example with Kohonen maps, to measure the quality of the quantification preserving the neighborhood relationships. If we assume that the parameter vector is in a compact Euclidean space and all it components are separated by a minimal distance, we show the strong consistency of the set of parameters almost realizing the minimum of the empirical extended variance.