A kernel method for hierarchical and non-hierarchical clustering
In cluster analysis, choosing a distance measure among given data is not straightforward. For defining a similarity matrix in hierarchical clustering and some types of non-hierarchical clustering such as spectral clustering ([1]), several distance measures such as Pearson correlation, Euclidean distance, and nonparametric measures have been widely used. The result of clustering, however, often depends on a specific choice of the measure. If our purpose is to extract a group of variables, for which we cannot presume the type of dependence, a specific choice of measure may not capture the dependence of variables. This paper proposes a novel approach to define a similarity matrix, which can incorporate any nonlinear correlation of variables. Our method uses kernel Hilbert spaces, which contain a sufficiently rich class of nonlinear functions of variables. We define the similarity of two variables based on the theory of kernel Hilbert spaces and covariance operators ([2]). This similarity represents the sum of all the nonlinear correlations in principle, and provides a more reasonable way for defining the similarity or dependence of two random variables than existing measures, which extract only partial information of variables.