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http://dx.doi.org/10.5351/CSAM.2015.22.4.321

Geodesic Clustering for Covariance Matrices  

Lee, Haesung (Department of Statistics, Pennsylvania State University)
Ahn, Hyun-Jung (Kantar Health)
Kim, Kwang-Rae (School of Mathematical Sciences, University of Nottingham)
Kim, Peter T. (Department of Mathematics and Statistics, University of Guelph)
Koo, Ja-Yong (Department of Statistics, Korea University)
Publication Information
Communications for Statistical Applications and Methods / v.22, no.4, 2015 , pp. 321-331 More about this Journal
Abstract
The K-means clustering algorithm is a popular and widely used method for clustering. For covariance matrices, we consider a geodesic clustering algorithm based on the K-means clustering framework in consideration of symmetric positive definite matrices as a Riemannian (non-Euclidean) manifold. This paper considers a geodesic clustering algorithm for data consisting of symmetric positive definite (SPD) matrices, utilizing the Riemannian geometric structure for SPD matrices and the idea of a K-means clustering algorithm. A K-means clustering algorithm is divided into two main steps for which we need a dissimilarity measure between two matrix data points and a way of computing centroids for observations in clusters. In order to use the Riemannian structure, we adopt the geodesic distance and the intrinsic mean for symmetric positive definite matrices. We demonstrate our proposed method through simulations as well as application to real financial data.
Keywords
Euclidean distance; extrinsic mean; geodesic distance; intrinsic mean; K-means; KOSPI; Riemannian geometry; SPD; stock data;
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