Finite Element Approach to Clustering of Multidimensional Time Series

• Authors: Illia Horenko
• Venue: SIAM Journal on Scientific Computing
• Year: 2010

We present a new approach to clustering of time series based on a minimization of the averaged clustering functional. The proposed functional describes the mean distance between observation data and its representation in terms of $\mathbf{K}$ abstract models of a certain predefined class (not necessarily given by some probability distribution). For a fixed time series $x(t)$ this functional depends on $\mathbf{K}$ sets of model parameters $\Theta=(\theta_1,\dots,\theta_\mathbf{K})$ and $\mathbf{K}$ functions of cluster affiliations $\Gamma=(\gamma_1(t),\dots,\gamma_{\mathbf{K}}(t))$ (characterizing the affiliation of any element $x(t)$ of the analyzed time series to one of the $\mathbf{K}$ clusters defined by the considered model parameters). We demonstrate that for a fixed set of model parameters $\Theta$ the appropriate Tykhonov-type regularization of this functional with some regularization factor $\epsilon^2$ results in a minimization problem similar to a variational problem usually associated with one-dimensional nonhomogeneous partial differential equations. This analogy allows us to apply the finite element framework to the problem of time series analysis and to propose a numerical scheme for time series clustering. We investigate the conditions under which the proposed scheme allows a monotone improvement of the initial parameter guess with respect to the minimization of the discretized version of the regularized functional. We also discuss the interpretation of the regularization factor in the Markovian case and show its connection to metastability and exit times. The computational performance of the resulting method is investigated numerically on multidimensional test data and is applied to the analysis of multidimensional historical stock market data.