Robust EM algorithm for model-based curve clustering

Model-based clustering approaches concern the paradigm of exploratory data analysis relying on the finite mixture model to automatically find a latent structure governing observed data. They are one of the most popular and successful approaches in cluster analysis. The mixture density estimation is generally performed by maximizing the observed-data log-likelihood by using the expectation-maximization (EM) algorithm. However, it is well-known that the EM algorithm initialization is crucial. In addition, the standard EM algorithm requires the number of clusters to be known a priori. Some solutions have been provided in [31, 12] for model-based clustering with Gaussian mixture models for multivariate data. In this paper we focus on model-based curve clustering approaches, when the data are curves rather than vectorial data, based on regression mixtures. We propose a new robust EM algorithm for clustering curves. We extend the model-based clustering approach presented in [31] for Gaussian mixture models, to the case of curve clustering by regression mixtures, including polynomial regression mixtures as well as spline or B-spline regressions mixtures. Our approach both handles the problem of initialization and the one of choosing the optimal number of clusters as the EM learning proceeds, rather than in a two-fold scheme. This is achieved by optimizing a penalized log-likelihood criterion. A simulation study confirms the potential benefit of the proposed algorithm in terms of robustness regarding initialization and funding the actual number of clusters.Comment: In Proceedings of the 2013 International Joint Conference on Neural Networks (IJCNN), 2013, Dallas, TX, US

A data driven equivariant approach to constrained Gaussian mixture modeling

Maximum likelihood estimation of Gaussian mixture models with different class-specific covariance matrices is known to be problematic. This is due to the unboundedness of the likelihood, together with the presence of spurious maximizers. Existing methods to bypass this obstacle are based on the fact that unboundedness is avoided if the eigenvalues of the covariance matrices are bounded away from zero. This can be done imposing some constraints on the covariance matrices, i.e. by incorporating a priori information on the covariance structure of the mixture components. The present work introduces a constrained equivariant approach, where the class conditional covariance matrices are shrunk towards a pre-specified matrix Psi. Data-driven choices of the matrix Psi, when a priori information is not available, and the optimal amount of shrinkage are investigated. The effectiveness of the proposal is evaluated on the basis of a simulation study and an empirical example

Flexible Mixture Modeling with the Polynomial Gaussian Cluster-Weighted Model

In the mixture modeling frame, this paper presents the polynomial Gaussian cluster-weighted model (CWM). It extends the linear Gaussian CWM, for bivariate data, in a twofold way. Firstly, it allows for possible nonlinear dependencies in the mixture components by considering a polynomial regression. Secondly, it is not restricted to be used for model-based clustering only being contextualized in the most general model-based classification framework. Maximum likelihood parameter estimates are derived using the EM algorithm and model selection is carried out using the Bayesian information criterion (BIC) and the integrated completed likelihood (ICL). The paper also investigates the conditions under which the posterior probabilities of component-membership from a polynomial Gaussian CWM coincide with those of other well-established mixture-models which are related to it. With respect to these models, the polynomial Gaussian CWM has shown to give excellent clustering and classification results when applied to the artificial and real data considered in the paper

Adaptive Seeding for Gaussian Mixture Models

We present new initialization methods for the expectation-maximization algorithm for multivariate Gaussian mixture models. Our methods are adaptions of the well-known $K$-means++ initialization and the Gonzalez algorithm. Thereby we aim to close the gap between simple random, e.g. uniform, and complex methods, that crucially depend on the right choice of hyperparameters. Our extensive experiments indicate the usefulness of our methods compared to common techniques and methods, which e.g. apply the original $K$-means++ and Gonzalez directly, with respect to artificial as well as real-world data sets.Comment: This is a preprint of a paper that has been accepted for publication in the Proceedings of the 20th Pacific Asia Conference on Knowledge Discovery and Data Mining (PAKDD) 2016. The final publication is available at link.springer.com (http://link.springer.com/chapter/10.1007/978-3-319-31750-2 24