Learning-based EM clustering for data on the unit hypersphere with application to exoplanet data

Highlights

• We construct a learning-based EM clustering algorithm for hyper-spherical data.

• The proposed method is robust to outliers without initializations, and not necessary to assign a number of clusters.

• Numerical and real examples with comparisons are given to demonstrate the effectiveness and superiority of the proposed method.

• The proposed method is applied to cluster exoplanet data in extrasolar planets.

Abstract

This study focuses on clustering algorithms for data on the unit hypersphere. This type of directional data lain on the surface of a unit hypersphere is used in geology, biology, meteorology, medicine and oceanography. The EM algorithm with mixtures of von Mises-Fisher distributions is often used for model-based clustering for data on the unit hypersphere. However, the EM algorithm is sensitive to initial values and outliers and a number of clusters must be assigned a priori. In this paper, we propose an effective approach, called a learning-based EM algorithm with von Mises-Fisher distributions, to cluster this type of hyper-spherical data. The proposed clustering method is robust to outliers, without the need for initialization, and automatically determines the number of clusters. Thus, it becomes a fully-unsupervised model-based clustering method for data on the unit hypersphere. Some numerical and real examples with comparisons are given to demonstrate the effectiveness and superiority of the proposed method. We also apply the proposed learning-based EM algorithm to cluster exoplanet data in extrasolar planets. The clustering results have several important implications for exoplanet data and allow an interpretation of exoplanet migration.

Introduction

In 1918, von Mises [1] first proposed a distribution on circular data and then Watson and Williams [2] studied inference problems for the von Mises distribution by constructing statistical methods for circular data. Studies involving (2-dimensional) circular data had been extended to (3-dimensional) spherical data and (high-dimensional) data on the unit hypersphere where they were also widely applied in geology, biology, meteorology, medicine and oceanography [3], [4], [5], [6], [7], [8], [9].

Clustering involves finding clusters within a data set that are characterized by the greatest similarity within the same cluster and the greatest dissimilarity between different clusters. It is a useful tool for data analysis that is a branch of statistical multivariate analysis and unsupervised learning for pattern recognition [10], [11]. From the statistical point of view, clustering methods can be divided into two categories: probability model-based approaches [12], [13], [14] and nonparametric approaches [15], [16]. For probability model-based approaches, which assume that a data set follows a mixture of probability distributions, and EM (expectation & maximization) algorithms [17], which allow the estimation of distribution parameters can be used for clustering in a data set. Nonparametric methods require an objective function of similarity or dissimilarity to partition clusters. In nonparametric approaches, an objective function of similarity or dissimilarity is generally required in which partition clustering methods are the most popular. Frequently used partition methods are k-means [18], [19], [20], fuzzy c-means (FCM) [21], [22], [23], mean shift [24], [25], [26] and possibilistic c-means [27], [28].

EM is a commonly used algorithm for clustering grouped data using a mixture model. It can be also applied to directional data. Banerjee et al. [6] proposed an EM algorithm using a mixture of von Mises-Fisher distributions, called it soft-moVMF. However, soft-moVMF [6] is sensitive to initial values. In the study by Banerjee et al. [6], no appropriate initialization technique was proposed and inappropriate initializations usually lead to bad clustering results for soft-moVMF. Soft-moVMF [6] does not work when there is an unknown number of clusters. It needs to assign a number, c, of clusters a priori. In general, the cluster number, c, is unknown. In this case, some validity functions, such as Akaike information criterion (AIC) [29], Bayesian information criterion (BIC) [30], [31], the gap statistic [32], [33] and fuzzy cluster validity [21], [34], can be used to find the number of clusters. However, they are supposed to be extra indices that are not embedded into the iterations of algorithms. There are many methods for determining the number of clusters, such as Peck et al. [35], Li et al. [36], Josse and Husson [37]. Although these algorithms can find a number of clusters, they are dependent on initialization and parameter selection. Recently, Yang et al. [38] proposed a robust EM algorithm for Gaussian mixture models that is robust to initialization and parameter selection and which automatically produces a suitable number of clusters. Since Yang et al. [38] proposed EM using Gaussian mixture models for data in Euclidean space, this study extends this method for clustering data on the unit hypersphere using mixtures of von Mises-Fisher distributions. This is termed a learning-based EM algorithm for data on the unit hypersphere. The proposed algorithm for this type of hyper-spherical data is free of initialization and automatically determines the number of clusters.

The remainder of this paper is organized as follows. Section 2 presents the proposed learning-based EM algorithm for data on the unit hypersphere. In Section 3, some experimental examples are used to compare the proposed learning-based EM algorithm with soft-moVMF and soft-moVMF + BIC. The results demonstrate the superiority and utility of the proposed method. Section 4 gives a real application of the learning-based EM algorithm to exoplanet data. Finally, conclusions are stated in Section 5.