On clustering shape data

ABSTRACT

Among the statistical methods to model stochastic behaviours of objects, clustering is a preliminary technique to recognize similar patterns within a group of observations in a data set. Various distances to measure differences among objects could be invoked to cluster data through numerous clustering methods. When variables in hand contain geometrical information of objects, such metrics should be adequately adapted. In fact, statistical methods for these typical data are endowed with a geometrical paradigm in a multivariate sense. In this paper, a procedure for clustering shape data is suggested employing appropriate metrics. Then, the best shape distance candidate as well as a suitable agglomerative method for clustering the simulated shape data are provided by considering cluster validation measures. The results are implemented in a real life application.     

K-medoids inverse regression

Abstract

K-means inverse regression was developed as an easy-to-use dimension reduction procedure for multivariate regression. This approach is similar to the original sliced inverse regression method, with the exception that the slices are explicitly produced by a K-means clustering of the response vectors. In this article, we propose K-medoids clustering as an alternative clustering approach for slicing and compare its performance to K-means in a simulation study. Although the two methods often produce comparable results, K-medoids tends to yield better performance in the presence of outliers. In addition to isolation of outliers, K-medoids clustering also has the advantage of accommodating a broader range of dissimilarity measures, which could prove useful in other graphical regression applications where slicing is required.     

Identifying Multiple Cluster Structures in a Data Matrix

Abstract

An aspect of cluster analysis which has been widely studied in recent years is the weighting and selection of variables. Procedures have been proposed which are able to identify the cluster structure present in a data matrix when that structure is confined to a subset of variables. Other methods assess the relative importance of each variable as revealed by a suitably chosen weight. But when a cluster structure is present in more than one subset of variables and is different from one subset to another, those solutions as well as standard clustering algorithms can lead to misleading results. Some very recent methodologies for finding consensus classifications of the same set of units can be useful also for the identification of cluster structures in a data matrix, but each one seems to be only partly satisfactory for the purpose at hand. Therefore a new more specific procedure is proposed and illustrated by analyzing two real data sets; its performances are evaluated by means of a simulation experiment.     

A new partitioning around medoids algorithm

Kaufman and Rousseeuw (1990) proposed a clustering algorithm Partitioning Around Medoids (PAM) which maps a distance matrix into a specified number of clusters. A particularly nice property is that PAM allows clustering with respect to any specified distance metric. In addition, the medoids are robust representations of the cluster centers, which is particularly important in the common context that many elements do not belong well to any cluster. Based on our experience in clustering gene expression data, we have noticed that PAM does have problems recognizing relatively small clusters in situations where good partitions around medoids clearly exist. In this paper, we propose to partition around medoids by maximizing a criteria "Average Silhouette" defined by Kaufman and Rousseeuw (1990). We also propose a fast-to-compute approximation of "Average Silhouette". We implement these two new partitioning around medoids algorithms and illustrate their performance relative to existing partitioning methods in simulations.     

Clustering and classification problems in genetics through U-statistics

ABSTRACT

Genetic data are frequently categorical and have complex dependence structures that are not always well understood. For this reason, clustering and classification based on genetic data, while highly relevant, are challenging statistical problems. Here we consider a versatile U-statistics-based approach for non-parametric clustering that allows for an unconventional way of solving these problems. In this paper we propose a statistical test to assess group homogeneity taking into account multiple testing issues and a clustering algorithm based on dissimilarities within and between groups that highly speeds up the homogeneity test. We also propose a test to verify classification significance of a sample in one of two groups. We present Monte Carlo simulations that evaluate size and power of the proposed tests under different scenarios. Finally, the methodology is applied to three different genetic data sets: global human genetic diversity, breast tumour gene expression and Dengue virus serotypes. These applications showcase this statistical framework's ability to answer diverse biological questions in the high dimension low sample size scenario while adapting to the specificities of the different datatypes.     

Composite likelihood for aggregate data from clustered multistate processes under intermittent observation

Abstract

Markov processes offer a useful basis for modeling the progression of organisms through successive stages of their life cycle. When organisms are examined intermittently in developmental studies, likelihoods can be constructed based on the resulting panel data in terms of transition probability functions. In some settings however, organisms cannot be tracked individually due to a difficulty in identifying distinct individuals, and in such cases aggregate counts of the number of organisms in different stages of development are recorded at successive time points. We consider the setting in which such aggregate counts are available for each of a number of tanks in a developmental study. We develop methods which accommodate clustering of the transition rates within tanks using a marginal modeling approach followed by robust variance estimation, and through use of a random effects model. Composite likelihood is proposed as a basis of inference in both settings. An extension which incorporates mortality is also discussed. The proposed methods are shown to perform well in empirical studies and are applied in an illustrative example on the growth of the Arabidopsis thaliana plant.     

Mixtures of general location model with factor analyzer covariance structure for clustering mixed type data

Cluster analysis is one of the most widely used method in statistical analyses, in which homogeneous subgroups are identified in a heterogeneous population. Due to the existence of the continuous and discrete mixed data in many applications, so far, some ordinary clustering methods such as, hierarchical methods, k-means and model-based methods have been extended for analysis of mixed data. However, in the available model-based clustering methods, by increasing the number of continuous variables, the number of parameters increases and identifying as well as fitting an appropriate model may be difficult. In this paper, to reduce the number of the parameters, for the model-based clustering mixed data of continuous (normal) and nominal data, a set of parsimonious models is introduced. Models in this set are extended, using the general location model approach, for modeling distribution of mixed variables and applying factor analyzer structure for covariance matrices. The ECM algorithm is used for estimating the parameters of these models. In order to show the performance of the proposed models for clustering, results from some simulation studies and analyzing two real data sets are presented.     

Cluster analysis of massive datasets in astronomy

Clusters of galaxies are a useful proxy to trace the distribution of mass in the universe. By measuring the mass of clusters of galaxies on different scales, one can follow the evolution of the mass distribution (Martínez and Saar, Statistics of the Galaxy Distribution, 2002). It can be shown that finding galaxy clusters is equivalent to finding density contour clusters (Hartigan, Clustering Algorithms, 1975): connected components of the level set S _c ≡{f>c} where f is a probability density function. Cuevas et al. (Can. J. Stat. 28, 367–382, 2000; Comput. Stat. Data Anal. 36, 441–459, 2001) proposed a nonparametric method for density contour clusters, attempting to find density contour clusters by the minimal spanning tree. While their algorithm is conceptually simple, it requires intensive computations for large datasets. We propose a more efficient clustering method based on their algorithm with the Fast Fourier Transform (FFT). The method is applied to a study of galaxy clustering on large astronomical sky survey data.     

Power and Sample Size for Fixed-Effects Inference in Reversible Linear Mixed Models

ABSTRACT

Despite the popularity of the general linear mixed model for data analysis, power and sample size methods and software are not generally available for commonly used test statistics and reference distributions. Statisticians resort to simulations with homegrown and uncertified programs or rough approximations which are misaligned with the data analysis. For a wide range of designs with longitudinal and clustering features, we provide accurate power and sample size approximations for inference about fixed effects in the linear models we call reversible. We show that under widely applicable conditions, the general linear mixed-model Wald test has noncentral distributions equivalent to well-studied multivariate tests. In turn, exact and approximate power and sample size results for the multivariate Hotelling–Lawley test provide exact and approximate power and sample size results for the mixed-model Wald test. The calculations are easily computed with a free, open-source product that requires only a web browser to use. Commercial software can be used for a smaller range of reversible models. Simple approximations allow accounting for modest amounts of missing data. A real-world example illustrates the methods. Sample size results are presented for a multicenter study on pregnancy. The proposed study, an extension of a funded project, has clustering within clinic. Exchangeability among the participants allows averaging across them to remove the clustering structure. The resulting simplified design is a single-level longitudinal study. Multivariate methods for power provide an approximate sample size. All proofs and inputs for the example are in the supplementary materials (available online).     

A local join counts methodology for spatial clustering in disease from relative risk models

ABSTRACT

This paper considers adaptation of hierarchical models for small area disease counts to detect disease clustering. A high risk area may be an outlier (in local terms) if surrounded by low risk areas, whereas a high risk cluster requires that both the focus area and surrounding areas demonstrate common elevated risk. A local join count method is suggested to detect local clustering of high disease risk in a single health outcome, and extends to assessing bivariate spatial clustering in relative risk. Applications include assessing spatial heterogeneity in effects of area predictors according to local clustering configuration, and gauging sensitivity of bivariate clustering to random effect assumptions.     

A comparative study of the K-means algorithm and the normal mixture model for clustering: Bivariate homoscedastic case

The K-means algorithm and the normal mixture model method are two common clustering methods. The K-means algorithm is a popular heuristic approach which gives reasonable clustering results if the component clusters are ball-shaped. Currently, there are no analytical results for this algorithm if the component distributions deviate from the ball-shape. This paper analytically studies how the K-means algorithm changes its classification rule as the normal component distributions become more elongated under the homoscedastic assumption and compares this rule with that of the Bayes rule from the mixture model method. We show that the classification rules of both methods are linear, but the slopes of the two classification lines change in the opposite direction as the component distributions become more elongated. The classification performance of the K-means algorithm is then compared to that of the mixture model method via simulation. The comparison, which is limited to two clusters, shows that the K-means algorithm provides poor classification performances consistently as the component distributions become more elongated while the mixture model method can potentially, but not necessarily, take advantage of this change and provide a much better classification performance.     

Unsupervised learning of regression mixture models with unknown number of components

ABSTRACT

We propose a new unsupervised learning algorithm to fit regression mixture models with unknown number of components. The developed approach consists in a penalized maximum likelihood estimation carried out by a robust expectation–maximization (EM)-like algorithm. We derive it for polynomial, spline, and B-spline regression mixtures. The proposed learning approach is unsupervised: (i) it simultaneously infers the model parameters and the optimal number of the regression mixture components from the data as the learning proceeds, rather than in a two-fold scheme as in standard model-based clustering using afterward model selection criteria, and (ii) it does not require accurate initialization unlike the standard EM for regression mixtures. The developed approach is applied to curve clustering problems. Numerical experiments on simulated and real data show that the proposed algorithm performs well and provides accurate clustering results, and confirm its benefit for practical applications.     

Characterizing and Approximating Infinite Scale Mixtures of Normals

ABSTRACT

The purpose of this study is to approximate and identify infinite scale mixtures of normals, SMN. A new method for approximating any infinite SMN with a known mixing measure by a finite SMN is presented. In the new method, the modulus of continuity of the normal family as a function of the scale is used to discretize the mixing measure. This method will be used to approximate univariate and multivariate SMN with mean 0. In the multivariate case, two different methods are used to approximate the infinite SMN. Several results related to SMN are proved and other known ones are presented. For example, SMN are characterized by their corresponding Laplace transforms.     

Strong Consistency of Reduced K‐means Clustering

Reduced k‐means clustering is a method for clustering objects in a low‐dimensional subspace. The advantage of this method is that both clustering of objects and low‐dimensional subspace reflecting the cluster structure are simultaneously obtained. In this paper, the relationship between conventional k‐means clustering and reduced k‐means clustering is discussed. Conditions ensuring almost sure convergence of the estimator of reduced k‐means clustering as unboundedly increasing sample size have been presented. The results for a more general model considering conventional k‐means clustering and reduced k‐means clustering are provided in this paper. Moreover, a consistent selection of the numbers of clusters and dimensions is described.     

Comparing Two-Stage Segmentation Methods for Choice Data with a One-Stage Latent Class Choice Analysis

Market segmentation is a key concept in marketing research. Identification of consumer segments helps in setting up and improving a marketing strategy. Hence, the need is to improve existing methods and to develop new segmentation methods. We introduce two new consumer indicators that can be used as segmentation basis in two-stage methods, the forces and the dfbetas. Both bases express a subject’s effect on the aggregate estimates of the parameters in a conditional logit model. Further, individual-level estimates, obtained by either estimating a conditional logit model for each individual separately with maximum likelihood or by hierarchical Bayes (HB) estimation of a mixed logit choice model, and the respondents’ raw choices are also used as segmentation basis. In the second stage of the methods the bases are classified into segments with cluster analysis or latent class models. All methods are applied to choice data because of the increasing popularity of choice experiments to analyze choice behavior. To verify whether two-stage segmentation methods can compete with a one-stage approach, a latent class choice model is estimated as well. A simulation study reveals the superiority of the two-stage method that clusters the HB estimates and the one-stage latent class choice model. Additionally, very good results are obtained for two-stage latent class cluster analysis of the choices as well as for the two-stage methods clustering the forces, the dfbetas and the choices.     

A Mann–Whitney scan statistic for continuous data

ABSTRACT

A new method is proposed for identifying clusters in continuous data indexed by time or by space. The scan statistic we introduce is derived from the well-known Mann–Whitney statistic. It is completely non parametric as it relies only on the ranks of the marks. This scan test seems to be very powerful against any clustering alternative. These results have applications in various fields, such as the study of climate data or socioeconomic data.     

Time-varying clustering of multivariate longitudinal observations

Abstract

We propose a statistical method for clustering multivariate longitudinal data into homogeneous groups. This method relies on a time-varying extension of the classical K-means algorithm, where a multivariate vector autoregressive model is additionally assumed for modeling the evolution of clusters' centroids over time. Model inference is based on a least-squares method and on a coordinate descent algorithm. To illustrate our work, we consider a longitudinal dataset on human development. Three variables are modeled, namely life expectancy, education and gross domestic product.     

Performance of Confidence Intervals in Regression Models with Unbalanced One-Fold Nested Error Structures

Abstract

In this article we consider the problem of constructing confidence intervals for a linear regression model with unbalanced nested error structure. A popular approach is the likelihood-based method employed by PROC MIXED of SAS. In this article, we examine the ability of MIXED to produce confidence intervals that maintain the stated confidence coefficient. Our results suggest that intervals for the regression coefficients work well, but intervals for the variance component associated with the primary level cannot be recommended. Accordingly, we propose alternative methods for constructing confidence intervals on the primary level variance component. Computer simulation is used to compare the proposed methods. A numerical example and SAS code are provided to demonstrate the methods.     

Clustering large number of extragalactic spectra of galaxies and quasars through canopies

Abstract

Cluster analysis is the distribution of objects into different groups or more precisely the partitioning of a data set into subsets (clusters) so that the data in subsets share some common trait according to some distance measure. Unlike classification, in clustering one has to first decide the optimum number of clusters and then assign the objects into different clusters. Solution of such problems for a large number of high dimensional data points is quite complicated and most of the existing algorithms will not perform properly. In the present work a new clustering technique applicable to large data set has been used to cluster the spectra of 702248 galaxies and quasars having 1,540 points in wavelength range imposed by the instrument. The proposed technique has successfully discovered five clusters from this 702,248X1,540 data matrix.     

Diagnostics for repeated measurements in generalized linear mixed effects models

ABSTRACT

As there is an extensive body of research on diagnostics in regression models, various outlier detection methods have been developed. These methods have been extended to mixed effects models and generalized linear models, but there exist intrinsic drawbacks and limitations. This paper presents two-dimensional plots to identify discordant subjects and observations in generalized linear mixed effects models, displaying discordance in two directions. The sTudentized Residual Sum of Squares is not an extension of any regression tools but a new approach designed to efficiently reflect the characteristics of repeated measures. And this noteworthy clustering of outliers is identified in the plot. Applications to real-life examples are presented to illustrate the favorable/beneficial performance of the new tool.