A Pitman measure of similarity in k-means for clustering heavy-tailed data

One of the most popular methods and algorithms to partition data to k clusters is k-means clustering algorithm. Since this method relies on some basic conditions such as, the existence of mean and finite variance, it is unsuitable for data that their variances are infinite such as data with heavy tailed distribution. Pitman Measure of Closeness (PMC) is a criterion to show how much an estimator is close to its parameter with respect to another estimator. In this article using PMC, based on k-means clustering, a new distance and clustering algorithm is developed for heavy tailed data.     

Simple Measures of Individual Cluster-Membership Certainty for Hard Partitional Clustering

We propose two probability-like measures of individual cluster-membership certainty that can be applied to a hard partition of the sample such as that obtained from the partitioning around medoids (PAM) algorithm, hierarchical clustering or k-means clustering. One measure extends the individual silhouette widths and the other is obtained directly from the pairwise dissimilarities in the sample. Unlike the classic silhouette, however, the measures behave like probabilities and can be used to investigate an individual’s tendency to belong to a cluster. We also suggest two possible ways to evaluate the hard partition using these measures. We evaluate the performance of both measures in individuals with ambiguous cluster membership, using simulated binary datasets that have been partitioned by the PAM algorithm or continuous datasets that have been partitioned by hierarchical clustering and k-means clustering. For comparison, we also present results from soft-clustering algorithms such as soft analysis clustering (FANNY) and two model-based clustering methods. Our proposed measures perform comparably to the posterior probability estimators from either FANNY or the model-based clustering methods. We also illustrate the proposed measures by applying them to Fisher’s classic dataset on irises.     

A tutorial on spectral clustering

In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.     

A novel fast heuristic to handle large-scale shape clustering

Clustering algorithms like types of k-means are fast, but they are inefficient for shape clustering. There are some algorithms, which are effective, but their time complexities are too high. This paper proposes a novel heuristic to solve large-scale shape clustering. The proposed method is effective and it solves large-scale clustering problems in fraction of a second.     

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.     

Comparison of various machine learning algorithms for estimating generalized propensity score

In this paper, we conducted a simulation study to evaluate the performance of four algorithms: multinomial logistic regression (MLR), bagging (BAG), random forest (RF), and gradient boosting (GB), for estimating generalized propensity score (GPS). Similar to the propensity score (PS), the ultimate goal of using GPS is to estimate unbiased average treatment effects (ATEs) in observational studies. We used the GPS estimates computed from these four algorithms with the generalized doubly robust (GDR) estimator to estimate ATEs in observational studies. We evaluated these ATE estimates in terms of bias and mean squared error (MSE). Simulation results show that overall, the GB algorithm produced the best ATE estimates based on these evaluation criteria. Thus, we recommend using the GB algorithm for estimating GPS in practice.     

Fuzzy clustering algorithm for latent class model

The expectation maximization (EM) algorithm is a widely used parameter approach for estimating the parameters of multivariate multinomial mixtures in a latent class model. However, this approach has unsatisfactory computing efficiency. This study proposes a fuzzy clustering algorithm (FCA) based on both the maximum penalized likelihood (MPL) for the latent class model and the modified penalty fuzzy c-means (PFCM) for normal mixtures. Numerical examples confirm that the FCA-MPL algorithm is more efficient (that is, requires fewer iterations) and more computationally effective (measured by the approximate relative ratio of accurate classification) than the EM algorithm.     

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.     

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.     

A new approach: Interrelated two-way clustering of gene expression data

The paper presents a new approach to interrelated two-way clustering of gene expression data. Clustering of genes has been effected using entropy and a correlation measure, whereas the samples have been clustered using the fuzzy C-means. The efficiency of this approach has been tested on two well known data sets: the colon cancer data set and the leukemia data set. Using this approach, we were able to identify the important co-regulated genes and group the samples efficiently at the same time.     

Automatic clustering algorithm for fuzzy data

Coppi et al. [7 R. Coppi, P. D'Urso, and P. Giordani, Fuzzy and possibilistic clustering for fuzzy data, Comput. Stat. Data Anal. 56 (2012), pp. 915–927. doi: 10.1016/j.csda.2010.09.013[Crossref], [Web of Science ®] , [Google Scholar]] applied Yang and Wu's [20 M.-S. Yang and K.-L. Wu, Unsupervised possibilistic clustering, Pattern Recognit. 30 (2006), pp. 5–21. doi: 10.1016/j.patcog.2005.07.005[Crossref], [Web of Science ®] , [Google Scholar]] idea to propose a possibilistic k-means (PkM) clustering algorithm for LR-type fuzzy numbers. The memberships in the objective function of PkM no longer need to satisfy the constraint in fuzzy k-means that of a data point across classes sum to one. However, the clustering performance of PkM depends on the initializations and weighting exponent. In this paper, we propose a robust clustering method based on a self-updating procedure. The proposed algorithm not only solves the initialization problems but also obtains a good clustering result. Several numerical examples also demonstrate the effectiveness and accuracy of the proposed clustering method, especially the robustness to initial values and noise. Finally, three real fuzzy data sets are used to illustrate the superiority of this proposed algorithm.     

Projection-based curve clustering

This paper focuses on unsupervised curve classification in the context of nuclear industry. At the Commissariat à l'Energie Atomique (CEA), Cadarache (France), the thermal-hydraulic computer code CATHARE is used to study the reliability of reactor vessels. The code inputs are physical parameters and the outputs are time evolution curves of a few other physical quantities. As the CATHARE code is quite complex and CPU time-consuming, it has to be approximated by a regression model. This regression process involves a clustering step. In the present paper, the CATHARE output curves are clustered using a k-means scheme, with a projection onto a lower dimensional space. We study the properties of the empirically optimal cluster centres found by the clustering method based on projections, compared with the ‘true’ ones. The choice of the projection basis is discussed, and an algorithm is implemented to select the best projection basis among a library of orthonormal bases. The approach is illustrated on a simulated example and then applied to the industrial problem.     

k-POD: A Method for k-Means Clustering of Missing Data

The k-means algorithm is often used in clustering applications but its usage requires a complete data matrix. Missing data, however, are common in many applications. Mainstream approaches to clustering missing data reduce the missing data problem to a complete data formulation through either deletion or imputation but these solutions may incur significant costs. Our k-POD method presents a simple extension of k-means clustering for missing data that works even when the missingness mechanism is unknown, when external information is unavailable, and when there is significant missingness in the data.

[Received November 2014. Revised August 2015.]     

Weighting variables in K-means clustering

The aim of this study is to assign weights w ₁, …, w _m to m clustering variables Z ₁, …, Z _m , so that k groups were uncovered to reveal more meaningful within-group coherence. We propose a new criterion to be minimized, which is the sum of the weighted within-cluster sums of squares and the penalty for the heterogeneity in variable weights w ₁, …, w _m . We will present the computing algorithm for such k-means clustering, a working procedure to determine a suitable value of penalty constant and numerical examples, among which one is simulated and the other two are real.     

Impact of Contamination on Training and Test Error Rates in Statistical Clustering

The k-means algorithm is one of the most common non hierarchical methods of clustering. It aims to construct clusters in order to minimize the within cluster sum of squared distances. However, as most estimators defined in terms of objective functions depending on global sums of squares, the k-means procedure is not robust with respect to atypical observations in the data. Alternative techniques have thus been introduced in the literature, e.g., the k-medoids method. The k-means and k-medoids methodologies are particular cases of the generalized k-means procedure. In this article, focus is on the error rate these clustering procedures achieve when one expects the data to be distributed according to a mixture distribution. Two different definitions of the error rate are under consideration, depending on the data at hand. It is shown that contamination may make one of these two error rates decrease even under optimal models. The consequence of this will be emphasized with the comparison of influence functions and breakdown points of these error rates.     

Clustering of Variables Based on Watson Distribution on Hypersphere: A Comparison of Algorithms

We consider n individuals described by p variables, represented by points of the surface of unit hypersphere. We suppose that the individuals are fixed and the set of variables comes from a mixture of bipolar Watson distributions. For the mixture identification, we use EM and dynamic clusters algorithms, which enable us to obtain a partition of the set of variables into clusters of variables.

Our aim is to evaluate the clusters obtained in these algorithms, using measures of within-groups variability and between-groups variability and compare these clusters with those obtained in other clustering approaches, by analyzing simulated and real data.     

Bayesian nonparametric clustering for large data sets

We propose two nonparametric Bayesian methods to cluster big data and apply them to cluster genes by patterns of gene–gene interaction. Both approaches define model-based clustering with nonparametric Bayesian priors and include an implementation that remains feasible for big data. The first method is based on a predictive recursion which requires a single cycle (or few cycles) of simple deterministic calculations for each observation under study. The second scheme is an exact method that divides the data into smaller subsamples and involves local partitions that can be determined in parallel. In a second step, the method requires only the sufficient statistics of each of these local clusters to derive global clusters. Under simulated and benchmark data sets the proposed methods compare favorably with other clustering algorithms, including k-means, DP-means, DBSCAN, SUGS, streaming variational Bayes and an EM algorithm. We apply the proposed approaches to cluster a large data set of gene–gene interactions extracted from the online search tool “Zodiac.”

     

k-Means Algorithm in Statistical Shape Analysis

In this work it is shown how the k-means method for clustering objects can be applied in the context of statistical shape analysis. Because the choice of the suitable distance measure is a key issue for shape analysis, the Hartigan and Wong k-means algorithm is adapted for this situation. Simulations on controlled artificial data sets demonstrate that distances on the pre-shape spaces are more appropriate than the Euclidean distance on the tangent space. Finally, results are presented of an application to a real problem of oceanography, which in fact motivated the current work.     

On the detection of transitive clusters in undirected networks

A network cluster is defined as a set of nodes with ‘strong’ within group ties and ‘weak’ between group ties. Most clustering methods focus on finding groups of ‘densely connected’ nodes, where the dyad (or tie between two nodes) serves as the building block for forming clusters. However, since the unweighted dyad cannot distinguish strong relationships from weak ones, it then seems reasonable to consider an alternative building block, i.e. one involving more than two nodes. In the simplest case, one can consider the triad (or three nodes), where the fully connected triad represents the basic unit of transitivity in an undirected network. In this effort we propose a clustering framework for finding highly transitive subgraphs in an undirected/unweighted network, where the fully connected triad (or triangle configuration) is used as the building block for forming clusters. We apply our methodology to four real networks with encouraging results. Monte Carlo simulation results suggest that, on average, the proposed method yields good clustering performance on synthetic benchmark graphs, relative to other popular methods.     

Sparse cluster analysis of large-scale discrete variables with application to single nucleotide polymorphism data

Currently, extreme large-scale genetic data present significant challenges for cluster analysis. Most of the existing clustering methods are typically built on the Euclidean distance and geared toward analyzing continuous response. They work well for clustering, e.g. microarray gene expression data, but often perform poorly for clustering, e.g. large-scale single nucleotide polymorphism (SNP) data. In this paper, we study the penalized latent class model for clustering extremely large-scale discrete data. The penalized latent class model takes into account the discrete nature of the response using appropriate generalized linear models and adopts the lasso penalized likelihood approach for simultaneous model estimation and selection of important covariates. We develop very efficient numerical algorithms for model estimation based on the iterative coordinate descent approach and further develop the expectation–maximization algorithm to incorporate and model missing values. We use simulation studies and applications to the international HapMap SNP data to illustrate the competitive performance of the penalized latent class model.