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.     

Cluster analysis using different correlation coefficients

Partitioning objects into closely related groups that have different states allows to understand the underlying structure in the data set treated. Different kinds of similarity measure with clustering algorithms are commonly used to find an optimal clustering or closely akin to original clustering. Using shrinkage-based and rank-based correlation coefficients, which are known to be robust, the recovery level of six chosen clustering algorithms is evaluated using Rand’s C values. The recovery levels using weighted likelihood estimate of correlation coefficient are obtained and compared to the results from using those correlation coefficients in applying agglomerative clustering algorithms. This work was supported by RIC(R) grants from Traditional and Bio-Medical Research Center, Daejeon University (RRC04713, 2005) by ITEP in Republic of Korea.     

Unsupervised Curve Clustering using B-Splines

Data in many different fields come to practitioners through a process naturally described as functional. Although data are gathered as finite vector and may contain measurement errors, the functional form have to be taken into account. We propose a clustering procedure of such data emphasizing the functional nature of the objects. The new clustering method consists of two stages: fitting the functional data by B‐splines and partitioning the estimated model coefficients using a k‐means algorithm. Strong consistency of the clustering method is proved and a real‐world example from food industry is given.     

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.     

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.     

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.     

Clustering confidence sets

In this article, we present a novel approach to clustering finite or infinite dimensional objects observed with different uncertainty levels. The novelty lies in using confidence sets rather than point estimates to obtain cluster membership and the number of clusters based on the distance between the confidence set estimates. The minimal and maximal distances between the confidence set estimates provide confidence intervals for the true distances between objects. The upper bounds of these confidence intervals can be used to minimize the within clustering variability and the lower bounds can be used to maximize the between clustering variability. We assign objects to the same cluster based on a min–max criterion and we separate clusters based on a max–min criterion. We illustrate our technique by clustering a large number of curves and evaluate our clustering procedure with a synthetic example and with a specific application.     

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.     

A sequential clustering algorithm with applications to gene expression data

Clustering algorithms are used in the analysis of gene expression data to identify groups of genes with similar expression patterns. These algorithms group genes with respect to a predefined dissimilarity measure without using any prior classification of the data. Most of the clustering algorithms require the number of clusters as input, and all the objects in the dataset are usually assigned to one of the clusters. We propose a clustering algorithm that finds clusters sequentially, and allows for sporadic objects, so there are objects that are not assigned to any cluster. The proposed sequential clustering algorithm has two steps. First it finds candidates for centers of clusters. Multiple candidates are used to make the search for clusters more efficient. Secondly, it conducts a local search around the candidate centers to find the set of objects that defines a cluster. The candidate clusters are compared using a predefined score, the best cluster is removed from data, and the procedure is repeated. We investigate the performance of this algorithm using simulated data and we apply this method to analyze gene expression profiles in a study on the plasticity of the dendritic cells.     

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.     

Selecting the precision parameter prior in Dirichlet process mixture models

We consider Dirichlet process mixture models in which the observed clusters in any particular dataset are not viewed as belonging to a finite set of possible clusters but rather as representatives of a latent structure in which objects belong to one of a potentially infinite number of clusters. As more information is revealed the number of inferred clusters is allowed to grow. The precision parameter of the Dirichlet process is a crucial parameter that controls the number of clusters. We develop a framework for the specification of the hyperparameters associated with the prior for the precision parameter that can be used both in the presence or absence of subjective prior information about the level of clustering. Our approach is illustrated in an analysis of clustering brands at the magazine Which?. The results are compared with the approach of Dorazio (2009) via a simulation study.     

Clustering objects on subsets of attributes (with discussion)

Summary.  A new procedure is proposed for clustering attribute value data. When used in conjunction with conventional distance-based clustering algorithms this procedure encourages those algorithms to detect automatically subgroups of objects that preferentially cluster on subsets of the attribute variables rather than on all of them simultaneously. The relevant attribute subsets for each individual cluster can be different and partially (or completely) overlap with those of other clusters. Enhancements for increasing sensitivity for detecting especially low cardinality groups clustering on a small subset of variables are discussed. Applications in different domains, including gene expression arrays, are presented.     

Model‐based linear clustering

The authors propose a profile likelihood approach to linear clustering which explores potential linear clusters in a data set. For each linear cluster, an errors‐in‐variables model is assumed. The optimization of the derived profile likelihood can be achieved by an EM algorithm. Its asymptotic properties and its relationships with several existing clustering methods are discussed. Methods to determine the number of components in a data set are adapted to this linear clustering setting. Several simulated and real data sets are analyzed for comparison and illustration purposes. The Canadian Journal of Statistics 38: 716–737; 2010 © 2010 Statistical Society of Canada     

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.]     

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.     

Distribution-free subset selection for incompletely ranked data

In market research and some other areas, it is common that a sample of n judges (consumers, evaluators, etc.) are asked to independently rank a series of k objects or candidates. It is usually difficult to obtain the judges' full cooperation to completely rank all k objects. A practical way to overcome this difficulty is to give each judge the freedom to choose the number of top candidates he is willing to rank. A frequently encountered question in this type of survey is how to select the best object or candidate from the incompletely ranked data. This paper proposes a subset selection procedure which constructs a random subset of all the k objects involved in the survey such that the best object is included in the subset with a prespecified confidence. It is shown that the proposed subset selection procedure is distribution-free over a very broad class of underlying distributions. An example from a market research study is used to illustrate the proposed procedure.     

Flexible modelling of simultaneously interval censored and truncated time‐to‐event data

This paper deals with the analysis of data from a HET‐CAM^VT experiment. From a statistical perspective, such data yield many challenges. First of all, the data are typically time‐to‐event like data, which are at the same time interval censored and right truncated. In addition, one has to cope with overdispersion as well as clustering. Traditional analysis approaches ignore overdispersion and clustering and summarize the data into a continuous score that can be analysed using simple linear models. In this paper, a novel combined frailty model is developed that simultaneously captures all of the aforementioned statistical challenges posed by the data. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

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 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.