Modified genetic algorithm-based clustering for probability density functions

This article modifies two internal validity measures and applies them to evaluate the quality of clustering for probability density functions (pdfs). Based on these measures, we propose a new modified genetic algorithm called GA-CDF to establish the suitable clusters for pdfs. The proposed algorithm is tested by four numerical examples including two synthetic data sets and two real data sets. These examples illustrate the superiority of proposed algorithm over some existing algorithms in evaluating the internal or external validity measures. It demonstrates the feasibility and applicability of the GA-CDF for practical problems in data mining.     

An automatic clustering algorithm for probability density functions

We propose an intuitive and computationally simple algorithm for clustering the probability density functions (pdfs). A data-driven learning mechanism is incorporated in the algorithm in order to determine the suitable widths of the clusters. The clustering results prove that the proposed algorithm is able to automatically group the pdfs and provide the optimal cluster number without any a priori information. The performance study also shows that the proposed algorithm is more efficient than existing ones. In addition, the clustering can serve as the intermediate compression tool in content-based multimedia retrieval that we apply the proposed algorithm to categorize a subset of COREL image database. And the clustering results indicate that the proposed algorithm performs well in colour image categorization.     

A Clustering Method for Categorical Ordinal Data

Often, categorical ordinal data are clustered using a well-defined similarity measure for this kind of data and then using a clustering algorithm not specifically developed for them. The aim of this article is to introduce a new clustering method suitably planned for ordinal data. Objects are grouped using a multinomial model, a cluster tree and a pruning strategy. Two types of pruning are analyzed through simulations. The proposed method allows to overcome two typical problems of cluster analysis: the choice of the number of groups and the scale invariance.     

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.     

Modeling probability density through ultraspherical polynomial transformations

We present a method for fitting parametric probability density models using an integrated square error criterion on a continuum of weighted Lebesgue spaces formed by ultraspherical polynomials. This approach is inherently suitable for creating mixture model representations of complex distributions and allows fully autonomous cluster analysis of high-dimensional datasets. The method is also suitable for extremely large sets, allowing post facto model selection and analysis even in the absence of the original data. Furthermore, the fitting procedure only requires the parametric model to be pointwise evaluable, making it trivial to fit user-defined models through a generic algorithm.     

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.     

Self-updating clustering algorithm for estimating the parameters in mixtures of von Mises distributions

The EM algorithm is the standard method for estimating the parameters in finite mixture models. Yang and Pan [25] proposed a generalized classification maximum likelihood procedure, called the fuzzy c-directions (FCD) clustering algorithm, for estimating the parameters in mixtures of von Mises distributions. Two main drawbacks of the EM algorithm are its slow convergence and the dependence of the solution on the initial value used. The choice of initial values is of great importance in the algorithm-based literature as it can heavily influence the speed of convergence of the algorithm and its ability to locate the global maximum. On the other hand, the algorithmic frameworks of EM and FCD are closely related. Therefore, the drawbacks of FCD are the same as those of the EM algorithm. To resolve these problems, this paper proposes another clustering algorithm, which can self-organize local optimal cluster numbers without using cluster validity functions. These numerical results clearly indicate that the proposed algorithm is superior in performance of EM and FCD algorithms. Finally, we apply the proposed algorithm to two real data sets.     

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.     

Using conditional independence for parsimonious model-based Gaussian clustering

In the framework of model-based cluster analysis, finite mixtures of Gaussian components represent an important class of statistical models widely employed for dealing with quantitative variables. Within this class, we propose novel models in which constraints on the component-specific variance matrices allow us to define Gaussian parsimonious clustering models. Specifically, the proposed models are obtained by assuming that the variables can be partitioned into groups resulting to be conditionally independent within components, thus producing component-specific variance matrices with a block diagonal structure. This approach allows us to extend the methods for model-based cluster analysis and to make them more flexible and versatile. In this paper, Gaussian mixture models are studied under the above mentioned assumption. Identifiability conditions are proved and the model parameters are estimated through the maximum likelihood method by using the Expectation-Maximization algorithm. The Bayesian information criterion is proposed for selecting the partition of the variables into conditionally independent groups. The consistency of the use of this criterion is proved under regularity conditions. In order to examine and compare models with different partitions of the set of variables a hierarchical algorithm is suggested. A wide class of parsimonious Gaussian models is also presented by parameterizing the component-variance matrices according to their spectral decomposition. The effectiveness and usefulness of the proposed methodology are illustrated with two examples based on real datasets.     

Efficient maximin distance designs for experiments in mixtures

In this paper, different dissimilarity measures are investigated to construct maximin designs for compositional data. Specifically, the effect of different dissimilarity measures on the maximin design criterion for two case studies is presented. Design evaluation criteria are proposed to distinguish between the maximin designs generated. An optimization algorithm is also presented. Divergence is found to be the best dissimilarity measure to use in combination with the maximin design criterion for creating space-filling designs for mixture variables.     

基于核密度估计的非线性时间序列聚类

本文研究的是时间序列的聚类问题。由于现实世界中时间序列多数是非线性的,而现有的时间序列聚类问题大都是基于线性时间序列模型进行聚类的,本文提出了可以用于非线性时间序列的聚类方法。以时间序列的二维核密度估计之间的相似性作为非线性时间序列的距离度量,该距离度量方式是一种非参数的距离度量方法,考虑到了时间序列自相关结构的差异,能够粗糙地识别时间序列形状和动态相关结构的相似性。与理论研究结果相一致,我们的模拟实验结果也验证了这种距离度量的有效性。     

Robust Detection of Multiple Outliers in Grouped Multivariate Data

Many methods have been developed for detecting multiple outliers in a single multivariate sample, but very few for the case where there may be groups in the data. We propose a method of simultaneously determining groups (as in cluster analysis) and detecting outliers, which are points that are distant from every group. Our method is an adaptation of the BACON algorithm proposed by Billor, Hadi and Velleman for the robust detection of multiple outliers in a single group of multivariate data. There are two versions of our method, depending on whether or not the groups can be assumed to have equal covariance matrices. The effectiveness of the method is illustrated by its application to two real data sets and further shown by a simulation study for different sample sizes and dimensions for 2 and 3 groups, with and without planted outliers in the data. When the number of groups is not known in advance, the algorithm could be used as a robust method of cluster analysis, by running it for various numbers of groups and choosing the best solution.     

Estimating standard error of intra-class correlation coefficients up to three level unbalanced nested clinical trials

ABSTRACT

Very often researchers plan a balanced design for cluster randomization clinical trials in conducting medical research, but unavoidable circumstances lead to unbalanced data. By adopting three or more levels of nested designs, they usually ignore the higher level of nesting and consider only two levels, this situation leads to underestimation of variance at higher levels. While calculating the sample size for three-level nested designs, in order to achieve desired power, intra-class correlation coefficients (ICCs) at individual level as well as higher levels need to be considered and must be provided along with respective standard errors. In the present paper, the standard errors of analysis of variance (ANOVA) estimates of ICCs for three-level unbalanced nested design are derived. To conquer the strong appeal of distributional assumptions, balanced design, equality of variances between clusters and large sample, general expressions for standard errors of ICCs which can be deployed in unbalanced cluster randomization trials are postulated. The expressions are evaluated on real data as well as highly unbalanced simulated data.     

Clustering of Variables Around Latent Components

Abstract

Clustering of variables around latent components is investigated as a means to organize multivariate data into meaningful structures. The coverage includes (i) the case where it is desirable to lump together correlated variables no matter whether the correlation coefficient is positive or negative; (ii) the case where negative correlation shows high disagreement among variables; (iii) an extension of the clustering techniques which makes it possible to explain the clustering of variables taking account of external data. The strategy basically consists in performing a hierarchical cluster analysis, followed by a partitioning algorithm. Both algorithms aim at maximizing the same criterion which reflects the extent to which variables in each cluster are related to the latent variable associated with this cluster. Illustrations are outlined using real data sets from sensory studies.     

Sample size determination for cluster randomization phase II trials of dichotomous data

One of the main goals for a phase II trial is to screen and select the best treatment to proceed onto further studies in a phase III trial. Under the flexible design proposed elsewhere, we discuss for cluster randomization trials sample size calculation with a given desired probability of correct selection to choose the best treatment when one treatment is better than all the others. We develop exact procedures for calculating the minimum required number of clusters with a given cluster size (or the minimum number of patients with a given number of repeated measurements) per treatment. An approximate sample size and the evaluation of its performance for two arms are also given. To help readers employ the results presented here, tables are provided to summarize the resulting minimum required sample sizes for cluster randomization trials with two arms and three arms in a variety of situations. Finally, to illustrate the sample size calculation procedures developed here, we use the data taken from a cluster randomization trial to study the association between the dietary sodium and the blood pressure.     

Model selection of generalized partially linear models with missing covariates

In this paper, a generalized partially linear model (GPLM) with missing covariates is studied and a Monte Carlo EM (MCEM) algorithm with penalized-spline (P-spline) technique is developed to estimate the regression coefficients and nonparametric function, respectively. As classical model selection procedures such as Akaike's information criterion become invalid for our considered models with incomplete data, some new model selection criterions for GPLMs with missing covariates are proposed under two different missingness mechanism, say, missing at random (MAR) and missing not at random (MNAR). The most attractive point of our method is that it is rather general and can be extended to various situations with missing observations based on EM algorithm, especially when no missing data involved, our new model selection criterions are reduced to classical AIC. Therefore, we can not only compare models with missing observations under MAR/MNAR settings, but also can compare missing data models with complete-data models simultaneously. Theoretical properties of the proposed estimator, including consistency of the model selection criterions are investigated. A simulation study and a real example are used to illustrate the proposed methodology.     

Stochastic correlation coefficient ensembles for variable selection

In this paper, we propose a novel Max-Relevance and Min-Common-Redundancy criterion for variable selection in linear models. Considering that the ensemble approach for variable selection has been proven to be quite effective in linear regression models, we construct a variable selection ensemble (VSE) by combining the presented stochastic correlation coefficient algorithm with a stochastic stepwise algorithm. We conduct extensive experimental comparison of our algorithm and other methods using two simulation studies and four real-life data sets. The results confirm that the proposed VSE leads to promising improvement on variable selection and regression accuracy.     

A Comparison of Hierarchical Methods for Clustering Functional Data

Functional data analysis (FDA)—the analysis of data that can be considered a set of observed continuous functions—is an increasingly common class of statistical analysis. One of the most widely used FDA methods is the cluster analysis of functional data; however, little work has been done to compare the performance of clustering methods on functional data. In this article, a simulation study compares the performance of four major hierarchical methods for clustering functional data. The simulated data varied in three ways: the nature of the signal functions (periodic, non periodic, or mixed), the amount of noise added to the signal functions, and the pattern of the true cluster sizes. The Rand index was used to compare the performance of each clustering method. As a secondary goal, clustering methods were also compared when the number of clusters has been misspecified. To illustrate the results, a real set of functional data was clustered where the true clustering structure is believed to be known. Comparing the clustering methods for the real data set confirmed the findings of the simulation. This study yields concrete suggestions to future researchers to determine the best method for clustering their functional data.     

Partially adaptive quantile estimators

This paper contrasts two approaches to estimating quantile regression models: traditional semi-parametric methods and partially adaptive estimators using flexible probability density functions (pdfs). While more general pdfs could have been used, the skewed Laplace was selected for pedagogical purposes. Monte Carlo simulations are used to compare the behavior of the semi-parametric and partially adaptive quantile estimators in the presence of possibly skewed and heteroskedastic data. Both approaches accommodate skewness and heteroskedasticity which are consistent with linear quantiles; however, the partially adaptive estimator considered allows for non linear quantiles and also provides simple tests for symmetry and heteroskedasticity. The methods are applied to the problem of estimating conditional quantile functions for wages corresponding to different levels of education.