Fuzzy Clusterwise Research Articles

GscaLCA in R: Fitting Fuzzy Clustering Analysis Incorporated with Generalized Structured Component Analysis

Clustering analysis identifying unknown heterogenous subgroups of a population (or a sample) has become increasingly popular along with the popularity of machine learning techniques. Although there are many software packages running clustering analysis, there is a lack of packages conducting clustering analysis within a structural equation modeling framework. The package, gscaLCA which is implemented in the R statistical computing environment, was developed for conducting clustering analysis and has been extended to a latent variable modeling. More specifically, by applying both fuzzy clustering (FC) algorithm and generalized structured component analysis (GSCA), the package gscaLCA computes membership prevalence and item response probabilities as posterior probabilities, which is applicable in mixture modeling such as latent class analysis in statistics. As a hybrid model between data clustering in classifications and model-based mixture modeling approach, fuzzy clusterwise GSCA, denoted as gscaLCA, encompasses many advantages from both methods: (1) soft partitioning from FC and (2) efficiency in estimating model parameters with bootstrap method via resolution of global optimization problem from GSCA. The main function, gscaLCA, works for both binary and ordered categorical variables. In addition, gscaLCA can be used for latent class regression as well. Visualization of profiles of latent classes based on the posterior probabilities is also available in the package gscaLCA. This paper contributes to providing a methodological tool, gscaLCA that applied researchers such as social scientists and medical researchers can apply clustering analysis in their research.

Combining hedonic information and CATA description for consumer segmentation

Check-all-that-apply (CATA) has become a popular method for obtaining a consumer-based sensory characterization. In most case studies, consumers are also asked to evaluate the set of products according to a liking scale with the aim to identify the key sensory attributes associated with the most liked, or disliked, products. The common approach consists, first, in the identification of consumer segments based on the preference profiles. Thereafter, the analysis of the CATA responses is performed within each segment. Our purpose herein is to investigate different ways to simultaneously identify clusters of preference profiles while taking into account the CATA attributes. These approaches are derived from strategies already proposed by the different co-authors, namely: Fuzzy Clusterwise Regression (FCR), Clustering around Latent Variables (CLV) approach with external data, CLUSCATA-liking and CLV3W. The first two approaches involve the aggregation of the individual CATA data into a contingency table, while the last two ones deal with the combination of liking and CATA data at the individual level. These four strategies are illustrated on the basis of a real case study. Results are compared with respect to cluster stability together with interpretability of liking profiles within each segment. The stability of the results, assessed by bootstrapping, differed according to the strategy used. Moreover, working at the individual level or with combined data lead to a somewhat different segmentation of the panel of consumers.

Application of the fuzzy clusterwise generalized structured component method to evaluate implementation of national education standard in Indonesia

Results of school accreditation and national examinations are two indicators that are often used to describe the achievement of quality in education in Indonesia. ‘Accreditation’ reflects the fulfillment of 8 national education standards (NES), while the national exam (NE, or UN in Bahasa) for students describes academic performance. Eight NES and academic performance are latent variables. The relationship between the two variables and the validity of its indicators can be evaluated by several methods. Path analysis with latent variables can be obtained through general structured component analysis (GSCA) with the assumption of homogeneity of variance. Since the data are not homogeneous, this study aims to apply the fuzzy clusterwise generalized structured component analysis (FCGSCA) to evaluate the relationship between the NES and the UN, and the validity of the indicators. The results showed that there were two school clusters in Indonesia. The evaluation of the measurement model indicated that some indicators of the accreditation instrument were not valid, i.e., 6 indicators in cluster 1 and 15 indicators in cluster 2. The structural model evaluation of the two clusters indicated that standard of process to the UN was not significant. Based on the overall goodness of the fit model, the total diversities of all variables that could be explained were 61.60% in cluster 1 and 59.90% in cluster 2.

Latent Class Regression Utilizing Fuzzy Clusterwise Generalized Structured Component Analysis

Latent class analysis (LCA) has been applied in many research areas to disentangle the heterogeneity of a population. Despite its popularity, its estimation method is limited to maximum likelihood estimation (MLE), which requires large samples to satisfy both the multivariate normality assumption and local independence assumption. Although many suggestions regarding adequate sample sizes were proposed, researchers continue to apply LCA with relatively smaller samples. When covariates are involved, the estimation issue is encountered more. In this study, we suggest a different estimating approach for LCA with covariates, also known as latent class regression (LCR), using a fuzzy clustering method and generalized structured component analysis (GSCA). This new approach is free from the distributional assumption and stable in estimating parameters. Parallel to the three-step approach used in the MLE-based LCA, we extend an algorithm of fuzzy clusterwise GSCA into LCR. This proposed algorithm has been demonstrated with an empirical data with both categorical and continuous covariates. Because the proposed algorithm can be used for a relatively small sample in LCR without requiring a multivariate normality assumption, the new algorithm is more applicable to social, behavioral, and health sciences.

Efficiency of Cluster Validity Indexes in Fuzzy Clusterwise Generalized Structured Component Analysis

Fuzzy clustering has been broadly applied to classify data into K clusters by assigning membership probabilities of each data point close to K centroids. Such a function has been applied into characterizing the clusters associated with a statistical model such as structural equation modeling. The characteristics identified by the statistical model further define the clusters as heterogeneous groups selected from a population. Recently, such statistical model has been formulated as fuzzy clusterwise generalized structured component analysis (fuzzy clusterwise GSCA). The same as in fuzzy clustering, the clusters are enumerated to infer the population and its parameters within the fuzzy clusterwise GSCA. However, the identification of clusters in fuzzy clustering is a difficult task because of the data-dependence of classification indexes, which is known as a cluster validity problem. We examined the cluster validity problem within the fuzzy clusterwise GSCA framework and proposed a new criterion for selecting the most optimal number of clusters using both fit indexes of the GSCA and the fuzzy validity indexes in fuzzy clustering. The criterion, named the FIT-FHV method combining a fit index, FIT, from GSCA and a cluster validation measure, FHV, from fuzzy clustering, performed better than any other indices used in fuzzy clusterwise GSCA.

Fuzzy Clusterwise Functional Extended Redundancy Analysis

Functional data refer to data that are assumed to be generated from an underlying smooth function varying over a continuum such as time or space. Functional linear models (FLMs) and functional extended redundancy analysis (FERA) are major regression analysis methods for investigating directional associations between predictor and dependent variables that can be functional In practice, functional data may often arise from heterogeneous subgroups of the population, which involve distinctive directional relationships between predictor and dependent variables. When such cluster-level heterogeneity is present, ignoring this heterogeneity would likely lead to biased statistical inferences. FLMs have been extended to capture cluster-level heterogeneity. Conversely, there has been no attempt to take into account cluster-level heterogeneity in FERA. In this paper, we propose to extend FERA to accommodate cluster-level heterogeneity by combining the method with fuzzy clusterwise regression (FCR) into a unified framework. The proposed method, called fuzzy clusterwise functional extended redundancy analysis (FCFERA), aims to estimate fuzzy memberships of individuals and clusterwise regression coefficient functions at the same time. A penalized least squares criterion is minimized to estimate these parameters by adopting an alternating least squares algorithm in combination with basis function expansions. We conduct simulation studies to investigate the performance of the proposed method. We also apply this method to real data to demonstrate its empirical usefulness.

Functional fuzzy clusterwise regression analysis

We propose a functional extension of fuzzy clusterwise regression, which estimates fuzzy memberships of clusters and regression coefficient functions for each cluster simultaneously. The proposed method permits dependent and/or predictor variables to be functional, varying over time, space, and other continua. The fuzzy memberships and clusterwise regression coefficient functions are estimated by minimizing an objective function that adopts a basis function expansion approach to approximating functional data. An alternating least squares algorithm is developed to minimize the objective function. We conduct simulation studies to demonstrate the superior performance of the proposed method compared to its non-functional counterpart and to examine the performance of various cluster validity measures for selecting the optimal number of clusters. We apply the proposed method to real datasets to illustrate the empirical usefulness of the proposed method.

A class of fuzzy clusterwise regression models

In this paper we introduce a class of fuzzy clusterwise regression models with LR fuzzy response variable and numeric explanatory variables, which embodies fuzzy clustering, into a fuzzy regression framework. The model bypasses the heterogeneity problem that could arise in fuzzy regression by subdividing the dataset into homogeneous clusters and performing separate fuzzy regression on each cluster. The integration of the clustering model into the regression framework allows us to simultaneously estimate the regression parameters and the membership degree of each observation to each cluster by optimizing a single objective function. The class of models proposed here includes, as special cases, the fuzzy clusterwise linear regression model and the fuzzy clusterwise polynomial regression model. We also introduce a set of goodness of fit indices to evaluate the fit of the regression model within each cluster as well as in the whole dataset. Finally, we consider some cluster validity criteria that are useful in identifying the “optimal” number of clusters. Several applications are provided in order to illustrate the approach.

Regularized fuzzy clusterwise ridge regression

Fuzzy clusterwise regression has been a useful method for investigating cluster-level heterogeneity of observations based on linear regression. This method integrates fuzzy clustering and ordinary least-squares regression, thereby enabling to estimate regression coefficients for each cluster and fuzzy cluster memberships of observations simultaneously. In practice, however, fuzzy clusterwise regression may suffer from multicollinearity as it builds on ordinary least-squares regression. To deal with this problem in fuzzy clusterwise regression, a new method, called regularized fuzzy clusterwise ridge regression, is proposed that combines ridge regression with regularized fuzzy clustering in a unified framework. In the proposed method, ridge regression is adopted to estimate clusterwise regression coefficients while handling potential multicollinearity among predictor variables. In addition, regularized fuzzy clustering based on maximizing entropy is utilized to systematically determine an optimal degree of fuzziness in memberships. A simulation study is conducted to evaluate parameter recovery of the proposed method as compared to the extant non-regularized counterpart. The usefulness of the proposed method is illustrated by an application concerning the relationship among the characteristics of used cars.

Fuzzy clusterwise linear regression analysis with symmetrical fuzzy output variable

The traditional regression analysis is usually applied to homogeneous observations. However, there are several real situations where the observations are not homogeneous. In these cases, by utilizing the traditional regression, we have a loss of performance in fitting terms. Then, for improving the goodness of fit, it is more suitable to apply the so-called clusterwise regression analysis. The aim of clusterwise linear regression analysis is to embed the techniques of clustering into regression analysis. In this way, the clustering methods are utilized for overcoming the heterogeneity problem in regression analysis. Furthermore, by integrating cluster analysis into the regression framework, the regression parameters (regression analysis) and membership degrees (cluster analysis) can be estimated simultaneously by optimizing one single objective function. In this paper the clusterwise linear regression has been analyzed in a fuzzy framework. In particular, a fuzzy clusterwise linear regression model (FCWLR model) with symmetrical fuzzy output and crisp input variables for performing fuzzy cluster analysis within a fuzzy linear regression framework is suggested. For measuring the goodness of fit of the suggested FCWLR model with fuzzy output, a fitting index is proposed. In order to illustrate the usefulness of FCWLR model in practice, several applications to artificial and real datasets are shown.

Fuzzy clusterwise regression in benefit segmentation: Application and investigation into its validity

This article describes a new technique for benefit segmentation, fuzzy clusterwise regression analysis (FCR). It combines cluistering with prediction and is based on multiattribute models of consumer behavior. FCR is especially useful when the number of observations per subject is small, when the relevant set of brands varies among subjects, when the researcher wants to take predictive fir into account in the segmentation procedure, or when subjects can belong to more than one segment. A cross-validation procedure is proposed to assess the validity of the results obtained with FCR. FCR is empirically illustrated for conjoint data on actual product involving price-quality tradeoffsl. The application supports the usefulness and validity of FCR as a benefit segmentation technique.

A Clusterwise Regression Method for Simultaneous Fuzzy Market Structuring and Benefit Segmentation

A generalized algorithm for fuzzy clusterwise regression (GFCR) is proposed that incorporates both benefit segmentation and market structuring within the framework of preference analysis. The method simultaneously estimates the models relating preferences to product dimensions within each of a number of clusters, and the degree of membership of brands and of subjects in those clusters. The performance of GFCR is assessed in a Monté Carlo study. An application to data on preferences for brands of margarine and butter is reported, the cross-validity of GFCR is assessed, and it is compared empirically with Kamakura's method. Managerial and research implications are discussed.

A fuzzy clusterwise regression approach to benefit segmentation

An algorithm for fuzzy clusterwise regression (FCR) is proposed which can be used for benefit segmentation within the framework of preference analysis. The method simultaneously estimates the models relating preference to product dimensions within each cluster, as well as the parameters indicating the degree of membership of subjects in these clusters. Information on the degree of competition within segments, opportunities for new products and indications for marketing strategies are provided, all from a consumers' perspective. Using synthetic data, the performance of the method is evaluated according to computational requirements, parameter recovery, and data reproduction. Comparisons with the clusterwise regression approach of DeSarbo, Oliver and Rangaswamy (1989) and Hagerty's (1985) optimal weighting method for conjoint analysis are given, as well as an application to the analysis of preferences for meat products. Special attention is paid to significance testing using Monte Carlo test procedures, and convergence to local optima.