Gaussian mixture models represent a conceptually and mathematically elegant class of models for c... more Gaussian mixture models represent a conceptually and mathematically elegant class of models for casting the density of a heterogeneous population where the observed data is collected from a population composed of a finite set of G homogeneous subpopulations with a Gaussian distribution. A limitation of these models is that they suffer from the curse of dimensionality, and the number of parameters becomes easily extremely large in the presence of high-dimensional data. In this paper, we propose a class of parsimonious Gaussian mixture models with constrained extended ultrametric covariance structures that are capable of exploring hierarchical relations among variables. The proposal shows to require a reduced number of parameters to be fit and includes constrained covariance structures across and within components that further reduce the number of parameters of the model.
The use of general descriptive names, registered names, trademarks, etc. in this publication does... more The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Advances in Data Analysis and Classification, 2022
Gaussian Mixture Models (GMMs) are one of the most widespread methodologies for model-based clust... more Gaussian Mixture Models (GMMs) are one of the most widespread methodologies for model-based clustering. They assume a multivariate Gaussian distribution for each component of the mixture, centered at the mean vector and with volume, shape and orientation derived by the covariance matrix. To reduce the large number of parameters produced by the covariance matrices, parsimonious parameterizations of the latter were proposed in literature, e.g., the eigen-decomposition and the parsimonious GMMs based on mixtures of probabilistic principal component analyzers and mixtures of factor analyzers. We introduce a new parameterization of a covariance matrix by defining an extended ultrametric covariance matrix and we implement it into a GMM. This structure can be used to describe multidimensional phenomena which are characterized by nested latent concepts having different levels of abstraction, from the most specific to the most general. The proposal is able to pinpoint a hierarchical structure on variables for each component of the GMM, thus identifying a different characterization of a multidimensional phenomenon for each component (cluster, subpopulation) of the mixture. At the same time, it defines a new parsimonious GMM since the ultrametric covariance structure reconstructs the relationships among variables with a limited number of parameters. The proposal is applied on synthetic and real data. On the former it shows good performance in terms of classification when compared to the other existing parameterizations, and on the latter it also provides insight into the hierarchical relationships among the variables for each cluster.
Studies in Classification, Data Analysis, and Knowledge Organization, 2017
In this work, a discrete model for clustering and a continuous factorial one for dimension reduct... more In this work, a discrete model for clustering and a continuous factorial one for dimension reduction are simultaneously fitted to categorical data, with the aim of identifying the best partition of the objects, described by the best orthogonal linear combinations of the factors, according to the least-squares criterion. This new methodology named multiple correspondence k-means is a useful alternative to the Tandem Analysis in the case of categorical data. Then, this approach has a double objective: data reduction and synthesis, simultaneously in the direction of rows and columns of the data matrix.
In this work, starting from the statistical knowledge about connection and correlation between co... more In this work, starting from the statistical knowledge about connection and correlation between couples of variables, the author carries out a systematic study and a critical examination on these two relations among couples of data matrices belonging to a three-way data matrix. Analogously to linear measures among variables, the measures of correlation between two matrices have to indicate the "direction" of dependence of the whole variables set of a data matrix with that of the other data matrix. In other words measures of correlation between couples of matrices have to evaluate whether all variables of a matrix are more frequently concordant or discordant with all variables of the other matrix. After defining the mathematical notation for the three-way data matrix, two different ways to measure the correlation between couples of matrices are examined and also conditions necessary to define it are given. The covariance and correlation between couples of matrices are matrices are introduces . These indices have two main requirements they become well-known measures of correlation among couples of variables when matrices degenerate into vectors; they measure the correlation among couples of matrices as a function of the measures of correlation among couples of variables, these last in the couple each one belonging to a matrix. The statistical meaning about connection between a couple of matrices can be explained according to the well-known meaning of this relation among variables. Two matrices are not connected with each other if any variable of one matrix is not connected with any variable of the other matrix. Also the measures of connection between couples of data matrices have two main properties they become well-known measures of connection between couples of variables when matrices degenerate into vectors; they measure the connection between two matrices as a function of measures of connection between couples of variables, these last each one belonging to a matrix.
In this paper after introducing the FActorial Matrices Analysis (FAMA ) in order to study a three... more In this paper after introducing the FActorial Matrices Analysis (FAMA ) in order to study a three-way matrix (Vichi, 1990), the Author examines techniques already known in literature, that can be derived from FAMA, adding new tools of analysis. Furthermore the Author traduces from FAMA new methodologies to overcome problems connected with some of the known techniques.
Methodologies in Classification: Clustering and Classification: J.D. Carroll, A. Chaturvedi: K-Mi... more Methodologies in Classification: Clustering and Classification: J.D. Carroll, A. Chaturvedi: K-Midranges Clustering.- A. Cerioli: A New Method for Detecting Influential Observations in Nonhierarchical Cluster Analysis.- J.C. Gower, G.J.S.Ross: Non-Probabilistic Classification.- A. Hardy, P. Andre: An Investigation of Nine Procedures for Detecting the Structure in a Data Set.- C. Hennig: Clustering and Outlier Identification: Fixed Point Cluster Analysis.- S. Korenjak-Cerne, V. Batagelj: Clustering Large Datasets of Mixed Units.- J.A. Martin-Fernandez, C. Barcelo-Vidal, V. Pawlowsky-Glahn: A Critical Approach to Non-Parametric Classification of Compositional Data.- F. Murtagh, J.L. Stark, M. Berry: Clustering Based on Wavelet Transform: Applications to Point Pattern Clustering and to High-Dimensional Data Analysis.- Comparison and Consensus of Classifications: C. Hayashi, K. Yamaoka: Beyond Simpon's Paradox: One Problem in Data Science.- F.J. Lapointe: For Consensus (With Branch Lengths).- B. Leclerc: Consensus of Classifications: The Case of Trees.- J. L. Thorley, M. Wilkinson, M. Charleston: The Information Content of Consensus Trees.- Fuzzy Clustering and Fuzzy Methods: S. Bodjanova: Compatible and Complementary Fuzzy Partitions.- S. Miyamoto, K. Umayahara: Two Methods of Fuzzy c-Means and Classification Functions.- M. Ryoke, Y. Nakamori, H. Tamura: Dynamic Determination of Mixing Parameters in Fuzzy Clustering.- M. Sato-Ilic, Y. Sato: A Dynamic Additive Fuzzy Clustering Model.- S. Turpin-Dhilly, C. Botte-Lecocq: Application of Fuzzy Mathematical Morphology for Pattern Classification.- Optimization in Classification and Constrained Classification: K.Bachar, I.-C. Lerman: Statistical Conditions for a Linear Complexity for an Algorithm of Hierarchical Classification Under Constraint of Contiguity.- V. Batagelj, A. Ferligoj: Constrained Clustering Problems.- T. Gastaldi, D. Vicari: A Constrained Clusterwise Procedure for Segmentation.- L. Gueguen, R. Vignes, J. Lebbe: Maximal Predictive Clustering with Order Constraint: a Linear and Optimal Algorithm.- I. Lari, M. Maravalle, B. Simeone: A Linear Programming Based Heuristic for a Hard Clustering Problem on Trees.- M. Mizuta: Two Principal Points of Symmetric Distributions.- N, Nicoloyannis, M. Terrenoire, D. Tounissoux: Pertinence for a Classification.- J. Trejos, A. Murillo, E. Piza: Global Stochastic Optimization Techniques Applied to Partitioning.- Probabilistic Modelling Methods in Classification and Pattern Recognition: H. Bensmail, J.J. Meulman: MCMC Inference for Model-Based Cluster Analysis.- L. Gyorfi, M. Horvath: On the Asymptotic Normality of a Resubstitution Error Estimate.- G. V. Orman: Stochastic Methods for Generative Systems Analysis.- H.J. Vos: Compensatory Rules for Optimal Classification with Mastery Scores.- Other Approaches for Classification: Discrimination, Neural Network, Regression Tree: Discrimination and Classification: V. Bertholet, J.P. Rasson, S. Lissoir: About the Automatic Detection of Training Sets for Multispectral Images Classification.- A.P.Duarte Silva: A 'Leaps and Bounds' Algorithm for Variable Selection in Two Group Discriminant Analysis.- M.-H. Huh, K-S. Yang: Canonical Discriminant Analysis of Multinomial Samples with Applications to Textual Data.- I.C. Lerman, J.F. Pinto da Costa: How to Extract Predictive Binary Attributes from a Categorical One.- L. Lizzani: A Density Distance Based Approach to Projection Pursuit Discriminant Analysis.- A. Montanari, D. G. Calo: Two Group Linear Discrimination Based on
Methodology is described for fitting a fuzzy partition and a parsimonious consensus hierarchy (ul... more Methodology is described for fitting a fuzzy partition and a parsimonious consensus hierarchy (ultrametric matrix) to a set of hierarchies of the same set of objects. A model defining a fuzzy partition of a set of hierarchical classifications, with every class of the partition synthesized by a parsimonious consensus hierarchy is described. Each consensus includes an optimal consensus hard partition of objects and all the hierarchical agglomerative aggregations among the clusters of the consensus partition. The performances of the methodology are illustrated by an extended simulation study and applications to real data. A discussion is provided on the new methodology and some interesting future developments are described.
Advanced Studies in Behaviormetrics and Data Science, 2020
Phenomena are usually multidimensional and their complexity cannot be directly explored via obser... more Phenomena are usually multidimensional and their complexity cannot be directly explored via observable variables. For this reason, a hierarchical structure of nested latent concepts representing different levels of abstraction of the phenomenon under study may be considered. In this paper, we provide a comparison between a procedure based on hierarchical clustering methods and a novelty model recently proposed, called Ultrametric Correlation Matrix (UCM) model. The latter aims at reconstructing the data correlation matrix via an ultrametric correlation matrix and supplies a parsimonious representation of multidimensional phenomena through a partition of the observable variables defining a reduced number of latent concepts. Moreover, the UCM model highlights two main features related to concepts: the correlation among concepts and the internal consistency of a concept. The performances of the UCM model and the procedure based on hierarchical clustering methods are illustrated by an application to the Holzinger data set which represents a real demonstration of a hierarchical factorial structure. The evaluation of the different methodological approaches—the UCM model and the procedure based on hierarchical clustering methods—is provided in terms of classification of variables and goodness of fit, other than of their suitability to analyse bottom-up latent structures of variables.
Advances in Data Analysis and Classification, 2020
Many relevant multidimensional phenomena are defined by nested latent concepts, which can be repr... more Many relevant multidimensional phenomena are defined by nested latent concepts, which can be represented by a tree-structure supposing a hierarchical relationship among manifest variables. The root of the tree is a general concept which includes more specific ones. The aim of the paper is to reconstruct an observed data correlation matrix of manifest variables through an ultrametric correlation matrix which is able to pinpoint the hierarchical nature of the phenomenon under study. With this scope, we introduce a novel model which detects consistent latent concepts and their relationships starting from the observed correlation matrix.
Computational Statistics & Data Analysis, 2020
The identification of different homogeneous groups of observations and their appropriate analysis... more The identification of different homogeneous groups of observations and their appropriate analysis in PLS-SEM has become a critical issue in many application fields. Usually, both SEM and PLS-SEM assume the homogeneity of units on which the model is applied. The approaches of segmentation proposed in the literature, consist of estimating separate models for each segment of statistical units, assigning these units to segments defined a priori. These approaches are not fully acceptable because no causal structure is postulated among variables. In other words, a model approach should be used, where the clusters obtained are homogeneous, both with respect to the structural causal relationships, and the mean differences between clusters. Therefore, a new methodology is proposed, where simultaneously non-hierarchical clustering and PLS-SEM is applied. This methodology is motivated by the fact that the sequential approach (i.e., the application, first, of SEM or PLS-SEM and subsequently the use of a clustering algorithm on the latent scores obtained) may fail to find the correct clustering structure of data. A simulation study and an application on real data are included to evaluate the performance of the proposed methodology.
Advances in Data Analysis and Classification, 2016
Disjoint factor analysis (DFA) is a new latent factor model that we propose here to identify fact... more Disjoint factor analysis (DFA) is a new latent factor model that we propose here to identify factors that relate to disjoint subsets of variables, thus simplifying the loading matrix structure. Similarly to exploratory factor analysis (EFA), the DFA does not hypothesize prior information on the number of factors and on the relevant relations between variables and factors. In DFA the population variance-covariance structure is hypothesized block diagonal after the proper permutation of variables and estimated by Maximum Likelihood, using an Coordinate Descent type algorithm. Inference on parameters on the number of factors and to confirm the hypothesized simple structure are provided. Properties such as scale equivariance, uniqueness, optimal simplification of loadings are satisfied by DFA. Relevant cross-loadings are also estimated in case they are detected from the best DFA solution. DFA has also the option to constrain a variable to load on a pre-specified factor so that the researcher can assume, a priori, some relations between variables and loadings. A simulation study shows performances of DFA and an application to optimally identify the dimensions of wellbeing is used to illustrate characteristics of the new methodology. A final discussion concludes the paper.
Some monotone transformations to map similarities into Euclidean distances are studied. The idea ... more Some monotone transformations to map similarities into Euclidean distances are studied. The idea developed in this work is to modify the usual transformation of the similarities into dissimilarities to obtain Euclidean distances. We use monotone mappings with an additive constant equal to the self-similarity and a multiplicative constant, i.e., actually a scale factor that reduces proportionally the similarities until the additive constant (self-similarity) is sufficient to embed similarities into Euclidean distances. For each transformation proposed, we give necessary and sufficient conditions to embed similarities into Euclidean space, while, until now, only sufficient conditions were known. To apply the proposed transformations no conditions on similarities are required (e.g., similarities to be positive semi-definite).
Gaussian mixture models represent a conceptually and mathematically elegant class of models for c... more Gaussian mixture models represent a conceptually and mathematically elegant class of models for casting the density of a heterogeneous population where the observed data is collected from a population composed of a finite set of G homogeneous subpopulations with a Gaussian distribution. A limitation of these models is that they suffer from the curse of dimensionality, and the number of parameters becomes easily extremely large in the presence of high-dimensional data. In this paper, we propose a class of parsimonious Gaussian mixture models with constrained extended ultrametric covariance structures that are capable of exploring hierarchical relations among variables. The proposal shows to require a reduced number of parameters to be fit and includes constrained covariance structures across and within components that further reduce the number of parameters of the model.
The use of general descriptive names, registered names, trademarks, etc. in this publication does... more The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Advances in Data Analysis and Classification, 2022
Gaussian Mixture Models (GMMs) are one of the most widespread methodologies for model-based clust... more Gaussian Mixture Models (GMMs) are one of the most widespread methodologies for model-based clustering. They assume a multivariate Gaussian distribution for each component of the mixture, centered at the mean vector and with volume, shape and orientation derived by the covariance matrix. To reduce the large number of parameters produced by the covariance matrices, parsimonious parameterizations of the latter were proposed in literature, e.g., the eigen-decomposition and the parsimonious GMMs based on mixtures of probabilistic principal component analyzers and mixtures of factor analyzers. We introduce a new parameterization of a covariance matrix by defining an extended ultrametric covariance matrix and we implement it into a GMM. This structure can be used to describe multidimensional phenomena which are characterized by nested latent concepts having different levels of abstraction, from the most specific to the most general. The proposal is able to pinpoint a hierarchical structure on variables for each component of the GMM, thus identifying a different characterization of a multidimensional phenomenon for each component (cluster, subpopulation) of the mixture. At the same time, it defines a new parsimonious GMM since the ultrametric covariance structure reconstructs the relationships among variables with a limited number of parameters. The proposal is applied on synthetic and real data. On the former it shows good performance in terms of classification when compared to the other existing parameterizations, and on the latter it also provides insight into the hierarchical relationships among the variables for each cluster.
Studies in Classification, Data Analysis, and Knowledge Organization, 2017
In this work, a discrete model for clustering and a continuous factorial one for dimension reduct... more In this work, a discrete model for clustering and a continuous factorial one for dimension reduction are simultaneously fitted to categorical data, with the aim of identifying the best partition of the objects, described by the best orthogonal linear combinations of the factors, according to the least-squares criterion. This new methodology named multiple correspondence k-means is a useful alternative to the Tandem Analysis in the case of categorical data. Then, this approach has a double objective: data reduction and synthesis, simultaneously in the direction of rows and columns of the data matrix.
In this work, starting from the statistical knowledge about connection and correlation between co... more In this work, starting from the statistical knowledge about connection and correlation between couples of variables, the author carries out a systematic study and a critical examination on these two relations among couples of data matrices belonging to a three-way data matrix. Analogously to linear measures among variables, the measures of correlation between two matrices have to indicate the "direction" of dependence of the whole variables set of a data matrix with that of the other data matrix. In other words measures of correlation between couples of matrices have to evaluate whether all variables of a matrix are more frequently concordant or discordant with all variables of the other matrix. After defining the mathematical notation for the three-way data matrix, two different ways to measure the correlation between couples of matrices are examined and also conditions necessary to define it are given. The covariance and correlation between couples of matrices are matrices are introduces . These indices have two main requirements they become well-known measures of correlation among couples of variables when matrices degenerate into vectors; they measure the correlation among couples of matrices as a function of the measures of correlation among couples of variables, these last in the couple each one belonging to a matrix. The statistical meaning about connection between a couple of matrices can be explained according to the well-known meaning of this relation among variables. Two matrices are not connected with each other if any variable of one matrix is not connected with any variable of the other matrix. Also the measures of connection between couples of data matrices have two main properties they become well-known measures of connection between couples of variables when matrices degenerate into vectors; they measure the connection between two matrices as a function of measures of connection between couples of variables, these last each one belonging to a matrix.
In this paper after introducing the FActorial Matrices Analysis (FAMA ) in order to study a three... more In this paper after introducing the FActorial Matrices Analysis (FAMA ) in order to study a three-way matrix (Vichi, 1990), the Author examines techniques already known in literature, that can be derived from FAMA, adding new tools of analysis. Furthermore the Author traduces from FAMA new methodologies to overcome problems connected with some of the known techniques.
Methodologies in Classification: Clustering and Classification: J.D. Carroll, A. Chaturvedi: K-Mi... more Methodologies in Classification: Clustering and Classification: J.D. Carroll, A. Chaturvedi: K-Midranges Clustering.- A. Cerioli: A New Method for Detecting Influential Observations in Nonhierarchical Cluster Analysis.- J.C. Gower, G.J.S.Ross: Non-Probabilistic Classification.- A. Hardy, P. Andre: An Investigation of Nine Procedures for Detecting the Structure in a Data Set.- C. Hennig: Clustering and Outlier Identification: Fixed Point Cluster Analysis.- S. Korenjak-Cerne, V. Batagelj: Clustering Large Datasets of Mixed Units.- J.A. Martin-Fernandez, C. Barcelo-Vidal, V. Pawlowsky-Glahn: A Critical Approach to Non-Parametric Classification of Compositional Data.- F. Murtagh, J.L. Stark, M. Berry: Clustering Based on Wavelet Transform: Applications to Point Pattern Clustering and to High-Dimensional Data Analysis.- Comparison and Consensus of Classifications: C. Hayashi, K. Yamaoka: Beyond Simpon's Paradox: One Problem in Data Science.- F.J. Lapointe: For Consensus (With Branch Lengths).- B. Leclerc: Consensus of Classifications: The Case of Trees.- J. L. Thorley, M. Wilkinson, M. Charleston: The Information Content of Consensus Trees.- Fuzzy Clustering and Fuzzy Methods: S. Bodjanova: Compatible and Complementary Fuzzy Partitions.- S. Miyamoto, K. Umayahara: Two Methods of Fuzzy c-Means and Classification Functions.- M. Ryoke, Y. Nakamori, H. Tamura: Dynamic Determination of Mixing Parameters in Fuzzy Clustering.- M. Sato-Ilic, Y. Sato: A Dynamic Additive Fuzzy Clustering Model.- S. Turpin-Dhilly, C. Botte-Lecocq: Application of Fuzzy Mathematical Morphology for Pattern Classification.- Optimization in Classification and Constrained Classification: K.Bachar, I.-C. Lerman: Statistical Conditions for a Linear Complexity for an Algorithm of Hierarchical Classification Under Constraint of Contiguity.- V. Batagelj, A. Ferligoj: Constrained Clustering Problems.- T. Gastaldi, D. Vicari: A Constrained Clusterwise Procedure for Segmentation.- L. Gueguen, R. Vignes, J. Lebbe: Maximal Predictive Clustering with Order Constraint: a Linear and Optimal Algorithm.- I. Lari, M. Maravalle, B. Simeone: A Linear Programming Based Heuristic for a Hard Clustering Problem on Trees.- M. Mizuta: Two Principal Points of Symmetric Distributions.- N, Nicoloyannis, M. Terrenoire, D. Tounissoux: Pertinence for a Classification.- J. Trejos, A. Murillo, E. Piza: Global Stochastic Optimization Techniques Applied to Partitioning.- Probabilistic Modelling Methods in Classification and Pattern Recognition: H. Bensmail, J.J. Meulman: MCMC Inference for Model-Based Cluster Analysis.- L. Gyorfi, M. Horvath: On the Asymptotic Normality of a Resubstitution Error Estimate.- G. V. Orman: Stochastic Methods for Generative Systems Analysis.- H.J. Vos: Compensatory Rules for Optimal Classification with Mastery Scores.- Other Approaches for Classification: Discrimination, Neural Network, Regression Tree: Discrimination and Classification: V. Bertholet, J.P. Rasson, S. Lissoir: About the Automatic Detection of Training Sets for Multispectral Images Classification.- A.P.Duarte Silva: A 'Leaps and Bounds' Algorithm for Variable Selection in Two Group Discriminant Analysis.- M.-H. Huh, K-S. Yang: Canonical Discriminant Analysis of Multinomial Samples with Applications to Textual Data.- I.C. Lerman, J.F. Pinto da Costa: How to Extract Predictive Binary Attributes from a Categorical One.- L. Lizzani: A Density Distance Based Approach to Projection Pursuit Discriminant Analysis.- A. Montanari, D. G. Calo: Two Group Linear Discrimination Based on
Methodology is described for fitting a fuzzy partition and a parsimonious consensus hierarchy (ul... more Methodology is described for fitting a fuzzy partition and a parsimonious consensus hierarchy (ultrametric matrix) to a set of hierarchies of the same set of objects. A model defining a fuzzy partition of a set of hierarchical classifications, with every class of the partition synthesized by a parsimonious consensus hierarchy is described. Each consensus includes an optimal consensus hard partition of objects and all the hierarchical agglomerative aggregations among the clusters of the consensus partition. The performances of the methodology are illustrated by an extended simulation study and applications to real data. A discussion is provided on the new methodology and some interesting future developments are described.
Advanced Studies in Behaviormetrics and Data Science, 2020
Phenomena are usually multidimensional and their complexity cannot be directly explored via obser... more Phenomena are usually multidimensional and their complexity cannot be directly explored via observable variables. For this reason, a hierarchical structure of nested latent concepts representing different levels of abstraction of the phenomenon under study may be considered. In this paper, we provide a comparison between a procedure based on hierarchical clustering methods and a novelty model recently proposed, called Ultrametric Correlation Matrix (UCM) model. The latter aims at reconstructing the data correlation matrix via an ultrametric correlation matrix and supplies a parsimonious representation of multidimensional phenomena through a partition of the observable variables defining a reduced number of latent concepts. Moreover, the UCM model highlights two main features related to concepts: the correlation among concepts and the internal consistency of a concept. The performances of the UCM model and the procedure based on hierarchical clustering methods are illustrated by an application to the Holzinger data set which represents a real demonstration of a hierarchical factorial structure. The evaluation of the different methodological approaches—the UCM model and the procedure based on hierarchical clustering methods—is provided in terms of classification of variables and goodness of fit, other than of their suitability to analyse bottom-up latent structures of variables.
Advances in Data Analysis and Classification, 2020
Many relevant multidimensional phenomena are defined by nested latent concepts, which can be repr... more Many relevant multidimensional phenomena are defined by nested latent concepts, which can be represented by a tree-structure supposing a hierarchical relationship among manifest variables. The root of the tree is a general concept which includes more specific ones. The aim of the paper is to reconstruct an observed data correlation matrix of manifest variables through an ultrametric correlation matrix which is able to pinpoint the hierarchical nature of the phenomenon under study. With this scope, we introduce a novel model which detects consistent latent concepts and their relationships starting from the observed correlation matrix.
Computational Statistics & Data Analysis, 2020
The identification of different homogeneous groups of observations and their appropriate analysis... more The identification of different homogeneous groups of observations and their appropriate analysis in PLS-SEM has become a critical issue in many application fields. Usually, both SEM and PLS-SEM assume the homogeneity of units on which the model is applied. The approaches of segmentation proposed in the literature, consist of estimating separate models for each segment of statistical units, assigning these units to segments defined a priori. These approaches are not fully acceptable because no causal structure is postulated among variables. In other words, a model approach should be used, where the clusters obtained are homogeneous, both with respect to the structural causal relationships, and the mean differences between clusters. Therefore, a new methodology is proposed, where simultaneously non-hierarchical clustering and PLS-SEM is applied. This methodology is motivated by the fact that the sequential approach (i.e., the application, first, of SEM or PLS-SEM and subsequently the use of a clustering algorithm on the latent scores obtained) may fail to find the correct clustering structure of data. A simulation study and an application on real data are included to evaluate the performance of the proposed methodology.
Advances in Data Analysis and Classification, 2016
Disjoint factor analysis (DFA) is a new latent factor model that we propose here to identify fact... more Disjoint factor analysis (DFA) is a new latent factor model that we propose here to identify factors that relate to disjoint subsets of variables, thus simplifying the loading matrix structure. Similarly to exploratory factor analysis (EFA), the DFA does not hypothesize prior information on the number of factors and on the relevant relations between variables and factors. In DFA the population variance-covariance structure is hypothesized block diagonal after the proper permutation of variables and estimated by Maximum Likelihood, using an Coordinate Descent type algorithm. Inference on parameters on the number of factors and to confirm the hypothesized simple structure are provided. Properties such as scale equivariance, uniqueness, optimal simplification of loadings are satisfied by DFA. Relevant cross-loadings are also estimated in case they are detected from the best DFA solution. DFA has also the option to constrain a variable to load on a pre-specified factor so that the researcher can assume, a priori, some relations between variables and loadings. A simulation study shows performances of DFA and an application to optimally identify the dimensions of wellbeing is used to illustrate characteristics of the new methodology. A final discussion concludes the paper.
Some monotone transformations to map similarities into Euclidean distances are studied. The idea ... more Some monotone transformations to map similarities into Euclidean distances are studied. The idea developed in this work is to modify the usual transformation of the similarities into dissimilarities to obtain Euclidean distances. We use monotone mappings with an additive constant equal to the self-similarity and a multiplicative constant, i.e., actually a scale factor that reduces proportionally the similarities until the additive constant (self-similarity) is sufficient to embed similarities into Euclidean distances. For each transformation proposed, we give necessary and sufficient conditions to embed similarities into Euclidean space, while, until now, only sufficient conditions were known. To apply the proposed transformations no conditions on similarities are required (e.g., similarities to be positive semi-definite).
Uploads
Papers by Maurizio Vichi