Roberto Rocci

Constrained monotone EM algorithms for finite mixture of multivariate Gaussians

The likelihood function for normal multivariate mixtures may present both local spurious maxima and also singularities and the latter may cause the failure of the optimization algorithms. Theoretical results assure that imposing some constraints on the eigenvalues of the covariance matrices of the multivariate normal components leads to a constrained parameter space with no singularities and at least a smaller number of local maxima of the likelihood function. Conditions assuring that an EM algorithm implementing such constraints maintains the monotonicity property of the usual EM algorithm are provided. Different approaches are presented and their performances are evaluated and compared using numerical experiments.

Simultaneous Component and Clustering Models for Three-way Data: Within and Between Approaches

In this paper two techniques for units clustering and factorial dimensionality reduction of variables and occasions of a three-mode data set are discussed. These techniques can be seen as the simultaneous version of two procedures based on the sequential application of k-means and Tucker2 algorithms and vice versa. The two techniques, T3Clus and 3Fk-means, have been compared theoretically and empirically by a simulation study. In the latter, it has been noted that neither T3Clus nor 3Fk-means outperforms the other in every case. From these results rises the idea to combine the two techniques in a unique general model, named CT3Clus, having T3Clus and 3Fk-means as special cases. A simulation study follows to show the effectiveness of the proposal.

Two-mode multi-partitioning

New methodologies for two-mode (objects and variables) multi-partitioning of two way data are presented. In particular, by reanalyzing the double k-means, that identifies a unique partition for each mode of the data, a relevant extension is discussed which allows to specify more partitions of one mode, conditionally to the partition of the other one. The performance of such generalized double k-means has been tested by both a simulation study and an application to gene microarray data.

Uniqueness of three-mode factor models with sparse cores: The 3 × 3 × 3 case

Three-Mode Factor Analysis (3MFA) and PARAFAC are methods to describe three-way data. Both methods employ models with components for the three modes of a three-way array; the 3MFA model also uses a three-way core array for linking all components to each other. The use of the core array makes the 3MFA model more general than the PARAFAC model (thus allowing a better fit), but also more complicated. Moreover, in the 3MFA model the components are not uniquely determined, and it seems hard to choose among all possible solutions. A particularly interesting feature of the PARAFAC model is that it does give unique components. The present paper introduces a class of 3MFA models in between 3MFA and PARAFAC that share the good properties of the 3MFA model and the PARAFAC model: They fit (almost) as well as the 3MFA model, they are relatively simple and they have the same uniqueness properties as the PARAFAC model.

New perspectives in statistical modeling and data analysis: proceedings of the 7th Conference of the Classification and data analysis group of the Italian statistical Society, Catania, September 9 - 11, 2009

Data Modeling for Evaluation.- Data Analysis in Economics.- Nonparametric Kernel Estimation.- Data Analysis in Industry and Services.- Visualization of Relationships.- Classification.- Analysis of Financial Data.- Functional Data Analysis.- Computer Intensive Methods.- Data Analysis in Environmental and Medical Sciences.- Analysis of Categorical Data.- Multivariate Analysis.

A New Dimension Reduction Method: Factor Discriminant K -means

Reduced K-means (RKM) and Factorial K-means (FKM) are two data reduction techniques incorporating principal component analysis and K-means into a unified methodology to obtain a reduced set of components for variables and an optimal partition for objects. RKM finds clusters in a reduced space by maximizing the between-clusters deviance without imposing any condition on the within-clusters deviance, so that clusters are isolated but they might be heterogeneous. On the other hand, FKM identifies clusters in a reduced space by minimizing the within-clusters deviance without imposing any condition on the between-clusters deviance. Thus, clusters are homogeneous, but they might not be isolated. The two techniques give different results because the total deviance in the reduced space for the two methodologies is not constant; hence the minimization of the within-clusters deviance is not equivalent to the maximization of the between-clusters deviance. In this paper a modification of the two techniques is introduced to avoid the afore mentioned weaknesses. It is shown that the two modified methods give the same results, thus merging RKM and FKM into a new methodology. It is called Factor Discriminant K-means (FDKM), because it combines Linear Discriminant Analysis and K-means. The paper examines several theoretical properties of FDKM and its performances with a simulation study. An application on real-world data is presented to show the features of FDKM.

Three-mode factor analysis with binary core and orthonormality constraints

A constrained version of Three-mode Factor Analysis model is considered in order to make its interpretation easier. The constraints are obtained by fixing some elements of the core to zero and requiring orthonormal factor loadings. An algorithm to solve the related minimization problem and an example of core constraints with theoretically interesting features, are given.

A simplification of a result by zellini on the maximal rank of symmetric three-way arrays

Zellini (1979, Theorem 3.1) has shown how to decompose an arbitrary symmetric matrix of ordern ×n as a linear combination of 1/2n(n+1) fixed rank one matrices, thus constructing an explicit tensor basis for the set of symmetricn ×n matrices. Zellinis decomposition is based on properties of persymmetric matrices. In the present paper, a simplified tensor basis is given, by showing that a symmetric matrix can also be decomposed in terms of 1/2n(n+1) fixed binary matrices of rank one. The decomposition implies that ann ×n ×p array consisting ofp symmetricn ×n slabs has maximal rank 1/2n(n+1). Likewise, an unconstrained INDSCAL (symmetric CANDECOMP/PARAFAC) decomposition of such an array will yield a perfect fit in 1/2n(n+1) dimensions. When the fitting only pertains to the off-diagonal elements of the symmetric matrices, as is the case in a version of PARAFAC where communalities are involved, the maximal number of dimensions can be further reduced to 1/2n(n−1). However, when the saliences in INDSCAL are constrained to be nonnegative, the tensor basis result does not apply. In fact, it is shown that in this case the number of dimensions needed can be as large asp, the number of matrices analyzed.

Methods for Asymmetric Three-way Scaling

A review of methods for asymmetric three-way scaling is presented focusing on their graphical capabilities. A general strategy of analysis is outlined with an example of application to import-export data.

Monotone Constrained EM Algorithms for Multinormal Mixture Models

We investigate the spectral decomposition of the covariance matrices of a multivariate normal mixture distribution in order to construct constrained EM algorithms which guarantee the monotonicity property. Furthermore we propose different set of constraints which can be simply implemented. These procedures have been tested on the ground of many numerical experiments.