Angela Montanari

Heteroscedastic factor mixture analysis

When data come from an unobserved heterogeneous population, common factor analysis is not appropriate to estimate the underlying constructs of interest. By replacing the traditional assumption of Gaussian distributed factors by a finite mixture of multivariate Gaussians, the unobserved heterogeneity can be modelled by latent classes. In so doing, we obtain a particular factor mixture analysis with heteroscedastic components. In this paper, the model is presented and a maximum likelihood estimation procedure via the expectation–maximization algorithm is developed. We also show that the approach well performs as a dimensionally reduced model-based clustering. Two real applications are illustrated and performances are compared to standard model-based clustering methods.

A skew-normal factor model for the analysis of student satisfaction towards university courses

Classical factor analysis relies on the assumption of normally distributed factors that guarantees the model to be estimated via the maximum likelihood method. Even when the assumption of Gaussian factors is not explicitly formulated and estimation is performed via the iterated principal factors’ method, the interest is actually mainly focussed on the linear structure of the data, since only moments up to the second ones are involved. In many real situations, the factors could not be adequately described by the first two moments only. For example, skewness characterizing most latent variables in social analysis can be properly measured by the third moment: the factors are not normally distributed and covariance is no longer a sufficient statistic. In this work we propose a factor model characterized by skew-normally distributed factors. Skew-normal refers to a parametric class of probability distributions, that extends the normal distribution by an additional shape parameter regulating the skewness. The model estimation can be solved by the generalized EM algorithm, in which the iterative Newthon–Raphson procedure is needed in the M-step to estimate the factor shape parameter. The proposed skew-normal factor analysis is applied to the study of student satisfaction towards university courses, in order to identify the factors representing different aspects of the latent overall satisfaction.

Urban Air Pollution Monitoring and Correlation Properties between Fixed-Site Stations

Abstract The rich regional air-monitoring network of the Emilia-Romagna region of Italy has been used to quantify the spatial variability of the main pollutants within urban environments and to analyze the correlations between stations. The spatial variability of the concentrations of the majority of pollutants within the city was very high, making it difficult to differentiate and characterize the urban environments and to apply legal limits with uniform criteria. On the other hand, the correlations between the fixed-site monitoring stations were high enough for their data to be retained generally very appropriately for controlling temporal trends. Starting from the high correlation level, a procedure was proposed and tested to derive pollution levels, using short-term measurements, such as passive samplers and mobile-station data. The importance of long-term statistics in urban air pollution mapping was emphasized. Treatment of missing data in time series and quality assurance were indicated as possible fields for applications for the correlation properties.

A projection pursuit approach to variable selection

A variable selection criterion based on projection pursuit is developed, exploiting the attractive property of projection pursuit methods to detect and ignore non informative variables in the cluster analysis context. Importance coefficients are introduced in order to measure the contribution of each variable to the definition of the projection pursuit solution. Each importance coefficient depends on the absolute value of the coefficient associated to each variable in the projection pursuit solution and on the variability of the corresponding variable. The selection criterion consists in retaining those variables which present an importance coefficient greater than a suitably chosen threshold. This is determined considering that in the no structure k-variate case the vectors of importance coefficients are uniformly distributed on the unit k-sphere. The good performances of the proposed method, tested both on real and simulated data, along with its simplicity, make it a valid competitor to the classical variable selection methods.

Comparison of Four Methods for Estimating Complete Life Tables from Abridged Life Tables Using Mortality Data Supplied to EUROCARE-3

ABSTRACT To estimate mortality due to cancer, it is necessary to have mortality data by year of age in the population of cancer patients. When such data are not available, estimating one-year (complete) life tables from five-year (abridged) life tables is necessary. Four such methods—Elandt–Johnson, Kostaki, Brass logit, and Akima spline methods—are compared with respect to 782 empirical complete life tables pertaining to 19 European regions or countries, from 1954 to 2000. Abridged life tables are first derived from the empirical ones, then used to produce one-year-life tables by each of the four methods. These reconstituted complete life tables are then compared with the empirical complete life tables. Among the four methods, the Elandt–Johnson demographic method produces the best reconstitutions at adult ages, specifically those ages at which observed cancer survival needs to be corrected.

Penalized factor mixture analysis for variable selection in clustered data

A model-based clustering approach which contextually performs dimension reduction and variable selection is presented. Dimension reduction is achieved by assuming that the data have been generated by a linear factor model with latent variables modeled as Gaussian mixtures. Variable selection is performed by shrinking the factor loadings though a penalized likelihood method with an L1 penalty. A maximum likelihood estimation procedure via the EM algorithm is developed and a modified BIC criterion to select the penalization parameter is illustrated. The effectiveness of the proposed model is explored in a Monte Carlo simulation study and in a real example.

Linear discriminant analysis and transvariation

In this paper, a two group linear discriminant function (LDF) is derived by minimizing Ginis transvariation probability. This solution is a special case of the projection pursuit method and improves the performance of Fishers LDF when the conditions for its optimality do not hold. Two groups are said to transvariate with respect to a variable Y (here a linear combination of the observed variables) if there exists at least one pair of units, belonging to different groups, in which the difference in sign between the Y values is opposite to that of the corresponding group mean values. As any difference satisfying this condition is called a transvariation, transvariation probability is defined as the ratio of the number of observed transvariations to its maximum possible value. When transvariation probability is used to measure group separation a linear discriminant function may be obtained as the linear combination along which transvariation probability is minimum. The performances of the proposed method are tested through a wide simulation study and on a real data set.

Maximum likelihood estimation of mixtures of factor analyzers

Mixtures of factor analyzers have been receiving wide interest in statistics as a tool for performing clustering and dimension reduction simultaneously. In this model it is assumed that, within each component, the data are generated according to a factor model. Therefore, the number of parameters on which the covariance matrices depend is reduced. Several estimation methods have been proposed for this model, both in the classical and in the Bayesian framework. However, so far, a direct maximum likelihood procedure has not been developed. This direct estimation problem, which simultaneously allows one to derive the information matrix for the mixtures of factor analyzers, is solved. The effectiveness of the proposed procedure is shown on a simulation study and on a toy example.

Independent factor discriminant analysis

In the general classification context the recourse to the so-called Bayes decision rule requires to estimate the class conditional probability density functions. A mixture model for the observed variables which is derived by assuming that the data have been generated by an independent factor model is proposed. Independent factor analysis is in fact a generative latent variable model whose structure closely resembles the one of the ordinary factor model, but it assumes that the latent variables are mutually independent and not necessarily Gaussian. The method therefore provides a dimension reduction together with a semiparametric estimate of the class conditional probability density functions. This density approximation is plugged into the classic Bayes rule and its performance is evaluated both on real and simulated data.

Model-based clustering of probability density functions

Complex data such as those where each statistical unit under study is described not by a single observation (or vector variable), but by a unit-specific sample of several or even many observations, are becoming more and more popular. Reducing these sample data by summary statistics, like the average or the median, implies that most inherent information (about variability, skewness or multi-modality) gets lost. Full information is preserved only if each unit is described by a whole distribution. This new kind of data, a.k.a. “distribution-valued data”, require the development of adequate statistical methods. This paper presents a method to group a set of probability density functions (pdfs) into homogeneous clusters, provided that the pdfs have to be estimated nonparametrically from the unit-specific data. Since elements belonging to the same cluster are naturally thought of as samples from the same probability model, the idea is to tackle the clustering problem by defining and estimating a proper mixture model on the space of pdfs. The issue of model building is challenging here because of the infinite-dimensionality and the non-Euclidean geometry of the domain space. By adopting a wavelet-based representation for the elements in the space, the task is accomplished by using mixture models for hyper-spherical data. The proposed solution is illustrated through a simulation experiment and on two real data sets.