Marcela Svarc

Principal components for multivariate functional data

A principal component method for multivariate functional data is proposed. Data can be arranged in a matrix whose elements are functions so that for each individual a vector of p functions is observed. This set of p curves is reduced to a small number of transformed functions, retaining as much information as possible. The criterion to measure the information loss is the integrated variance. Under mild regular conditions, it is proved that if the original functions are smooth this property is inherited by the principal components. A numerical procedure to obtain the smooth principal components is proposed and the goodness of the dimension reduction is assessed by two new measures of the proportion of explained variability. The method performs as expected in various controlled simulated data sets and provides interesting conclusions when it is applied to real data sets.

Selection of Variables for Cluster Analysis and Classification Rules

In this article we introduce two procedures for variable selection in cluster analysis and classification rules. One is mainly aimed at detecting the “noisy” noninformative variables, while the other also deals with multicolinearity and general dependence. Both methods are designed to be used after a “satisfactory” grouping procedure has been carried out. A forward–backward algorithm is proposed to make such procedures feasible in large datasets. A small simulation is performed and some real data examples are analyzed.

Pattern recognition via projection-based kNN rules

A new procedure for pattern recognition is introduced based on the concepts of random projections and nearest neighbors. It can be considered as an improvement of the classical nearest neighbor classification rules. Besides the concept of neighbors, the notion of district, a larger set into which the data will be projected, is introduced. Then a one-dimensional kNN method is applied to the projected data on randomly selected directions. This method, which is more accurate to handle high-dimensional data, has some robustness properties. The procedure is also universally consistent. Moreover, the method is challenged with the Isolet data set where a very high classification score is obtained.

Deprivation and the Dimensionality of Welfare: A Variable‐Selection Cluster‐Analysis Approach

In this paper we tackle the problems of dimensionality of welfare and that of identifying the multidimensionally poor by first finding the poor using the original space of attributes, and then reducing the welfare space. The starting point is the notion that the ‘poor’ constitutes a group of individuals that are essentially different from the ‘non-poor’ in a truly multidimensional framwework. Once this group has been identified, we propose reducing the dimension of the original welfare space by solving the problem of finding the smallest set of attributes that can reproduce as accurately as possible the ‘poor/non-poor’ classification in the first stage.

Classifying speech sonority functional data using a projected Kolmogorov-Smirnov approach

Abstract This paper addresses a linguistically motivated question of classification of functional data, namely the statistical classification of languages according to their rhythmic features. This is an important open problem in phonology. The analysis is based on the information provided by the sonority, which is an index of local regularity of the speech signal. Our main tool is the projected Kolmogorov–Smirnov test. This is a new goodness of fit test for functional data. The result obtained supports the linguistic conjecture of the existence of three rhythmic classes.

Resistant estimates for high dimensional and functional data based on random projections

We herein propose a new robust estimation method based on random projections that is adaptive and automatically produces a robust estimate, while enabling easy computations for high or infinite dimensional data. Under some restricted contamination models, the procedure is robust and attains full efficiency. We tested the method using both simulated and real data.

A divisive clustering method for functional data with special consideration of outliers

This paper presents DivClusFD, a new divisive hierarchical method for the non-supervised classification of functional data. Data of this type present the peculiarity that the differences among clusters may be caused by changes as well in level as in shape. Different clusters can be separated in different subregion and there may be no subregion in which all clusters are separated. In each step of division, the DivClusFD method explores the functions and their derivatives at several fixed points, seeking the subregion in which the highest number of clusters can be separated. The number of clusters is estimated via the gap statistic. The functions are assigned to the new clusters by combining the k-means algorithm with the use of functional boxplots to identify functions that have been incorrectly classified because of their atypical local behavior. The DivClusFD method provides the number of clusters, the classification of the observed functions into the clusters and guidelines that may be for interpreting the clusters. A simulation study using synthetic data and tests of the performance of the DivClusFD method on real data sets indicate that this method is able to classify functions accurately.

The Choice of the Initial Estimate for Computing MM-Estimates

We show, using a Monte Carlo study, that MM-estimates with projection estimates as starting point of an iterative weighted least squares algorithm, behave more robustly than MM-estimates starting at an S-estimate and similar Gaussian efficiency. Moreover the former have a robustness behavior close to the P-estimates with an additional advantage: they are asymptotically normal making statistical inference possible.