arXiv subjects

M. Rosário Oliveira

Publications and source records attributed to M. Rosário Oliveira.

8 recordsLinked to original sources

arXiv2026

Minimum Covariance Determinant Estimator and Outlier Detection for Interval-valued Data

Interval-valued data are one of the most common symbolic data types, which enables the preservation of the underlying variability of the data. The interval mean and covariance matrix can be estimated using the barycenter approach based on the Mallows distance. However, as for conventional data, classical estimates can be significantly affected by anomalous data points, frequently present in real-life datasets. To address this problem, we develop a robust alternative which estimates location and scale by extending the Minimum Covariance Determinant estimator to interval-valued data. The algorithm yields a robust Interval-Mahalanobis distance, which can be used to detect anomalous observations based on adaptive cutoff values. Through extensive simulation studies across various contamination levels, we demonstrate that the interval-valued robust estimator consistently outperforms classical methods in covariance matrix estimation and achieves superior outlier detection accuracy. Finally, the applicability and effectiveness of the proposed method are illustrated through real-world datasets.

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arXiv2026

Explainable Outlier Detection for Interval-valued Data

Explainability is increasingly recognized as a key aspect of outlier detection. However, for complex data structures such as interval-valued data, it remains largely unexplored. Building on an outlier detection framework based on the Interval Minimum Covariance Determinant estimator, we propose a novel approach to explain the outlyingness of interval-valued observations using the concept of the Shapley value. We derive a closed-form expression for the Shapley value of the squared robust Interval-Mahalanobis distance, enabling efficient computation of variable contributions. This formulation allows for a fine-grained interpretation of outliers, providing a detailed decomposition into contributions from centers, ranges, and cross-terms of the interval-valued observations. Moreover, the Shapley value is closely connected to the concept of cellwise outliers, as it can help identify variable-specific outliers that may not be evident at multivariate level. We further extend the framework through the Shapley interaction index to capture pairwise variable interactions driving atypical behavior. The practical utility of the proposed approach is illustrated through two real-world datasets.

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arXiv2025

Interval Fisher's Discriminant Analysis and Visualisation

In Data Science, entities are typically represented by single valued measurements. Symbolic Data Analysis extends this framework to more complex structures, such as intervals and histograms, that express internal variability. We propose an extension of multiclass Fisher's Discriminant Analysis to interval-valued data, using Moore's interval arithmetic and the Mallows' distance. Fisher's objective function is generalised to consider simultaneously the contributions of the centres and the ranges of intervals and is numerically maximised. The resulting discriminant directions are then used to classify interval-valued observations.To support visual assessment, we adapt the class map, originally introduced for conventional data, to classifiers that assign labels through minimum distance rules. We also extend the silhouette plot to this setting and use stacked mosaic plots to complement the visual display of class assignments. Together, these graphical tools provide insight into classifier performance and the strength of class membership. Applications to real datasets illustrate the proposed methodology and demonstrate its value in interpreting classification results for interval-valued data.

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arXiv2025

Location and association measures for interval-valued data based on Mallows' distance

The growing demand to analyse large and complex datasets has spurred the development of Symbolic Data Analysis as a promising approach to address contemporary data challenges. Amongst these, interval-valued data introduces new theoretical and methodological questions that remain open. In this paper, we generalise measures of location and association for interval-valued random variables using Mallows' distance. Departing from restrictive assumptions such as uniform distributions over microdata, our proposal extends the barycentre approach to any absolutely continuous distribution with finite second moment. A key contribution is the derivation of explicit formulas for Mallows' distance in p-dimensional interval spaces. These formulas decompose into components for centres, ranges, and a novel cross-term that captures their interaction. This decomposition leads to a new theoretical symbolic covariance matrix that explicitly accounts for the dependence between centres and ranges - a relation often obscured in current definitions of symbolic covariance. Theoretical developments are supported by empirical studies on diverse real-world datasets, each reflecting different degrees of information about the underlying microdata. These applications highlight both the flexibility of the proposed methodology and the interpretability of its results.

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arXiv2023

On the trace ratio method and Fisher's discriminant analysis for robust multigroup classification

We compare two different linear dimensionality reduction strategies for the multigroup classification problem: the trace ratio method and Fisher's discriminant analysis. Recently, trace ratio optimization has gained in popularity due to its computational efficiency, as well as the occasionally better classification results. However, a statistical understanding is still incomplete. We study and compare the properties of the two methods. Then, we propose a robust version of the trace ratio method, to handle the presence of outliers in the data. We reinterpret an asymptotic perturbation bound for the solution to the trace ratio, in a contamination setting. Finally, we compare the performance of the trace ratio method and Fisher's discriminant analysis on both synthetic and real datasets, using classical and robust estimators.

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arXiv2021

Association measures for interval variables

Symbolic Data Analysis (SDA) is a relatively new field of statistics that extends conventional data analysis by taking into account intrinsic data variability and structure. Unlike conventional data analysis, in SDA the features characterizing the data can be multi-valued, such as intervals or histograms. SDA has been mainly approached from a sampling perspective. In this work, we propose a model that links the micro-data and macro-data of interval-valued symbolic variables, which takes a populational perspective. Using this model, we derive the micro-data assumptions underlying the various definitions of symbolic covariance matrices proposed in the literature, and show that these assumptions can be too restrictive, raising applicability concerns. We analyze the various definitions using worked examples and four datasets. Our results show that the existence/absence of correlations in the macro-data may not be correctly captured by the definitions of symbolic covariance matrices and that, in real data, there can be a strong divergence between these definitions. Thus, in order to select the most appropriate definition, one must have some knowledge about the micro-data structure.

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arXiv2018

Theoretical Foundations of Forward Feature Selection Methods based on Mutual Information

Feature selection problems arise in a variety of applications, such as microarray analysis, clinical prediction, text categorization, image classification and face recognition, multi-label learning, and classification of internet traffic. Among the various classes of methods, forward feature selection methods based on mutual information have become very popular and are widely used in practice. However, comparative evaluations of these methods have been limited by being based on specific datasets and classifiers. In this paper, we develop a theoretical framework that allows evaluating the methods based on their theoretical properties. Our framework is grounded on the properties of the target objective function that the methods try to approximate, and on a novel categorization of features, according to their contribution to the explanation of the class; we derive upper and lower bounds for the target objective function and relate these bounds with the feature types. Then, we characterize the types of approximations taken by the methods, and analyze how these approximations cope with the good properties of the target objective function. Additionally, we develop a distributional setting designed to illustrate the various deficiencies of the methods, and provide several examples of wrong feature selections. Based on our work, we identify clearly the methods that should be avoided, and the methods that currently have the best performance.

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arXiv2016

Theoretical Evaluation of Feature Selection Methods based on Mutual Information

Feature selection methods are usually evaluated by wrapping specific classifiers and datasets in the evaluation process, resulting very often in unfair comparisons between methods. In this work, we develop a theoretical framework that allows obtaining the true feature ordering of two-dimensional sequential forward feature selection methods based on mutual information, which is independent of entropy or mutual information estimation methods, classifiers, or datasets, and leads to an undoubtful comparison of the methods. Moreover, the theoretical framework unveils problems intrinsic to some methods that are otherwise difficult to detect, namely inconsistencies in the construction of the objective function used to select the candidate features, due to various types of indeterminations and to the possibility of the entropy of continuous random variables taking null and negative values.

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