Francisco de A. T. de Carvalho

Clustering of interval data based on city-block distances

The recording of interval data has become a common practice with the recent advances in database technologies. This paper introduces clustering methods for interval data based on the dynamic cluster algorithm. Two methods are considered: one with adaptive distances and the other without.

Proximity Coefficients between Boolean symbolic objects

The aim of this paper is to present an approach to calculate the proximity between Boolean symbolic objects (BSO), that take into account simultaneously the variability, as range of values, and some kinds of logical dependencies between variables. A BSO is described by a logical conjunction of properties, each property being a disjunction of values on a variable. Our approach is based on both a comparison function and an aggregation fuction A comparison function is a proximity index based on a positive measure, called description potential of a Boolean elementary event (cardinal of the disjunction of values on a variable of a BSO), and on the proximity indices related to data matrix of binary variables. An aggregation function is a proximity index, related to Minkowsky distance, that aggregate the p results given by the comparison functions.

Selected Contributions in Data Analysis and Classification

This volume presents recent methodological developments in data analysis and classification. A wide range of topics is covered that includes methods for classification and clustering, dissimilarity analysis, graph analysis, consensus methods, conceptual analysis of data, analysis of symbolic data, statistical multivariate methods, data mining and knowledge discovery in databases. Besides structural and theoretical results, the book presents a wide variety of applications, in fields such as biology, micro-array analysis, cyber traffic, bank fraud detection, and text analysis. Combining new methodological advances with a wide variety of real applications, this volume is certainly of special value for researchers and practitioners, providing new analytical tools that are useful in theoretical research and daily practice in classification and data analysis.

A new method to fit a linear regression model for interval-valued data

This paper introduces a new approach to fit a linear regression model on interval-valued data. Each example of the learning set is described by a feature vector where each feature value is an interval. In the proposed approach, it is fitted two linear regression models, respectively, on the mid-point and range of the interval values assumed by the variables on the learning set. The prediction of the lower and upper bound of the interval value of the dependent variable is accomplished from its mid-point and range which are estimated from the fitted linear regression models applied to the mid-point and range of each interval values of the independent variables. The evaluation of the proposed prediction method is based on the estimation of the average behaviour of root mean squared error and of the determination coefficient in the framework of a Monte Carlo experience in comparison with the method proposed by Billard and Diday [3].

Extension based proximities between constrained Boolean symbolic objects

In conventional exploratory data analysis each variable takes a single value. In real life applications, the data will be more general spreading from single values to interval or set of values and including constraints between variables. Such data set are identified as Boolean symbolic data. The purpose of this paper is to present two extension based approaches to calculate proximities between constrained Boolean symbolic objects. Both approaches compares a pair of these objects at the level of the whole set of variables by functions based on the description potential of its join, union and conjunctions. The first comparison function is inspired on a function proposed by Ichino and Yaguchi (1994) while the others are based on the proximity indices related to arrays of binary variables.

Statistical proximity functions of Boolean symbolic objects based on histograms

In this paper we make a synthesis between the Ichino and Yaguchi (1994) and Moore (1991) metrics to obtain a new logical proximity function between Boolean symbolic objects. Then, we use histograms defined on these objects to obtain a statistical one. For both logical and statistical proximity functions, we study its properties and we present examples to illustrate the usefulness of our approach.

Univariate and multivariate linear regression methods to predict interval-valued features

This paper introduces two new approaches to fit a linear regression model on interval-valued data Each example of the learning set is described by a feature vector where each feature value is an interval In the first proposed approach, it is fitted two independent linear regression models, respectively, on the mid-point and range of the interval values assumed by the variables on the learning set In the second approach, is fitted a multivariate linear regression models on these mid-point and range The prediction of the lower and upper bound of the interval value of the dependent variable is accomplished from its mid-point and range which are estimated from the fitted linear regression models applied to the mid-point and range of each interval values of the independent variables The evaluation of the proposed prediction methods is based on the average behavior of the root mean squared error and the determination coefficient in the framework of a Monte Carlo experiment in comparison with the method proposed by Billard and Diday [2].

Applying constrained linear regression models to predict interval-valued data

Billard and Diday [2] were the first to present a regression method for interval-value data. De Carvalho et al [5] presented a new approach that incorporated the information contained in the ranges of the intervals and that presented a better performance when compared with the Billard and Diday method. However, both methods do not guarantee that the predicted values of the lower bounds (ŷLi) will be lower than the predicted values of the upper bounds (ŷUi). This paper presents two approaches based on regression models with inequality constraints that guarantee the mathematical coherence between the predicted values ŷLi and ŷUi. The performance of these approaches, in relation with the methods proposed by Billard and Diday [2] and De Carvalho et al [2], will be evaluated in framework of Monte Carlo experiments.

Classification of SAR Images Through a Convex Hull Region Oriented Approach

This paper presents a new symbolic classifier based on a region oriented approach. Concerning the learning step, each class is described by a region (or a set of regions) in R p defined by the convex hull of the objects belonging to this class. In the allocation step, the assignment of a new object to a class is based on a dissimilarity matching function which compares the class description (a region or a set of regions) with a point in R p . To show the usefulness of this approach, experiments with simulated SAR images were considered. The evaluation of the proposed classifier is based on the prediction accuracy and it is achieved in the framework of a Monte Carlo experience.

Clustering of Interval-Valued Data Using Adaptive Squared Euclidean Distances

This paper presents a clustering method for interval-valued data using a dynamic cluster algorithm with adaptive squared Euclidean distances. This method furnishes a partition and a prototype to each cluster by optimizing an adequacy criterion that measures the fitting between the clusters and their representatives. To compare a class with its representative, the method uses an adaptive version of a squared Euclidean distance to interval-valued data. Experiments with real and artificial interval-valued data sets shows the usefulness of the this method.