Rosanna Verde

Dynamic clustering of interval data using a Wasserstein-based distance

Interval data allow statistical units to be described by means of intervals of values, whereas their representation by means of a single value appears to be too reductive or inconsistent. In the present paper, we present a Wasserstein-based distance for interval data, and we show its interesting properties in the context of clustering techniques. We show that the proposed distance generalizes a wide set of distances proposed for interval data by different approaches or in different contexts of analysis. An application on real data is performed to illustrate the impact of using different metrics and the proposed one using a dynamic clustering algorithm.

A New Wasserstein Based Distance for the Hierarchical Clustering of Histogram Symbolic Data

Symbolic Data Analysis (SDA) aims to to describe and analyze complex and structured data extracted, for example, from large databases. Such data, which can be expressed as concepts, are modeled by symbolic objects described by multivalued variables. In the present paper we present a new distance, based on the Wasserstein metric, in order to cluster a set of data described by distributions with finite continue support, or, as called in SDA, by “histograms”. The proposed distance permits us to define a measure of inertia of data with respect to a barycenter that satisfies the Huygens theorem of decomposition of inertia. We propose to use this measure for an agglomerative hierarchical clustering of histogram data based on the Ward criterion. An application to real data validates the procedure.

Dynamic Clustering of Histogram Data: Using the Right Metric

In this paper we present a review of some metrics to be proposed as allocation functions in the Dynamic Clustering Algorithm (DCA) when data are distribution or histograms of values. The choice of the most suitable distance plays a central role in the DCA because it is related to the criterion function that is optimized. Moreover, it has to be consistent with the prototype which represents the cluster. In such a way, for each proposed metric, we identify the corresponding prototype according to the minimization of the criterion function and then to the best fitting between the partition and the best representation of the clusters. Finally, we focus our attention on a Wassertein based distance showing its optimality in partitioning a set of histogram data with respect to a representation of the clusters by means of their barycenter expressed in terms of distributions.

Comparing Histogram Data Using a Mahalanobis–Wasserstein Distance

In this paper, we present a new distance for comparing data described by histograms. The distance is a generalization of the classical Mahalanobis distance for data described by correlated variables. We define a way to extend the classical concept of inertia and codeviance from a set of points to a set of data described by histograms. The same results are also presented for data described by continuous density functions (empiric or estimated). An application to real data is performed to illustrate the effects of the new distance using dynamic clustering.

A Dynamical Clustering Algorithm for Multi-nominal Data

In this paper we present a dynamical clustering algorithm in order to partition a set of multi-nominal data in k classes. This kind of data can be considered as a particular description of symbolic objects. In this algorithm, the representation of the classes are given by prototypes that generalize the characteristics of the elements belonging to each class. A suitable allocation function (context dependent) is considered in this context to assign an object to a class. The final classes are described by the distributions associated to the multi-nominal variables of the elements belonging to each class. That representation corresponds to the usual description of the so called modal symbolic objects.

Ordinary Least Squares for Histogram Data Based on Wasserstein Distance

Histogram data is a kind of symbolic representation which allows to describe an individual by an empirical frequency distribution. In this paper we introduce a linear regression model for histogram variables. We present a new Ordinary Least Squares approach for a linear model estimation, using the Wasserstein metric between histograms. In this paper we suppose that the regression coefficient are scalar values. After having illustrated the concurrent approaches, we corroborate the proposed estimation method by an application on a real dataset.

Classification and Multivariate Analysis for Complex Data Structures

Key Notes.- Classification and Discrimination.- Data Mining.- Robustness and Classification.- Categorical Data and Latent Class Approach.- Latent Variables and Related Methods.- Symbolic, Multivalued and Conceptual Data Analysis.- Spatial, Temporal, Streaming and Functional Data Analysis.- Bio and Health Science.

Clustering Spatio-functional data: a model based approach

In many environmental sciences, such as, in agronomy, in metereology, in oceanography, data analysis has to take into account both spatial and functional components. In this paper we present a strategy for clustering spatio-functional data. The proposed methodology is based on concepts of spatial statistics theory, such as variogram and covariogram when data are curves. Moreover a summarizing spatio-functional model for each cluster is obtained. The assessment of the method is carried out with a study on real data.

Dynamic clustering of histogram data based on adaptive squared Wasserstein distances

Histogram-valued data are treating differently from bar-count data.A new clustering method for histogram-valued data is proposed.Two adaptive clustering strategy are proposed.A set of quality-of-partition indices are proposed.No other clustering method exist for histogram-valued data. This paper presents a Dynamic Clustering Algorithm for histogram data with an automatic weighting step of the variables by using adaptive distances. The Dynamic Clustering Algorithm is a k-means-like algorithm for clustering a set of objects into a predefined number of classes. Histogram data are realizations of particular set-valued descriptors defined in the context of Symbolic Data Analysis. We propose to use the ? 2 Wasserstein distance for clustering histogram data and two novel adaptive distance based clustering schemes. The ? 2 Wasserstein distance allows to express the variability of a set of histograms in two components: the first related to the variability of their averages and the second to the variability of the histograms related to different size and shape. The weighting step aims to take into account global and local adaptive distances as well as two components of the variability of a set of histograms. To evaluate the clustering results, we extend some classic partition quality indexes when the proposed adaptive distances are used in the clustering criterion function. Examples on synthetic and real-world datasets corroborate the proposed clustering procedure.

Multiclass Generalized Eigenvalue Proximal Support Vector Machines

Support Vector Machines represent state of the art in supervised learning. Recently, the Regularized Generalized Eigenvalue Classifier (ReGEC) extension has been proposed to solve binary classification problems. In the present work we describe MultiReGEC, a novel technique that generalizes ReGEC to multiclass classification problems. This method is based on statistical and geometrical considerations, providing strong fundamentals to the proposed extension. After a detailed description of the MultiReGEC algorithm, we show, through extensive numerical experiments, that the accuracy of the proposed algorithm well compares with other de facto standard techniques.