Linear regression for numeric symbolic variables: an ordinary least squares approach based on Wasserstein Distance

In this paper we present a linear regression model for modal symbolic data. The observed variables are histogram variables according to the definition given in the framework of Symbolic Data Analysis and the parameters of the model are estimated using the classic Least Squares method. An appropriate metric is introduced in order to measure the error between the observed and the predicted distributions. In particular, the Wasserstein distance is proposed. Some properties of such metric are exploited to predict the response variable as direct linear combination of other independent histogram variables. Measures of goodness of fit are discussed. An application on real data corroborates the proposed method

Trajectory clustering using adaptive squared distances

The paper deals with the clustering of trajectories of moving objects. A k-means-like algorithm based on a Euclidean distance between piece-wise linear curves is used. The main novelty of the paper is the opportunity of considering in the clustering procedure a step that automatically weights the importance of subtrajectories of the original ones. The algorithm uses an adaptive distances approach and a cluster-wise weighting. The proposed algorithm is tested against some workbench trajectory datasets

‘‘Spaghetti’’ PCA analysis: An extension of principal components analysis to time dependent interval data

We introduce a special type of interval description depending on time. Each observation is characterized by an oriented interval of values with a starting and an ending value for each period of observation. Several factorial techniques have been developed in order to treat interval data, but not yet for oriented intervals. We present an extension of PCA to time dependent interval data, or, in general, to oriented intervals

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 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 con-text of clustering techniques. We show that the proposed distance generalizes a wide set of distances pro-posed 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