# Linear regression for numeric symbolic variables: a least squares approach based on Wasserstein Distance

@article{Irpino2015LinearRF,
title={Linear regression for numeric symbolic variables: a least squares approach based on Wasserstein Distance},
author={Antonio Irpino and Rosanna Verde},
journal={Advances in Data Analysis and Classification},
year={2015},
volume={9},
pages={81-106}
}
• Published 7 February 2012
• Computer Science, Mathematics
• Advances in Data Analysis and Classification
In this paper we present a new linear regression technique for distributional symbolic variables, i.e., variables whose realizations can be histograms, empirical distributions or empirical estimates of parametric distributions. Such data are known as numerical modal data according to the Symbolic Data Analysis definitions. In order to measure the error between the observed and the predicted distributions, the $$\ell _2$$ℓ2 Wasserstein distance is proposed. Some properties of such a metric are…
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## References

SHOWING 1-10 OF 39 REFERENCES
Univariate and Multivariate Linear Regression Methods to Predict Interval-Valued Features
• Mathematics
Australian Conference on Artificial Intelligence
• 2004
Two new approaches to fit a linear regression model on interval-valued data are introduced and 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.
A new linear regression model for histogram-valued variables
• Mathematics
• 2011
In classical data analysis, each individual takes one single “value” on each descriptive variable. Symbolic Data Analysis ([Bock and Diday (2000)], [Billard and Diday (2007)]) generalizes this
Ordinary Least Squares for Histogram Data Based on Wasserstein Distance
• Computer Science, Mathematics
COMPSTAT
• 2010
A linear regression model for histogram variables is introduced and a new Ordinary Least Squares approach for a linear model estimation, using the Wasserstein metric between histograms is presented, assuming that the regression coefficient are scalar values.
A New Wasserstein Based Distance for the Hierarchical Clustering of Histogram Symbolic Data
• Computer Science
Data Science and Classification
• 2006
A new distance is presented, 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”, a measure of inertia of data with respect to a barycenter that satisfies the Huygens theorem of decomposition of inertia.
Descriptive Statistics for Symbolic Data
• Mathematics
• 2000
The intention of this chapter is to extend the concept of frequency distribution, and the standard definitions of descriptive statistics for real-valued data, such as the empirical mean the empirical
Symbolic Data Analysis: Conceptual Statistics and Data Mining (Wiley Series in Computational Statistics)
• Computer Science
• 2007
This chapter discusses Descriptive Statistics: Two or More Variates, which focuses on the part of the model concerned with Hierarchy-Divisive Clustering and Cluster Analysis.
Regression Shrinkage and Selection via the Lasso
A new method for estimation in linear models called the lasso, which minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant, is proposed.