Abstract
We propose a new linear regression model for interval-valued variables. The model uses quantile functions to represent the intervals, thereby considering the distributions within them. In this paper we study the special case where the Uniform distribution is assumed in each observed interval, and we analyze the extension to the Symmetric Triangular distribution. The parameters of the model are obtained solving a constrained quadratic optimization problem that uses the Mallows distance between quantile functions. As in the classical case, a goodness-of-fit measure is deduced. Two applications on up-to-date fields are presented: one predicting duration of unemployment and the other allowing forecasting burned area by forest fires.

Abstract
Histogram-valued variables are a particular kind of variables studied in Symbolic Data Analysis where to each entity under analysis corresponds a distribution that may be represented by a histogram or by a quantile function. Linear regression models for this type of data are necessarily more complex than a simple generalization of the classical model: the parameters cannot be negative; still the linear relation between the variables must be allowed to be either direct or inverse. In this work, we propose a new linear regression model for histogram-valued variables that solves this problem, named Distribution and Symmetric Distribution Regression Model. To determine the parameters of this model, it is necessary to solve a quadratic optimization problem, subject to non-negativity constraints on the unknowns; the error measure between the predicted and observed distributions uses the Mallows distance. As in classical analysis, the model is associated with a goodness-of-fit measure whose values range between 0 and 1. Using the proposed model, applications with real and simulated data are presented.

Abstract
Experimental incapacity to track microbe-microbe interactions in structures like biofilms, and the complexity inherent to the mathematical modelling of those interactions, raises the need for feasible, alternative modelling approaches. This work proposes an agent-based representation of the diffusion of N-acyl homoserine lactones (AHL) in a multicellular environment formed by Pseudomonas aeruginosa and Candida albicans. Depending on the spatial location, C. albicans cells were variably exposed to AHLs, an observation that might help explain why phenotypic switching of individual cells in biofilms occurred at different time points. The simulation and algebraic results were similar for simpler scenarios, although some statistical differences could be observed (p<0.05). The model was also successfully applied to a more complex scenario representing a small multicellular environment containing C. albicans and P. aeruginosa cells encased in a 3-D matrix. Further development of this model may help create a predictive tool to depict biofilm heterogeneity at the single-cell level.

Abstract
We address classification of distributional data, where units are described by histogram or interval-valued variables. The proposed approach uses a linear discriminant function where distributions or intervals are represented by quantile functions, under specific assumptions. This discriminant function allows defining a score for each unit, in the form of a quantile function, which is used to classify the units in two a priori groups, using the Mallows distance. There is a diversity of application areas for the proposed linear discriminant method. In this work we classify the airline companies operating in NY airports based on air time and arrival/departure delays, using a full year flights.

Abstract

Abstract