As a common problem in data clustering applications, how to identify a suitable proximity measure between data instances is still an open problem. Especially when interval-valued data is becoming more and more popular, it is expected to have a suitable distance for intervals. Existing distance measures only consider the lower and upper bounds of intervals, but overlook the overlapped area betwe...

This paper employs the axiomatic approach underpinning the literature on income inequality measurement to analyze measures of dispersion in interval data. We …nd that some widely employed measures fail to properly measure dispersion when data are not of the ratio type. We go on to prove that, under reasonable conditions, variance is the only decomposable measure that can be used to consistently...

In this paper we address the problem of clustering interval data, adopting a model-based approach. To this purpose, parametric models for intervalvalued variables are used which consider configurations for the variancecovariance matrix that take the nature of the interval data directly into account. Results, both on synthetic and empirical data, clearly show the well-founding of the proposed ap...

In this paper we present a mathematical model for archetypal analysis of data represented by means of intervals of real numbers. We extend the model for single-valued data proposed in the pioneering work of Cutler and Breiman on this topic. The core problem is a non-convex optimization one, which we solve by means of a sequential quadratic programming method. We show numerical experiments perfo...

We consider a binary, linear classification problem in which the data points are assumed to be unknown, but bounded within given hyper-rectangles, i.e., the covariates are bounded within intervals explicitly given for each data point separately. We address the problem of designing a robust classifier in this setting by minimizing the worst-case value of a given loss function, over all possible ...

In this paper we discuss some issues which arise when applying classical data analysis techniques to interval data, focusing on the notions of dispersion, association and linear combinations of interval variables. We present some methods that have been proposed for analysing this kind of data, namely for clustering, discriminant analysis, linear regression and interval time series analysis.

IXSQL, an extension to SQL, is proposed for the management of interval data. IXSQL is syntactically and semantically upwards consistent with SQL2. Its specification has been based both on theoretical results and actual user requirements for the management of temporal data, a special case of interval data. Design decisions and implementation issues are also discussed.