An Ordinal Approach to Risk Measurement

In this short note, we aim at a qualitative framework for modeling multivariate risk. To this extent, we consider completely distributive lattices as underlying universes, and make use of lattice functions to formalize the notion of risk measure. Several properties of risk measures are translated into this general setting, and used to provide axiomatic characterizations. Moreover, a notion of quantile of a lattice-valued random variable is proposed, which shown to retain several desirable properties of its real-valued counterpart.lattice; risk measure; Sugeno integral; quantile.

How regular can maxitive measures be?

We examine domain-valued maxitive measures defined on the Borel subsets of a topological space. Several characterizations of regularity of maxitive measures are proved, depending on the structure of the topological space. Since every regular maxitive measure is completely maxitive, this yields sufficient conditions for the existence of a cardinal density. We also show that every outer-continuous maxitive measure can be decomposed as the supremum of a regular maxitive measure and a maxitive measure that vanishes on compact subsets under appropriate conditions.Comment: 24 page

Report of Meeting: The Fifteenth Katowice–Debrecen Winter Seminar Będlewo (Poland), January 28–31, 2015

Sprawozdanie z konferencji: The Fifteenth Katowice–Debrecen Winter Seminar Będlewo (Poland), January 28–31, 2015

Data granulation by the principles of uncertainty

Researches in granular modeling produced a variety of mathematical models, such as intervals, (higher-order) fuzzy sets, rough sets, and shadowed sets, which are all suitable to characterize the so-called information granules. Modeling of the input data uncertainty is recognized as a crucial aspect in information granulation. Moreover, the uncertainty is a well-studied concept in many mathematical settings, such as those of probability theory, fuzzy set theory, and possibility theory. This fact suggests that an appropriate quantification of the uncertainty expressed by the information granule model could be used to define an invariant property, to be exploited in practical situations of information granulation. In this perspective, a procedure of information granulation is effective if the uncertainty conveyed by the synthesized information granule is in a monotonically increasing relation with the uncertainty of the input data. In this paper, we present a data granulation framework that elaborates over the principles of uncertainty introduced by Klir. Being the uncertainty a mesoscopic descriptor of systems and data, it is possible to apply such principles regardless of the input data type and the specific mathematical setting adopted for the information granules. The proposed framework is conceived (i) to offer a guideline for the synthesis of information granules and (ii) to build a groundwork to compare and quantitatively judge over different data granulation procedures. To provide a suitable case study, we introduce a new data granulation technique based on the minimum sum of distances, which is designed to generate type-2 fuzzy sets. We analyze the procedure by performing different experiments on two distinct data types: feature vectors and labeled graphs. Results show that the uncertainty of the input data is suitably conveyed by the generated type-2 fuzzy set models.Comment: 16 pages, 9 figures, 52 reference

Scientific Endeavors of A.M. Mathai: An Appraisal on the Occasion of his Eightieth Birthday, April 2015

A.M. Mathai is Emeritus Professor of Mathematics and Statistics at McGill University, Canada, and Director of the Centre for Mathematical and Statistical Sciences, India. He has published over 300 research papers and more than 25 books on topics in mathematics, statistics, physics, astrophysics, chemistry, and biology. He is a Fellow of the Institute of Mathematical Statistics, National Academy of Sciences of India, President of the Mathematical Society of India, and a Member of the International Statistical Institute. He is the founder of the Canadian Journal of Statistics and the Statistical Society of Canada. He is instrumental in the implementation of the United Nations Basic Space Science Initiative. The paper is an attempt to capture the broad spectrum of scientific endeavors of Professor A.M. Mathai at the occasion of his anniversary.Comment: 21 pages, LaTe

Characterizing and Reasoning about Probabilistic and Non-Probabilistic Expectation

Expectation is a central notion in probability theory. The notion of expectation also makes sense for other notions of uncertainty. We introduce a propositional logic for reasoning about expectation, where the semantics depends on the underlying representation of uncertainty. We give sound and complete axiomatizations for the logic in the case that the underlying representation is (a) probability, (b) sets of probability measures, (c) belief functions, and (d) possibility measures. We show that this logic is more expressive than the corresponding logic for reasoning about likelihood in the case of sets of probability measures, but equi-expressive in the case of probability, belief, and possibility. Finally, we show that satisfiability for these logics is NP-complete, no harder than satisfiability for propositional logic.Comment: To appear in Journal of the AC