Catálogo de publicaciones - libros

Uncertainty is an attribute of information. The path-breaking work of Shannon has led to a universal acceptance of the thesis that information is statistical in nature. Concomitantly, existing theories of uncertainty are based on probability theory. The generalized theory of uncertainty (GTU) departs from existing theories in essential ways. First, the thesis that information is statistical in nature is replaced by a much more general thesis that information is a generalized constraint, with statistical uncertainty being a special, albeit important case. Equating information to a generalized constraint is the fundamental thesis of GTU.

In this lecture, I expound and comment on a model, or even more ambitiously, a theory, for representing, and drawing inferences from, vague probability assessments. The details of this theory have been published in two papers, the first [3] dealing with its behavioural underpinnings, and the second [1, 2] with its deeper mathematical aspects.

The narrowly defined technical problems that occupied civil engineers during the last century and a half, such as the mechanics of the materials steel, concrete and water, have for most practical purposes been solved. The outstanding challenges relate to interactions between technological systems, the natural environment and human society, at a range of scales up to the global. Management of these coupled systems is obviously a problem of decision making under uncertainty, informed by, on the one hand, sometimes quite dense datasets but, on the other, perhaps only the vaguest of intuitions about the behaviour of the systems in question. An extension of the scope of engineering from a narrowly focussed technical activity to one that more consciously engages with society and the natural environment means that approaches based upon the strictures of individual decision rationality may have to be modified as part of collective and perhaps highly contested decision processes.

In many real-life situations, we are interested in the value of a physical quantity that is difficult or impossible to measure directly. Examples of such quantities are the distance to a star and the amount of oil in a given well. Since we cannot measure directly, a natural idea is to measure . Specifically, we find some easier-to-measure quantities ,…, which are related to by a known relation = (,…,); this relation may be a simple functional transformation, or complex algorithm (e.g., for the amount of oil, numerical solution to an inverse problem).

Statistical analysis of dependencies existing in data sets is now one of the most important applications of statistics. It is also a core part of data mining - a rapidly developing in recent years part of information technology. Statistical methods that have been proposed for the analysis of dependencies in data sets can be roughly divided into two groups: tests of statistical independence and statistical measures of the strength of dependence.

The conditional variance of random variables plays an important role for wellknown variance decomposition formulas. In this paper, the conditional variance for fuzzy random variables and some properties of it are considered. Moreover possible applications of the variance decomposition formula are presented.

The is a fundamental concept in statistics. Specifying the density function of a random variable on gives a natural description of the distribution of on the universe . When it cannot be specified, an estimate of this density may be performed by using a sample of n observations independent and identically distributed (,…,) of .

Three methods are outlined aiming at the (partial) ranking of the elements of a poset respecting the poset order. First, we obtain a partial ranking by applying first degree stochastic dominance to the rank probability distributions of the poset elements. Then we use the minimum, product or copula as an artefact for pairwisely coupling rank probability distributions into bivariate distributions. They serve as a basis for generating a probabilistic relation which constitutes a graded version of stochastic dominance. The transitivity of the probabilistic relations being characterizable in the framework of cycle-transitivity, a cutting level is specified that provides us with a strict partial order. Finally, we apply the graded stochastic dominance principle directly to the mutual rank probabilities. Based on exhaustive experiments, a conjecture on the transitivity of the associated probabilistic relation is made, and a (partial) ranking of the poset elements is extracted.

We show that the best test for fuzzy hypotheses in the Bayesian framework is equivalent to Neyman-Pearson lemma in the classical statistics.

As a matter of fact in many real situations uncertainty is not only present in form of randomness (stochastic uncertainty) but also in form of fuzziness (imprecision), for instance due to the inexactness of measurements of continuous quantities. From the probabilistic point of view the unavoidable fuzziness of measurements has (amongst others) the following far-reaching consequence: According to the classical Strong Law of Large Numbers (SLLN), the probability of an event can be regarded as the limit of the relative frequencies of induced by a sequence of identically distributed, independent, integrable random variables () (with probability one).