Inés Couso

A survey of concepts of independence for imprecise probabilities

Our aim in this paper is to clarify the notion of independence for imprecise probabilities. Suppose that two marginal experiments are each described by an imprecise probability model, i.e., by a convex set of probability distributions or an equivalent model such as upper and lower probabilities or previsions. Then there are several ways to define independence of the two experiments and to construct an imprecise probability model for the joint experiment. We survey and compare six definitions of independence. To clarify the meaning of the definitions and the relationships between them, we give simple examples which involve drawing balls from urns. For each concept of independence, we give a mathematical definition, an intuitive or behavioural interpretation, assumptions under which the definition is justified, and an example of an urn model to which the definition is applicable. Each of the independence concepts we study appears to be useful in some kinds of application. The concepts of strong independence and epistemic independence appear to be the most frequently applicable.

Combining GP operators with SA search to evolve fuzzy rule based classifiers

Abstract The genotype–phenotype encoding of fuzzy rule bases in GA, along with their corresponding crossover and mutation operators, can be used by other search schemes, improving the behavior of these last ones. As a practical consequence of this, a simulated annealing-based method for inducting both parameters and structure of a fuzzy classifier has been developed. The adjacency operator in SA has been replaced with a macromutation taken from tree-shaped genotype GAs. We will show that results of SA search are similar to those of GP in both the efficiency of the learned classifiers and in its linguistic interpretability, while the memory consumption of the learning process is lower.

Genetic learning of fuzzy rules based on low quality data

Genetic fuzzy systems (GFS) are based on the use of genetic algorithms for designing fuzzy systems, and for providing them with learning and adaptation capabilities. In this context, fuzzy sets represent linguistic granules of information, contained in the antecedents and consequents of the rules, whereas the data used in the genetic learning is assumed to be crisp. GFS seldom deal with fuzzy-valued data. In this paper we address this problem, and propose a set of techniques that can be incorporated to different GFS in order to learn a knowledge base (KB) from interval and fuzzy data for regression problems. Details will be given about the representation of non-standard data with fuzzy sets, about the needed changes in the reasoning method of the fuzzy rule-based system, and also about a new generalization of the mean squared error to vague data. In addition, we will show that the learning process requires a genetic algorithm that must be capable of optimizing a multicriteria fitness function, containing both crisp and interval-valued criteria. Lastly, we benchmark our procedures with some machine learning related datasets and a real-world problem of marketing, and the techniques proposed here are shown to improve the generalization properties of other KBs obtained from crisp training data.

Joint propagation of probability and possibility in risk analysis: Towards a formal framework

This paper discusses some models of Imprecise Probability Theory obtained by propagating uncertainty in risk analysis when some input parameters are stochastic and perfectly observable, while others are either random or deterministic, but the information about them is partial and is represented by possibility distributions. Our knowledge about the probability of events pertaining to the output of some function of interest from the risk analysis model can be either represented by a fuzzy probability or by a probability interval. It is shown that this interval is the average cut of the fuzzy probability of the event, thus legitimating the propagation method. Besides, several independence assumptions underlying the joint probability-possibility propagation methods are discussed and illustrated by a motivating example.

Advocating the Use of Imprecisely Observed Data in Genetic Fuzzy Systems

In our opinion, and in accordance with current literature, the precise contribution of genetic fuzzy systems to the corpus of the machine learning theory has not been clearly stated yet. In particular, we question the existence of a set of problems for which the use of fuzzy rules, in combination with genetic algorithms, produces more robust models, or classifiers that are inherently better than those arising from the Bayesian point of view. We will show that this set of problems actually exists, and comprises interval and fuzzy valued datasets, but it is not being exploited. Current genetic fuzzy classifiers deal with crisp classification problems, where the role of fuzzy sets is reduced to give a parametric definition of a set of discriminant functions, with a convenient linguistic interpretation. Provided that the customary use of fuzzy sets in statistics is vague data, we propose to test genetic fuzzy classifiers over imprecisely measured data and design experiments well suited to these problems. The same can be said about genetic fuzzy models: the use of a scalar fitness function assumes crisp data, where fuzzy models, a priori, do not have advantages over statistical regression.

Statistical reasoning with set-valued information: Ontic vs. epistemic views

In information processing tasks, sets may have a conjunctive or a disjunctive reading. In the conjunctive reading, a set represents an object of interest and its elements are subparts of the object, forming a composite description. In the disjunctive reading, a set contains mutually exclusive elements and refers to the representation of incomplete knowledge. It does not model an actual object or quantity, but partial information about an underlying object or a precise quantity. This distinction between what we call ontic vs. epistemic sets remains valid for fuzzy sets, whose membership functions, in the disjunctive reading are possibility distributions, over deterministic or random values. This paper examines the impact of this distinction in statistics. We show its importance because there is a risk of misusing basic notions and tools, such as conditioning, distance between sets, variance, regression, etc. when data are set-valued. We discuss several examples where the ontic and epistemic points of view yield different approaches to these concepts.

Higher order models for fuzzy random variables

A fuzzy random variable is viewed as the imprecise observation of the outcomes in a random experiment. Since randomness and vagueness coexist in the same framework, it seems reasonable to integrate fuzzy random variables into imprecise probabilities theory. Nevertheless, fuzzy random variables are commonly presented in the literature as classical measurable functions associated to a classical probability measure. We present here a higher order possibility model that represents the imprecise information provided by a fuzzy random variable. We compare it with previous classical models in the literature. First, some aspects about the acceptability function associated to a fuzzy random variable are investigated. Secondly, we present three different higher order possibility models, all of them arising in a natural way. We investigate their similarities and differences, and observe that the first one (the fuzzy probability envelope) is the most informative. Finally we compare the fuzzy probability envelope with the (classical) probability measure induced by the fuzzy random variable. We conclude that the classical probability measure does not always contain all relevant information provided by a fuzzy random variable.

Divergence measure between fuzzy sets

In this paper we propose a way of measuring the difference between two fuzzy sets by means of a function which we will call divergence. We define this concept by means of a group of natural axioms and we study in detail the most important classes of such measures, those which have the local property.

On the Variability of the Concept of Variance for Fuzzy Random Variables

Fuzzy random variables possess several interpretations. Historically, they were proposed either as a tool for handling linguistic label information in statistics or to represent uncertainty about classical random variables. Accordingly, there are two different approaches to the definition of the variance of a fuzzy random variable. In the first one, the variance of the fuzzy random variable is defined as a crisp number, which makes it easier to handle in further processing. In the second case, the variance is defined as a fuzzy interval, thus offering a gradual description of our incomplete knowledge about the variance of an underlying, imprecisely observed, classical random variable. In this paper, we also discuss another view of fuzzy random variables, which comes down to a set of random variables induced by a fuzzy relation describing an ill-known conditional probability. This view leads to yet another definition of the variance of a fuzzy random variable in the context of the theory of imprecise probabilities. The new variance is a real interval, which achieves a compromise between both previous definitions in terms of representation simplicity. Our main objective is to demonstrate, with the help of simple examples, the practical significance of these definitions of variance induced by various existing views of fuzzy random variables.

Mutual information-based feature selection and partition design in fuzzy rule-based classifiers from vague data

Algorithms for preprocessing databases with incomplete and imprecise data are seldom studied. For the most part, we lack numerical tools to quantify the mutual information between fuzzy random variables. Therefore, these algorithms (discretization, instance selection, feature selection, etc.) have to use crisp estimations of the interdependency between continuous variables, whose application to vague datasets is arguable. In particular, when we select features for being used in fuzzy rule-based classifiers, we often use a mutual information-based ranking of the relevance of inputs. But, either with crisp or fuzzy data, fuzzy rule-based systems route the input through a fuzzification interface. The fuzzification process may alter this ranking, as the partition of the input data does not need to be optimal. In our opinion, to discover the most important variables for a fuzzy rule-based system, we want to compute the mutual information between the fuzzified variables, and we should not assume that the ranking between the crisp variables is the best one. In this paper we address these problems, and propose an extended definition of the mutual information between two fuzzified continuous variables. We also introduce a numerical algorithm for estimating the mutual information from a sample of vague data. We will show that this estimation can be included in a feature selection algorithm, and also that, in combination with a genetic optimization, the same definition can be used to obtain the most informative fuzzy partition for the data. Both applications will be exemplified with the help of some benchmark problems.