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

International audienceIn 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

Reasoning with random sets: An agenda for the future

In this paper, we discuss a potential agenda for future work in the theory of random sets and belief functions, touching upon a number of focal issues: the development of a fully-fledged theory of statistical reasoning with random sets, including the generalisation of logistic regression and of the classical laws of probability; the further development of the geometric approach to uncertainty, to include general random sets, a wider range of uncertainty measures and alternative geometric representations; the application of this new theory to high-impact areas such as climate change, machine learning and statistical learning theory.Comment: 94 pages, 17 figure

A fuzzified BRAIN algorithm for learning DNF from incomplete data

Aim of this paper is to address the problem of learning Boolean functions from training data with missing values. We present an extension of the BRAIN algorithm, called U-BRAIN (Uncertainty-managing Batch Relevance-based Artificial INtelligence), conceived for learning DNF Boolean formulas from partial truth tables, possibly with uncertain values or missing bits. Such an algorithm is obtained from BRAIN by introducing fuzzy sets in order to manage uncertainty. In the case where no missing bits are present, the algorithm reduces to the original BRAIN

Automatic synthesis of fuzzy systems: An evolutionary overview with a genetic programming perspective

Studies in Evolutionary Fuzzy Systems (EFSs) began in the 90s and have experienced a fast development since then, with applications to areas such as pattern recognition, curve‐fitting and regression, forecasting and control. An EFS results from the combination of a Fuzzy Inference System (FIS) with an Evolutionary Algorithm (EA). This relationship can be established for multiple purposes: fine‐tuning of FIS's parameters, selection of fuzzy rules, learning a rule base or membership functions from scratch, and so forth. Each facet of this relationship creates a strand in the literature, as membership function fine‐tuning, fuzzy rule‐based learning, and so forth and the purpose here is to outline some of what has been done in each aspect. Special focus is given to Genetic Programming‐based EFSs by providing a taxonomy of the main architectures available, as well as by pointing out the gaps that still prevail in the literature. The concluding remarks address some further topics of current research and trends, such as interpretability analysis, multiobjective optimization, and synthesis of a FIS through Evolving methods

Building an interpretable fuzzy rule base from data using Orthogonal Least Squares Application to a depollution problem

In many fields where human understanding plays a crucial role, such as bioprocesses, the capacity of extracting knowledge from data is of critical importance. Within this framework, fuzzy learning methods, if properly used, can greatly help human experts. Amongst these methods, the aim of orthogonal transformations, which have been proven to be mathematically robust, is to build rules from a set of training data and to select the most important ones by linear regression or rank revealing techniques. The OLS algorithm is a good representative of those methods. However, it was originally designed so that it only cared about numerical performance. Thus, we propose some modifications of the original method to take interpretability into account. After recalling the original algorithm, this paper presents the changes made to the original method, then discusses some results obtained from benchmark problems. Finally, the algorithm is applied to a real-world fault detection depollution problem.Comment: pre-print of final version published in Fuzzy Sets and System

Knowledge Acquisition from Data Bases

Centre for Intelligent Systems and their ApplicationsGrant No.6897502Knowledge acquisition from databases is a research frontier for both data base technology and machine learning (ML) techniques,and has seen sustained research over recent years.It also acts as a link between the two fields,thus offering a dual benefit. Firstly, since database technology has already found wide application in many fields ML research obviously stands to gain from this greater exposure and established technological foundation. Secondly, ML techniques can augment the ability of existing database systems to represent acquire,and process a collection of expertise such as those which form part of the semantics of many advanced applications (e.gCAD/CAM).The major contribution of this thesis is the introduction of an effcient induction algorithm to facilitate the acquisition of such knowledge from databases. There are three typical families of inductive algorithms: the generalisation- specialisation based AQ11-like family, the decision tree based ID3-like family,and the extension matrix based family. A heuristic induction algorithm, HCV based on the newly-developed extension matrix approach is described in this thesis. By dividing the positive examples (PE) of a specific class in a given example set into intersect in groups and adopting a set of strategies to find a heuristic conjunctive rule in each group which covers all the group's positiv examples and none of the negativ examples(NE),HCV can find rules in the form of variable-valued logic for PE against NE in low-order polynomial time. The rules generated in HCV are shown empirically to be more compact than the rules produced by AQ1-like algorithms and the decision trees produced by the ID3-like algorithms. KEshell2, an intelligent learning database system, which makes use of the HCV algorithm and couples ML techniques with database and knowledgebase technology, is also described

Fuzzy set covering as a new paradigm for the induction of fuzzy classification rules

In 1965 Lofti A. Zadeh proposed fuzzy sets as a generalization of crisp (or classic) sets to address the incapability of crisp sets to model uncertainty and vagueness inherent in the real world. Initially, fuzzy sets did not receive a very warm welcome as many academics stood skeptical towards a theory of imprecise'' mathematics. In the middle to late 1980's the success of fuzzy controllers brought fuzzy sets into the limelight, and many applications using fuzzy sets started appearing. In the early 1970's the first machine learning algorithms started appearing. The AQ family of algorithms pioneered by Ryszard S. Michalski is a good example of the family of set covering algorithms. This class of learning algorithm induces concept descriptions by a greedy construction of rules that describe (or cover) positive training examples but not negative training examples. The learning process is iterative, and in each iteration one rule is induced and the positive examples covered by the rule removed from the set of positive training examples. Because positive instances are separated from negative instances, the term separate-and-conquer has been used to contrast the learning strategy against decision tree induction that use a divide-and-conquer learning strategy. This dissertation proposes fuzzy set covering as a powerful rule induction strategy. We survey existing fuzzy learning algorithms, and conclude that very few fuzzy learning algorithms follow a greedy rule construction strategy and no publications to date made the link between fuzzy sets and set covering explicit. We first develop the theoretical aspects of fuzzy set covering, and then apply these in proposing the first fuzzy learning algorithm that apply set covering and make explicit use of a partial order for fuzzy classification rule induction. We also investigate several strategies to improve upon the basic algorithm, such as better search heuristics and different rule evaluation metrics. We then continue by proposing a general unifying framework for fuzzy set covering algorithms. We demonstrate the benefits of the framework and propose several further fuzzy set covering algorithms that fit within the framework. We compare fuzzy and crisp rule induction, and provide arguments in favour of fuzzy set covering as a rule induction strategy. We also show that our learning algorithms outperform other fuzzy rule learners on real world data. We further explore the idea of simultaneous concept learning in the fuzzy case, and continue to propose the first fuzzy decision list induction algorithm. Finally, we propose a first strategy for encoding the rule sets generated by our fuzzy set covering algorithms inside an equivalent neural network

Fuzzy set covering as a new paradigm for the induction of fuzzy classification rules

In 1965 Lofti A. Zadeh proposed fuzzy sets as a generalization of crisp (or classic) sets to address the incapability of crisp sets to model uncertainty and vagueness inherent in the real world. Initially, fuzzy sets did not receive a very warm welcome as many academics stood skeptical towards a theory of imprecise'' mathematics. In the middle to late 1980's the success of fuzzy controllers brought fuzzy sets into the limelight, and many applications using fuzzy sets started appearing. In the early 1970's the first machine learning algorithms started appearing. The AQ family of algorithms pioneered by Ryszard S. Michalski is a good example of the family of set covering algorithms. This class of learning algorithm induces concept descriptions by a greedy construction of rules that describe (or cover) positive training examples but not negative training examples. The learning process is iterative, and in each iteration one rule is induced and the positive examples covered by the rule removed from the set of positive training examples. Because positive instances are separated from negative instances, the term separate-and-conquer has been used to contrast the learning strategy against decision tree induction that use a divide-and-conquer learning strategy. This dissertation proposes fuzzy set covering as a powerful rule induction strategy. We survey existing fuzzy learning algorithms, and conclude that very few fuzzy learning algorithms follow a greedy rule construction strategy and no publications to date made the link between fuzzy sets and set covering explicit. We first develop the theoretical aspects of fuzzy set covering, and then apply these in proposing the first fuzzy learning algorithm that apply set covering and make explicit use of a partial order for fuzzy classification rule induction. We also investigate several strategies to improve upon the basic algorithm, such as better search heuristics and different rule evaluation metrics. We then continue by proposing a general unifying framework for fuzzy set covering algorithms. We demonstrate the benefits of the framework and propose several further fuzzy set covering algorithms that fit within the framework. We compare fuzzy and crisp rule induction, and provide arguments in favour of fuzzy set covering as a rule induction strategy. We also show that our learning algorithms outperform other fuzzy rule learners on real world data. We further explore the idea of simultaneous concept learning in the fuzzy case, and continue to propose the first fuzzy decision list induction algorithm. Finally, we propose a first strategy for encoding the rule sets generated by our fuzzy set covering algorithms inside an equivalent neural network