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Aisbett J.,University of Newcastle | Rickard J.T.,Distributed Infinity Inc | Morgenthaler D.,Lockheed Martin
Fuzzy Sets and Systems | Year: 2011

This paper explores the link between type-2 fuzzy sets and multivariate modeling. Elements of a space X are treated as observations fuzzily associated with values in a multivariate feature space. A category or class is likewise treated as a fuzzy allocation of feature values (possibly dependent on values in X). We observe that a type-2 fuzzy set on X generated by these two fuzzy allocations captures imprecision in the class definition and imprecision in the observations. In practice many type-2 fuzzy sets are in fact generated in this way and can therefore be interpreted as the output of a classification task. We then show that an arbitrary type-2 fuzzy set can be so constructed, by taking as a feature space a set of membership functions on X. This construction presents a new perspective on the Representation Theorem of Mendel and John. The multivariate modeling underpinning the type-2 fuzzy sets can also constrain realizable forms of membership functions. Because averaging operators such as centroid and subsethood on type-2 fuzzy sets involve a search for optima over membership functions, constraining this search can make computation easier and tighten the results. We demonstrate how the construction can be used to combine representations of concepts and how it therefore provides an additional tool, alongside standard operations such as intersection and subsethood, for concept fusion and computing with words. © 2010 Elsevier B.V. All rights reserved.

Rickard J.T.,Distributed Infinity Inc | Aisbett J.,University of Newcastle | Yager R.R.,The College of New Rochelle
IEEE Transactions on Fuzzy Systems | Year: 2015

We introduce a new structure for fuzzy cognitive maps (FCM) where the traditional fan-in structure involving an inner product followed by a squashing function to describe the causal influences of antecedent nodes to a particular consequent node is replaced with a weighted mean type operator. In this paper, we employ the weighted power mean (WPM). Through appropriate selection of the weights and exponents in the WPM operators, we can both account for the relative importance of different antecedent nodes in the dynamics of a particular node, as well as take a perspective ranging continuously from the most pessimistic (minimum) to the most optimistic (maximum) on the normalized aggregation of antecedents for each node. We consider this FCM structure to be more intuitive than the traditional one, as the nonlinearity involved in the WPM is more scrutable with regard to the aggregation of its inputs. We provide examples of this new FCM structure to illustrate its behavior, including its convergence, and compare it with a traditional FCM architecture on a scenario presented in previous works. © 2015 IEEE.

Aisbett J.,University of Newcastle | Rickard J.T.,Distributed Infinity Inc | Morgenthaler D.G.,Lockheed Martin
IEEE Transactions on Fuzzy Systems | Year: 2010

For many readers and potential authors, type-2 (T2) fuzzy sets might be more readily understood if expressed by the use of standard mathematical notation and terminology. This paper, therefore, translates constructs associated with T2 fuzzy sets to the language of functions on spaces. Such translations may encourage researchers in different disciplines to investigate T2 fuzzy sets, thereby potentially broadening their application and strengthening the underlying theory. © 2006 IEEE.

Aisbett J.,University of Newcastle | Rickard J.T.,Distributed Infinity Inc
IEEE Transactions on Fuzzy Systems | Year: 2014

Centroids are practically important in type-1 and type-2 fuzzy logic systems as a method of defuzzification and type reduction. However, computational problems arise when membership functions (MF) have singleton spikes. The novel thresholding aggregation operators that were described in our companion paper ' New Classes of Threshold Aggregation Functions Based Upon the Tsallis q-Exponential with Applications to Perceptual Computing' produce such MFs with spikes. Such spikes may occur when modeling concepts defined on a real-valued domain, and they are also formed in unions of fuzzy sets in which some have MFs with discrete support and others have support defined on an interval. This paper presents a modified definition of the centroid of a fuzzy set that avoids the computational problems associated with the usual definition and reduces to this definition when MFs are continuous and normal (i.e., of unit height) on some interval. We also present an enhanced Karnik-Mendel-type algorithm to compute the modified centroid of interval type-2 fuzzy sets whose MFs have spikes. © 2014 IEEE.

Rickard J.T.,Distributed Infinity Inc | Aisbett J.,University of Newcastle
IEEE Transactions on Fuzzy Systems | Year: 2014

We introduce two new classes of single-parameter aggregation functions based upon the Tsallis $q$-exponential (QE) function of nonextensive statistical mechanics. These aggregation functions (denoted QE aggregation) facilitate simple modeling of the common human reasoning trait of 'threshold' inference, where either 1) at least one input must exceed a threshold in order to achieve a nonzero aggregation output; or 2) if any one of the inputs exceeds a different threshold, the aggregation output takes its maximum value. We illustrate the thresholding behavior of these functions on interval type-2 fuzzy inputs using an example known in the literature as the Investment Judgment Advisor. We believe that the new QE class of aggregation operators will prove useful in extending the range of options available for the design of perceptual computing systems. © 2014 IEEE.

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