Cross product kernels for fuzzy set similarity

2020 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2020

This paper explores the connection between fuzzy inference systems and kernel methods. Particularly, we explore the connection between the fuzzy-basis-function expansion of non-singleton fuzzy systems and regularized kernel methods with positive-definite kernels. We show that positive-definite kernel on fuzzy sets, i.e., a real-valued function defined on the set of fuzzy sets, is the core concept enabling such a connection. Furthermore, we rewrite the fuzzy-basis-function (FBF) expansion of non-singleton fuzzy systems in terms of a kernel-basis-function expansion with positive-definite kernels on fuzzy sets. The consequence of doing this is that fuzzy systems can be trained by regularized kernel methods, and the advantages of doing this are: 1) fuzzy systems can have more resistance to the curse of dimensionality, 2) they can explicitly regularize the function being approximated, 3) they can satisfy the Representer Theorem of kernel methods, which bounds the number of learned rules ...

Information Processing and Management of Uncertainty in Knowledge-Based Systems

Convolution kernels are essential tools in signal processing: they are used to filter noisy signal, interpolate discrete signals, .... However, in a given application, it is often hard to select an optimal shape of the kernel. This is why, in practice, it may be useful to possess efficient tools to perform a robustness analysis, talking the form in our case of an imprecise convolution. When convolution kernels are positive, their formal equivalence with probability distributions allows one to use imprecise probability theory to achieve such an imprecise convolution. However, many kernels can have negative values, in which case the previous equivalence does not hold anymore. Yet, we show mathematically in this paper that, while the formal equivalence is lost, the computational tools used to describe sets of probabilities by intervals on the singletons still retain their key properties when used to approximate sets of (possibly) non-positive kernels. We then illustrate their use on a single application that consists of filtering a human electrocardiogram signal by using a low-pass filter whose order is imprecisely known. We show, in this experiment, that the proposed approach leads to tighter bounds than previously proposed approaches.

Research Journal of Applied Sciences, Engineering and Technology, 2016

The characteristics of a fuzzy set are decided by its membership function. This work aims to provide a geometric approach for enhancing the design and performance of fuzzy systems. Similarity Estimator (SimE) evaluates the membership functions of fuzzy sets on Euclidean space based on geometric area. The overlapping regions between the sets are partitioned into geometric structures. The area of overlapping is computed by summing the area of polygons and integrating the area under curves. Similarity between fuzzy sets is directly proportional to the area of overlapping between them. SimE was tested over a range of real numbers with finite intervals. Fuzzy sets using different membership functions were created for the same data distribution. From the test results it can be inferred that fuzzy sets defined using triangular membership functions have a minimum overlapping area when compared to fuzzy sets defined using other membership function. Optimal overlapping area of fuzzy sets improves the semantic representation and the performance of the system. SimE can be used by knowledge engineers to design efficient fuzzy systems.

Reasoning with fuzzy sets can be achieved through measures such as similarity and distance. However, these measures can often give misleading results when considered independently, for example giving the same value for two different pairs of fuzzy sets. This is particularly a problem where many fuzzy sets are generated from real data, and while two different measures may be used to automatically compare such fuzzy sets, it is difficult to interpret two different results. This is especially true where a large number of fuzzy sets are being compared as part of a reasoning system. This paper introduces a method for combining the results of multiple measures into a single measure for the purpose of analysing and comparing fuzzy sets. The combined measure alleviates ambiguous results and aids in the automatic comparison of fuzzy sets. The properties of the combined measure are given, and demonstrations are presented with discussions on the advantages over using a single measure.

The choice of a similarity measure is very important to com-pare between two samples. In addition, the similarity between many fuzzy sets (i.e., many linguistic variables) needs aggregation operators which can influence the fuzzy similarity results. In this paper we com-pare results of fuzzy implications aggregated to a fuzzy similarity mea-sure applied to shapes recognition which are described by an extended curvature scale space (CSS) descriptor. We present experimental results on the vision Speech and Signal Processing Surrey University database.

2009

In this paper, we survey the relationship between the similarity measure and dissimilarity measure for fuzzy sets. First, we design a similarity measure using a distance measure for fuzzy sets and prove its usefulness. From this result, we assert that the similarity between two complementary fuzzy sets satisfies the fuzzy entropy definition. We also show that the summation of the similarity and dissimilarity measures between two membership functions of fuzzy sets constitute all the information of the fuzzy set itself.

Journal of Intelligent & Fuzzy Systems, 2017

An important problem in fuzzy theory is how to determine if two elements are similar or not. There are different definitions to measure how similar or close are elements. One of the most important concepts in this context is that one of similarity. This paper aims at studying fuzzy similarities defined by fuzzy implications, logical equivalences expressed by fuzzy bi-implication operators and aggregation operators. We propose a definition of proximity between two fuzzy finite sets by using the previous operators and we study when this type of operators satisfy the formal definition of similarity.

Studies in Classification, Data Analysis, and Knowledge Organization, 2008

We consider distance-based similarity measures for real-valued vectors of interest in kernel-based machine learning algorithms. In particular, a truncated Euclidean similarity measure and a self-normalized similarity measure related to the Canberra distance. It is proved that they are positive semi-definite (p.s.d.), thus facilitating their use in kernel-based methods, like the Support Vector Machine, a very popular machine learning tool. These kernels may be better suited than standard kernels (like the RBF) in certain situations, that are described in the paper. Some rather general results concerning positivity properties are presented in detail as well as some interesting ways of proving the p.s.d. property.

SAMI 2005, 3rd Slovakian-Hungarian Joint Symposium on Applied Machine Intelligence, 2005

In case of fuzzy reasoning in sparse fuzzy rule bases, the question of selecting the suitable fuzzy similarity measure is essential. The rule antecedents of the sparse fuzzy rule bases are not fully covering the input universe therefore fuzzyreasoning methods applied for sparse fuzzy rule bases requires similaritymeasures able to distinguish the similarity of non-overlapping fuzzy sets, too. The goal of this paper is enumerating some of these distance based similaritymeasures and briefly introducing them. Keywords: similarity measure, distance of fuzzy sets, vague environment

2018 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2018

Similarity measures are useful for reasoning about fuzzy sets. Hence, many classical set-theoretic similarity measures have been extended for comparing fuzzy sets. In previous work, a set-theoretic similarity measure considering the bidirectional subsethood for intervals was introduced. The measure addressed specific concerns of many common similarity measures, and it was shown to be bounded above and below by Jaccard and Dice measures respectively. Herein, we extend our prior measure from similarity on intervals to fuzzy sets. Specifically, we propose a vertical-slice extension where two fuzzy sets are compared based on their membership values. We show that the proposed extension maintains all common properties (i.e., reflexivity, symmetry, transitivity, and overlapping) of the original fuzzy similarity measure. We demonstrate and contrast its behaviour along with common fuzzy set-theoretic measures using different types of fuzzy sets (i.e., normal, non-normal, convex, and non-convex) in respect to different discretization levels.