fuzzycreator: a python-based toolkit for automatically generating and analysing data-driven fuzzy sets

This paper presents a toolkit for automatic generation and analysis of fuzzy sets (FS) from data. Toolkits are vital for the wider dissemination, accessibility and implementation of theoretic work and applications on FSs. There are currently several toolkits in the literature that focus on knowledge representation and fuzzy inference, but there are few that focus on the automatic generation and comparison of FSs. As there are several methods of constructing FSs from data, it is important to have the tools to use these methods. This paper presents an open-source, python-based toolkit, named fuzzycreator, that facilitates the creation of both conventional and non-conventional (non-normal and non-convex) type-1, interval type-2 and general type-2 FSs from data. These FSs may then be analysed and compared through a series of tools and measures (included in the toolkit), such as evaluating their similarity and distance. An overview of the key features of the toolkit are given and demonstrations which provide rapid access to cutting-edge methodologies in FSs to both expert and non-expert users

SyFSeL: generating synthetic fuzzy sets made simple

Empirical tests can help determine if methods developed for fuzzy sets work correctly. However, finding a large enough data set with suitable properties to conduct thorough tests can be challenging. This paper presents a new library named SyFSeL (Synthetic Fuzzy Set Library) which automatically generates synthetic fuzzy sets with specified characteristics and fuzzy set type. SyFSeL generates as many sets as desired, with adjustable parameters to enable users to emulate real data. Generated fuzzy sets are exported so users can import them into their own fuzzy systems software. SyFSeL can also create graphical plots of the generated sets, examples of which are shown in this paper. The library is cross-platform and open-source under the GNU General Public License, and users are free to develop upon and adapt the code. However, SyFSeL has been designed so that no understanding of the code is required to use it

On the Choice of Similarity Measures for Type-2 Fuzzy Sets

Similarity measures are among the most common methods of comparing type-2 fuzzy sets and have been used in numerous applications. However, deciding how to measure similarity and choosing which existing measure to use can be difficult. Whilst some measures give results that highly correlate with each other, others give considerably different results. We evaluate all of the current similarity measures on type-2 fuzzy sets to discover which measures have common properties of similarity and, for those that do not, we discuss why the properties are different, demonstrate whether and what effect this has in applications, and discuss how a measure can avoid missing a property that is required. We analyse existing measures in the context of computing with words using a comprehensive collection of data-driven fuzzy sets. Specifically, we highlight and demonstrate how each method performs at clustering words of similar meaning

Extending Similarity Measures of Interval Type-2 Fuzzy Sets to General Type-2 Fuzzy Sets

Similarity measures provide one of the core tools that enable reasoning about fuzzy sets. While many types of similarity measures exist for type-1 and interval type-2 fuzzy sets, there are very few similarity measures that enable the comparison of general type-2 fuzzy sets. In this paper, we introduce a general method for extending existing interval type-2 similarity measures to similarity measures for general type-2 fuzzy sets. Specifically, we show how similarity measures for interval type-2 fuzzy sets can be employed in conjunction with the zSlices based general type-2 representation for fuzzy sets to provide measures of similarity which preserve all the common properties (i.e. reflexivity, symmetry, transitivity and overlapping) of the original interval type-2 similarity measure. We demonstrate examples of such extended fuzzy measures and provide comparisons between (different types of) interval and general type-2 fuzzy measures.Comment: International Conference on Fuzzy Systems 2013 (Fuzz-IEEE 2013

SyFSeL: generating synthetic fuzzy sets made simple

Empirical tests can help determine if methods developed for fuzzy sets work correctly. However, finding a large enough data set with suitable properties to conduct thorough tests can be challenging. This paper presents a new library named SyFSeL (Synthetic Fuzzy Set Library) which automatically generates synthetic fuzzy sets with specified characteristics and fuzzy set type. SyFSeL generates as many sets as desired, with adjustable parameters to enable users to emulate real data. Generated fuzzy sets are exported so users can import them into their own fuzzy systems software. SyFSeL can also create graphical plots of the generated sets, examples of which are shown in this paper. The library is cross-platform and open-source under the GNU General Public License, and users are free to develop upon and adapt the code. However, SyFSeL has been designed so that no understanding of the code is required to use it

Maximal functions and related weight classes

The famous result of Muckenhoupt on the connection between weights $\omega$ in $A_p$-classes and the boundedness of the maximal operator in $L_p(\omega)$ is extended to the case $p=\infty$ by the introduction of the geometrical maximal operator. Estimates of the norm of the maximal operators are given in terms of the $A_p$-constants. The equality of two differently defined $A_{\infty}$-constants is proved. Thereby an answer is given to a question posed by R. Johnson. For non-increasing functions on the positive real line a parallel theory to the $A_p$-theory is established for the connection between weights in $B_p$-classes and maximal functions, thereby extending and developing the recent results of Ari o and Muckenhoupt

Extending similarity measures of interval type-2 fuzzy sets to general type-2 fuzzy sets

Similarity measures provide one of the core tools that enable reasoning about fuzzy sets. While many types of similarity measures exist for type-1 and interval type-2 fuzzy sets, there are very few similarity measures that enable the comparison of general type-2 fuzzy sets. In this paper, we introduce a general method for extending existing interval type-2 similarity measures to similarity measures for general type-2 fuzzy sets. Specifically, we show how similarity measures for interval type-2 fuzzy sets can be employed in conjunction with the zSlices based general type-2 representation for fuzzy sets to provide measures of similarity which preserve all the common properties (i.e. reflexivity, symmetry, transitivity and overlapping) of the original interval type-2 similarity measure. We demonstrate examples of such extended fuzzy measures and provide comparisons between (different types of) interval and general type-2 fuzzy measures

Measuring the directional distance between fuzzy sets

The measure of distance between two fuzzy sets is a fundamental tool within fuzzy set theory. However, current distance measures within the literature do not account for the direction of change between fuzzy sets; a useful concept in a variety of applications, such as Computing With Words. In this paper, we highlight this utility and introduce a distance measure which takes the direction between sets into account. We provide details of its application for normal and non-normal, as well as convex and non-convex fuzzy sets. We demonstrate the new distance measure using real data from the MovieLens dataset and establish the benefits of measuring the direction between fuzzy sets

Extending similarity measures of interval type-2 fuzzy sets to general type-2 fuzzy sets

Similarity measures provide one of the core tools that enable reasoning about fuzzy sets. While many types of similarity measures exist for type-1 and interval type-2 fuzzy sets, there are very few similarity measures that enable the comparison of general type-2 fuzzy sets. In this paper, we introduce a general method for extending existing interval type-2 similarity measures to similarity measures for general type-2 fuzzy sets. Specifically, we show how similarity measures for interval type-2 fuzzy sets can be employed in conjunction with the zSlices based general type-2 representation for fuzzy sets to provide measures of similarity which preserve all the common properties (i.e. reflexivity, symmetry, transitivity and overlapping) of the original interval type-2 similarity measure. We demonstrate examples of such extended fuzzy measures and provide comparisons between (different types of) interval and general type-2 fuzzy measures

Novel methods of measuring the similarity and distance between complex fuzzy sets

This thesis develops measures that enable comparisons of subjective information that is represented through fuzzy sets. Many applications rely on information that is subjective and imprecise due to varying contexts and so fuzzy sets were developed as a method of modelling uncertain data. However, making relative comparisons between data-driven fuzzy sets can be challenging. For example, when data sets are ambiguous or contradictory, then the fuzzy set models often become non-normal or non-convex, making them difficult to compare. This thesis presents methods of comparing data that may be represented by such (complex) non-normal or non-convex fuzzy sets. The developed approaches for calculating relative comparisons also enable fusing methods of measuring similarity and distance between fuzzy sets. By using multiple methods, more meaningful comparisons of fuzzy sets are possible. Whereas if only a single type of measure is used, ambiguous results are more likely to occur. This thesis provides a series of advances around the measuring of similarity and distance. Based on them, novel applications are possible, such as personalised and crowd-driven product recommendations. To demonstrate the value of the proposed methods, a recommendation system is developed that enables a person to describe their desired product in relation to one or more other known products. Relative comparisons are then used to find and recommend something that matches a person's subjective preferences. Demonstrations illustrate that the proposed method is useful for comparing complex, non-normal and non-convex fuzzy sets. In addition, the recommendation system is effective at using this approach to find products that match a given query