Josie McCulloch

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.

Analysing fuzzy sets through combining measures of similarity and distance

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.

Juzzy online: An online toolkit for the design, implementation, execution and sharing of Type-1 and Type-2 fuzzy logic systems

In this paper we present an online fuzzy logic toolkit for the design, implementation, execution and sharing of type-1 (T1), interval type-2 (T2) and (zSlices based) general T2 fuzzy logic system (FLSs). The motivation to develop the toolkit stems from the desire to provide a free-to-use fuzzy logic toolkit available which is platform-independent, easily accessible and which does not require any background knowledge of programming. This toolkit aims to help expand the accessibility of FLSs, in particular of T2 FLSs, to both research and industrial applications outside of the fuzzy logic community and computer science more generally. We review the features currently available through the JuzzyOnline toolkit (including a complete, previously unseen visualisation of the inference steps for zSlices based general T2 FLSs) and demonstrate a sample Fuzzy Logic System implementation of the toolkit. Finally, we conclude with some future developments and a call for feedback and contributions to aid in further development.

A similarity-based inference engine for non-singleton fuzzy logic systems

In non-singleton fuzzy logic systems (NSFLSs) input uncertainties are modelled with input fuzzy sets in order to capture input uncertainty such as sensor noise. The performance of NSFLSs in handling such uncertainties depends both on the actual input fuzzy sets (and their inherent model of uncertainty) and on the way that they affect the inference process. This paper proposes a novel type of NSFLS by replacing the composition-based inference method of type-1 fuzzy relations with a similarity-based inference method that makes NSFLSs more sensitive to changes in the inputs uncertainty characteristics. The proposed approach is based on using the Jaccard ratio to measure the similarity between input and antecedent fuzzy sets, then using the measured similarity to determine the firing strength of each individual fuzzy rule. The standard and novel approaches to NSFLSs are experimentally compared for the well-known problem of Mackey-Glass time series predictions, where the NSFLSs inputs have been perturbed with different levels of Gaussian noise. The experiments are repeated for system training under both noisy and noise-free conditions. Analyses of the results show that the new method outperforms the standard approach by substantially reducing the prediction errors.

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.

Measuring the similarity between zSlices general type-2 fuzzy sets with non-normal secondary membership functions

This paper presents a method of measuring the similarity between general type-2 fuzzy sets that may have non-normal secondary membership functions. Such fuzzy sets are increasingly common in applications such as the modelling of the subjective meaning of linguistic terms by groups of people. By building upon existing similarity measures in the literature, which thus far cannot compare such fuzzy sets, we derive an extended similarity measure which can be applied to both normal and non-normal (in terms of the secondary membership functions) general type-2 fuzzy sets. We provide proofs that the proposed method follows all of the common properties of a similarity measure and demonstrations are given to compare the proposed method with others in the literature.

“Give me what I want” - enabling complex queries on rich multi-attribute data

Consumer and more generally, human preferences are highly complex, depending on a multitude of factors, most of which are not crisp, but uncertain/fuzzy in nature. Thus, user selection amongst a set of items is dependent on the complex comparison of items based on a large number of imprecise item-attributes such as price, size, colour, etc. This paper proposes the mechanisms to underpin the digital replication of such complex preference-based item selection with the view to enabling improved digital item search and recommendation systems. For example, a user may query “I would like a product of similar size but at a cheaper price.” The proposed method involves splitting query-attributes into two categories; those to remain similar (e.g., size) and those to be changed in a specific direction (e.g., price - to be lower). A combination of similarity and distance measures is then used to compare and rank recommendations. Initial results are presented indicating that the proposed method is effective at ranking items according to intuition and expected user preferences.

Interval-valued sensory evaluation for customized beverage product formulation and continuous manufacturing

Understanding of consumer preferences and perceptions is a vital challenge for the food and beverage industry. Food and beverage product development is a very complex process that deals with highly uncertain factors, including consumer perceptions and manufacturing complexity. Sensory evaluation is widely used in the food industry for product design and defining market segments. Here, we develop a two-step approach to minimize uncertainty in the food and beverage product development, including consumers as co-creators. First, we develop interval-valued questionnaires to capture sensory perceptions of consumers for the corresponding sensory attributes. The data captured is modelled with fuzzy sets in order to then facilitate the design of new consumer-tailored products. Then, we demonstrate the real-world manufacture of a personalized beverage product with a continuous food formulation system. Finally, we highlight consumers” perceptions for the corresponding sensory attributes and their fuzzy set generated agreement models to capture product acceptance for the formulated and commercial orange juice drinks, and consequently to establish that continuous beverage formulator is capable of making similar commercial products for individuals.

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 (nonnormal 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.

A fuzzy directional distance measure

The measure of distance between two fuzzy sets is a fundamental tool within fuzzy set theory, however, distance measures currently within the literature use a crisp value to represent the distance between fuzzy sets. A real valued distance measure is developed into a fuzzy distance measure which better reflects the uncertainty inherent in fuzzy sets and a fuzzy directional distance measure is presented, which accounts for the direction of change between fuzzy sets. A multiplicative version is explored as a full maximal assignment is computationally intractable so an intermediate solution is offered.