Exploring Constrained Type-2 fuzzy sets

Fuzzy logic has been widely used to model human reasoning thanks to its inherent capability of handling uncertainty. In particular, the introduction of Type-2 fuzzy sets added the possibility of expressing uncertainty even on the definition of the membership functions. Type-2 sets, however, don’t pose any restrictions on the continuity or convexity of their embedded sets while these properties may be desirable in certain contexts. To overcome this problem, Constrained Type-2 fuzzy sets have been proposed. In this paper, we focus on Interval Constrained Type-2 sets to see how their unique structure can be exploited to build a new inference process. This will set some ground work for future developments, such as the design of a new defuzzification process for Constrained Type-2 fuzzy systems

On transitioning from type-1 to interval type-2 fuzzy logic systems

Capturing the uncertainty arising from system noise has been a core feature of fuzzy logic systems (FLSs) for many years. This paper builds on previous work and explores the methodological transition of type-l (Tl) to interval type-2 fuzzy sets (IT2 FSs) for given "levels" of uncertainty. Specifically, we propose to transition from Tl to IT2 FLSs through varying the size of the Footprint Of Uncertainty (FOU) of their respective FSs while maintaining the original FS shape (e.g., triangular) and keeping the size of the FOU over the FS as constant as possible. The latter is important as it enables the systematic relating of FOU size to levels of uncertainty and vice versa, while the former enables an intuitive comparison between the Tl and T2 FSs. The effectiveness of the proposed method is demonstrated through a series of experiments using the well-known Mackey-Glass (MG) time series prediction problem. The results are compared with the results of the IT2 FS creation method introduced in [1] which follows a similar methodology as the proposed approach but does not maintain the membership function (MF) shape

Contrasting singleton type-1 and interval type-2 non-singleton type-1 fuzzy logic systems

Most applications of both type-1 and type-2 fuzzy logic systems are employing singleton fuzzification due to its simplicity and reduction in its computational speed. However, using singleton fuzzification assumes that the input data (i.e., measurements) are precise with no uncertainty associated with them. This paper explores the potential of combining the uncertainty modelling capacity of interval type-2 fuzzy sets with the simplicity of type-1 fuzzy logic systems (FLSs) by using interval type-2 fuzzy sets solely as part of the non-singleton input fuzzifier. This paper builds on previous work and uses the methodological design of the footprint of uncertainty (FOU) of interval type-2 fuzzy sets for given levels of uncertainty. We provide a detailed investigation into the ability of both types of fuzzy sets (type-1 and interval type-2) to capture and model different levels of uncertainty/noise through varying the size of the FOU of the underlying input fuzzy sets from type-1 fuzzy sets to very “wide” interval type-2 fuzzy sets as part of type-1 non-singleton FLSs using interval type-2 input fuzzy sets. By applying the study in the context of chaotic time-series prediction, we show how, as uncertainty/noise increases, interval type-2 input fuzzy sets with FOUs of increasing size become more and more viable

Improved uncertainty capture for nonsingleton fuzzy systems

In non-singleton fuzzy logic systems (NSFLSs), input uncertainties are modelled with input fuzzy sets in order to capture input uncertainty (e.g., sensor noise). The performance of NSFLSs in handling such uncertainties depends on both: the appropriate modelling in the input fuzzy sets of the uncertainties present in the system’s inputs, and on how the input fuzzy sets (and their inherent model of uncertainty) interact with the antecedent and thus affect the inference within the remainder of the NSFLS. This paper proposes a novel development on the latter. Specifically, an alteration to the standard composition method of type-1 fuzzy relations is proposed, and applied to build a new type of NSFLS. The proposed approach is based on employing the centroid of the intersection of input and antecedent sets as origin of the firing degree, rather than the traditional maximum of their intersection, thus making the NSFLS more sensitive to changes in the input’s uncertainty characteristics. The traditional and novel approach to NSFLSs are experimentally compared for two well-known problems of Mackey-Glass and Lorenz chaotic time series predictions, where the NSFLSs’ inputs have been perturbed with different levels of Gaussian noise. Experiments are repeated for system training under noisy and noise-free conditions. Analyses of the results show that the new method outperforms the traditional approach. Moreover, it is shown that while formally more complex, in practice, the new method has no significant computational overhead compared to the standard approach

Constrained interval type-2 fuzzy sets

In many contexts, type-2 fuzzy sets are obtained from a type-1 fuzzy set to which we wish to add uncertainty. However, in the current type-2 representation there is no restriction on the shape of the footprint of uncertainty and the embedded sets that can be considered acceptable. This leads, usually, to the loss of the semantic relationship between the type-2 fuzzy set and the concept it models. As a consequence, the interpretability of some of the embedded sets and the explainability of the uncertainty measures obtained from them can decrease. To overcome these issues, constrained type-2 fuzzy sets have been proposed. However, no formal definitions for some of their key components (e.g. acceptable embedded sets) and constrained operations have been given. The goal of this paper is to provide some theoretical underpinning for the definition of constrained type-2 sets, their inferencing and defuzzification method. To conclude, the constrained inference framework is presented, applied to two real world cases and briefly compared to the standard interval type-2 inference and defuzzification method

Towards a framework for capturing interpretability of hierarchical fuzzy systems - a participatory design approach

Hierarchical fuzzy systems (HFSs) have been shown to have the potential to improve the interpretability of fuzzy logic systems (FLSs). However, challenges remain, such as: "How can we measure their interpretability?", "How can we make an informed assessment of how HFSs should be designed to enhance interpretability?". The challenges of measuring the interpretability of HFSs include issues such as their topological structure, the number of layers, the meaning of intermediate variables, and so on. In this paper, an initial framework to measure the interpretability of HFSs is proposed, combined with a participatory user design process to create a specific instance of the framework for an application context. This approach enables the subjective views of a range of practitioners, experts in the design and creation of FLSs, to be taken into account in shaping the design of a generic framework for measuring interpretability in HFSs. This design process and framework are demonstrated through two classification application examples, showing the ability of the resulting index to appropriately capture interpretability as perceived by system design experts

The UK Clinical Research Collaboration (UKCRC) Tissue Directory and Coordination Centre: the UK’s centre for facilitating the usage of human samples for medical research

The UKCRC Tissue Directory and Coordination Centre was established to improve access to and utilisation of UK human tissue samples for medical research. The key output of the Centre is the creation of the UK’s first pan-disease Tissue Directory (https://directory.biobankinguk.org/). Any researcher can search the Directory based on a series of simple key words including disease classification, age, sex, sample type, preservation details, quality indicators and datasets available. The Directory as of April 2017 contains 100 Bioresources. Researchers seeking fresh samples can also search for facilities that offer bespoke collection services. Future work of the Centre will be to explore greater standardisation of biobanking activities across the UK and to facilitate an inter-connected research infrastructure related to the use of human biosamples

On aggregating uncertain information by type-2 OWA operators for soft decision making

Yager's ordered weighted averaging (OWA) operator has been widely used in soft decision making to aggregate experts. individual opinions or preferences for achieving an overall decision. The traditional Yager's OWA operator focuses exclusively on the aggregation of crisp numbers. However, human experts usually tend to express their opinions or preferences in a very natural way via linguistic terms. Type-2 fuzzy sets provide an efpcient way of knowledge representation for modeling linguistic terms. In order to aggregate linguistic opinions via OWA mechanism, we propose a new type of OWA operator, termed type-2 OWA operator, to aggregate the linguistic opinions or preferences in human decision making modeled by type-2 fuzzy sets. A Direct Approach to aggregating interval type-2 fuzzy sets by type-2 OWA operator is suggested in this paper. Some examples are provided to delineate the proposed technique. © 2010 Wiley Periodicals, Inc

Alpha-level aggregation: A practical approach to type-1 OWA operation for aggregating uncertain information with applications to breast cancer treatments

Type-1 Ordered Weighted Averaging (OWA) operator provides us with a new technique for directly aggregating uncertain information with uncertain weights via OWA mechanism in soft decision making and data mining, in which uncertain objects are modeled by fuzzy sets. The Direct Approach to performing type-1 OWA operation involves high computational overhead. In this paper, we define a type-1 OWA operator based on the α-cuts of fuzzy sets. Then, we prove a Representation Theorem of type-1 OWA operators, by which type-1 OWA operators can be decomposed into a series of α-level type-1 OWA operators. Furthermore, we suggest a fast approach, called Alpha-Level Approach, to implementing the type-1 OWA operator. A practical application of type-1 OWA operators to breast cancer treatments is addressed. Experimental results and theoretical analyses show that: 1) the Alpha-Level Approach with linear order complexity can achieve much higher computing efficiency in performing type-1 OWA operation than the existing Direct Approach, 2) the type-1 OWA operators exhibit different aggregation behaviors from the existing fuzzy weighted averaging (FWA) operators, and 3) the type-1 OWA operators demonstrate the ability to efficiently aggregate uncertain information with uncertain weights in solving real-world soft decision-making problems. © 2011 IEEE

Type-1 OWA operators for aggregating uncertain information with uncertain weights induced by type-2 linguistic quantifiers

The OWA operator proposed by Yager has been widely used to aggregate experts' opinions or preferences in human decision making. Yager's traditional OWA operator focuses exclusively on the aggregation of crisp numbers. However, experts usually tend to express their opinions or preferences in a very natural way via linguistic terms. These linguistic terms can be modelled or expressed by (type-1) fuzzy sets. In this paper, we define a new type of OWA operator, the type-1 OWA operator that works as an uncertain OWA operator to aggregate type-1 fuzzy sets with type-1 fuzzy weights, which can be used to aggregate the linguistic opinions or preferences in human decision making with linguistic weights. The procedure for performing type-1 OWA operations is analysed. In order to identify the linguistic weights associated to the type-1 OWA operator, type-2 linguistic quantifiers are proposed. The problem of how to derive linguistic weights used in type-1 OWA aggregation given such type of quantifier is solved. Examples are provided to illustrate the proposed concepts. Crown Copyright © 2008