Browsing by Author "Greenfield, Sarah"

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    Accuracy and complexity evaluation of defuzzification strategies for the discretised interval type-2 fuzzy set.
    (Elsevier, 2013) Greenfield, Sarah; Chiclana, Francisco
    The work reported in this paper addresses the challenge of the efficient and accurate defuzzification of discretised interval type-2 fuzzy sets. The exhaustive method of defuzzification for type-2 fuzzy sets is extremely slow, owing to its enormous computational complexity. Several approximate methods have been devised in response to this bottleneck. In this paper we survey four alternative strategies for defuzzifying an interval type-2 fuzzy set: 1. The Karnik-Mendel Iterative Procedure, 2. the Wu-Mendel Approximation, 3. the Greenfield-Chiclana Collapsing Defuzzifier, and 4. the Nie-Tan Method. We evaluated the different methods experimentally for accuracy, by means of a comparative study using six representative test sets with varied characteristics, using the exhaustive method as the standard. A preliminary ranking of the methods was achieved using a multi-criteria decision making methodology based on the assignment of weights according to performance. The ranking produced, in order of decreasing accuracy, is 1. the Collapsing Defuzzifier, 2. the Nie-Tan Method, 3. the Karnik-Mendel Iterative Procedure, and 4. the Wu-Mendel Approximation. Following that, a more rigorous analysis was undertaken by means of the Wilcoxon Nonparametric Test, in order to validate the preliminary test conclusions. It was found that there was no evidence of a significant difference between the accuracy of the Collapsing and Nie-Tan Methods, and between that of the Karnik-Mendel Iterative Procedure and the Wu-Mendel Approximation. However, there was evidence to suggest that the collapsing and Nie-Tan Methods are more accurate than the Karnik-Mendel Iterative Procedure and the Wu-Mendel Approximation. In relation to efficiency, each method’s computational complexity was analysed, resulting in a ranking (from least computationally complex to most computationally complex) as follows: 1. the Nie-Tan Method, 2. the Karnik-Mendel Iterative Procedure (lowest complexity possible), 3. the Greenfield-Chiclana Collapsing Defuzzifier, 4. the Karnik-Mendel Iterative Procedure (highest complexity possible), and 5. the Wu-Mendel Approximation.
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    The Collapsing Defuzzifier for discretised generalised type-2 fuzzy sets
    (Elsevier, 2018-08-01) Greenfield, Sarah; Chiclana, Francisco
    The Greenfield–Chiclana Collapsing Defuzzifier is an established efficient accurate technique for the defuzzification of the interval type-2 fuzzy set. This paper reports on the extension of the Collapsing Defuzzifier to the generalised type-2 fuzzy set. Existing techniques for the defuzzification of generalised type-2 fuzzy sets are presented after which the interval Collapsing Defuzzifier is summarised. The collapsing technique is then extended to generalised type-2 fuzzy sets, giving the Generalised Greenfield–Chiclana Collapsing Defuzzifier. This is contrasted experimentally with both the benchmark Exhaustive Defuzzifier and the α-Planes/Karnik–Mendel Iterative Procedure approach in relation to efficiency and accuracy. The GGCCD is demonstrated to be many times faster than the Exhaustive Defuzzifier and its accuracy is shown to be excellent. In relation to the α-Planes/Karnik–Mendel Iterative Procedure approach it is shown to be comparable in accuracy, but faster.
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    The collapsing method of defuzzification for discretised interval type-2 fuzzy sets.
    (Elsevier, 2009-06) Greenfield, Sarah; Chiclana, Francisco; Coupland, Simon; John, Robert, 1955-
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    The Collapsing method of defuzzification for discretised interval type-2 fuzzy sets.
    (2007) Greenfield, Sarah; Chiclana, Francisco; John, Robert, 1955-; Coupland, Simon
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    The collapsing method: Does the direction of collapse affect accuracy?
    (European Society of Fuzzy logic and Technology, 2009) Greenfield, Sarah; Chiclana, Francisco; John, Robert, 1955-
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    Combining the alpha-planes representation with an Interval Defuzzification Method
    (2011) Greenfield, Sarah; Chiclana, Francisco
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    Defuzzification of the Discretised Generalised Type-2 Fuzzy Set: Experimental Evaluation
    (Elsevier, 2013) Greenfield, Sarah; Chiclana, Francisco
    The work reported in this paper addresses the challenge of the efficient and accurate defuzzification of discretised generalised type-2 fuzzy sets as created by the inference stage of a Mamdani Fuzzy Inferencing System. The exhaustive method of defuzzification for type-2 fuzzy sets is extremely slow, owing to its enormous computational complexity. Several approximate methods have been devised in response to this defuzzification bottleneck. In this paper we begin by surveying the main alternative strategies for defuzzifying a generalised type-2 fuzzy set: (1) Vertical Slice Centroid Type-Reduction; (2) the sampling method; (3) the elite sampling method; and (4) the $\alpha$-planes method. We then evaluate the different methods experimentally for accuracy and efficiency. For accuracy the exhaustive method is used as the standard. The test results are analysed statistically by means of the Wilcoxon Nonparametric Test and the elite sampling method shown to be the most accurate. In regards to efficiency, Vertical Slice Centroid Type-Reduction is demonstrated to be the fastest technique.
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    Fuzzy in 3-D: Contrasting Complex Fuzzy Sets with Type-2 Fuzzy Sets
    (2013-06) Greenfield, Sarah; Chiclana, Francisco
    Complex fuzzy sets come in two forms, the standard form, postulated in 2002 by Ramot et al., and the 2011 innovation of pure complex fuzzy sets, proposed by Tamir et al.. In this paper we compare and contrast both forms of complex fuzzy set with type-2 fuzzy sets, as regards their rationales, applications, definitions, and structures. In addition, pure complex fuzzy sets are compared with type-2 fuzzy sets in relation to their inferencing operations. Complex fuzzy sets and type-2 fuzzy sets differ in their roles and applications. Their definitions differ also, though there is equivalence between those of a pure complex fuzzy set and a type-2 fuzzy set. Structural similarity is evident between these three-dimensional sets. Complex fuzzy sets are represented by a line, and type-2 fuzzy sets by a surface, but a surface is simply a generalisation of a line. This similarity is particularly evident between pure complex fuzzy sets and type-2 fuzzy sets, which are both mappings from the domain onto the unit square. Type-2 fuzzy sets were found not to be isomorphic to pure complex fuzzy sets.
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    Fuzzy in 3-D: Two Contrasting Paradigms
    (European Centre for Soft Computing, 2015-12-31) Greenfield, Sarah; Chiclana, Francisco
    Type-2 fuzzy sets and complex fuzzy sets are both three dimensional extensions of type-1 fuzzy sets. Complex fuzzy sets come in two forms, the standard form, postulated in 2002 by Ramot et al., and the 2011 innovation of pure complex fuzzy sets, proposed by Tamir et al.. In this paper we compare and contrast both forms of complex fuzzy set with type-2 fuzzy sets, as regards their rationales, applications, definitions, and structures. In addition, pure complex fuzzy sets are compared with type-2 fuzzy sets in relation to their inferencing operations. Complex fuzzy sets and type-2 fuzzy sets differ in their roles and applications; complex fuzzy sets are pertinent to inferencing where there is seasonality, and type-2 fuzzy sets are applicable to reasoning under uncertainty. Their definitions differ also, though there is equivalence between those of a pure complex fuzzy set and a type-2 fuzzy set. Structural similarity is evident between these three- dimensional sets. Complex fuzzy sets are represented by a 3–D line, and type- 2 fuzzy sets by a 3–D surface, but a surface is simply a generalisation of a line. This similarity is particularly apparent between pure complex fuzzy sets and type- 2 fuzzy sets, which are both mappings from the domain onto the unit square. However type-2 fuzzy sets were found not to be isomorphic to pure complex fuzzy sets. The mechanisms by which complex fuzzy sets model and quantify periodicity, and type-2 fuzzy sets model and quantify uncertainty are discussed. A type-2 fuzzy set can be represented as the union of its type-2 embedded set. An embedded type-2 fuzzy set is a type-2 fuzzy set in itself, whose geomet- rical representation is a 3-D line. Thus, geometrically an embedded type-2 fuzzy set can be seen as equivalent to a pure complex fuzzy set and therefore a type-2 fuzzy set can be represented as the union of a collection pure complex fuzzy sets, which in turn can be regarded as embedded complex fuzzy sets of a type-2 fuzzy set. This relationship is exploited to provide a complex definition of a type-2 fuzzy set.
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    Fuzzy Photo Project
    (2011-09-28) Greenfield, Sarah
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    Geometric Defuzzification Revisited
    (Elsevier, 2018-08-01) Greenfield, Sarah
    In this paper the Geometric Defuzzification strategy for type-2 fuzzy sets is reappraised. For both discretised and geometric fuzzy sets the techniques for type-1, interval type-2, and generalised type-2 defuzzification are presented in turn. In the type-2 case the accuracy of Geometric Defuzzification is assessed through a series of test runs on interval type-2 fuzzy sets, using Exhaustive Defuzzification as the benchmark method. These experiments demonstrate the Geometric Defuzzifier to be wildly inaccurate. The test sets take many shapes; they are not confined to those type-2 sets with rotational symmetry that have previously been acknowledged by the technique’s developers to be problematic as regards accuracy. Type-2 Geometric Defuzzification is then examined theoretically. The defuzzification strategy is demonstrated to be built upon a fallacious application of the concept of centroid. This explains the markedly inaccurate experimental results. Thus the accuracy issues of type-2 Geometric Defuzzification are revealed to be inevitable, fundamental and significant.
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    The Grid Method of Discretisation for Type-2 Fuzzy Sets
    (2012-09) Greenfield, Sarah
    In order to perform fuzzy inferencing, it is normal practice to discretise fuzzy sets. For type-1 fuzzy sets there is only one method of discretisation, but type-2 fuzzy sets may be discretised in more than one way. This paper introduces the grid method of type-2 discretisation — a simpler, more convenient alternative to the established technique.
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    Interval-Valued Complex Fuzzy Logic
    (IEEE, 2016) Greenfield, Sarah; Chiclana, Francisco; Dick, Scott
    Data is frequently characterised by both uncertainty and seasonality. Type-2 fuzzy sets are an extension of type-1 fuzzy sets offering a conceptual scheme within which the effects of uncertainties in fuzzy inferencing may be modelled and minimised. Complex fuzzy sets are type-1 fuzzy sets extended by an additional phase term which permits them to intuitively represent the seasonal aspect of fuzziness in time-series applications. Type-2 fuzzy sets take two forms, generalised, and the simpler interval. Interval-valued fuzzy sets are type-1 fuzzy sets whose behaviour and properties are equivalent to interval type-2 fuzzy sets. This paper is concerned with the combination of interval-valued fuzzy sets and complex fuzzy sets to develop interval-valued complex fuzzy sets, an adaption of complex fuzzy sets such that the membership function assigns each point on the domain to an interval. From the definition of the interval-valued complex fuzzy set, the principles of interval-valued complex fuzzy logic are developed.
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    Join and Meet Operations for Interval-Valued Complex Fuzzy Logic
    (2016) Greenfield, Sarah; Chiclana, Francisco; Dick, Scott
    Interval-valued complex fuzzy logic is able to handle scenarios where both seasonality and uncertainty feature. The interval-valued complex fuzzy set is defined, and the interval valued complex fuzzy inferencing system outlined. Highly pertinent to complex fuzzy logic operations is the concept of rotational invariance, which is an intuitive and desirable characteristic. Interval-valued complex fuzzy logic is driven by interval-valued join and meet operations. Four pairs of alternative algorithms for these operations are specified; three pairs possesses the attribute of rotational invariance, whereas the other pair lacks this characteristic.
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    A novel sampling method for type-2 defuzzification
    (2005) Greenfield, Sarah; John, Robert, 1955-; Coupland, Simon
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    Optimised generalised type-2 join and meet operations
    (2007) Greenfield, Sarah; John, Robert, 1955-
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    The sampling method of defuzzification for Type-2 fuzzy sets: experimental evaluation.
    (Elsevier, 2012) Greenfield, Sarah; Chiclana, Francisco; John, Robert, 1955-; Coupland, Simon
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    Slicing Strategies for the Generalised Type-2 Mamdani Fuzzy Inferencing System
    (Springer, 2016) Greenfield, Sarah; Chiclana, Francisco
    As a three-dimensional object, there are a number of ways of slicing a generalised type-2 fuzzy set. In the context of the Mamdani Fuzzy Inferencing System, this paper concerns three accepted slicing strategies, the vertical slice, the wavy slice, and the horizontal slice or alpha -plane. Two ways of de ning the generalised type-2 fuzzy set, vertical slices and wavy slices, are presented. Fuzzi cation and inferencing is presented in terms of vertical slices. After that, the application of all three slicing strategies to defuzzi cation is described, and their strengths and weaknesses assessed.
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    The Stratic Defuzzifier for Discretised General Type-2 Fuzzy Sets
    (Elsevier, 2020-11-21) Greenfield, Sarah; Chiclana, Francisco
    Stratification is a feature of the type-reduced set of the general type-2 fuzzy set, from which a new technique for general type-2 defuzzification, Stratic Defuzzification, may be derived. Existing defuzzification strategies are summarised. The stratified structure is described, after which the Stratic Defuzzifier is presented and contrasted experimentally for accuracy and efficiency with both the Exhaustive Method of Defuzzification (to benchmark accuracy) and the alpha-Planes/Karnik–Mendel Iterative Procedure strategy, employing 5, 11, 21, 51 and 101 alpha-planes. The Stratic Defuzzifier is shown to be much faster than the Exhaustive Defuzzifier. In fact the Stratic Defuzzifier and the alpha-Planes/Karnik–Mendel Iterative Procedure Method are comparably speedy; the speed of execution correlates with the number of planes participating in the defuzzification process. The accuracy of the Stratic Defuzzifier is shown to be excellent. It is demonstrated to be more accurate than the alpha-Planes/Karnik–Mendel Iterative Procedure Method in four of six test cases, regardless of the number of -planes employed. In one test case, it is less accurate than the alpha-Planes/Karnik–Mendel Iterative Procedure Method, regardless of the number of alpha-planes employed. In the remaining test case, the alpha-Planes/Karnik–Mendel Iterative Procedure Method with 11 alpha-Planes gives the most accurate result, with the Stratic Defuzzifier coming second.
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