Interval-Valued Methods in Classifications and Decisions

A new concept of equivalence between intervals and the induced indistinguishability between interval‐valued (IV) fuzzy sets are proposed and considered. A new notion of the degree of IV equivalence is presented where partial or linear orders and the width of intervals are involved reflecting uncertainty. Furthermore, construction methods of the considered equivalences are provided and the relation between them and other notions of IV equivalences are studied. Finally, a methodology to apply the proposed equivalences in the field of image processing is shown, along with an illustrative example.

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“…). More examples of IV aggregation functions in the case of linear orders are provided in the literature 33,34 …”

Section: Fuzzy Settingmentioning

confidence: 99%

Interval‐valued equivalence measures respecting uncertainty in image processing

et al. 2021

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“…The upper bounds of these aggregations were built with the Lehmer means (Bullen et al, 57 p. 185) for which the standard monotonicity (4) is not fulfilled, so

A_{3}

A_{4}

do not satisfy (5) but they satisfy (6). 19

A_{5} (α_{1}, …, α_{n}) = (][, \sqrt{\frac{{\underset{α}{true}}_{1}^{2} + \dots + {\underset{α}{true}}_{n}^{2}}{n}}, \sqrt[3]{\frac{{\overset{α}{true}}_{1}^{3} + \dots + {\overset{α}{true}}_{n}^{3}}{n}}),

A_{6} (α_{1}, …, α_{n}) = (][, \sqrt[3]{\frac{{\underset{α}{true}}_{1}^{3} + \dots + {\underset{α}{true}}_{n}^{3}}{n}}, \sqrt[4]{\frac{{\overset{α}{true}}_{1}^{4} + \dots + {\overset{α}{true}}_{n}^{4}}{n}}),

where

A_{5}

and

A_{6}

are representable aggregation functions, (built with the use of adequate root‐power means, Hardy et al, 58 p. 26) possible aggregation functions and aggregation functions with respect to the admissible linear orders

\leq_{L e x 1}

and

\leq_{L e x 2}

, 16 however they are not necessary aggregation functions 19

A_{7}

…”

Section: Interval‐valued Aggregationsmentioning

confidence: 99%

“…where

A_{10}

is a representable aggregation function, a necessary aggregation function and a possible aggregation function 19 . Furthermore, it is an aggregation function with respect to the admissible linear orders

\leq_{L e x 1}

and

\leq_{L e x 2}

16 …”

Section: Interval‐valued Aggregationsmentioning

confidence: 99%

“…Aggregation functions are useful in many application areas 13 . In this contribution we will use interval‐valued aggregation functions which were developed by several authors and applied with a success 13–18 . Here we present the results on aggregation methods connected with possible and necessary aggregation functions 19 .…”

Section: Introductionmentioning

confidence: 99%

“…13 In this contribution we will use interval-valued aggregation functions which were developed by several authors and applied with a success. [13][14][15][16][17][18] Here we present the results on aggregation methods connected with possible and necessary aggregation functions. 19 The first approach to use these functions and interval modelling in the case of large number of attributes for microarrays was presented in Bentkowska 16 (Chapter 5), where the aim was to find the best classifiers with respect to the proposed there measure based on accuracy at the specific points on the receiver operating characteristic (ROC) curve.…”

mentioning

confidence: 99%

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Interval modelling in optimization of k‐NN classifiers for large number of attributes in data sets on an example of DNA microarrays

Bentkowska

Bazan

Zaręba

et al. 2021

Int J of Intelligent Sys

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In this contribution there are considered interval‐valued methods of improving the quality of classification by the k‐ N N binary classifiers in the case of large number of attributes in data sets. One of the possible applications of the introduced method are the microarray data so the presented results may be applied in medical diagnosis support, for example, in identification of marker genes. The proposed algorithm involving reduction methods of time complexity and interval modeling of data using interval‐valued aggregation functions has a significantly higher classification quality than the other considered algorithm with the aggregation method involving the numerical arithmetic mean. The former algorithm is based on the interval‐valued aggregation of uncertainty intervals determined on the basis of certainty coefficients of individual k‐ N N classifiers while the latter is based on the aggregation of certainty coefficients of individual k‐ N N classifiers with the use of the numerical arithmetic mean. Moreover, the performance of the new algorithm based on interval‐valued methods was compared with the known from the literature classifiers in cancer diagnosis based on gene expression microarrays. The obtained results prove that in practical applications the newly proposed algorithm can be successfully used.

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On the Question of Modifying Membership Functions

Barchev

Sudakov

2020

Software Engineering Perspectives in Intelligent Systems

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Interval-Valued Methods in Classifications and Decisions

Cited by 14 publications

References 10 publications

Interval‐valued equivalence measures respecting uncertainty in image processing

Interval‐valued equivalence measures respecting uncertainty in image processing

Interval modelling in optimization of k‐NN classifiers for large number of attributes in data sets on an example of DNA microarrays

On the Question of Modifying Membership Functions

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