Why Triangular and Trapezoid Membership Functions: A Simple Explanation

Kreinovich, Vladik; Kosheleva, Olga; Shahbazova, Shahnaz N.

doi:10.1007/978-3-030-38893-5_2

Cited by 32 publications

(28 citation statements)

References 8 publications

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“…The qualitative risk factors in this study were split into three attributes: positive, average, and negative, which served as inputs to the fuzzy logic approach, with bankrupt and non-bankrupt as outputs. The trapezoidal and triangular membership functions were used to define the data's input and output (Kreinovich et al, 2020). The data was graded depending on the membership meaning.…”

Section: Fuzzificationmentioning

confidence: 99%

Bankruptcy Prediction Model with Risk Factors using Fuzzy Logic Approach

Kin

Aizam

Hasan

et al. 2021

JCRINN

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Forecasting bankruptcy remains crucial, especially during this pandemic. Managers, financial institutions, and government agencies rely on the information regarding an impending bankruptcy threat to make decisions. This paper developed a straightforward bankruptcy prediction model using the fuzzy logic approach for individuals and companies to evaluate their performance and analyse the tendency of getting bankrupt. A sample of 250 respondents from banks and financial firms were tested using the qualitative risk factors, namely, industrial risk, management risk, financial flexibility, credibility, competitiveness, and operational risk. This study provides a comprehensive analysis using the Fuzzy Inference System (FIS) editor in the MATLAB software, where the model's accuracy is compared to the actual results. The results show an accuracy rate of 99.20%, indicating that this approach can determine the likelihood of bankruptcy. The fuzzy logic approach can improve prediction accuracy while also guiding decision-makers in detecting and preventing possible financial crises in their early phases.

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Section: Fuzzificationmentioning

confidence: 99%

Bankruptcy Prediction Model with Risk Factors using Fuzzy Logic Approach

Kin

Aizam

Hasan

et al. 2021

JCRINN

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show abstract

“…The theoretical explanation which proves the practicality of trapMFs is discussed in [18]. An intuitive explanation on how trapMF is functioning well is explained in simple terms in [19]. In short, a trapMF ( Figure 1) is defined by a lower limit a, an upper limit d, a lower support limit b, and an upper support limit c, where a < b < c < d [18].…”

Section: Literature Review Trapezoidal Mfmentioning

confidence: 99%

“…In a simple explanation, as an example, for a property such as "small", different values namely x are assigned, along with their degrees, μ(x). The main motivation in fuzzy implementation is to ensure the value x and x' are close, along with their corresponding MFs μ(x) and μ(x') which should be close too [19].…”

Section: Literature Review Trapezoidal Mfmentioning

confidence: 99%

Integration of Cluster Centers and Gaussian Distributions in Fuzzy C-Means for the Construction of Trapezoidal Membership Function

Khairuddin¹,

Hasan²,

Hashmani³

2020

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Fuzzy C-Means (FCM) is one of the mostly used techniques for fuzzy clustering and proven to be robust and more efficient based on various applications. Image segmentation, stock market and web analytics are examples of popular applications which use FCM. One limitation of FCM is that it only produces Gaussian membership function (MF). The literature shows that different types of membership functions may perform better than other types based on the data used. This means that, by only having Gaussian membership function as an option, it limits the capability of fuzzy systems to produce accurate outcomes. Hence, this paper presents a method to generate another popular shape of MF, the trapezoidal shape (trapMF) from FCM to allow more flexibility to FCM in producing outputs. The construction of trapMF is using mathematical theory of Gaussian distributions, confidence interval and inflection points. The cluster centers or mean (μ) and standard deviation (σ) from the Gaussian output are fully used to determine four trapezoidal parameters; lower limit a, upper limit d, lower support limit b, and upper support limit c with the assistance of function trapmf() in Matlab fuzzy toolbox. The result shows that the mathematical theory of Gaussian distributions can be applied to generate trapMF from FCM.

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“…In [18][19][20], it is shown that these membership functions are the most robust-in the sense that a given change in the input x to x ≈ x leads to the smallest possible change in the value A(x) of the corresponding membership function; specifically:…”

Section: How This Empirical Fact Is Explained Nowmentioning

confidence: 99%

“…In [20], we describe this requirement in crisp terms-as minimizing the difference between the values of A(x) and A(x ).…”

mentioning

confidence: 99%

Why Triangular Membership Functions Are Successfully Used in F-Transform Applications: A Global Explanation to Supplement the Existing Local Ones

Kosheleva

Kreinovich

Nguyen

2019

Axioms

Self Cite

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The main ideas of F-transform came from representing expert rules. It would be therefore reasonable to expect that the more accurately the membership functions describe human reasoning, the more successful will be the corresponding F-transform formulas. We know that an adequate description of our reasoning corresponds to complicated membership functions-however, somewhat surprisingly, many successful applications of F-transform use the simplest possible triangular membership functions. There exist some explanations for this phenomenon, which are based on local behavior of the signal. In this paper, we supplement these local explanations by a global one: namely, we prove that triangular membership functions are the only one that provide the exact reconstruction of the appropriate global characteristic of the signal.

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Why Triangular and Trapezoid Membership Functions: A Simple Explanation

Cited by 32 publications

References 8 publications

Bankruptcy Prediction Model with Risk Factors using Fuzzy Logic Approach

Bankruptcy Prediction Model with Risk Factors using Fuzzy Logic Approach

Integration of Cluster Centers and Gaussian Distributions in Fuzzy C-Means for the Construction of Trapezoidal Membership Function

Why Triangular Membership Functions Are Successfully Used in F-Transform Applications: A Global Explanation to Supplement the Existing Local Ones

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