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

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## Abstract

**:**

## 1. Formulation of the Problem

#### 1.1. F-Transforms: A Brief Reminder

- which is equal to 0 outside the interval $[-1,1]$,
- which, starting at $t=-1$, increases to 1 until it reaches $t=0$,
- which then decreases to 0, and
- for which$$\sum _{i}A\left(\frac{t-{t}_{i}}{h}\right)=1\phantom{\rule{2.em}{0ex}}$$

#### 1.2. A Somewhat Unexpected Empirical Fact

#### 1.3. It Is Desirable to Have Theoretical Explanations for This Empirical Fact

#### 1.4. How This Empirical Fact Is Explained Now

- In [20], we describe this requirement in crisp terms—as minimizing the difference between the values of $A\left(x\right)$ and $A\left({x}^{\prime}\right)$.
- In contrast, in [18,19], we consider this requirement in fuzzy terms—as the requirement that the degree of closeness between $A\left(x\right)$ and $A\left({x}^{\prime}\right)$ should be the largest possible. In [18], this is done with type-1 fuzzy techniques, and in [19], with type-2 fuzzy techniques.

- either values $A\left(x\right)$ at nearby points x
- or the effect of noise—which is also added locally, for each x separately.

#### 1.5. What We Do in This Paper

- first, we consider the selection of an appropriate global characteristic, solve the resulting optimization problem, and thus find the global characteristic that is optimal in some reasonable sense;
- second, we prove that the triangular membership functions are the only ones that allow us to uniquely reconstruct the optimal global characteristic of the original signal.

#### 1.6. The Structure of the Paper

## 2. Local vs. Global Characteristics: Main Idea

#### 2.1. What We Mean by Local and Global Characteristics

#### 2.2. Resulting Idea

## 3. Which Global Characteristics Should We Represent: Discussion

#### 3.1. Need for Linearization

#### 3.2. Which Linear Quantities Should We Select?

#### 3.3. How to Define What Is Most Appropriate?

## 4. Selecting the Most Adequate Global Characteristic: Towards Precise Formulation of the Problem

#### 4.1. Towards Describing What Is More Appropriate and What Is Less Appropriate

**Definition**

**1.**

#### 4.2. Discussion

- what it means for the alternative a to be better than the alternative b (we will denote it by $a\prec b$), and
- what it means for the alternatives a and b to be of the same quality (we will denote it by $a\sim b$).

- we have $a\prec b$ if and only if $f\left(a\right)<f\left(b\right)$; and
- we have $a\sim b$ if and only if $f\left(a\right)=f\left(b\right)$.

- either a is better than b with respect to the original optimality criterion, i.e.,$$f\left(a\right)<f\left(b\right),\phantom{\rule{2.em}{0ex}}$$
- or with respect to the original optimality criterion, the alternatives a and b are of equal quality, but the alternative a is better with respect to the second objective function, i.e., if$$f\left(a\right)=f\left(b\right)\mathrm{and}g\left(a\right)g\left(b\right).$$

- we have $a\prec b$ if and only if$$f\left(a\right)<f\left(b\right)\mathrm{or}\left(f\right(a)=f(b\left)\mathrm{and}g\right(a)g(b\left)\right);\phantom{\rule{2.em}{0ex}}$$
- we have $a\sim b$ if and only if$$f\left(a\right)=f\left(b\right)\mathrm{and}g\left(a\right)=g\left(b\right).\phantom{\rule{2.em}{0ex}}$$

- we can say that a is better than b in the sense of this criterion; we will denote this by$$a\prec b;\phantom{\rule{2.em}{0ex}}$$
- we can say that b is better than a in the sense of the given criterion; we will denote this by$$b\prec a;\phantom{\rule{2.em}{0ex}}$$
- or we can say that the two alternatives are equally good with respect to the given criterion; we will denote this by $a\sim b$.

**Definition**

**2**

**.**Let $\mathcal{A}$ be a set; its elements will called alternatives. By an optimality criterion, we mean a pair of relations $\langle \prec ,\sim \rangle $ of the set A that satisfies the following properties:

- for every two alternatives a and b, we have one and only one of three options:$a\prec b$,$b\prec a$, and$a\sim b$;
- if$a\prec b$and$b\prec c$, then$a\prec c$;
- if$a\prec b$and$b\sim c$, then$a\prec c$;
- if$a\sim b$and$b\prec c$, then$a\prec c$;
- if$a\sim b$and$b\sim c$, then$a\sim c$;
- $a\sim a$, and
- if$a\sim b$, then$b\sim a$.

**Definition**

**3.**

#### 4.3. Discussion

- either a is better than b according to the original optimality criterion,
- or a is equivalent to b in terms of the original optimality criterion but better according to the additional optimality criterion.

**Definition**

**4.**

#### 4.4. Need for Scale-Invariance

**Definition**

**5.**

- if${\{c\xb7q\left(t\right)\}}_{c}\prec {\{c\xb7r\left(t\right)\}}_{c}$, then${\{c\xb7q(\lambda \xb7t)\}}_{c}\prec {\{c\xb7r(\lambda \xb7t)\}}_{c}$;
- if${\{c\xb7q\left(t\right)\}}_{c}\sim {\{c\xb7r\left(t\right)\}}_{c}$, then${\{c\xb7q(\lambda \xb7t)\}}_{c}\sim {\{c\xb7r(\lambda \xb7t)\}}_{c}$.

## 5. Which Characteristics Are the Most Adequate: Auxiliary Result

#### Discussion

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**1.**

- if $F\prec G$, then ${T}_{\lambda}\left(F\right)\prec {T}_{\lambda}\left(G\right)$; and
- if $F\sim G$, then ${T}_{\lambda}\left(F\right)\sim {T}_{\lambda}\left(G\right)$.

## 6. Main Result: A New Justification of Triangular Membership Functions

**Definition**

**6.**

#### 6.1. Case of $\beta =0$

#### 6.2. General Case

**Proposition**

**2.**

**Proof**

**of**

**Proposition**

**2.**

- the value ${x}_{0}={\int}_{0}^{1}A\left(t\right)\xb7x\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt$, and
- the value ${x}_{1}={\int}_{0}^{1}A(t-1)\xb7x\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt$.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Kosheleva, O.; Kreinovich, V.; Nguyen, T.N.
Why Triangular Membership Functions Are Successfully Used in F-Transform Applications: A Global Explanation to Supplement the Existing Local Ones. *Axioms* **2019**, *8*, 95.
https://doi.org/10.3390/axioms8030095

**AMA Style**

Kosheleva O, Kreinovich V, Nguyen TN.
Why Triangular Membership Functions Are Successfully Used in F-Transform Applications: A Global Explanation to Supplement the Existing Local Ones. *Axioms*. 2019; 8(3):95.
https://doi.org/10.3390/axioms8030095

**Chicago/Turabian Style**

Kosheleva, Olga, Vladik Kreinovich, and Thach Ngoc Nguyen.
2019. "Why Triangular Membership Functions Are Successfully Used in F-Transform Applications: A Global Explanation to Supplement the Existing Local Ones" *Axioms* 8, no. 3: 95.
https://doi.org/10.3390/axioms8030095