# Bipolar Dissimilarity and Similarity Correlations of Numbers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Correlation Functions (Association Measures)

**Example**

**1.**

#### 2.2. Similarity and Dissimilarity Functions (Fuzzy Relations)

**Proposition**

**1.**

**Proposition**

**2**

**.**If the similarity (dissimilarity) function is bipolar, then it is co-symmetric and consistent.

#### 2.3. Constructing Correlation Functions from Similarity and Dissimilarity Functions

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

- (a)
- Correlation between x and y is positive if the similarity between x and y is greater than the dissimilarity between them. In the opposite case, the correlation between x and y is negative.
- (b)
- Correlation between x and y is positive if they are “similar” and negative if they are “different”.

**Proposition**

**3**

**.**Let F(x) be a p-transformation of elements of the set $\Omega $ into n-tuples $F\left(x\right)=\left(F{\left(x\right)}_{1},\dots ,F{\left(x\right)}_{n}\right)$ then the function

## 3. Results

#### 3.1. Constructing Pearson’s Linear Correlation Coefficient Using Bipolar Dissimilarity Function

**Proposition**

**4.**

**Proof.**

#### 3.2. Non-Bipolar Similarity, Dissimilarity, and Correlation Functions for Real Numbers

**Proposition 5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

#### 3.3. Bipolar Similarity, Dissimilarity, and Correlation Functions for Real Numbers

**Proposition**

**8.**

**Proof.**

## 4. Bipolar Dissimilarity and Similarity Correlation in Risk Assessment

- Measured values (see Figure 6) are stored in a database.
- Histogram is created based on the stored data as illustrated in Figure 7.
- Fuzzy set is fitted to the histogram, (see in [29]). This set represents the normal reactions of the patient under the same conditions. Medical recommendations for the specific patient should be available in the database as well, or the age- and sex-specific values from the literature can be used instead. Figure 8. shows the fuzzy sets generated based on the measurements and medical recommendations for the above case study.

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**The correlation function (31) on the set of real numbers [−100,100]\{0} was obtained by rescaling the values of the bipolar similarity function (30) using the formula (13).

**Figure 6.**Previously measured values (23-year-old man, HRrest = 55 bpm, HRmax = 200 bpm, Weight = 69 kg, duration = 1 h, sampling frequency = 5 s).

**Figure 7.**Histogram based on previous measurements (23-year-old man, HRrest = 55 bpm, HRmax = 200 bpm, Weight = 69 kg, duration = 1 h, sampling frequency = 5 s).

**Figure 8.**Medical recommendation-based set and histogram-based fuzzy set based on previous measurements (23-year-old man, HRrest = 55 bpm, HRmax = 200 bpm, Weight = 69 kg, duration = 1 h, sampling frequency = 5 s).

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**MDPI and ACS Style**

Batyrshin, I.Z.; Tóth-Laufer, E.
Bipolar Dissimilarity and Similarity Correlations of Numbers. *Mathematics* **2022**, *10*, 797.
https://doi.org/10.3390/math10050797

**AMA Style**

Batyrshin IZ, Tóth-Laufer E.
Bipolar Dissimilarity and Similarity Correlations of Numbers. *Mathematics*. 2022; 10(5):797.
https://doi.org/10.3390/math10050797

**Chicago/Turabian Style**

Batyrshin, Ildar Z., and Edit Tóth-Laufer.
2022. "Bipolar Dissimilarity and Similarity Correlations of Numbers" *Mathematics* 10, no. 5: 797.
https://doi.org/10.3390/math10050797