# An Application of Fuzzy Multiple Linear Regression in Biological Paradigm

Authors: Saima Mustafa; Shumaila Ghaffar; Murrium Bibi; Muhammad Ghaffar Khan; Qaisara Praveen; Harish Garg; Mahamane Saminou
Journal: Complexity
 (2022)

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## Abstract

The regression model is generally utilized in several fields of study because of its applications. Regression is an extremely incredible approach; it builds up a connection between dependent and independent variables. We have addressed a powerful computational model by utilizing dengue information joined with fuzzy multiple linear regression. Information is accumulated on dengue fever through the survey. This paper is centered on the comparison of the crisp method with fuzzy multiple linear regression, and then, the utilization of a fuzzy multiple regression method is explained after the comparison. We have used multiple regression and then converted the said technique into three fuzzy cases. The effectiveness of the fuzzy multiple regression model is measured by numerical computation and comparison of both techniques. 2020 Mathematics Subject Classification. Primary 30C45; 30C50; 30C80; Secondary 11B65, 47B38.


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## Body

## 1. Introduction

Regression is a method for determining the statistical relationship between two or more variables where a change in a dependent variable is associated with a change in one or more independent variables [1]. Multiple linear regression describes the relationship of one dependent variable with more than one independent variable. This is a statistical technique that is used in examining how multiple independent variables and dependent variables are related. Certain assumptions need to be fulfilled for achieving better results from multiple regression such as linearity, normality, no multicollinearity, and homoscedasticity. Fuzzy set theory was first introduced by Zadeh [2] in 1965, and it is a technique used to handle vague, uncertain, imprecise, or unclear information. This technique is appropriate in the case of vague information [1]. Many recent developments of fuzzy and its applications have been explored by different researchers [3–5]. They have applied different fuzzy tools, and the applications of these tools have been explored in different fields such as decision-making and logistics processes. Fuzzy regression was proposed by the Japanese researcher Tanaka (1982) [6]. The model of fuzzy linear regression has been treated from diverse points of view which depend upon the type of data given in the input and data try to be achieved from the output [7]. Fuzzy regression can be used in very complex systems in the real world such as economy, marketing, finance, ecology, and industry, see [7]. Dengue fever is a viral illness caused by mosquito bites that are responsible for infecting approximately 96 million infected people in America yearly [8]. In recent decades, dengue has increased in geographic incidence and distribution. The impact of climatic factors on transmission was investigated by researchers in [9, 10]. Zadeh et al. [2] gave the idea of fuzzy set theory to deal with the vagueness and uncertainties occurring in decision making. It is one of the applications which is in multisensing paradigm-based urban air quality monitoring and hazardous gas source analyzing [11]. Multiple linear regressions are often used to explain the relationship of multiple independent variables. This method is particularly appropriate for disease models and is widely used in the health sciences.In this paper, we discuss the problems that occur in different situations of multiple linear regression. First, it involves a linear relationship between the independent and dependent variables which do not give clear results based on the given information. Second, it specifies the distribution of errors normally distributed between the observed and expected values which cause impreciseness. Third, it assumes the data do not contain multicollinearity. Another difficulty can be caused by nonpreciseness and vague observations that occur frequently in practice. Due to these facts, we used the term “fuzzy” in multiple linear regression which overcome all these difficulties. This paper focuses on multiple linear regressions in a biological paradigm. Here is the numerical computation depending on the dengue study.


## 2. Fuzzy Sets and Numbers

The fuzzy set is defined as a set that contracts with vague boundaries [2]. The set consists of fuzzy logic and linguistic variables. Fuzzy sets are represented as(1)A=μ1X1+μ2X2+⋯+μnXn,where μ1 and n express the membership function of Xi, i=1,2,…,n in A, and the union is denoted by the plus sign [12]. Fuzzy sets are scientific, mathematical models of unclear quantitative or qualitative data, as often as possible, which are generated from natural languages. The model depends upon the generalization of the characteristics function of a set and the classical concept.


## 3. Multiple Linear Regression Model

Sir Francis Galton [1], an English Victorian, introduced the term regression. The general parametric equation is(2)Y=fX+ϵ,Y=τ0+τ1X1+τ2X2+⋯+τqXq+ϵ,where Y and X represent the dependent and independent variables. The coefficients τ1,τ2,…,τq represent slopes, and ϵ is the random error.A fuzzy regression technique was first proposed by Tanaka [6]. We have considered the following cases of dependent and independent variables.Case 1.
(3)z˜i=τ0+τ1k˜1+ϵi.Case 2.
(4)z˜i=τ˜0+τ˜1k1+ϵi.Case 3.
(5)z˜i=τ˜0+τ˜1k˜1+ϵi.
In the above models, here,τ0 and τ1 are the intercept and slope of the regression line respectively. Zi are the fuzzy responses. In case 1, the parameters τ0 and τ1 are crisp parameters, and Ki are fuzzy. In Case 2, the parameters τ0 and τ1 are fuzzy but Ki are crisp. In case 3, the predictor and parameters are all fuzzy.
Consider the multiple fuzzy regression model which can be generalized as follows:(6)y˜i=τ˜0+τ˜1x˜j1+τ˜2x˜j2+⋯+τ˜px˜jp+ϵj.
Using the centered values of the crisp predictor, the above equation can be written in matrix form as follows:(7)y˜=x˜τ˜+ϵ,where y˜ is a n×1 fuzzy vector, x˜ is a n×p matrix of p fuzzy predictors, and τ˜ is a p×1 vector of unknown p fuzzy parameters. As a result of the lack of linearity of FcRp, ϵ is reduced to nonfuzzy random variable FRVϵ.(8)yja=xj1aτ1a+xj2aτ2a+⋯+xjpaτpa,yjb=xj1bτ1b+xj2bτ2b+⋯+xjpbτpb,yjc=xj1cτ1c+xj2cτ2c+⋯+xjpcτpc,where yja,yjb,yjc are left, middle, and right values, respectively.
Theτ^ja is as follows:(9)τ^ja=xjaTxja−1xjaTyja.
On the same lines, the above equations can be simplified as(10)τ^jb=xjbTxjb−1xjbTyjb,τ^jc=xjcTxjc−1xjcTyjc.
Consider the multiple fuzzy regression model is generalized as follows:(11)y˜i=τ˜0+τ˜1x˜i1+τ˜2x˜i2+⋯+τ˜px˜ip+ϵi,where y˜ is a n×1 fuzzy vector, x is a n×p matrix of p crisp predictors, and τ˜ is a p×1 vector of unknown p fuzzy parameters. As a result of the lack of linearity of FcRp, ϵ is reduced to a nonfuzzy random variable FRVϵ.(12)yja=xj1τ1a+xj2τ2a+⋯+xjpτpa,yjb=xj1τ1b+xj2τ2b+⋯+xjpτpb,yjc=xj1τ1c+xj2τ2c+⋯+xjpτpc,yjd=xj1τ1d+xj2τ2d+⋯+xjpτpd,where yja,yjb,yjc,yd is lower, middle-lower, middle-upper, and upper values correspondingly.
Theτ^ja is as follows:(13)τ^ja=xTx−1xTyja.
On the same lines above, the equation is simplified as(14)τ^jb=xTx−1xTyjb,τ^jc=xTx−1xTyjc,τ^jd=xTx−1xTyjd.


## 4. Numerical Computation

This research depends on the primary data of dengue fever patients collected from two public sector hospitals in Rawalpindi namedBenazir,Holyfamily, respectively. After taking the data, we applied principal component analysis to eliminate insignificant variables and consider those variables that have a significant impact on our study. We have implemented the Keiser-Meyer-Olkin and Bartlett’s test for the reduction of data.


### 4.1. Normality Test

H0: the data are normal.H1: the data are normal.If thep value is greater than 0.05, then this is normal.Table1 indicates that all the p values are greater than 0.05, so we conclude that our data are normal.Table 1
Normality test.
VariablesP valueAge0.178Suffering0.322Checkup0.334BPU0.103BPL0.212Figure1 indicates that all the p values are greater than 0.05 so we concluded that our data are normal.Figure 1
Data are normal.


### 4.2. Multicollinearity

Here, using the VIF (variance inflection factor) values, we have seen that each value of VIF is below 10, and the assumption is fulfilled.The multiple linear regression results are explained in Table2 and represented by Figure 2.Table 2
Multiple regression.
VariablesConstantSEtP valueIntercept108.4736.35017.0830.001Age0.0430.0520.8410.403Suffering fever−0.1040.289−0.3600.720Check up−0.0040.668−0.0060.996BPU−0.0450.060−0.7560.452BPL0.0880.075−1.1670.246Figure 2
Best Fitted Graph of Multiple Linear regression.


### 4.3. Interpretation

The estimated values in Table3 represent the regression coefficient which shows that the values of dengue fever increased by 0.043 units for one element increase in age, decreased 0.104 units for a unit increase in suffering fever, decreased 0.004 units for one element add-in checkup, decreased by 0.045 units for the unit increase in BP U, and decreased 0.088 units for one unit add-in BP L.Table 3
KMO and Bartlett’s test.
Kaiser-Meyar-Olkin measure adequacyMeasure of sampling[.660]Bartlett’s test of sphericityApprox. chi-square[53.67]DF[10]Sig.[.000]These results are represented in Figure2 which shows the best fitted classical multiple regression model.


#### 4.3.1. Case 1

Fuzzy multiple linear regression model with fuzzy independent variables and crisp dependent variable.The fuzzy multiple linear regression results are explained in Table4 and represented by Figure 3.Table 4
Result of fuzzy multiple regression when fuzzy independent variables and crisp-dependent variable in triangular membership function.
VariablesConstantSEtP valueIntercept0.5130.0638.0730.001Age0.0370.0550.6790.499Suffering fever0.1140.0502.2940.024Check up0.0410.0820.4990.619BPU0.0640.0820.7790.438BPL0.0650.0441.4950.138Figure 3
Fuzzy multiple regression using triangular membership function.


### 4.4. Interpretation

The estimated fuzzy regression coefficient shows that rates of Y increase by 0.037 units for one element add-inX1, 0.114 divisions on behalf of the unit increase in X2, 0.014 entities for the unit increase in X3, 0.064 units for the unit increase in X4, and increased 0.065 units for the unit rise in X5.


#### 4.4.1. Case 1

Fuzzy multiple linear regression model with fuzzy independent variables and crisp dependent variable.The multiple linear regression results are explained in Table5 and represented by Figure 4.Table 5
Result of fuzzy multiple regression when fuzzy independent variables and crisp-dependent variable in trapezoidal membership function.
VariablesConstantSEtP valueIntercept2.1140.14714.4200.001Age0.0530.0820.6470.519Suffering fever0.1600.0682.3580.020Check up0.0990.0581.7000.092BPU−0.1560.103−1.5140.133BPL0.2890.1122.5840.011Figure 4
Fuzzy multiple regression using trapezoidal membership function.


### 4.5. Interpretation

The estimated fuzzy regression coefficient indicates that the value of Y increases by 0.200836 units for a unit increase inX1, 0.277264 units for the unit rise in X2, 0.024512 units for the unit increasingly in X3, decreases by 0.09249 units for the unit increase in X4, and increases 0.43941 units for the unit increase in X5.


## 4.1. Normality Test

H0: the data are normal.H1: the data are normal.If thep value is greater than 0.05, then this is normal.Table1 indicates that all the p values are greater than 0.05, so we conclude that our data are normal.Table 1
Normality test.
VariablesP valueAge0.178Suffering0.322Checkup0.334BPU0.103BPL0.212Figure1 indicates that all the p values are greater than 0.05 so we concluded that our data are normal.Figure 1
Data are normal.


## 4.2. Multicollinearity

Here, using the VIF (variance inflection factor) values, we have seen that each value of VIF is below 10, and the assumption is fulfilled.The multiple linear regression results are explained in Table2 and represented by Figure 2.Table 2
Multiple regression.
VariablesConstantSEtP valueIntercept108.4736.35017.0830.001Age0.0430.0520.8410.403Suffering fever−0.1040.289−0.3600.720Check up−0.0040.668−0.0060.996BPU−0.0450.060−0.7560.452BPL0.0880.075−1.1670.246Figure 2
Best Fitted Graph of Multiple Linear regression.


## 4.3. Interpretation

The estimated values in Table3 represent the regression coefficient which shows that the values of dengue fever increased by 0.043 units for one element increase in age, decreased 0.104 units for a unit increase in suffering fever, decreased 0.004 units for one element add-in checkup, decreased by 0.045 units for the unit increase in BP U, and decreased 0.088 units for one unit add-in BP L.Table 3
KMO and Bartlett’s test.
Kaiser-Meyar-Olkin measure adequacyMeasure of sampling[.660]Bartlett’s test of sphericityApprox. chi-square[53.67]DF[10]Sig.[.000]These results are represented in Figure2 which shows the best fitted classical multiple regression model.


### 4.3.1. Case 1

Fuzzy multiple linear regression model with fuzzy independent variables and crisp dependent variable.The fuzzy multiple linear regression results are explained in Table4 and represented by Figure 3.Table 4
Result of fuzzy multiple regression when fuzzy independent variables and crisp-dependent variable in triangular membership function.
VariablesConstantSEtP valueIntercept0.5130.0638.0730.001Age0.0370.0550.6790.499Suffering fever0.1140.0502.2940.024Check up0.0410.0820.4990.619BPU0.0640.0820.7790.438BPL0.0650.0441.4950.138Figure 3
Fuzzy multiple regression using triangular membership function.


## 4.3.1. Case 1

Fuzzy multiple linear regression model with fuzzy independent variables and crisp dependent variable.The fuzzy multiple linear regression results are explained in Table4 and represented by Figure 3.Table 4
Result of fuzzy multiple regression when fuzzy independent variables and crisp-dependent variable in triangular membership function.
VariablesConstantSEtP valueIntercept0.5130.0638.0730.001Age0.0370.0550.6790.499Suffering fever0.1140.0502.2940.024Check up0.0410.0820.4990.619BPU0.0640.0820.7790.438BPL0.0650.0441.4950.138Figure 3
Fuzzy multiple regression using triangular membership function.


## 4.4. Interpretation

The estimated fuzzy regression coefficient shows that rates of Y increase by 0.037 units for one element add-inX1, 0.114 divisions on behalf of the unit increase in X2, 0.014 entities for the unit increase in X3, 0.064 units for the unit increase in X4, and increased 0.065 units for the unit rise in X5.


### 4.4.1. Case 1

Fuzzy multiple linear regression model with fuzzy independent variables and crisp dependent variable.The multiple linear regression results are explained in Table5 and represented by Figure 4.Table 5
Result of fuzzy multiple regression when fuzzy independent variables and crisp-dependent variable in trapezoidal membership function.
VariablesConstantSEtP valueIntercept2.1140.14714.4200.001Age0.0530.0820.6470.519Suffering fever0.1600.0682.3580.020Check up0.0990.0581.7000.092BPU−0.1560.103−1.5140.133BPL0.2890.1122.5840.011Figure 4
Fuzzy multiple regression using trapezoidal membership function.


## 4.4.1. Case 1

Fuzzy multiple linear regression model with fuzzy independent variables and crisp dependent variable.The multiple linear regression results are explained in Table5 and represented by Figure 4.Table 5
Result of fuzzy multiple regression when fuzzy independent variables and crisp-dependent variable in trapezoidal membership function.
VariablesConstantSEtP valueIntercept2.1140.14714.4200.001Age0.0530.0820.6470.519Suffering fever0.1600.0682.3580.020Check up0.0990.0581.7000.092BPU−0.1560.103−1.5140.133BPL0.2890.1122.5840.011Figure 4
Fuzzy multiple regression using trapezoidal membership function.


## 4.5. Interpretation

The estimated fuzzy regression coefficient indicates that the value of Y increases by 0.200836 units for a unit increase inX1, 0.277264 units for the unit rise in X2, 0.024512 units for the unit increasingly in X3, decreases by 0.09249 units for the unit increase in X4, and increases 0.43941 units for the unit increase in X5.


## 5. Performance Comparison

The following section describes the performance of multiple linear regression and fuzzy multiple linear regression.It is important to know which method from classical multiple linear regression and fuzzy multiple linear regression method is performing best and gives significant results. Comparison between these methods is made by using different evaluation criteria. Results obtained by using different evaluation techniques are given in Table6. The empirical analysis shows that the MSE, RMAE, BIC, and RAE in the case of fuzzy multiple linear regression are all smaller than the classical multiple linear regression method. This shows that fuzzy multiple regression can smooth the defuzzified forecast, and it is more consistent.Table 6
Error comparison of multiple regression and fuzzy multiple regression.
Error computationMultiple regressionFuzzy regressionMSE49.034730.086875BIC944.26−20.168RMSE7.002480.294746RAE4903.4738.687522It is important to know which method is performing best and gives significant results among classical multiple linear regression and fuzzy multiple linear regression method. Comparison between these methods is evaluated by using different evaluation criteria. Results obtained by using different evaluation techniques are given in Table7. The empirical analysis shows that the MSE, RMAE, BIC, and RAE in the case of fuzzy multiple linear regression are all smaller than the classical multiple linear regression method. This shows that fuzzy multiple regression can smooth the defuzzified forecast, and it is more consistent.Table 7
Error comparison of multiple regression and fuzzy multiple regression.
Error computationMultiple regressionFuzzy regressionMSE49.034730.025824BIC944.26−225.783RMSE7.002480.1607RAE4903.4732.582449We have compared the fuzzy regression results of triangular and trapezoidal membership functions in Figure4 which shows that the triangular membership function results were lower than that of trapezoidal so the triangular membership function is more efficient compared to the trapezoidal membership function and can be used in further studies for comparison.


## 6. Conclusion

There are numeral classical methods that are used to distribute accurate information, but in a lot of circumstances, accurate quantities cannot be achieved. This paper is based on the basic idea of the multiple linear regression method and fuzzy multiple regression method. In the proposed work, the fuzzy multiple regression method is done by using the triangular and trapezoidal membership function and evaluating the computation of dengue fever data to express the efficiency of the proposed fuzzy multiple regression with the existing multiple regression model. The realistic result of the mean square errorMSE, Bayesian information criteria BIC, root absolute error RAE, and root mean square error RMSE of fuzzy multiple regression with triangular and trapezoidal membership function is smaller compared to the multiple linear regression which indicates that the proposed method has a better performance as compared to multiple regression [13–17].



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