# A New Partial Correlation Coefficient Technique Based on Intuitionistic Fuzzy Information and Its Pattern Recognition Application

Authors: Paul Augustine Ejegwa; Idoko Charles Onyeke; Nasreen Kausar; Parameshwari Kattel
Journal: International Journal of Intelligent Systems
 (2023)

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## Abstract

Computation of correlation coefficient among attributes of ordinary database is important especially in the classification and analysis of data. Due to the hesitations in the process of data classification, the idea of intuitionistic fuzzy data (IFD) is appropriate for a reliable classification. To achieve a dependable correlation, the construct of partial correlation coefficient based on IFD has been considered. The construct of partial correlation coefficient of intuitionistic fuzzy sets (PCCIFSs) is reasonable since correlation coefficients of intuitionistic fuzzy sets (CCIFSs) are limited in the sense that it only expressed linear association and direction of such relation between IFD without minding the effect of other IFD. On the contrary, partial correlation coefficient finds the exact association between any two IFD by muting the effect of other IFD which could sway the result of the correlation coefficient. In previous works, the idea of PCCIFSs was introduced based on the multivariate correlation model using empirical logit transform. Besides the fact that the outputs of multivariate correlation model are not always easy to interpret, the approach also never considered the three parameters of IFSs and does not use the values of CCIFSs for the computational process. With these setbacks, we are motivated to propose a novel approach of finding PCCIFSs by incorporating the three parameters of IFD based on a modified CCIFSs approach. A comparative analysis of the robust PCCIFSs approach and the existing approach is considered to justify the novel approach. An application of the new approach of PCCIFSs is considered in the case of pattern recognition where the patterns are represented as intuitionistic fuzzy data.


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## Body

## 1. Introduction

The multiplicity of objectives in most human decision-making cases necessitates the use of multiple-criteria decision-making (MCDM) approaches. MCDM is mostly applicable in large-scale systems tasks. Fundamental to MCDM is the modelling of proper association between the individual components criteria of the decision function. In most instances, human beings are capable to express the suitable relationship between the criteria of components, however, with some hesitations. Albeit, the introduction of fuzzy sets [1] give an appropriate framework for the construction of MCDM since fuzzy set possesses the ability to offer a link between a linguistic expression and a mathematical modelling. Fuzzy set is very inadequate since it considers membership grade without minding the significance of nonmembership grade and hesitation.In order to ameliorate the limitation of fuzzy sets, Atanassov [2] developed the concept of intuitionistic fuzzy sets (IFSs) by incorporating the nonmembership degree (NMD) together with the membership degree (MD) to enhance the modelling of practical problems. The structure of IFS consists of MD represented by β and NMD represented by γ with the possibility of the existent of hesitation represented by δ, whereby their aggregate is always one and β+γ≤1. Many practical decision-making cases have been solved using IFSs. Boran and Akay [3] developed a biparametric similarity approach for IFSs with pattern recognition application, and in [4, 5], some novel distance measures under intuitionistic fuzzy information were introduced and applied in pattern recognition. In [6, 7], the idea of IFSs was used to discuss disease diagnosis based on composite relation. In [8], Heronian aggregation operators on IFSs were applied in group decision-making problem. An approach of the cluster algorithm using IFSs was discussed in [9], and decision-making processes were executed using correlation operators on IFSs [10, 11]. In fact, assorted application examples have been discussed from the perspective of IFSs [12–15]. Sundry discussions of decision-making in uncertain frameworks have been presented [16–18].In the analysis and classification of data, it is usual to employ correlation coefficient to find the connections among datasets. Correlation coefficient is a useful device for the measurement of association between two datasets. Correlation coefficient is positive when two datasets are directly associated and is negative when two datasets are not directly associated. In addition, zero correlation coefficient shows there is no relation between the datasets. The fact is that most datasets are imprecise, and the idea of correlation coefficient under fuzzy data has been discussed [19, 20]. In the same way, the idea of correlation measure based on intuitionistic fuzzy data (IFD) has been studied [21–27] because of the setback of fuzzy correlation coefficient.The concept of correlation coefficient is not sufficient for data analysis and classification in the sense that it only expressed linear association and direction of such relation between datasets without minding the effect of other datasets. Because of the limitation of correlation coefficient, the idea of partial correlation coefficient was introduced. In partial correlation coefficient, the exact association between any two datasets is found by muting the effect of other datasets which could sway the result of the correlation coefficient. The idea of partial correlation coefficient under fuzzy sets while muting the effects of other associating fuzzy sets had been proposed in [28] based on the correlation measure approach in [19]. In an attempt to align the partial correlation coefficient of fuzzy sets to normal framework, Hung and Wu [29] studied partial correlation coefficient of fuzzy sets using empirical logit transformation. Hung [30] proposed an approach for finding a partial correlation coefficient of IFSs (PCCIFSs), which considered only MD and NMD of the IFD without minding the influence of hesitation. The fact is that the approach in [30] does not take account of hesitation, and its result cannot be reliable because the beauty of IFD is in the hesitancy grade. In addition, the approach of PCCIFSs [30] was introduced based on the multivariate correlation model, which produces outputs that are difficult to interpret. More so, the approach in [30] does not use the values of CCIFSs by means of mathematical statistics. All these setbacks constitute the motivation to introduce a new approach of finding PCCIFSs which incorporates the three parameters of IFSs based on CCIFSs.In this work, a new approach of computing PCCIFSs among IFD is developed as an extension of the method under fuzzy context [28] by considering the three parameters of IFSs based on a modified approach of CCIFSs discussed in [23]. This novel approach takes into account the fundamental parameters of IFSs to avoid information loss experienced in the existing approach [30]. With the proposed approach, we will be able to perform the following activities:(i)
Find a linear relationship between two IFD(ii)
Compute a partial correlation coefficient between any two IFD while removing the effect of other associated IFD to enhance precise correlation measureAfter demonstrating the theoretical properties of the proposed approach, we apply the method in pattern recognition of building materials. For the application, we carry out the following functions:(i)
Computing the correlation coefficients between the building materials(ii)
Utilize the correlation coefficient values to calculate the precise relationship among the building materials via the proposed approach of PCCIFSsThe main objectives of the study are as follows:(1)
Modify a method of finding CCIFSs via statistical viewpoint in [23] with a comparison analysis(2)
Propose a novel technique of computing PCCIFSs based on the modified approach of CCIFSs(3)
Present a comparative analysis to show the effectiveness of the new PCCIFSs approach over the existing PCCIFSs approach(4)
Apply the new technique of computing PCCIFSs to the pattern recognition problemThe outline of the rest of the work is as follows: Section2 presents the concept of IFSs and its measures of correlation coefficient. Section 3 discusses PCCIFSs with some theoretical results and comparison. Section 4 dwells on the application of the new approach of PCCIFSs involving pattern recognition problems, and Section 5 concludes the work with future research direction.


## 2. Preliminaries

This section presents some basic ideas of IFSs and CCIFSs as discussed in [22, 23, 25] with a modified version.


### 2.1. Notion of Intuitionistic Fuzzy Sets

Some basic concepts of IFS are given to be used in the main work ( [2, 31]). Take X to be a nonempty set X.Definition 1.
An IFSR in X is defined by the structure.(1)R=βRx,γRxxx∈X,orR=x,βRx,γRx∣x∈X,where the functions βR,γR:X⟶0,1 define MD and NMD of x∈X in which(2)0≤βRx+γRx≤1.
For any IFSR in X, δRx=1−βRx−γRx represents the hesitation margin of R.Definition 2.
Intuitionistic fuzzy pairs (IFPs) or intuitionistic fuzzy values (IFVs) are characterized by the formx,y, where x+y≤1 for x,y∈0,1. IFPs/IFVs weigh IFS for which the components (x and y) are construed as MD and NMD.Definition 3.
AssumeR and I are IFSs in X, then we get the following expressions:(1)
R=I iff βRx=βIx and γRx=γIx for all x∈X(2)
R⊆I iff βRx≤βIx and γRx≥γIx for all x∈X(3)
R≺_I iff βRx≤βIx and γRx≤γIx for all x∈X(4)
R¯=γRx,βRx/xx∈X(5)
R∪I=maxβRx,βIx/x,minγRx,γIx/xx∈X(6)
R∩I=minβRx,βIx/x,maxγRx,γIx/xx∈X


### 2.2. Correlation Coefficient of IFSs

Some methods of calculating CCIFSs by means of mathematical statistical have been discussed in literature [22, 23, 25]. Let R and I be IFSs in X=x1,…,xn, n∈1,∞, and let ∂R,I be a correlation coefficient between R and I. Then, the following definition defines CCIFSs.Definition 4.
(see [22]). The correlation coefficient ∂R,I is a measuring function ∂:IFS×IFS⟶−1,1 if and only if the following conditions are satisfied:(1)
∂R,I∈−1,1(2)
∂R,I=∂I,R(3)
IfR and I are equal, then ∂R,I=1
WhenσR,I approaches 1, it shows that R and I are more correlated with better performance index. Again, if σR,I approaches −1, then R and I are more/less not correlated. Now, we present some existing methods of calculating CCIFSs. Before the presentation, we defined the mean values of IFSs as follows:Definition 5..
LetR and I be IFSs in X=x1,…,xn, n∈1,∞. Then, the mean values of R and I are represented as follows:(3)βR¯,βI¯=Σi=1nβRxin,Σi=1nβIxinγR¯,γI¯=Σi=1nγRxin,Σi=1nγIxinδR¯,δI¯=Σi=1nδRxin,Σi=1nδIxin.
Some existing approaches of computing CCIFSs are as follows:(i)
Approach in [23]is expressed as follows:(4)∂1R,I=Σi=1n∇iR∇iIΣi=1n∇i2RΣi=1n∇i2I,where∇iR,∇iI are the deviations of R and I defined by the following equation:(5)∇iR=βRxi−βR¯−γRxi−γR¯∇iI=βIxi−βI¯−γIxi−γI¯.(ii)
Approach in [22]is expressed as follows:(6)∂2R,I=∆1R,I+∆2R,I2,where(7)∆1R,I=Σi=1nβRxi−βR¯βIxi−βI¯Σi=1nβRxi−βR¯2Σi=1nβIxi−βI¯20.5∆2R,I=Σi=1nγRxi−γR¯γIxi−γI¯Σi=1nγRxi−γR¯2Σi=1nγIxi−γI¯20.5.(iii)
Approach in [25]is expressed as follows:(8)∂3R,I=ΦR,IΨRΨI,where(9)ΦR,I=Σi=1nβRxi−βR¯βIxi−βI¯+γRxi−γR¯γIxi−γI¯ΨR=Σi=1nβRxi−βR¯2+γRxi−γR¯2ΨI=Σi=1nβIxi−βI¯2+γIxi−γI¯2.


#### 2.2.1. Modified Intuitionistic Fuzzy Correlation Coefficient Approach

Because the method presented in [23] does not consider the hesitation grades of R and I, certainly the output would not be reliable due to information loss. By incorporating the complete parameters of IFSs, the following new correlation coefficient between IFSs R and I is obtained:(10)∂˜R,I=Σi=1n∇iR∇iIΣi=1n∇i2RΣi=1n∇i2I0.5,where the deviations of R and I can be computed by the following equation:(11)∇iR=βRxi−βR¯−γRxi−γR¯−δRxi−δR¯∇iI=βIxi−βI¯−γIxi−γI¯−δIxi−δI¯.Equation (10) can be rewritten as follows:(12)∂˜R,I=ΦR,IΨRΨI,where(13)ΦR,I=1n−1Σi=1n∇iR∇iI,is the covariance of R and I, and(14)ΨR=1n−1Σi=1n∇i2R,ΨI=1n−1Σi=1n∇i2I,are the variances of R and I, respectively.Certainly,∂˜R,I=∂˜I,R since(15)∂˜R,I=Σi=1n∇iR∇iIΣi=1n∇i2RΣi=1n∇i2I0.5=Σi=1n∇iI∇iRΣi=1n∇i2IΣi=1n∇i2R0.5=σ˜I,R.We intend to show that equation (10) is more reliable than the approaches in [22, 23, 25] because it considers MD, NMD, and HM of the concerned IFSs. To see this, we consider the following illustrations.


#### 2.2.2. Numerical Illustrations

To show that the modified approach of correlation coefficient is more dependable compare to the listed ones [22, 23, 25], we consider an example involving dissimilar IFSs.Example 1.
SupposeR and I are IFSs in X=x1,x2,x3, where(16)R=0.1,0.2x1,0.2,0.1x2,0.3,0x3,I=0.3,0x1,0.2,0.2x2,0.1,0.6x3.
By mere inspection, the correlation coefficient ofR and I is perfectly negative since R≠I, R⊈I, and I⊈R. We use equations (4), (6), (8), and the modified approach, i.e., equation (10) to show these facts.
Using the approach in [23], we obtain ∂1R,I=−0.9898. Using the approach in [22], we get ∂2R,I=−0.9970. Using the approach in [25], we get ∂3R,I=−0.8800. With the modified approach given as equation (10), we get the correlation coefficient ∂˜R,I=−1. Table 1 contains the results for nippy comparison.
From Table1, the results of equations (4), (6), and (8) show that the correlation coefficient of R and I is negative but not perfectly negative although R and I are dissimilar. On the contrary, the modified approach, i.e., equation (10) shows that R and I are perfectly negative correlated in agreement to the dissimilarity of R and I. With this, we can say that the modified approach has a better performance index compared to the approaches in [22, 23, 25].
Now, we employ the enlisted CCIFSs approaches to discuss the car purchasing process.Table 1
Correlation coefficient values.
ApproachesHung [22]Liu et al. [23]Thao et al. [25]Modified methodExample1−0.9970−0.9898−0.8800−1Example 2.
A car buyer wants to buy a car whose descriptions are represented by the following set.(17)D=D1fuelintake,D2price,D3levelofcomfort,D4design,D5security.
The buyer’s descriptions are fuzzified intuitionistically as follows:(18)R=0.8,0.1D1,0.6,0.2D2,0.7,0.2D3,0.65,0.1D4,0.85,0.15D5.
On reaching the cars’ stand, the car dealer gives the descriptions of the available car models in intuitionistic fuzzy features as follows:(19)R1=0.4,0.4D1,0.5,0.1D2,0.6,0.2D3,0.4,0.5D4,0.3,0.2D5,R2=0.79,0.1D1,0.7,0.2D2,0.5,0.3D3,0.55,0.2D4,0.65,0.2D5,R3=0.3,0.5D1,0.4,0.3D2,0.8,0.1D3,0.1,0.6D4,0.5,0.4D5,R4=0.7,0.3D1,0.6,0.1D2,0.7,0.1D3,0.6,0.2D4,0.7,0.2D5,R5=0.5,0.2D1,0.6,0.3D2,0.7,0.1D3,0.6,0.2D4,0.5,0.3D5.
The decision of the buyer is to buy the car model that suits his/her preference. Using the modified CCIFSs approach, we get the correlation coefficients of the buyer’s preference and the available car models as follows:(20)∂˜Iˇ1,Rˇ=0.1677,∂˜Iˇ2,Rˇ=0.3158,∂˜Iˇ3,Rˇ=−0.6133,∂˜Iˇ4,Rˇ=0.8364,∂˜Iˇ5,Rˇ=−0.6628.
From the results, the buyer should go for the car modelI4 because it fits the buyer’s preference the most. This decision tallies with the mere observation, which indicates that the car model I4 is more related to R than all the other car models.
By employing the enlisted CCIFSs approaches, the correlation coefficients of the buyer’s preference and the available car models are presented in Table2.
By comparison, the new approach that includes the hesitation parameter gives better interpretations of the correlations which exist between the buyer’s preference and the car models. From Table2, we see that the correlation coefficient values obtained from the approaches in [22, 23, 25] indicate that the buyer should go for the car model I2, which does not actually tally with the mere observation. Certainly, this irregularity is due to the exclusion of one of the descriptive parameters of IFSs. This justifies the reason why the hesitation margins of the considered IFSs should be taken into consideration while computing CCIFSs in agreement with [32].
The reason why we discuss the modified approach of intuitionistic fuzzy correlation coefficient is to enable us to present a new partial correlation coefficient of IFSs. The concept of partial correlation coefficient of IFSs has been initiated in [30]. This work proposes a new approach of computing partial correlation coefficient of IFSs based on equation (10).Table 2
Correlation coefficient values.
ApproachesIˇ1,RˇIˇ2,RˇIˇ3,RˇIˇ4,RˇIˇ5,RˇHung [22]−0.37110.5114−0.7631−0.0300−0.3314Liu et al. [23]−0.29730.4456−0.6746−0.4125−0.4551Thao et al. [25]−0.11450.4027−0.64070.0869−0.4222Modified method0.16770.3158−0.61330.8364−0.6628


## 2.1. Notion of Intuitionistic Fuzzy Sets

Some basic concepts of IFS are given to be used in the main work ( [2, 31]). Take X to be a nonempty set X.Definition 1.
An IFSR in X is defined by the structure.(1)R=βRx,γRxxx∈X,orR=x,βRx,γRx∣x∈X,where the functions βR,γR:X⟶0,1 define MD and NMD of x∈X in which(2)0≤βRx+γRx≤1.
For any IFSR in X, δRx=1−βRx−γRx represents the hesitation margin of R.Definition 2.
Intuitionistic fuzzy pairs (IFPs) or intuitionistic fuzzy values (IFVs) are characterized by the formx,y, where x+y≤1 for x,y∈0,1. IFPs/IFVs weigh IFS for which the components (x and y) are construed as MD and NMD.Definition 3.
AssumeR and I are IFSs in X, then we get the following expressions:(1)
R=I iff βRx=βIx and γRx=γIx for all x∈X(2)
R⊆I iff βRx≤βIx and γRx≥γIx for all x∈X(3)
R≺_I iff βRx≤βIx and γRx≤γIx for all x∈X(4)
R¯=γRx,βRx/xx∈X(5)
R∪I=maxβRx,βIx/x,minγRx,γIx/xx∈X(6)
R∩I=minβRx,βIx/x,maxγRx,γIx/xx∈X


## 2.2. Correlation Coefficient of IFSs

Some methods of calculating CCIFSs by means of mathematical statistical have been discussed in literature [22, 23, 25]. Let R and I be IFSs in X=x1,…,xn, n∈1,∞, and let ∂R,I be a correlation coefficient between R and I. Then, the following definition defines CCIFSs.Definition 4.
(see [22]). The correlation coefficient ∂R,I is a measuring function ∂:IFS×IFS⟶−1,1 if and only if the following conditions are satisfied:(1)
∂R,I∈−1,1(2)
∂R,I=∂I,R(3)
IfR and I are equal, then ∂R,I=1
WhenσR,I approaches 1, it shows that R and I are more correlated with better performance index. Again, if σR,I approaches −1, then R and I are more/less not correlated. Now, we present some existing methods of calculating CCIFSs. Before the presentation, we defined the mean values of IFSs as follows:Definition 5..
LetR and I be IFSs in X=x1,…,xn, n∈1,∞. Then, the mean values of R and I are represented as follows:(3)βR¯,βI¯=Σi=1nβRxin,Σi=1nβIxinγR¯,γI¯=Σi=1nγRxin,Σi=1nγIxinδR¯,δI¯=Σi=1nδRxin,Σi=1nδIxin.
Some existing approaches of computing CCIFSs are as follows:(i)
Approach in [23]is expressed as follows:(4)∂1R,I=Σi=1n∇iR∇iIΣi=1n∇i2RΣi=1n∇i2I,where∇iR,∇iI are the deviations of R and I defined by the following equation:(5)∇iR=βRxi−βR¯−γRxi−γR¯∇iI=βIxi−βI¯−γIxi−γI¯.(ii)
Approach in [22]is expressed as follows:(6)∂2R,I=∆1R,I+∆2R,I2,where(7)∆1R,I=Σi=1nβRxi−βR¯βIxi−βI¯Σi=1nβRxi−βR¯2Σi=1nβIxi−βI¯20.5∆2R,I=Σi=1nγRxi−γR¯γIxi−γI¯Σi=1nγRxi−γR¯2Σi=1nγIxi−γI¯20.5.(iii)
Approach in [25]is expressed as follows:(8)∂3R,I=ΦR,IΨRΨI,where(9)ΦR,I=Σi=1nβRxi−βR¯βIxi−βI¯+γRxi−γR¯γIxi−γI¯ΨR=Σi=1nβRxi−βR¯2+γRxi−γR¯2ΨI=Σi=1nβIxi−βI¯2+γIxi−γI¯2.


### 2.2.1. Modified Intuitionistic Fuzzy Correlation Coefficient Approach

Because the method presented in [23] does not consider the hesitation grades of R and I, certainly the output would not be reliable due to information loss. By incorporating the complete parameters of IFSs, the following new correlation coefficient between IFSs R and I is obtained:(10)∂˜R,I=Σi=1n∇iR∇iIΣi=1n∇i2RΣi=1n∇i2I0.5,where the deviations of R and I can be computed by the following equation:(11)∇iR=βRxi−βR¯−γRxi−γR¯−δRxi−δR¯∇iI=βIxi−βI¯−γIxi−γI¯−δIxi−δI¯.Equation (10) can be rewritten as follows:(12)∂˜R,I=ΦR,IΨRΨI,where(13)ΦR,I=1n−1Σi=1n∇iR∇iI,is the covariance of R and I, and(14)ΨR=1n−1Σi=1n∇i2R,ΨI=1n−1Σi=1n∇i2I,are the variances of R and I, respectively.Certainly,∂˜R,I=∂˜I,R since(15)∂˜R,I=Σi=1n∇iR∇iIΣi=1n∇i2RΣi=1n∇i2I0.5=Σi=1n∇iI∇iRΣi=1n∇i2IΣi=1n∇i2R0.5=σ˜I,R.We intend to show that equation (10) is more reliable than the approaches in [22, 23, 25] because it considers MD, NMD, and HM of the concerned IFSs. To see this, we consider the following illustrations.


### 2.2.2. Numerical Illustrations

To show that the modified approach of correlation coefficient is more dependable compare to the listed ones [22, 23, 25], we consider an example involving dissimilar IFSs.Example 1.
SupposeR and I are IFSs in X=x1,x2,x3, where(16)R=0.1,0.2x1,0.2,0.1x2,0.3,0x3,I=0.3,0x1,0.2,0.2x2,0.1,0.6x3.
By mere inspection, the correlation coefficient ofR and I is perfectly negative since R≠I, R⊈I, and I⊈R. We use equations (4), (6), (8), and the modified approach, i.e., equation (10) to show these facts.
Using the approach in [23], we obtain ∂1R,I=−0.9898. Using the approach in [22], we get ∂2R,I=−0.9970. Using the approach in [25], we get ∂3R,I=−0.8800. With the modified approach given as equation (10), we get the correlation coefficient ∂˜R,I=−1. Table 1 contains the results for nippy comparison.
From Table1, the results of equations (4), (6), and (8) show that the correlation coefficient of R and I is negative but not perfectly negative although R and I are dissimilar. On the contrary, the modified approach, i.e., equation (10) shows that R and I are perfectly negative correlated in agreement to the dissimilarity of R and I. With this, we can say that the modified approach has a better performance index compared to the approaches in [22, 23, 25].
Now, we employ the enlisted CCIFSs approaches to discuss the car purchasing process.Table 1
Correlation coefficient values.
ApproachesHung [22]Liu et al. [23]Thao et al. [25]Modified methodExample1−0.9970−0.9898−0.8800−1Example 2.
A car buyer wants to buy a car whose descriptions are represented by the following set.(17)D=D1fuelintake,D2price,D3levelofcomfort,D4design,D5security.
The buyer’s descriptions are fuzzified intuitionistically as follows:(18)R=0.8,0.1D1,0.6,0.2D2,0.7,0.2D3,0.65,0.1D4,0.85,0.15D5.
On reaching the cars’ stand, the car dealer gives the descriptions of the available car models in intuitionistic fuzzy features as follows:(19)R1=0.4,0.4D1,0.5,0.1D2,0.6,0.2D3,0.4,0.5D4,0.3,0.2D5,R2=0.79,0.1D1,0.7,0.2D2,0.5,0.3D3,0.55,0.2D4,0.65,0.2D5,R3=0.3,0.5D1,0.4,0.3D2,0.8,0.1D3,0.1,0.6D4,0.5,0.4D5,R4=0.7,0.3D1,0.6,0.1D2,0.7,0.1D3,0.6,0.2D4,0.7,0.2D5,R5=0.5,0.2D1,0.6,0.3D2,0.7,0.1D3,0.6,0.2D4,0.5,0.3D5.
The decision of the buyer is to buy the car model that suits his/her preference. Using the modified CCIFSs approach, we get the correlation coefficients of the buyer’s preference and the available car models as follows:(20)∂˜Iˇ1,Rˇ=0.1677,∂˜Iˇ2,Rˇ=0.3158,∂˜Iˇ3,Rˇ=−0.6133,∂˜Iˇ4,Rˇ=0.8364,∂˜Iˇ5,Rˇ=−0.6628.
From the results, the buyer should go for the car modelI4 because it fits the buyer’s preference the most. This decision tallies with the mere observation, which indicates that the car model I4 is more related to R than all the other car models.
By employing the enlisted CCIFSs approaches, the correlation coefficients of the buyer’s preference and the available car models are presented in Table2.
By comparison, the new approach that includes the hesitation parameter gives better interpretations of the correlations which exist between the buyer’s preference and the car models. From Table2, we see that the correlation coefficient values obtained from the approaches in [22, 23, 25] indicate that the buyer should go for the car model I2, which does not actually tally with the mere observation. Certainly, this irregularity is due to the exclusion of one of the descriptive parameters of IFSs. This justifies the reason why the hesitation margins of the considered IFSs should be taken into consideration while computing CCIFSs in agreement with [32].
The reason why we discuss the modified approach of intuitionistic fuzzy correlation coefficient is to enable us to present a new partial correlation coefficient of IFSs. The concept of partial correlation coefficient of IFSs has been initiated in [30]. This work proposes a new approach of computing partial correlation coefficient of IFSs based on equation (10).Table 2
Correlation coefficient values.
ApproachesIˇ1,RˇIˇ2,RˇIˇ3,RˇIˇ4,RˇIˇ5,RˇHung [22]−0.37110.5114−0.7631−0.0300−0.3314Liu et al. [23]−0.29730.4456−0.6746−0.4125−0.4551Thao et al. [25]−0.11450.4027−0.64070.0869−0.4222Modified method0.16770.3158−0.61330.8364−0.6628


## 2.2.1. Modified Intuitionistic Fuzzy Correlation Coefficient Approach

Because the method presented in [23] does not consider the hesitation grades of R and I, certainly the output would not be reliable due to information loss. By incorporating the complete parameters of IFSs, the following new correlation coefficient between IFSs R and I is obtained:(10)∂˜R,I=Σi=1n∇iR∇iIΣi=1n∇i2RΣi=1n∇i2I0.5,where the deviations of R and I can be computed by the following equation:(11)∇iR=βRxi−βR¯−γRxi−γR¯−δRxi−δR¯∇iI=βIxi−βI¯−γIxi−γI¯−δIxi−δI¯.Equation (10) can be rewritten as follows:(12)∂˜R,I=ΦR,IΨRΨI,where(13)ΦR,I=1n−1Σi=1n∇iR∇iI,is the covariance of R and I, and(14)ΨR=1n−1Σi=1n∇i2R,ΨI=1n−1Σi=1n∇i2I,are the variances of R and I, respectively.Certainly,∂˜R,I=∂˜I,R since(15)∂˜R,I=Σi=1n∇iR∇iIΣi=1n∇i2RΣi=1n∇i2I0.5=Σi=1n∇iI∇iRΣi=1n∇i2IΣi=1n∇i2R0.5=σ˜I,R.We intend to show that equation (10) is more reliable than the approaches in [22, 23, 25] because it considers MD, NMD, and HM of the concerned IFSs. To see this, we consider the following illustrations.


## 2.2.2. Numerical Illustrations

To show that the modified approach of correlation coefficient is more dependable compare to the listed ones [22, 23, 25], we consider an example involving dissimilar IFSs.Example 1.
SupposeR and I are IFSs in X=x1,x2,x3, where(16)R=0.1,0.2x1,0.2,0.1x2,0.3,0x3,I=0.3,0x1,0.2,0.2x2,0.1,0.6x3.
By mere inspection, the correlation coefficient ofR and I is perfectly negative since R≠I, R⊈I, and I⊈R. We use equations (4), (6), (8), and the modified approach, i.e., equation (10) to show these facts.
Using the approach in [23], we obtain ∂1R,I=−0.9898. Using the approach in [22], we get ∂2R,I=−0.9970. Using the approach in [25], we get ∂3R,I=−0.8800. With the modified approach given as equation (10), we get the correlation coefficient ∂˜R,I=−1. Table 1 contains the results for nippy comparison.
From Table1, the results of equations (4), (6), and (8) show that the correlation coefficient of R and I is negative but not perfectly negative although R and I are dissimilar. On the contrary, the modified approach, i.e., equation (10) shows that R and I are perfectly negative correlated in agreement to the dissimilarity of R and I. With this, we can say that the modified approach has a better performance index compared to the approaches in [22, 23, 25].
Now, we employ the enlisted CCIFSs approaches to discuss the car purchasing process.Table 1
Correlation coefficient values.
ApproachesHung [22]Liu et al. [23]Thao et al. [25]Modified methodExample1−0.9970−0.9898−0.8800−1Example 2.
A car buyer wants to buy a car whose descriptions are represented by the following set.(17)D=D1fuelintake,D2price,D3levelofcomfort,D4design,D5security.
The buyer’s descriptions are fuzzified intuitionistically as follows:(18)R=0.8,0.1D1,0.6,0.2D2,0.7,0.2D3,0.65,0.1D4,0.85,0.15D5.
On reaching the cars’ stand, the car dealer gives the descriptions of the available car models in intuitionistic fuzzy features as follows:(19)R1=0.4,0.4D1,0.5,0.1D2,0.6,0.2D3,0.4,0.5D4,0.3,0.2D5,R2=0.79,0.1D1,0.7,0.2D2,0.5,0.3D3,0.55,0.2D4,0.65,0.2D5,R3=0.3,0.5D1,0.4,0.3D2,0.8,0.1D3,0.1,0.6D4,0.5,0.4D5,R4=0.7,0.3D1,0.6,0.1D2,0.7,0.1D3,0.6,0.2D4,0.7,0.2D5,R5=0.5,0.2D1,0.6,0.3D2,0.7,0.1D3,0.6,0.2D4,0.5,0.3D5.
The decision of the buyer is to buy the car model that suits his/her preference. Using the modified CCIFSs approach, we get the correlation coefficients of the buyer’s preference and the available car models as follows:(20)∂˜Iˇ1,Rˇ=0.1677,∂˜Iˇ2,Rˇ=0.3158,∂˜Iˇ3,Rˇ=−0.6133,∂˜Iˇ4,Rˇ=0.8364,∂˜Iˇ5,Rˇ=−0.6628.
From the results, the buyer should go for the car modelI4 because it fits the buyer’s preference the most. This decision tallies with the mere observation, which indicates that the car model I4 is more related to R than all the other car models.
By employing the enlisted CCIFSs approaches, the correlation coefficients of the buyer’s preference and the available car models are presented in Table2.
By comparison, the new approach that includes the hesitation parameter gives better interpretations of the correlations which exist between the buyer’s preference and the car models. From Table2, we see that the correlation coefficient values obtained from the approaches in [22, 23, 25] indicate that the buyer should go for the car model I2, which does not actually tally with the mere observation. Certainly, this irregularity is due to the exclusion of one of the descriptive parameters of IFSs. This justifies the reason why the hesitation margins of the considered IFSs should be taken into consideration while computing CCIFSs in agreement with [32].
The reason why we discuss the modified approach of intuitionistic fuzzy correlation coefficient is to enable us to present a new partial correlation coefficient of IFSs. The concept of partial correlation coefficient of IFSs has been initiated in [30]. This work proposes a new approach of computing partial correlation coefficient of IFSs based on equation (10).Table 2
Correlation coefficient values.
ApproachesIˇ1,RˇIˇ2,RˇIˇ3,RˇIˇ4,RˇIˇ5,RˇHung [22]−0.37110.5114−0.7631−0.0300−0.3314Liu et al. [23]−0.29730.4456−0.6746−0.4125−0.4551Thao et al. [25]−0.11450.4027−0.64070.0869−0.4222Modified method0.16770.3158−0.61330.8364−0.6628


## 3. Partial Correlation Coefficient of Intuitionistic Fuzzy Attributes

To start with, we first reiterate the existing technique of finding partial correlation coefficient (PCC) of IFSs [30] and then present the novel technique based on equation (10) between intuitionistic fuzzy attributes (IFAs).


### 3.1. Existing Technique of Partial Correlation Coefficient of IFAs

Hung [30] developed a PCC between of IFAs, R1,R2,…,Rp by the means of the normal correlation model. The IFA was adapted to fit into normal framework using logit transform [33] as follows:(21)ZiRj=logβRjxiγRjxi,for i=1,…,n and j=1,…,p. ZiRj was approximated via empirical logit transform [34] to avoid the case when ZiRj=log0 or ZiRj=log∞ as follows:(22)ZiRj=logβRjxi+1/2nγRjxi+1/2n,for i=1,…,n, j=1,…,p.Suppose we have two IFAsRi and Rj, then the correlation is determined as follows:(23)∂2Ri,Rj=SSRRjSSTORi,where SSTORi=∑i=1nβRixi−βRi¯2+γRixi−γRi¯2 (total sum of squares) and SSRRj=SSTORi−SSERj (regression sum of squares) for SSERj=∑i=1ne2 (error sum of squares).Hence, the correlation coefficient betweenRi and Rj is represented as follows.(24)∂Ri,Rj=SSRRjSSTORi.Thus, the PCC between IFAs,R1 and R2, when R3,…,Rp are muted is represented as follows:(25)∂R1,R2R3,…,Rp=SSER3,…,Rp−SSER2,…,RpSSER3,…,Rp.


### 3.2. New Partial Correlation Coefficient of IFAs

We present a new approach of computing PCC of IFSs by the means of mathematical statistics based on equation (10). Assume there are random samples of three IFAs R1, R2, and R3 in X. The linear correlation between R1 and R2 can be computed by using equation (10). The fact is that R1, R2, and R3 are IFAs drawn from the space X, and the linear correlation between R1 and R2 could be impacted by R3. Thus, to compute the precise linear correlation between R1 and R2, the effect of R3 must be removed. Such a precise linear correlation between R1 and R2 can be determined by partial correlation coefficient.


#### 3.2.1. First-Order Partial Correlation Coefficient of IFAs

Definition 6.
LetR1, R2, and R3 be IFAs in the sample space X. Then, the PCC between IFAs R1 and R2 without the influence of IFA R3 is defined as follows:(26)∂˜R1,R2R3=∂˜R1,R2−∂˜R1,R3∂˜R2,R31−∂˜2R1,R31−∂˜2R2,R31/2,where ∂˜R1,R2,∂˜R1,R3, and ∂˜R2,R3 can be calculated by using equation (12). We can rewrite equation (26) as follows:(27)∂˜2C,DE=∂˜R1,R2−∂˜R1,R3∂˜R2,R321−∂˜2R1,R31−∂˜2R2,R3.
Now, we delineate some characteristics of the first-order PCC of IFAs as follows:Theorem 1.
SupposeR1,R2, and R3 are IFAs drawn from X=x1,…,xn for n∈⌈1,∞⌉. Then, ∂˜R1,R2R3=∂˜R2,R1R3.Proof.
We know that∂˜R1,R2=∂˜R2,R1 by equation (12). Thus,(28)∂˜R1,R2R3=∂˜R1,R2−∂˜R1,R3∂˜R2,R31−∂˜2R1,R31−∂˜2R2,R31/2=∂˜R2,R1−∂˜R2,R3∂˜R1,R31−∂˜2R2,R31−∂˜2R1,R31/2=∂˜R2,R1R3,which completes the proof.Proposition 1.
With the same hypothesis in Theorem1, a PCC ∂˜R1,R2R3becomesacorrelationcoefficient∂˜R1,R2ifandonlyif∂˜R1,R3=0=∂˜R2,R3.Proof.
Assume that∂˜R1,R3=∂˜R2,R3=0; then, we have the following expression:(29)∂˜R1,R2R3=∂˜R1,R2=∂˜R2,R1=∂˜R2,R1R3.
The converse is straightforward.Remark 1.
With the same hypothesis in Theorem1, the following statements are valid:(i)
IfR1 and R3 have a perfect positive linear correlation, then ∂˜R1,R2R3=∂˜R2,R1R3=∞(ii)
If∂˜R1,R3=0, then(30)∂˜R1,R2R3=∂˜R1,R21−∂˜2R2,R31/2=∂˜R2,R1R3.Also, if∂˜R2,R3=0, then(31)∂˜R1,R2R3=∂˜R1,R21−∂˜2R1,R31/2=∂˜R2,R1R3.(iii)
If∂˜R1,R2=0, ∂˜R1,R3≠0, and ∂˜R2,R3≠0, then(32)∂˜R1,R2R3=−∂˜R1,R3∂˜R2,R31−∂˜2R1,R31−∂˜2R2,R31/2.


#### 3.2.2.nth-Order Partial Correlation Coefficient of IFAs

So far, we have been considering PCC between two IFAs by keeping the third IFA muted. The concept ofnth-order PCC of IFAs is conceivable in the case that there are random samples of more than three IFAs.LetR1,…,Rr,Rr+1,…,Rr+s be IFAs in X, where each IFA is represented as follows:(33)Ri=x,βRix,γRixx∈X,where βRi,γRi:X⟶0,1 for i=1,…,r+s. Also, Ri can be written as follows:(34)Ri=xk,βR1xk,γR1xk,…,βRr+sxk,γRr+sxkxk∈X,for k=1,…,n.We designate the covariances of IFAsR1,…,Rr,Rr+1,…,Rr+s as covariance matrix N, which we define as follows:(35)N=ΦRi,Rji=1,…,r+s,j=1,…,r+s,where(36)ΦRi,Rj=Σk=1n∇kRi∇kRjn−1,is the covariance of Ri,Rj, in which(37)∇kRi=βRi2xk−βRi¯2−γRi2xk−γRi¯2−δRi2xk−δRi¯2∇kRj=βRj2xk−βRj¯2−γRj2xk−γRj¯2−δRj2xk−δRj¯2,are the deviations of Ri and Rj, respectively.Definition 7.
The PCC ofRi,Rj, i,j=1,…,r, by removing the effects of Rr+1,…,Rr+s can be obtained as follows:
Suppose we have four parts partition ofN,(38)N=N11N12N21N22,where N11 is the r×r covariance matrix of Ri,Rj for i,j=1,…,r, N12 is the r×s covariance matrix of Ri,Rj for i=1,…,r, j=r+1,…,r+s, N21 is the s×r covariance matrix of Ri,Rj for i=r+1,…,r+s, j=1,…,r, and N22 is the s×s covariance matrix of Ri,Rj for i,j=r+1,…,r+s. Then, certainly, N12 and N21 transpose each other.
By keepingRr+1,…,Rr+s constant, the PCC of Ri,Rj is defined as follows:(39)∂˜Ri,RjRr+1,…,Rr+s=ΦRi,RjRr+1,…,Rr+sΦRi,RiRr+1,…,Rr+sΦRj,RjRr+1,…,Rr+s1/2,where i,j=1,…,r and ΦRi,RjRr+1,…,Rr+s is the i,jth element of partial covariance matrix of N112=N11−N12N22−1N21.
It follows that(40)∂˜Ri,RjRr+1,…,Rr+s=∂˜Rj,RiRr+1,…,Rr+s.
SinceN112 is a symmetry matrix and(41)ΦRi,RjRr+1,…,Rr+s=ΦRj,RiRr+1,…,Rr+s.Theorem 2.
Ifr=2 and s=1, then the nth-order PCC of IFAs is equivalent to the first-order PCC of IFAs.Proof.
The proof of Theorem2 is in the appendix section.Remark 2.
Ifr=2 and s=2, then the nth-order PCC of PFAs is as follows:(42)∂˜R1,R2R3,R4=∂˜R1,R2R4−∂˜R2,R3R4∂˜R1,R3R41−∂˜2R2,R3R41−∂˜2R1,R3R41/2,and redrafted as(43)∂˜1234=∂˜124−∂˜234∂˜1341−∂˜23421−∂˜13421/2.
Indeed,∂˜1234=∂˜1243 since(44)∂˜1234=∂˜124−∂˜234∂˜1341−∂˜23421−∂˜13421/2=∂˜123−∂˜243∂˜1431−∂˜24321−∂˜14321/2=∂˜1243.


#### 3.2.3. Comparative Analysis of the PCC Approaches

We present the comparison of the approaches to show the edge of the new approach of PCC of IFAs over the approach in [30].Example 3.
LetR1, R2, R3, and R4 be IFSs in X=x1,x2,x3,x4,x5 and are defined as follows:(45)R1=0.1,0.8x1,0.3,0.5x2,0.6,0.2x3,0.9,0x4,1,0x5,R2=0.01,0.96x1,0.09,0.75x2,0.36,0.36x3,0.81,0x4,1,0x5,R3=0.96,0.01x1,0.75,0.09x2,0.36,0.36x3,0,0.81x4,0,1x5,R4=0,1x1,0.01,0.94x2,0.13,0.59x3,0.66,0x4,1,0x5.
We aim to compute the PCC between the IFAs by removing the effect of IFAR1. Using the approach in [30], we obtain the following results:(46)∂R2R2R1=1.000,∂R2R3R1=−1.000,∂R2R4R1=0.983,∂R3R3R1=1.000,∂R3R4R1=−0.983,∂R4R4R1=1.000.
With the novel approach, we obtain the following results:(47)∂˜R2R2R1=1.000,∂˜R2R3R1=0.508,∂˜R2R4R1=0.975,∂˜R3R3R1=1.000,∂˜R3R4R1=0.672,∂˜R4R4R1=1.000.
The PCC values are compiled in Table3.
From Table3, it is observed that the new approach and the approach in [30] yield the same result for PCC between two equal IFAs when the effect of R1 is removed. The new approach gives reasonable results for the correlations between R2,R3 and R3,R4 when the effect of R1 is removed except for R2,R4. From the approach in [30], the correlation coefficient between R2,R3 when the effect of R1 is removed shows that R2 and R3 are negatively perfect correlated, which is not true. Similarly, the correlation coefficient between R3,R4 when the effect of R1 is removed shows that R3 and R4 are negatively correlated, which is also not true. However, the new approach shows that positive correlations exist between R2,R3 and R3,R4 when the effect of R1 is removed. The limitation observed with the existing approach is due to information loss because of the omission of intuitionistic fuzzy indexes.Table 3
PCC values.
ApproachesR2R2R1R2R3R1R2R4R1R3R3R1R3R4R1R4R4R1Hung [30]1.000−1.0000.9831.000−0.9831.000New method1.0000.5080.9751.0000.6721.000


## 3.1. Existing Technique of Partial Correlation Coefficient of IFAs

Hung [30] developed a PCC between of IFAs, R1,R2,…,Rp by the means of the normal correlation model. The IFA was adapted to fit into normal framework using logit transform [33] as follows:(21)ZiRj=logβRjxiγRjxi,for i=1,…,n and j=1,…,p. ZiRj was approximated via empirical logit transform [34] to avoid the case when ZiRj=log0 or ZiRj=log∞ as follows:(22)ZiRj=logβRjxi+1/2nγRjxi+1/2n,for i=1,…,n, j=1,…,p.Suppose we have two IFAsRi and Rj, then the correlation is determined as follows:(23)∂2Ri,Rj=SSRRjSSTORi,where SSTORi=∑i=1nβRixi−βRi¯2+γRixi−γRi¯2 (total sum of squares) and SSRRj=SSTORi−SSERj (regression sum of squares) for SSERj=∑i=1ne2 (error sum of squares).Hence, the correlation coefficient betweenRi and Rj is represented as follows.(24)∂Ri,Rj=SSRRjSSTORi.Thus, the PCC between IFAs,R1 and R2, when R3,…,Rp are muted is represented as follows:(25)∂R1,R2R3,…,Rp=SSER3,…,Rp−SSER2,…,RpSSER3,…,Rp.


## 3.2. New Partial Correlation Coefficient of IFAs

We present a new approach of computing PCC of IFSs by the means of mathematical statistics based on equation (10). Assume there are random samples of three IFAs R1, R2, and R3 in X. The linear correlation between R1 and R2 can be computed by using equation (10). The fact is that R1, R2, and R3 are IFAs drawn from the space X, and the linear correlation between R1 and R2 could be impacted by R3. Thus, to compute the precise linear correlation between R1 and R2, the effect of R3 must be removed. Such a precise linear correlation between R1 and R2 can be determined by partial correlation coefficient.


### 3.2.1. First-Order Partial Correlation Coefficient of IFAs

Definition 6.
LetR1, R2, and R3 be IFAs in the sample space X. Then, the PCC between IFAs R1 and R2 without the influence of IFA R3 is defined as follows:(26)∂˜R1,R2R3=∂˜R1,R2−∂˜R1,R3∂˜R2,R31−∂˜2R1,R31−∂˜2R2,R31/2,where ∂˜R1,R2,∂˜R1,R3, and ∂˜R2,R3 can be calculated by using equation (12). We can rewrite equation (26) as follows:(27)∂˜2C,DE=∂˜R1,R2−∂˜R1,R3∂˜R2,R321−∂˜2R1,R31−∂˜2R2,R3.
Now, we delineate some characteristics of the first-order PCC of IFAs as follows:Theorem 1.
SupposeR1,R2, and R3 are IFAs drawn from X=x1,…,xn for n∈⌈1,∞⌉. Then, ∂˜R1,R2R3=∂˜R2,R1R3.Proof.
We know that∂˜R1,R2=∂˜R2,R1 by equation (12). Thus,(28)∂˜R1,R2R3=∂˜R1,R2−∂˜R1,R3∂˜R2,R31−∂˜2R1,R31−∂˜2R2,R31/2=∂˜R2,R1−∂˜R2,R3∂˜R1,R31−∂˜2R2,R31−∂˜2R1,R31/2=∂˜R2,R1R3,which completes the proof.Proposition 1.
With the same hypothesis in Theorem1, a PCC ∂˜R1,R2R3becomesacorrelationcoefficient∂˜R1,R2ifandonlyif∂˜R1,R3=0=∂˜R2,R3.Proof.
Assume that∂˜R1,R3=∂˜R2,R3=0; then, we have the following expression:(29)∂˜R1,R2R3=∂˜R1,R2=∂˜R2,R1=∂˜R2,R1R3.
The converse is straightforward.Remark 1.
With the same hypothesis in Theorem1, the following statements are valid:(i)
IfR1 and R3 have a perfect positive linear correlation, then ∂˜R1,R2R3=∂˜R2,R1R3=∞(ii)
If∂˜R1,R3=0, then(30)∂˜R1,R2R3=∂˜R1,R21−∂˜2R2,R31/2=∂˜R2,R1R3.Also, if∂˜R2,R3=0, then(31)∂˜R1,R2R3=∂˜R1,R21−∂˜2R1,R31/2=∂˜R2,R1R3.(iii)
If∂˜R1,R2=0, ∂˜R1,R3≠0, and ∂˜R2,R3≠0, then(32)∂˜R1,R2R3=−∂˜R1,R3∂˜R2,R31−∂˜2R1,R31−∂˜2R2,R31/2.


### 3.2.2.nth-Order Partial Correlation Coefficient of IFAs

So far, we have been considering PCC between two IFAs by keeping the third IFA muted. The concept ofnth-order PCC of IFAs is conceivable in the case that there are random samples of more than three IFAs.LetR1,…,Rr,Rr+1,…,Rr+s be IFAs in X, where each IFA is represented as follows:(33)Ri=x,βRix,γRixx∈X,where βRi,γRi:X⟶0,1 for i=1,…,r+s. Also, Ri can be written as follows:(34)Ri=xk,βR1xk,γR1xk,…,βRr+sxk,γRr+sxkxk∈X,for k=1,…,n.We designate the covariances of IFAsR1,…,Rr,Rr+1,…,Rr+s as covariance matrix N, which we define as follows:(35)N=ΦRi,Rji=1,…,r+s,j=1,…,r+s,where(36)ΦRi,Rj=Σk=1n∇kRi∇kRjn−1,is the covariance of Ri,Rj, in which(37)∇kRi=βRi2xk−βRi¯2−γRi2xk−γRi¯2−δRi2xk−δRi¯2∇kRj=βRj2xk−βRj¯2−γRj2xk−γRj¯2−δRj2xk−δRj¯2,are the deviations of Ri and Rj, respectively.Definition 7.
The PCC ofRi,Rj, i,j=1,…,r, by removing the effects of Rr+1,…,Rr+s can be obtained as follows:
Suppose we have four parts partition ofN,(38)N=N11N12N21N22,where N11 is the r×r covariance matrix of Ri,Rj for i,j=1,…,r, N12 is the r×s covariance matrix of Ri,Rj for i=1,…,r, j=r+1,…,r+s, N21 is the s×r covariance matrix of Ri,Rj for i=r+1,…,r+s, j=1,…,r, and N22 is the s×s covariance matrix of Ri,Rj for i,j=r+1,…,r+s. Then, certainly, N12 and N21 transpose each other.
By keepingRr+1,…,Rr+s constant, the PCC of Ri,Rj is defined as follows:(39)∂˜Ri,RjRr+1,…,Rr+s=ΦRi,RjRr+1,…,Rr+sΦRi,RiRr+1,…,Rr+sΦRj,RjRr+1,…,Rr+s1/2,where i,j=1,…,r and ΦRi,RjRr+1,…,Rr+s is the i,jth element of partial covariance matrix of N112=N11−N12N22−1N21.
It follows that(40)∂˜Ri,RjRr+1,…,Rr+s=∂˜Rj,RiRr+1,…,Rr+s.
SinceN112 is a symmetry matrix and(41)ΦRi,RjRr+1,…,Rr+s=ΦRj,RiRr+1,…,Rr+s.Theorem 2.
Ifr=2 and s=1, then the nth-order PCC of IFAs is equivalent to the first-order PCC of IFAs.Proof.
The proof of Theorem2 is in the appendix section.Remark 2.
Ifr=2 and s=2, then the nth-order PCC of PFAs is as follows:(42)∂˜R1,R2R3,R4=∂˜R1,R2R4−∂˜R2,R3R4∂˜R1,R3R41−∂˜2R2,R3R41−∂˜2R1,R3R41/2,and redrafted as(43)∂˜1234=∂˜124−∂˜234∂˜1341−∂˜23421−∂˜13421/2.
Indeed,∂˜1234=∂˜1243 since(44)∂˜1234=∂˜124−∂˜234∂˜1341−∂˜23421−∂˜13421/2=∂˜123−∂˜243∂˜1431−∂˜24321−∂˜14321/2=∂˜1243.


### 3.2.3. Comparative Analysis of the PCC Approaches

We present the comparison of the approaches to show the edge of the new approach of PCC of IFAs over the approach in [30].Example 3.
LetR1, R2, R3, and R4 be IFSs in X=x1,x2,x3,x4,x5 and are defined as follows:(45)R1=0.1,0.8x1,0.3,0.5x2,0.6,0.2x3,0.9,0x4,1,0x5,R2=0.01,0.96x1,0.09,0.75x2,0.36,0.36x3,0.81,0x4,1,0x5,R3=0.96,0.01x1,0.75,0.09x2,0.36,0.36x3,0,0.81x4,0,1x5,R4=0,1x1,0.01,0.94x2,0.13,0.59x3,0.66,0x4,1,0x5.
We aim to compute the PCC between the IFAs by removing the effect of IFAR1. Using the approach in [30], we obtain the following results:(46)∂R2R2R1=1.000,∂R2R3R1=−1.000,∂R2R4R1=0.983,∂R3R3R1=1.000,∂R3R4R1=−0.983,∂R4R4R1=1.000.
With the novel approach, we obtain the following results:(47)∂˜R2R2R1=1.000,∂˜R2R3R1=0.508,∂˜R2R4R1=0.975,∂˜R3R3R1=1.000,∂˜R3R4R1=0.672,∂˜R4R4R1=1.000.
The PCC values are compiled in Table3.
From Table3, it is observed that the new approach and the approach in [30] yield the same result for PCC between two equal IFAs when the effect of R1 is removed. The new approach gives reasonable results for the correlations between R2,R3 and R3,R4 when the effect of R1 is removed except for R2,R4. From the approach in [30], the correlation coefficient between R2,R3 when the effect of R1 is removed shows that R2 and R3 are negatively perfect correlated, which is not true. Similarly, the correlation coefficient between R3,R4 when the effect of R1 is removed shows that R3 and R4 are negatively correlated, which is also not true. However, the new approach shows that positive correlations exist between R2,R3 and R3,R4 when the effect of R1 is removed. The limitation observed with the existing approach is due to information loss because of the omission of intuitionistic fuzzy indexes.Table 3
PCC values.
ApproachesR2R2R1R2R3R1R2R4R1R3R3R1R3R4R1R4R4R1Hung [30]1.000−1.0000.9831.000−0.9831.000New method1.0000.5080.9751.0000.6721.000


## 3.2.1. First-Order Partial Correlation Coefficient of IFAs

Definition 6.
LetR1, R2, and R3 be IFAs in the sample space X. Then, the PCC between IFAs R1 and R2 without the influence of IFA R3 is defined as follows:(26)∂˜R1,R2R3=∂˜R1,R2−∂˜R1,R3∂˜R2,R31−∂˜2R1,R31−∂˜2R2,R31/2,where ∂˜R1,R2,∂˜R1,R3, and ∂˜R2,R3 can be calculated by using equation (12). We can rewrite equation (26) as follows:(27)∂˜2C,DE=∂˜R1,R2−∂˜R1,R3∂˜R2,R321−∂˜2R1,R31−∂˜2R2,R3.
Now, we delineate some characteristics of the first-order PCC of IFAs as follows:Theorem 1.
SupposeR1,R2, and R3 are IFAs drawn from X=x1,…,xn for n∈⌈1,∞⌉. Then, ∂˜R1,R2R3=∂˜R2,R1R3.Proof.
We know that∂˜R1,R2=∂˜R2,R1 by equation (12). Thus,(28)∂˜R1,R2R3=∂˜R1,R2−∂˜R1,R3∂˜R2,R31−∂˜2R1,R31−∂˜2R2,R31/2=∂˜R2,R1−∂˜R2,R3∂˜R1,R31−∂˜2R2,R31−∂˜2R1,R31/2=∂˜R2,R1R3,which completes the proof.Proposition 1.
With the same hypothesis in Theorem1, a PCC ∂˜R1,R2R3becomesacorrelationcoefficient∂˜R1,R2ifandonlyif∂˜R1,R3=0=∂˜R2,R3.Proof.
Assume that∂˜R1,R3=∂˜R2,R3=0; then, we have the following expression:(29)∂˜R1,R2R3=∂˜R1,R2=∂˜R2,R1=∂˜R2,R1R3.
The converse is straightforward.Remark 1.
With the same hypothesis in Theorem1, the following statements are valid:(i)
IfR1 and R3 have a perfect positive linear correlation, then ∂˜R1,R2R3=∂˜R2,R1R3=∞(ii)
If∂˜R1,R3=0, then(30)∂˜R1,R2R3=∂˜R1,R21−∂˜2R2,R31/2=∂˜R2,R1R3.Also, if∂˜R2,R3=0, then(31)∂˜R1,R2R3=∂˜R1,R21−∂˜2R1,R31/2=∂˜R2,R1R3.(iii)
If∂˜R1,R2=0, ∂˜R1,R3≠0, and ∂˜R2,R3≠0, then(32)∂˜R1,R2R3=−∂˜R1,R3∂˜R2,R31−∂˜2R1,R31−∂˜2R2,R31/2.


## 3.2.2.nth-Order Partial Correlation Coefficient of IFAs

So far, we have been considering PCC between two IFAs by keeping the third IFA muted. The concept ofnth-order PCC of IFAs is conceivable in the case that there are random samples of more than three IFAs.LetR1,…,Rr,Rr+1,…,Rr+s be IFAs in X, where each IFA is represented as follows:(33)Ri=x,βRix,γRixx∈X,where βRi,γRi:X⟶0,1 for i=1,…,r+s. Also, Ri can be written as follows:(34)Ri=xk,βR1xk,γR1xk,…,βRr+sxk,γRr+sxkxk∈X,for k=1,…,n.We designate the covariances of IFAsR1,…,Rr,Rr+1,…,Rr+s as covariance matrix N, which we define as follows:(35)N=ΦRi,Rji=1,…,r+s,j=1,…,r+s,where(36)ΦRi,Rj=Σk=1n∇kRi∇kRjn−1,is the covariance of Ri,Rj, in which(37)∇kRi=βRi2xk−βRi¯2−γRi2xk−γRi¯2−δRi2xk−δRi¯2∇kRj=βRj2xk−βRj¯2−γRj2xk−γRj¯2−δRj2xk−δRj¯2,are the deviations of Ri and Rj, respectively.Definition 7.
The PCC ofRi,Rj, i,j=1,…,r, by removing the effects of Rr+1,…,Rr+s can be obtained as follows:
Suppose we have four parts partition ofN,(38)N=N11N12N21N22,where N11 is the r×r covariance matrix of Ri,Rj for i,j=1,…,r, N12 is the r×s covariance matrix of Ri,Rj for i=1,…,r, j=r+1,…,r+s, N21 is the s×r covariance matrix of Ri,Rj for i=r+1,…,r+s, j=1,…,r, and N22 is the s×s covariance matrix of Ri,Rj for i,j=r+1,…,r+s. Then, certainly, N12 and N21 transpose each other.
By keepingRr+1,…,Rr+s constant, the PCC of Ri,Rj is defined as follows:(39)∂˜Ri,RjRr+1,…,Rr+s=ΦRi,RjRr+1,…,Rr+sΦRi,RiRr+1,…,Rr+sΦRj,RjRr+1,…,Rr+s1/2,where i,j=1,…,r and ΦRi,RjRr+1,…,Rr+s is the i,jth element of partial covariance matrix of N112=N11−N12N22−1N21.
It follows that(40)∂˜Ri,RjRr+1,…,Rr+s=∂˜Rj,RiRr+1,…,Rr+s.
SinceN112 is a symmetry matrix and(41)ΦRi,RjRr+1,…,Rr+s=ΦRj,RiRr+1,…,Rr+s.Theorem 2.
Ifr=2 and s=1, then the nth-order PCC of IFAs is equivalent to the first-order PCC of IFAs.Proof.
The proof of Theorem2 is in the appendix section.Remark 2.
Ifr=2 and s=2, then the nth-order PCC of PFAs is as follows:(42)∂˜R1,R2R3,R4=∂˜R1,R2R4−∂˜R2,R3R4∂˜R1,R3R41−∂˜2R2,R3R41−∂˜2R1,R3R41/2,and redrafted as(43)∂˜1234=∂˜124−∂˜234∂˜1341−∂˜23421−∂˜13421/2.
Indeed,∂˜1234=∂˜1243 since(44)∂˜1234=∂˜124−∂˜234∂˜1341−∂˜23421−∂˜13421/2=∂˜123−∂˜243∂˜1431−∂˜24321−∂˜14321/2=∂˜1243.


## 3.2.3. Comparative Analysis of the PCC Approaches

We present the comparison of the approaches to show the edge of the new approach of PCC of IFAs over the approach in [30].Example 3.
LetR1, R2, R3, and R4 be IFSs in X=x1,x2,x3,x4,x5 and are defined as follows:(45)R1=0.1,0.8x1,0.3,0.5x2,0.6,0.2x3,0.9,0x4,1,0x5,R2=0.01,0.96x1,0.09,0.75x2,0.36,0.36x3,0.81,0x4,1,0x5,R3=0.96,0.01x1,0.75,0.09x2,0.36,0.36x3,0,0.81x4,0,1x5,R4=0,1x1,0.01,0.94x2,0.13,0.59x3,0.66,0x4,1,0x5.
We aim to compute the PCC between the IFAs by removing the effect of IFAR1. Using the approach in [30], we obtain the following results:(46)∂R2R2R1=1.000,∂R2R3R1=−1.000,∂R2R4R1=0.983,∂R3R3R1=1.000,∂R3R4R1=−0.983,∂R4R4R1=1.000.
With the novel approach, we obtain the following results:(47)∂˜R2R2R1=1.000,∂˜R2R3R1=0.508,∂˜R2R4R1=0.975,∂˜R3R3R1=1.000,∂˜R3R4R1=0.672,∂˜R4R4R1=1.000.
The PCC values are compiled in Table3.
From Table3, it is observed that the new approach and the approach in [30] yield the same result for PCC between two equal IFAs when the effect of R1 is removed. The new approach gives reasonable results for the correlations between R2,R3 and R3,R4 when the effect of R1 is removed except for R2,R4. From the approach in [30], the correlation coefficient between R2,R3 when the effect of R1 is removed shows that R2 and R3 are negatively perfect correlated, which is not true. Similarly, the correlation coefficient between R3,R4 when the effect of R1 is removed shows that R3 and R4 are negatively correlated, which is also not true. However, the new approach shows that positive correlations exist between R2,R3 and R3,R4 when the effect of R1 is removed. The limitation observed with the existing approach is due to information loss because of the omission of intuitionistic fuzzy indexes.Table 3
PCC values.
ApproachesR2R2R1R2R3R1R2R4R1R3R3R1R3R4R1R4R4R1Hung [30]1.000−1.0000.9831.000−0.9831.000New method1.0000.5080.9751.0000.6721.000


## 4. Application of Partial Correlation Coefficient Involving Pattern Recognitions

Pattern recognition is the process of data categorization based on information extracted from pattern and/or their representation. Though pattern recognition has important applications in many areas, the presence of uncertainties most often affect the process of pattern recognition. As a result, approaching the process of pattern recognition based on intuitionistic fuzzy information will help in controlling the uncertainties that will adversely affect the outcome of the pattern recognition. In this section, we recognize and classify patterns of building materials in some feature spaces (five to be precise) as random samples of IFAs using the new approach of partial correlation measure.To apply the approach, we first compute the correlation coefficients of the patterns, find the first partial correlation coefficients of the patterns, and compute the second partial correlation coefficients of the patterns to determine the precise linear correlation among the building patterns.


### 4.1. Experimental Illustration

Suppose there are building patterns represented by IFAsIi for i=1,2,3,4 in the feature space F=f1,f2,f3,f4,f5 as follows:(48)I1=0.800,0.100f1,0.700,0.200f2,0.900,0.000f3,0.600,0.300f4,0.800,0.100f5,I2=0.900,0.100f1,0.800,0.100f2,0.800,0.100f3,0.500,0.300f4,0.700,0.200f5,I3=0.500,0.300f1,0.500,0.200f2,0.900,0.000f3,0.500,0.400f4,0.700,0.100f5,I4=0.700,0.200f1,0.500,0.400f2,0.900,0.100f3,0.600,0.300f4,0.800,0.000f5.Our first assignment is to compute the correlation coefficient between each of the building patterns using the modified approach (i.e., equation (12)) after computing the hesitation margins of Ii for i=1,2,3,4.


#### 4.1.1. Algorithm for Computing Correlation Coefficient between Building Patterns

The algorithm based on the modified approach (i.e., equation (12)) for computing the correlation coefficient between the building patterns is made up of the following steps:(1)
Set value forn for the feature space F(2)
Initialize the intuitionistic fuzzy pairs of the building patterns(3)
Find the mean values of the intuitionistic fuzzy pairs of the building patterns(4)
Compute the correlation coefficient between the building patterns based on equation (10)(5)
Classify the building patterns based on the results of Step 4, for which the correlation coefficient is the greatest.


#### 4.1.2. Results and Discussion

By following the steps of the algorithm, we have the following results in the correlation coefficient matrix:(49)From the matrix, we see that patternI1 can be associated with pattern I4, pattern I2 can be associated with pattern I1, and patterns I3 and I4 can be associated with each other. We observe that the matrix is symmetric and ∂˜IiIi+1=∂˜Ii+1Ii. The correlation coefficient results show that the four random IFAs have positive linear relationship with each other with the exceptions of ∂˜I1I2, ∂˜I2I3, and ∂˜I2I4.To get a precise correlation coefficient among the building patterns, we find the first-order PCC of the patternsIi. By equation (27) and the corresponding correlation coefficient values, we obtain the following values of the first-order PCC by removing the effect of another IFA:(50)∂˜I1I2I3=−0.0895,∂˜I1I2I4=−0.0616,∂˜I1I3I2=0.1822,∂˜I1I3I4=0.1091,∂˜I1I4I2=0.2607,∂˜I1I4I3=0.2268,∂˜I2I3I1=−0.1280,∂˜I2I3I4=−0.0805,∂˜I2I4I1=−0.1849,∂˜I2I4I3=−0.1690,∂˜I3I4I1=0.3176,∂˜I3I4I2=0.3338.By juxtaposing these results with the correlation coefficient matrix, we see that when the effect of another IFA is removed from the correlation coefficients, the partial correlation coefficients either increase or decrease. From the outputs, the following classifications are made: patternI1 has a negative effect on the correlation between patterns I2, I3 and I2,I4 and a positive effect on pattern I3,I4; pattern I2 has only positive effect on the correlation between patterns I1,I3, I1,I4, and I3,I4; pattern I3 has a negative effect on the correlation between patterns I1,I2 and I2,I4 and a positive influence on pattern I1,I4; pattern I4 has a negative effect on the correlation between patterns I1,I2 and I2,I3 and a positive effect on pattern I1,I3.By removing the effect of another IFA from the first-order partial correlation coefficients, we obtain the following expressions:(51)∂˜I1I2I3I4=−0.0533,∂˜I1I3I2I4=0.1047,∂˜I1I4I2I3=0.2157,∂˜I2I3I1I4=−0.0744,∂˜I2I4PI1I3=−0.1534,∂˜I3I4I1I2=0.3016.By comparing these results with the first-order partial correlation coefficients, we see that when the effect of another IFA is removed from the first-order partial correlation coefficients, the second-order partial correlation coefficients either increase or decrease, and the effects are as follows: patternI1 has a negative effect on the first-order partial correlation among patterns I2,I4I3 and I2,I3I4 and a positive effect on pattern I3,I4I2; pattern I2 has only positive effect on the first-order partial correlation among patterns I1,I3I4, I1,I4I3, and I3,I4I1; pattern I3 has a negative effect on the first-order partial correlation among patterns I1,I2I4 and I2,I4I1 and a positive influence on pattern I1,I4I2; pattern I4 has a negative effect on the first-order partial correlation among patterns I1,I2I3 and I2,I3I1 and a positive effect on pattern I1,I3I2.We discover that patternI2 has positive influence on both the simple correlations and the first-order partial correlations. In a generic form, the index of a partial correlation coefficient of IFAs either increase or decrease when the effect of other IFA(s) is/are removed depending on the direction of the linear relationship between the removed IFA(s) and the observed IFAs.The negative linear relationship of the partial correlation coefficients, namely,∂˜I1I2I3I4, ∂˜I2I3I1P4, and ∂˜I2I4I1I3, imply that the linear relationships between I1,I2, I2,I3, and I2,I4 are actually negative. The same reason hold for the positive partial correlation coefficients ∂˜I1I3I2I4, ∂˜I1I4I2I3, and ∂˜I3I4I1I2.


## 4.1. Experimental Illustration

Suppose there are building patterns represented by IFAsIi for i=1,2,3,4 in the feature space F=f1,f2,f3,f4,f5 as follows:(48)I1=0.800,0.100f1,0.700,0.200f2,0.900,0.000f3,0.600,0.300f4,0.800,0.100f5,I2=0.900,0.100f1,0.800,0.100f2,0.800,0.100f3,0.500,0.300f4,0.700,0.200f5,I3=0.500,0.300f1,0.500,0.200f2,0.900,0.000f3,0.500,0.400f4,0.700,0.100f5,I4=0.700,0.200f1,0.500,0.400f2,0.900,0.100f3,0.600,0.300f4,0.800,0.000f5.Our first assignment is to compute the correlation coefficient between each of the building patterns using the modified approach (i.e., equation (12)) after computing the hesitation margins of Ii for i=1,2,3,4.


### 4.1.1. Algorithm for Computing Correlation Coefficient between Building Patterns

The algorithm based on the modified approach (i.e., equation (12)) for computing the correlation coefficient between the building patterns is made up of the following steps:(1)
Set value forn for the feature space F(2)
Initialize the intuitionistic fuzzy pairs of the building patterns(3)
Find the mean values of the intuitionistic fuzzy pairs of the building patterns(4)
Compute the correlation coefficient between the building patterns based on equation (10)(5)
Classify the building patterns based on the results of Step 4, for which the correlation coefficient is the greatest.


### 4.1.2. Results and Discussion

By following the steps of the algorithm, we have the following results in the correlation coefficient matrix:(49)From the matrix, we see that patternI1 can be associated with pattern I4, pattern I2 can be associated with pattern I1, and patterns I3 and I4 can be associated with each other. We observe that the matrix is symmetric and ∂˜IiIi+1=∂˜Ii+1Ii. The correlation coefficient results show that the four random IFAs have positive linear relationship with each other with the exceptions of ∂˜I1I2, ∂˜I2I3, and ∂˜I2I4.To get a precise correlation coefficient among the building patterns, we find the first-order PCC of the patternsIi. By equation (27) and the corresponding correlation coefficient values, we obtain the following values of the first-order PCC by removing the effect of another IFA:(50)∂˜I1I2I3=−0.0895,∂˜I1I2I4=−0.0616,∂˜I1I3I2=0.1822,∂˜I1I3I4=0.1091,∂˜I1I4I2=0.2607,∂˜I1I4I3=0.2268,∂˜I2I3I1=−0.1280,∂˜I2I3I4=−0.0805,∂˜I2I4I1=−0.1849,∂˜I2I4I3=−0.1690,∂˜I3I4I1=0.3176,∂˜I3I4I2=0.3338.By juxtaposing these results with the correlation coefficient matrix, we see that when the effect of another IFA is removed from the correlation coefficients, the partial correlation coefficients either increase or decrease. From the outputs, the following classifications are made: patternI1 has a negative effect on the correlation between patterns I2, I3 and I2,I4 and a positive effect on pattern I3,I4; pattern I2 has only positive effect on the correlation between patterns I1,I3, I1,I4, and I3,I4; pattern I3 has a negative effect on the correlation between patterns I1,I2 and I2,I4 and a positive influence on pattern I1,I4; pattern I4 has a negative effect on the correlation between patterns I1,I2 and I2,I3 and a positive effect on pattern I1,I3.By removing the effect of another IFA from the first-order partial correlation coefficients, we obtain the following expressions:(51)∂˜I1I2I3I4=−0.0533,∂˜I1I3I2I4=0.1047,∂˜I1I4I2I3=0.2157,∂˜I2I3I1I4=−0.0744,∂˜I2I4PI1I3=−0.1534,∂˜I3I4I1I2=0.3016.By comparing these results with the first-order partial correlation coefficients, we see that when the effect of another IFA is removed from the first-order partial correlation coefficients, the second-order partial correlation coefficients either increase or decrease, and the effects are as follows: patternI1 has a negative effect on the first-order partial correlation among patterns I2,I4I3 and I2,I3I4 and a positive effect on pattern I3,I4I2; pattern I2 has only positive effect on the first-order partial correlation among patterns I1,I3I4, I1,I4I3, and I3,I4I1; pattern I3 has a negative effect on the first-order partial correlation among patterns I1,I2I4 and I2,I4I1 and a positive influence on pattern I1,I4I2; pattern I4 has a negative effect on the first-order partial correlation among patterns I1,I2I3 and I2,I3I1 and a positive effect on pattern I1,I3I2.We discover that patternI2 has positive influence on both the simple correlations and the first-order partial correlations. In a generic form, the index of a partial correlation coefficient of IFAs either increase or decrease when the effect of other IFA(s) is/are removed depending on the direction of the linear relationship between the removed IFA(s) and the observed IFAs.The negative linear relationship of the partial correlation coefficients, namely,∂˜I1I2I3I4, ∂˜I2I3I1P4, and ∂˜I2I4I1I3, imply that the linear relationships between I1,I2, I2,I3, and I2,I4 are actually negative. The same reason hold for the positive partial correlation coefficients ∂˜I1I3I2I4, ∂˜I1I4I2I3, and ∂˜I3I4I1I2.


## 4.1.1. Algorithm for Computing Correlation Coefficient between Building Patterns

The algorithm based on the modified approach (i.e., equation (12)) for computing the correlation coefficient between the building patterns is made up of the following steps:(1)
Set value forn for the feature space F(2)
Initialize the intuitionistic fuzzy pairs of the building patterns(3)
Find the mean values of the intuitionistic fuzzy pairs of the building patterns(4)
Compute the correlation coefficient between the building patterns based on equation (10)(5)
Classify the building patterns based on the results of Step 4, for which the correlation coefficient is the greatest.


## 4.1.2. Results and Discussion

By following the steps of the algorithm, we have the following results in the correlation coefficient matrix:(49)From the matrix, we see that patternI1 can be associated with pattern I4, pattern I2 can be associated with pattern I1, and patterns I3 and I4 can be associated with each other. We observe that the matrix is symmetric and ∂˜IiIi+1=∂˜Ii+1Ii. The correlation coefficient results show that the four random IFAs have positive linear relationship with each other with the exceptions of ∂˜I1I2, ∂˜I2I3, and ∂˜I2I4.To get a precise correlation coefficient among the building patterns, we find the first-order PCC of the patternsIi. By equation (27) and the corresponding correlation coefficient values, we obtain the following values of the first-order PCC by removing the effect of another IFA:(50)∂˜I1I2I3=−0.0895,∂˜I1I2I4=−0.0616,∂˜I1I3I2=0.1822,∂˜I1I3I4=0.1091,∂˜I1I4I2=0.2607,∂˜I1I4I3=0.2268,∂˜I2I3I1=−0.1280,∂˜I2I3I4=−0.0805,∂˜I2I4I1=−0.1849,∂˜I2I4I3=−0.1690,∂˜I3I4I1=0.3176,∂˜I3I4I2=0.3338.By juxtaposing these results with the correlation coefficient matrix, we see that when the effect of another IFA is removed from the correlation coefficients, the partial correlation coefficients either increase or decrease. From the outputs, the following classifications are made: patternI1 has a negative effect on the correlation between patterns I2, I3 and I2,I4 and a positive effect on pattern I3,I4; pattern I2 has only positive effect on the correlation between patterns I1,I3, I1,I4, and I3,I4; pattern I3 has a negative effect on the correlation between patterns I1,I2 and I2,I4 and a positive influence on pattern I1,I4; pattern I4 has a negative effect on the correlation between patterns I1,I2 and I2,I3 and a positive effect on pattern I1,I3.By removing the effect of another IFA from the first-order partial correlation coefficients, we obtain the following expressions:(51)∂˜I1I2I3I4=−0.0533,∂˜I1I3I2I4=0.1047,∂˜I1I4I2I3=0.2157,∂˜I2I3I1I4=−0.0744,∂˜I2I4PI1I3=−0.1534,∂˜I3I4I1I2=0.3016.By comparing these results with the first-order partial correlation coefficients, we see that when the effect of another IFA is removed from the first-order partial correlation coefficients, the second-order partial correlation coefficients either increase or decrease, and the effects are as follows: patternI1 has a negative effect on the first-order partial correlation among patterns I2,I4I3 and I2,I3I4 and a positive effect on pattern I3,I4I2; pattern I2 has only positive effect on the first-order partial correlation among patterns I1,I3I4, I1,I4I3, and I3,I4I1; pattern I3 has a negative effect on the first-order partial correlation among patterns I1,I2I4 and I2,I4I1 and a positive influence on pattern I1,I4I2; pattern I4 has a negative effect on the first-order partial correlation among patterns I1,I2I3 and I2,I3I1 and a positive effect on pattern I1,I3I2.We discover that patternI2 has positive influence on both the simple correlations and the first-order partial correlations. In a generic form, the index of a partial correlation coefficient of IFAs either increase or decrease when the effect of other IFA(s) is/are removed depending on the direction of the linear relationship between the removed IFA(s) and the observed IFAs.The negative linear relationship of the partial correlation coefficients, namely,∂˜I1I2I3I4, ∂˜I2I3I1P4, and ∂˜I2I4I1I3, imply that the linear relationships between I1,I2, I2,I3, and I2,I4 are actually negative. The same reason hold for the positive partial correlation coefficients ∂˜I1I3I2I4, ∂˜I1I4I2I3, and ∂˜I3I4I1I2.


## 5. Conclusion

In the process of computing the correlation coefficient of any two IFAs, other different associated IFAs could influence the correlation outcome, and so, the conventional methods of computing correlation coefficient of IFAs [22, 23, 25] cannot give a precise result. This limitation is the justification for the concept of partial correlation coefficient of random IFAs, which keeps the influence of the other IFA(s) ineffective. We have proposed a new technique of computing partial correlation coefficient of random intuitionistic fuzzy data by considering all the descriptive parameters of IFAs unlike the approach discussed in [30]. In fact, this new approach extended the work of Chiang Lin [28] in fuzzy context to intuitionistic fuzzy environment. The approach of computing correlation coefficient of IFAs in [23] was modified to include all the descriptive parameters of IFAs to enhance reliable results, which we used to compute the partial correlation coefficients among the considered random IFAs. We experimented the application of the proposed partial correlation coefficient on pattern recognition of building patterns, where patterns were represented by random IFAs collected hypothetically. From the obtained results, we discovered that when we remove the effect of another IFA, the index of the partial correlation coefficient among the IFAs either increases or decreases subject to the linear relationship and the direction of the correlation between the impassive/muted IFA and the observed IFAs. In conclusion, we assert that when determining the linear relationship between any two random IFAs, the concept of simple correlation coefficient would not be sufficient because it does not give information on the influence of other associated IFAs and thus leading to unreliable results. Albeit, this approach cannot measure multiple partial correlation of random IFAs. Thus, investigating the concept of partial correlation coefficient among random IFAs on the basis of multiple correlation coefficients of random IFAs with application in decision-making could be exploited in our future endeavour. The developed PCCIFSs could be used to discuss group decision-making using consensus-based TOPSIS-Sort-B for multiple-criteria sorting [35], consensus based on consistency control [36], and individual and local world opinion centered on opinion dynamics [37], among others.



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*Source: 5540085-2023-03-06.xml*