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

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.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 $\beta$ and NMD represented by $\gamma$ with the possibility of the existent of hesitation represented by $\delta$, whereby their aggregate is always one and $\beta +\gamma \le 1$. 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 measure

After 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 PCCIFSs

The 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 problem

The outline of the rest of the work is as follows: Section 2 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 IFS $\mathfrak{R}$ in $X$ is defined by the structure.$$\begin{array}{}\text{(1)}& \begin{array}{c}\mathfrak{R}=\frac{{\beta }_{\mathfrak{R}}x,{\gamma }_{\mathfrak{R}}x}{x}x\in X,\mathrm{o}\mathrm{r}\\ \mathfrak{R}=x,{\beta }_{\mathfrak{R}}x,{\gamma }_{\mathfrak{R}}x\mid x\in X\end{array}\text{,}\end{array}$$where the functions ${\beta }_{\mathfrak{R}},{\gamma }_{\mathfrak{R}}:X⟶\mathrm{0,1}$ define MD and NMD of $x\in X$ in which$$\begin{array}{}\text{(2)}& 0\le {\beta }_{\mathfrak{R}}x+{\gamma }_{\mathfrak{R}}x\le 1\text{.}\end{array}$$

For any IFS $\mathfrak{R}$ in $X$, ${\delta }_{\mathfrak{R}}x=1-{\beta }_{\mathfrak{R}}x-{\gamma }_{\mathfrak{R}}x$ represents the hesitation margin of $\mathfrak{R}$.

Definition 2.

Intuitionistic fuzzy pairs (IFPs) or intuitionistic fuzzy values (IFVs) are characterized by the form $x,y$, where $x+y\le 1$ for $x,y\in \mathrm{0,1}$. IFPs/IFVs weigh IFS for which the components ($x$ and $y$) are construed as MD and NMD.

Definition 3.

Assume $\mathfrak{R}$ and $\mathfrak{I}$ are IFSs in $X$, then we get the following expressions:

(1) $\mathfrak{R}=\mathfrak{I}$ iff ${\beta }_{\mathfrak{R}}x={\beta }_{\mathfrak{I}}x$ and ${\gamma }_{\mathfrak{R}}x={\gamma }_{\mathfrak{I}}x$ for all $x\in X$

(2) $\mathfrak{R}\subseteq \mathfrak{I}$ iff ${\beta }_{\mathfrak{R}}x\le {\beta }_{\mathfrak{I}}x$ and ${\gamma }_{\mathfrak{R}}x\ge {\gamma }_{\mathfrak{I}}x$ for all $x\in X$

(3) $\mathfrak{R}\underset{\_}{\prec }\mathfrak{I}$ iff ${\beta }_{\mathfrak{R}}x\le {\beta }_{\mathfrak{I}}x$ and ${\gamma }_{\mathfrak{R}}x\le {\gamma }_{\mathfrak{I}}x$ for all $x\in X$

(4) $\overline{\mathfrak{R}}={\gamma }_{\mathfrak{R}}x,{\beta }_{\mathfrak{R}}x/xx\in X$

(5) $\mathfrak{R}\cup \mathfrak{I}=\max {\beta }_{\mathfrak{R}}x,{\beta }_{\mathfrak{I}}x/x,\min {\gamma }_{\mathfrak{R}}x,{\gamma }_{\mathfrak{I}}x/xx\in X$

(6) $\mathfrak{R}\cap \mathfrak{I}=\min {\beta }_{\mathfrak{R}}x,{\beta }_{\mathfrak{I}}x/x,\max {\gamma }_{\mathfrak{R}}x,{\gamma }_{\mathfrak{I}}x/xx\in X$

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 $\mathfrak{R}$ and $\mathfrak{I}$ be IFSs in $X=x_1,\dots ,x_n$, $n\in 1,\infty$, and let $\partial \mathfrak{R},\mathfrak{I}$ be a correlation coefficient between $\mathfrak{R}$ and $\mathfrak{I}$. Then, the following definition defines CCIFSs.

Definition 4.

(see [22]). The correlation coefficient $\partial \mathfrak{R},\mathfrak{I}$ is a measuring function $\partial :IFS\times IFS⟶-\mathrm{1,1}$ if and only if the following conditions are satisfied:

(1) $\partial \mathfrak{R},\mathfrak{I}\in -\mathrm{1,1}$

(2) $\partial \mathfrak{R},\mathfrak{I}=\partial \mathfrak{I},\mathfrak{R}$

(3) If $\mathfrak{R}$ and $\mathfrak{I}$ are equal, then $\partial \mathfrak{R},\mathfrak{I}=1$

When $\sigma \mathfrak{R},\mathfrak{I}$ approaches 1, it shows that $\mathfrak{R}$ and $\mathfrak{I}$ are more correlated with better performance index. Again, if $\sigma \mathfrak{R},\mathfrak{I}$ approaches $-1$, then $\mathfrak{R}$ and $\mathfrak{I}$ 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..

Let $\mathfrak{R}$ and $\mathfrak{I}$ be IFSs in $X=x_1,\dots ,x_n$, $n\in 1,\infty$. Then, the mean values of $\mathfrak{R}$ and $\mathfrak{I}$ are represented as follows:$$\begin{array}{}\text{(3)}& \begin{array}{c}\overline{{\beta }_{\mathfrak{R}}},\overline{{\beta }_{\mathfrak{I}}}=\frac{{\mathrm{\Sigma }}_{i=1}^n{\beta }_{\mathfrak{R}}{x}_i}{n},\frac{{\mathrm{\Sigma }}_{i=1}^n{\beta }_{\mathfrak{I}}{x}_i}{n}\\ \overline{{\gamma }_{\mathfrak{R}}},\overline{{\gamma }_{\mathfrak{I}}}=\frac{{\mathrm{\Sigma }}_{i=1}^n{\gamma }_{\mathfrak{R}}{x}_i}{n},\frac{{\mathrm{\Sigma }}_{i=1}^n{\gamma }_{\mathfrak{I}}{x}_i}{n}\\ \overline{{\delta }_{\mathfrak{R}}},\overline{{\delta }_{\mathfrak{I}}}=\frac{{\mathrm{\Sigma }}_{i=1}^n{\delta }_{\mathfrak{R}}{x}_i}{n},\frac{{\mathrm{\Sigma }}_{i=1}^n{\delta }_{\mathfrak{I}}{x}_i}{n}\end{array}\text{.}\end{array}$$

Some existing approaches of computing CCIFSs are as follows:

(i) Approach in [23]is expressed as follows:$$\begin{array}{}\text{(4)}& {\partial }_1\mathfrak{R},\mathfrak{I}=\frac{{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i\mathfrak{R}{\nabla }_i\mathfrak{I}}{\sqrt{{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i^2\mathfrak{R}}\sqrt{{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i^2\mathfrak{I}}}\text{,}\end{array}$$

  where ${\nabla }_i\mathfrak{R},{\nabla }_i\mathfrak{I}$ are the deviations of $\mathfrak{R}$ and $\mathfrak{I}$ defined by the following equation:$$\begin{array}{}\text{(5)}& \begin{array}{c}{\nabla }_i\mathfrak{R}={\beta }_{\mathfrak{R}}{x}_i-\overline{{\beta }_{\mathfrak{R}}}-{\gamma }_{\mathfrak{R}}{x}_i-\overline{{\gamma }_{\mathfrak{R}}}\\ {\nabla }_i\mathfrak{I}={\beta }_{\mathfrak{I}}{x}_i-\overline{{\beta }_{\mathfrak{I}}}-{\gamma }_{\mathfrak{I}}{x}_i-\overline{{\gamma }_{\mathfrak{I}}}\end{array}\text{.}\end{array}$$

(ii) Approach in [22]is expressed as follows:$$\begin{array}{}\text{(6)}& {\partial }_2\mathfrak{R},\mathfrak{I}=\frac{{∆}_1\mathfrak{R},\mathfrak{I}+{∆}_2\mathfrak{R},\mathfrak{I}}{2}\text{,}\end{array}$$

  where$$\begin{array}{}\text{(7)}& \begin{array}{c}{∆}_1\mathfrak{R},\mathfrak{I}=\frac{{\mathrm{\Sigma }}_{i=1}^n{\beta }_{\mathfrak{R}}{x}_i-\overline{{\beta }_{\mathfrak{R}}}{\beta }_{\mathfrak{I}}{x}_i-\overline{{\beta }_{\mathfrak{I}}}}{{{\mathrm{\Sigma }}_{i=1}^n{\beta }_{\mathfrak{R}}{x}_i-\overline{{\beta }_{\mathfrak{R}}}{}^2{\mathrm{\Sigma }}_{i=1}^n{\beta }_{\mathfrak{I}}{x}_i-\overline{{\beta }_{\mathfrak{I}}}{}^2}^{0.5}}\\ {∆}_2\mathfrak{R},\mathfrak{I}=\frac{{\mathrm{\Sigma }}_{i=1}^n{\gamma }_{\mathfrak{R}}{x}_i-\overline{{\gamma }_{\mathfrak{R}}}{\gamma }_{\mathfrak{I}}{x}_i-\overline{{\gamma }_{\mathfrak{I}}}}{{{\mathrm{\Sigma }}_{i=1}^n{\gamma }_{\mathfrak{R}}{x}_i-\overline{{\gamma }_{\mathfrak{R}}}{}^2{\mathrm{\Sigma }}_{i=1}^n{\gamma }_{\mathfrak{I}}{x}_i-\overline{{\gamma }_{\mathfrak{I}}}{}^2}^{0.5}}\end{array}\text{.}\end{array}$$

(iii) Approach in [25]is expressed as follows:$$\begin{array}{}\text{(8)}& {\partial }_3\mathfrak{R},\mathfrak{I}=\frac{\mathrm{\Phi }\mathfrak{R},\mathfrak{I}}{\sqrt{\mathrm{\Psi }\mathfrak{R}}\sqrt{\mathrm{\Psi }\mathfrak{I}}}\text{,}\end{array}$$

  where$$\begin{array}{c}\text{(9)}& \mathrm{\Phi }\mathfrak{R},\mathfrak{I}={\mathrm{\Sigma }}_{i=1}^n{\beta }_{\mathfrak{R}}{x}_i-\overline{{\beta }_{\mathfrak{R}}}{\beta }_{\mathfrak{I}}{x}_i-\overline{{\beta }_{\mathfrak{I}}}+{\gamma }_{\mathfrak{R}}{x}_i-\overline{{\gamma }_{\mathfrak{R}}}{\gamma }_{\mathfrak{I}}{x}_i-\overline{{\gamma }_{\mathfrak{I}}}\begin{array}{c}\mathrm{\Psi }\mathfrak{R}={\mathrm{\Sigma }}_{i=1}^n{\beta }_{\mathfrak{R}}{x}_i-\overline{{\beta }_{\mathfrak{R}}}{}^2+{\gamma }_{\mathfrak{R}}{x}_i-\overline{{\gamma }_{\mathfrak{R}}}{}^2\\ \mathrm{\Psi }\mathfrak{I}={\mathrm{\Sigma }}_{i=1}^n{\beta }_{\mathfrak{I}}{x}_i-\overline{{\beta }_{\mathfrak{I}}}{}^2+{\gamma }_{\mathfrak{I}}{x}_i-\overline{{\gamma }_{\mathfrak{I}}}{}^2\end{array}\text{.}\end{array}$$

2.2.1. Modified Intuitionistic Fuzzy Correlation Coefficient Approach

Because the method presented in [23] does not consider the hesitation grades of $\mathfrak{R}$ and $\mathfrak{I}$, 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 $\mathfrak{R}$ and $\mathfrak{I}$ is obtained:$$\begin{array}{}\text{(10)}& \tilde{\partial }\mathfrak{R},\mathfrak{I}=\frac{{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i\mathfrak{R}{\nabla }_i\mathfrak{I}}{{{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i^2\mathfrak{R}{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i^2\mathfrak{I}}^{0.5}}\text{,}\end{array}$$where the deviations of $\mathfrak{R}$ and $\mathfrak{I}$ can be computed by the following equation:$$\begin{array}{}\text{(11)}& \begin{array}{c}{\nabla }_i\mathfrak{R}={\beta }_{\mathfrak{R}}{x}_i-\overline{{\beta }_{\mathfrak{R}}}-{\gamma }_{\mathfrak{R}}{x}_i-\overline{{\gamma }_{\mathfrak{R}}}-{\delta }_{\mathfrak{R}}{x}_i-\overline{{\delta }_{\mathfrak{R}}}\\ {\nabla }_i\mathfrak{I}={\beta }_{\mathfrak{I}}{x}_i-\overline{{\beta }_{\mathfrak{I}}}-{\gamma }_{\mathfrak{I}}{x}_i-\overline{{\gamma }_{\mathfrak{I}}}-{\delta }_{\mathfrak{I}}{x}_i-\overline{{\delta }_{\mathfrak{I}}}\end{array}\text{.}\end{array}$$

Equation (10) can be rewritten as follows:$$\begin{array}{}\text{(12)}& \tilde{\partial }\mathfrak{R},\mathfrak{I}=\frac{\mathrm{\Phi }\mathfrak{R},\mathfrak{I}}{\sqrt{\mathrm{\Psi }\mathfrak{R}\mathrm{\Psi }\mathfrak{I}}}\text{,}\end{array}$$where$$\begin{array}{}\text{(13)}& \mathrm{\Phi }\mathfrak{R},\mathfrak{I}=\frac{1}{n-1}{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i\mathfrak{R}{\nabla }_i\mathfrak{I},\end{array}$$is the covariance of $\mathfrak{R}$ and $\mathfrak{I}$, and$$\begin{array}{}\text{(14)}& \mathrm{\Psi }\mathfrak{R}=\frac{1}{n-1}{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i^2\mathfrak{R},\mathrm{\Psi }\mathfrak{I}=\frac{1}{n-1}{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i^2\mathfrak{I},\end{array}$$are the variances of $\mathfrak{R}$ and $\mathfrak{I}$, respectively.

Certainly, $\tilde{\partial }\mathfrak{R},\mathfrak{I}=\tilde{\partial }\mathfrak{I},\mathfrak{R}$ since$$\begin{array}{c}\text{(15)}& \tilde{\partial }\mathfrak{R},\mathfrak{I}=\frac{{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i\mathfrak{R}{\nabla }_i\mathfrak{I}}{{{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i^2\mathfrak{R}{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i^2\mathfrak{I}}^{0.5}}=\frac{{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i\mathfrak{I}{\nabla }_i\mathfrak{R}}{{{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i^2\mathfrak{I}{\mathrm{\Sigma }}_{i=1}^n{\nabla }_i^2\mathfrak{R}}^{0.5}}\\ =\tilde{\sigma }\mathfrak{I},\mathfrak{R}\text{.}\end{array}$$

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.

Suppose $\mathfrak{R}$ and $\mathfrak{I}$ are IFSs in $X=x_1,x_2,x_3$, where$$\begin{array}{}\text{(16)}& \mathfrak{R}=\frac{\mathrm{0.1,0.2}}{x_1},\frac{\mathrm{0.2,0.1}}{x_2},\frac{\mathrm{0.3,0}}{x_3},\mathfrak{I}=\frac{\mathrm{0.3,0}}{x_1},\frac{\mathrm{0.2,0.2}}{x_2},\frac{\mathrm{0.1,0.6}}{x_3}\text{.}\end{array}$$

By mere inspection, the correlation coefficient of $\mathfrak{R}$ and $\mathfrak{I}$ is perfectly negative since $\mathfrak{R}\ne \mathfrak{I}$, $\mathfrak{R}⊈\mathfrak{I}$, and $\mathfrak{I}⊈\mathfrak{R}$. 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 ${\partial }_1\mathfrak{R},\mathfrak{I}=-0.9898$. Using the approach in [22], we get ${\partial }_2\mathfrak{R},\mathfrak{I}=-0.9970$. Using the approach in [25], we get ${\partial }_3\mathfrak{R},\mathfrak{I}=-0.8800$. With the modified approach given as equation (10), we get the correlation coefficient $\tilde{\partial }\mathfrak{R},\mathfrak{I}=-1$. Table 1 contains the results for nippy comparison.

From Table 1, the results of equations (4), (6), and (8) show that the correlation coefficient of $\mathfrak{R}$ and $\mathfrak{I}$ is negative but not perfectly negative although $\mathfrak{R}$ and $\mathfrak{I}$ are dissimilar. On the contrary, the modified approach, i.e., equation (10) shows that $\mathfrak{R}$ and $\mathfrak{I}$ are perfectly negative correlated in agreement to the dissimilarity of $\mathfrak{R}$ and $\mathfrak{I}$. 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.

Approaches	Hung [22]	Liu et al. [23]	Thao et al. [25]	Modified method
Example 1	$-0.9970$	$-0.9898$	$-0.8800$	$-1$

Example 2.

A car buyer wants to buy a car whose descriptions are represented by the following set.$$\begin{array}{}\text{(17)}& D=D_1\mathrm{f}\mathrm{u}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{k}\mathrm{e},{\mathrm{D}}_2\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e},{\mathrm{D}}_3\mathrm{l}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{f}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{t},{\mathrm{D}}_4\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n},{\mathrm{D}}_5\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}\text{.}\end{array}$$

The buyer’s descriptions are fuzzified intuitionistically as follows:$$\begin{array}{}\text{(18)}& \mathfrak{R}=\frac{\mathrm{0.8,0.1}}{D_1},\frac{\mathrm{0.6,0.2}}{D_2},\frac{\mathrm{0.7,0.2}}{D_3},\frac{\mathrm{0.65,0.1}}{D_4},\frac{\mathrm{0.85,0.15}}{D_5}\text{.}\end{array}$$

On reaching the cars’ stand, the car dealer gives the descriptions of the available car models in intuitionistic fuzzy features as follows:$$\begin{array}{c}\text{(19)}& {\mathfrak{R}}_1=\frac{\mathrm{0.4,0.4}}{D_1},\frac{\mathrm{0.5,0.1}}{D_2},\frac{\mathrm{0.6,0.2}}{D_3},\frac{\mathrm{0.4,0.5}}{D_4},\frac{\mathrm{0.3,0.2}}{D_5}\text{,}{\mathfrak{R}}_2=\frac{\mathrm{0.79,0.1}}{D_1},\frac{\mathrm{0.7,0.2}}{D_2},\frac{\mathrm{0.5,0.3}}{D_3},\frac{\mathrm{0.55,0.2}}{D_4},\frac{\mathrm{0.65,0.2}}{D_5}\text{,}\\ {\mathfrak{R}}_3=\frac{\mathrm{0.3,0.5}}{D_1},\frac{\mathrm{0.4,0.3}}{D_2},\frac{\mathrm{0.8,0.1}}{D_3},\frac{\mathrm{0.1,0.6}}{D_4},\frac{\mathrm{0.5,0.4}}{D_5}\text{,}\\ {\mathfrak{R}}_4=\frac{\mathrm{0.7,0.3}}{D_1},\frac{\mathrm{0.6,0.1}}{D_2},\frac{\mathrm{0.7,0.1}}{D_3},\frac{\mathrm{0.6,0.2}}{D_4},\frac{\mathrm{0.7,0.2}}{D_5}\text{,}\\ {\mathfrak{R}}_5=\frac{\mathrm{0.5,0.2}}{D_1},\frac{\mathrm{0.6,0.3}}{D_2},\frac{\mathrm{0.7,0.1}}{D_3},\frac{\mathrm{0.6,0.2}}{D_4},\frac{\mathrm{0.5,0.3}}{D_5}\text{.}\end{array}$$

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:$$\begin{array}{c}\text{(20)}& \tilde{\partial }{\check{\mathfrak{I}}}_1,\check{\mathfrak{R}}=0.1677,\tilde{\partial }{\check{\mathfrak{I}}}_2,\check{\mathfrak{R}}=0.3158,\tilde{\partial }{\check{\mathfrak{I}}}_3,\check{\mathfrak{R}}=-0.6133\text{,}\tilde{\partial }{\check{\mathfrak{I}}}_4,\check{\mathfrak{R}}=0.8364,\tilde{\partial }{\check{\mathfrak{I}}}_5,\check{\mathfrak{R}}=-0.6628\text{.}\end{array}$$

From the results, the buyer should go for the car model ${\mathfrak{I}}_4$ because it fits the buyer’s preference the most. This decision tallies with the mere observation, which indicates that the car model ${\mathfrak{I}}_4$ is more related to $\mathfrak{R}$ 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 Table 2.

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 Table 2, 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 ${\mathfrak{I}}_2$, 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.

Approaches	${\check{\mathfrak{I}}}_1,\check{\mathfrak{R}}$	${\check{\mathfrak{I}}}_2,\check{\mathfrak{R}}$	${\check{\mathfrak{I}}}_3,\check{\mathfrak{R}}$	${\check{\mathfrak{I}}}_4,\check{\mathfrak{R}}$	${\check{\mathfrak{I}}}_5,\check{\mathfrak{R}}$
Hung [22]	$-0.3711$	0.5114	$-0.7631$	$-0.0300$	$-0.3314$
Liu et al. [23]	$-0.2973$	0.4456	$-0.6746$	$-0.4125$	$-0.4551$
Thao et al. [25]	$-0.1145$	0.4027	$-0.6407$	0.0869	$-0.4222$
Modified method	0.1677	0.3158	$-0.6133$	0.8364	$-0.6628$

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, ${\mathfrak{R}}_1,{\mathfrak{R}}_2,\dots ,{\mathfrak{R}}_p$ by the means of the normal correlation model. The IFA was adapted to fit into normal framework using logit transform [33] as follows:$$\begin{array}{}\text{(21)}& Z_i{\mathfrak{R}}_j=\log \frac{{\beta }_{{\mathfrak{R}}_j}{x}_i}{{\gamma }_{{\mathfrak{R}}_j}{x}_i}\text{,}\end{array}$$for $i=1,\dots ,n$ and $j=1,\dots ,p$. $Z_i{\mathfrak{R}}_j$ was approximated via empirical logit transform [34] to avoid the case when $Z_i{\mathfrak{R}}_j=\log 0$ or $Z_i{\mathfrak{R}}_j=\log \infty$ as follows:$$\begin{array}{}\text{(22)}& Z_i{\mathfrak{R}}_j=\log \frac{{\beta }_{{\mathfrak{R}}_j}{x}_i+1/2n}{{\gamma }_{{\mathfrak{R}}_j}{x}_i+1/2n}\text{,}\end{array}$$for $i=1,\dots ,n$, $j=1,\dots ,p$.

Suppose we have two IFAs ${\mathfrak{R}}_i$ and ${\mathfrak{R}}_j$, then the correlation is determined as follows:$$\begin{array}{}\text{(23)}& {\partial }^2{\mathfrak{R}}_i,{\mathfrak{R}}_j=\frac{\mathrm{S}\mathrm{S}\mathrm{R}{\mathfrak{R}}_j}{\mathrm{S}\mathrm{S}\mathrm{T}\mathrm{O}{\mathfrak{R}}_i}\text{,}\end{array}$$where $\mathrm{S}\mathrm{S}\mathrm{T}\mathrm{O}{\mathfrak{R}}_i={\sum }_{i=1}^n{\beta }_{{\mathfrak{R}}_i}{x}_i-\overline{{\beta }_{{\mathfrak{R}}_i}}{}^2+{\gamma }_{{\mathfrak{R}}_i}{x}_i-\overline{{\gamma }_{{\mathfrak{R}}_i}}{}^2$ (total sum of squares) and $\mathrm{S}\mathrm{S}\mathrm{R}{\mathfrak{R}}_j=\mathrm{S}\mathrm{S}\mathrm{T}\mathrm{O}{\mathfrak{R}}_i-\mathrm{S}\mathrm{S}\mathrm{E}{\mathfrak{R}}_j$ (regression sum of squares) for $\mathrm{S}\mathrm{S}\mathrm{E}{\mathfrak{R}}_j={\sum }_{i=1}^n{e}^2$ (error sum of squares).

Hence, the correlation coefficient between ${\mathfrak{R}}_i$ and ${\mathfrak{R}}_j$ is represented as follows.$$\begin{array}{}\text{(24)}& \partial {\mathfrak{R}}_i,{\mathfrak{R}}_j=\sqrt{\frac{\mathrm{S}\mathrm{S}\mathrm{R}{\mathfrak{R}}_j}{\mathrm{S}\mathrm{S}\mathrm{T}\mathrm{O}{\mathfrak{R}}_i}}\text{.}\end{array}$$

Thus, the PCC between IFAs, ${\mathfrak{R}}_1$ and ${\mathfrak{R}}_2$, when ${\mathfrak{R}}_3,\dots ,{\mathfrak{R}}_p$ are muted is represented as follows:$$\begin{array}{}\text{(25)}& \partial {\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3,\dots ,{\mathfrak{R}}_p=\frac{\mathrm{S}\mathrm{S}\mathrm{E}{\mathfrak{R}}_3,\dots ,{\mathfrak{R}}_p-\mathrm{S}\mathrm{S}\mathrm{E}{\mathfrak{R}}_2,\dots ,{\mathfrak{R}}_p}{\mathrm{S}\mathrm{S}\mathrm{E}{\mathfrak{R}}_3,\dots ,{\mathfrak{R}}_p}\text{.}\end{array}$$

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 ${\mathfrak{R}}_1$, ${\mathfrak{R}}_2$, and ${\mathfrak{R}}_3$ in $X$. The linear correlation between ${\mathfrak{R}}_1$ and ${\mathfrak{R}}_2$ can be computed by using equation (10). The fact is that ${\mathfrak{R}}_1$, ${\mathfrak{R}}_2$, and ${\mathfrak{R}}_3$ are IFAs drawn from the space $X$, and the linear correlation between ${\mathfrak{R}}_1$ and ${\mathfrak{R}}_2$ could be impacted by ${\mathfrak{R}}_3$. Thus, to compute the precise linear correlation between ${\mathfrak{R}}_1$ and ${\mathfrak{R}}_2$, the effect of ${\mathfrak{R}}_3$ must be removed. Such a precise linear correlation between ${\mathfrak{R}}_1$ and ${\mathfrak{R}}_2$ can be determined by partial correlation coefficient.

3.2.1. First-Order Partial Correlation Coefficient of IFAs

Definition 6.

Let ${\mathfrak{R}}_1$, ${\mathfrak{R}}_2$, and ${\mathfrak{R}}_3$ be IFAs in the sample space $X$. Then, the PCC between IFAs ${\mathfrak{R}}_1$ and ${\mathfrak{R}}_2$ without the influence of IFA ${\mathfrak{R}}_3$ is defined as follows:$$\begin{array}{}\text{(26)}& \tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3=\frac{\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2-\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_3\tilde{\partial }{\mathfrak{R}}_2,\mathfrak{R}3}{{1-{\tilde{\partial }}^2{\mathfrak{R}}_1,{\mathfrak{R}}_{3}1-{\tilde{\partial }}^2{\mathfrak{R}}_2,{\mathfrak{R}}_3}^{1/2}}\text{,}\end{array}$$where $\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2$,$\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_3$, and $\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_3$ can be calculated by using equation (12). We can rewrite equation (26) as follows:$$\begin{array}{}\text{(27)}& {\tilde{\partial }}^{2}C,DE=\frac{{\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2-\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_3\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_3}^2}{1-{\tilde{\partial }}^2{\mathfrak{R}}_1,{\mathfrak{R}}_{3}1-{\tilde{\partial }}^2{\mathfrak{R}}_2,{\mathfrak{R}}_3}\text{.}\end{array}$$

Now, we delineate some characteristics of the first-order PCC of IFAs as follows:

Theorem 1.

Suppose ${\mathfrak{R}}_1,{\mathfrak{R}}_2$, and ${\mathfrak{R}}_3$ are IFAs drawn from $X=x_1,\dots ,x_n$ for $n\in \lceil 1,\infty \rceil$. Then, $\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3=\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_1{\mathfrak{R}}_3$.

Proof.

We know that $\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2=\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_1$ by equation (12). Thus,$$\begin{array}{c}\text{(28)}& \tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3=\frac{\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2-\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_3\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_3}{{1-{\tilde{\partial }}^2{\mathfrak{R}}_1,{\mathfrak{R}}_{3}1-{\tilde{\partial }}^2{\mathfrak{R}}_2,{\mathfrak{R}}_3}^{1/2}}=\frac{\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_1-\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_3\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_3}{{1-{\tilde{\partial }}^2{\mathfrak{R}}_2,{\mathfrak{R}}_{3}1-{\tilde{\partial }}^2{\mathfrak{R}}_1,{\mathfrak{R}}_3}^{1/2}}\\ =\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_1{\mathfrak{R}}_3\text{,}\end{array}$$which completes the proof.

Proposition 1.

With the same hypothesis in Theorem 1, a PCC $\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3\mathrm{b}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2\mathrm{i}\mathrm{f}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{f}\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_3=0=\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_3$.

Proof.

Assume that $\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_3=\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_3=0$; then, we have the following expression:$$\begin{array}{}\text{(29)}& \tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3=\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2=\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_1=\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_1{\mathfrak{R}}_3\text{.}\end{array}$$

The converse is straightforward.

Remark 1.

With the same hypothesis in Theorem 1, the following statements are valid:

(i) If ${\mathfrak{R}}_1$ and ${\mathfrak{R}}_3$ have a perfect positive linear correlation, then $\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3=\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_1{\mathfrak{R}}_3=\infty$

(ii) If $\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_3=0$, then$$\begin{array}{}\text{(30)}& \tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3=\frac{\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2}{{1-{\tilde{\partial }}^2{\mathfrak{R}}_2,{\mathfrak{R}}_3}^{1/2}}=\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_1{\mathfrak{R}}_3\text{.}\end{array}$$

  Also, if $\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_3=0$, then$$\begin{array}{}\text{(31)}& \tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3=\frac{\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2}{{1-{\tilde{\partial }}^2{\mathfrak{R}}_1,{\mathfrak{R}}_3}^{1/2}}=\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_1{\mathfrak{R}}_3\text{.}\end{array}$$

(iii) If $\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2=0$, $\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_3\ne 0$, and $\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_3\ne 0$, then$$\begin{array}{}\text{(32)}& \tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3=\frac{-\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_3\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_3}{{1-{\tilde{\partial }}^2{\mathfrak{R}}_1,{\mathfrak{R}}_{3}1-{\tilde{\partial }}^2{\mathfrak{R}}_2,{\mathfrak{R}}_3}^{1/2}}\text{.}\end{array}$$

3.2.2. $n^{\mathrm{t}\mathrm{h}}$-Order Partial Correlation Coefficient of IFAs

So far, we have been considering PCC between two IFAs by keeping the third IFA muted. The concept of $n^{\mathrm{t}\mathrm{h}}$-order PCC of IFAs is conceivable in the case that there are random samples of more than three IFAs.

Let ${\mathfrak{R}}_1,\dots ,{\mathfrak{R}}_r,{\mathfrak{R}}_{r+1},\dots ,{\mathfrak{R}}_{r+s}$ be IFAs in $X$, where each IFA is represented as follows:$$\begin{array}{}\text{(33)}& {\mathfrak{R}}_i=x,{\beta }_{{\mathfrak{R}}_i}x,{\gamma }_{{\mathfrak{R}}_i}xx\in X\text{,}\end{array}$$where ${\beta }_{{\mathfrak{R}}_i},{\gamma }_{{\mathfrak{R}}_i}:X⟶\mathrm{0,1}$ for $i=1,\dots ,r+s$. Also, ${\mathfrak{R}}_i$ can be written as follows:$$\begin{array}{}\text{(34)}& {\mathfrak{R}}_i=x_k,{\beta }_{{\mathfrak{R}}_1}{x}_k,{\gamma }_{{\mathfrak{R}}_1}{x}_k,\dots ,{\beta }_{{\mathfrak{R}}_{r+s}}{x}_k,{\gamma }_{{\mathfrak{R}}_{r+s}}{x}_k{x}_k\in X\text{,}\end{array}$$for $k=1,\dots ,n$.

We designate the covariances of IFAs ${\mathfrak{R}}_1,\dots ,{\mathfrak{R}}_r,{\mathfrak{R}}_{r+1},\dots ,{\mathfrak{R}}_{r+s}$ as covariance matrix $\mathcal{N}$, which we define as follows:$$\begin{array}{}\text{(35)}& \mathcal{N}=\mathrm{\Phi }{\mathfrak{R}}_i,{\mathfrak{R}}_{j}i=1,\dots ,r+s,j=1,\dots ,r+s\text{,}\end{array}$$where$$\begin{array}{}\text{(36)}& \mathrm{\Phi }{\mathfrak{R}}_i,{\mathfrak{R}}_j=\frac{{\mathrm{\Sigma }}_{k=1}^n{\nabla }_k{\mathfrak{R}}_i{\nabla }_k{\mathfrak{R}}_j}{n-1},\end{array}$$is the covariance of ${\mathfrak{R}}_i,{\mathfrak{R}}_j$, in which$$\begin{array}{}\text{(37)}& \begin{array}{c}{\nabla }_k{\mathfrak{R}}_i={\beta }_{{\mathfrak{R}}_i}^2{x}_k-{\overline{{\beta }_{{\mathfrak{R}}_i}}}^2-{\gamma }_{{\mathfrak{R}}_i}^2{x}_k-{\overline{{\gamma }_{{\mathfrak{R}}_i}}}^2-{\delta }_{{\mathfrak{R}}_i}^2{x}_k-{\overline{{\delta }_{{\mathfrak{R}}_i}}}^2\\ {\nabla }_k{\mathfrak{R}}_j={\beta }_{{\mathfrak{R}}_j}^2{x}_k-{\overline{{\beta }_{{\mathfrak{R}}_j}}}^2-{\gamma }_{{\mathfrak{R}}_j}^2{x}_k-{\overline{{\gamma }_{{\mathfrak{R}}_j}}}^2-{\delta }_{{\mathfrak{R}}_j}^2{x}_k-{\overline{{\delta }_{{\mathfrak{R}}_j}}}^2\end{array},\end{array}$$are the deviations of ${\mathfrak{R}}_i$ and ${\mathfrak{R}}_j$, respectively.

Definition 7.

The PCC of ${\mathfrak{R}}_i,{\mathfrak{R}}_j$, $i,j=1,\dots ,r$, by removing the effects of ${\mathfrak{R}}_{r+1},\dots ,{\mathfrak{R}}_{r+s}$ can be obtained as follows:

Suppose we have four parts partition of $\mathcal{N}$,$$\begin{array}{}\text{(38)}& \mathcal{N}=\begin{array}{cc}{\mathcal{N}}_{11}& {\mathcal{N}}_{12}\\ {\mathcal{N}}_{21}& {\mathcal{N}}_{22}\end{array}\text{,}\end{array}$$where ${\mathcal{N}}_{11}$ is the $r\times r$ covariance matrix of ${\mathfrak{R}}_i,{\mathfrak{R}}_j$ for $i,j=1,\dots ,r$, ${\mathcal{N}}_{12}$ is the $r\times s$ covariance matrix of ${\mathfrak{R}}_i,{\mathfrak{R}}_j$ for $i=1,\dots ,r$, $j=r+1,\dots ,r+s$, ${\mathcal{N}}_{21}$ is the $s\times r$ covariance matrix of ${\mathfrak{R}}_i,{\mathfrak{R}}_j$ for $i=r+1,\dots ,r+s$, $j=1,\dots ,r$, and ${\mathcal{N}}_{22}$ is the $s\times s$ covariance matrix of ${\mathfrak{R}}_i,{\mathfrak{R}}_j$ for $i,j=r+1,\dots ,r+s$. Then, certainly, ${\mathcal{N}}_{12}$ and ${\mathcal{N}}_{21}$ transpose each other.

By keeping ${\mathfrak{R}}_{r+1},\dots ,{\mathfrak{R}}_{r+s}$ constant, the PCC of ${\mathfrak{R}}_i,{\mathfrak{R}}_j$ is defined as follows:$$\begin{array}{}\text{(39)}& \tilde{\partial }{\mathfrak{R}}_i,{\mathfrak{R}}_j{\mathfrak{R}}_{r+1},\dots ,{\mathfrak{R}}_{r+s}=\frac{\mathrm{\Phi }{\mathfrak{R}}_i,{\mathfrak{R}}_j{\mathfrak{R}}_{r+1},\dots ,{\mathfrak{R}}_{r+s}}{{\mathrm{\Phi }{\mathfrak{R}}_i,{\mathfrak{R}}_i{{\mathfrak{R}}_{r+1},\mathfrak{\dots },\mathfrak{R}}_{r+s}\mathrm{\Phi }{\mathfrak{R}}_j,{\mathfrak{R}}_j{\mathfrak{R}}_{r+1},\dots ,{\mathfrak{R}}_{r+s}}^{1/2}}\text{,}\end{array}$$where $i,j=1,\dots ,r$ and $\mathrm{\Phi }{\mathfrak{R}}_i,{\mathfrak{R}}_j{\mathfrak{R}}_{r+1},\dots ,{\mathfrak{R}}_{r+s}$ is the $i,j^{\mathrm{t}\mathrm{h}}$ element of partial covariance matrix of ${\mathcal{N}}_{112}={\mathcal{N}}_{11}-{\mathcal{N}}_{12}{\mathcal{N}}_{22}^{-1}{\mathcal{N}}_{21}$.

It follows that$$\begin{array}{}\text{(40)}& \tilde{\partial }{\mathfrak{R}}_i,{\mathfrak{R}}_j{\mathfrak{R}}_{r+1},\dots ,{\mathfrak{R}}_{r+s}=\tilde{\partial }{\mathfrak{R}}_j,{\mathfrak{R}}_i{\mathfrak{R}}_{r+1},\dots ,{\mathfrak{R}}_{r+s}\text{.}\end{array}$$

Since ${\mathcal{N}}_{112}$ is a symmetry matrix and$$\begin{array}{}\text{(41)}& \mathrm{\Phi }{\mathfrak{R}}_i,{\mathfrak{R}}_j{\mathfrak{R}}_{r+1},\dots ,{\mathfrak{R}}_{r+s}=\mathrm{\Phi }{\mathfrak{R}}_j,{\mathfrak{R}}_i{\mathfrak{R}}_{r+1},\dots ,{\mathfrak{R}}_{r+s}\text{.}\end{array}$$

Theorem 2.

If $r=2$ and $s=1$, then the $n^{\mathrm{t}\mathrm{h}}$-order PCC of IFAs is equivalent to the first-order PCC of IFAs.

Proof.

The proof of Theorem 2 is in the appendix section.

Remark 2.

If $r=2$ and $s=2$, then the $n^{\mathrm{t}\mathrm{h}}$-order PCC of PFAs is as follows:$$\begin{array}{}\text{(42)}& \tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3,{\mathfrak{R}}_4=\frac{\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_4-\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_3{\mathfrak{R}}_4\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_3{\mathfrak{R}}_4}{{1-{\tilde{\partial }}^2{\mathfrak{R}}_2,{\mathfrak{R}}_3{\mathfrak{R}}_{4}1-{\tilde{\partial }}^2{\mathfrak{R}}_1,{\mathfrak{R}}_3{\mathfrak{R}}_4}^{1/2}}\text{,}\end{array}$$and redrafted as$$\begin{array}{}\text{(43)}& {\tilde{\partial }}_{1234}=\frac{{\tilde{\partial }}_{124}-{\tilde{\partial }}_{234}{\tilde{\partial }}_{134}}{{1-{\tilde{\partial }}_{234}^{2}1-{\tilde{\partial }}_{134}^2}^{1/2}}\text{.}\end{array}$$

Indeed, ${\tilde{\partial }}_{1234}={\tilde{\partial }}_{1243}$ since$$\begin{array}{c}\text{(44)}& {\tilde{\partial }}_{1234}=\frac{{\tilde{\partial }}_{124}-{\tilde{\partial }}_{234}{\tilde{\partial }}_{134}}{{1-{\tilde{\partial }}_{234}^{2}1-{\tilde{\partial }}_{134}^2}^{1/2}}=\frac{{\tilde{\partial }}_{123}-{\tilde{\partial }}_{243}{\tilde{\partial }}_{143}}{{1-{\tilde{\partial }}_{243}^{2}1-{\tilde{\partial }}_{143}^2}^{1/2}}\\ {\widetilde{=\partial }}_{1243}\text{.}\end{array}$$

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.

Let ${\mathfrak{R}}_1$, ${\mathfrak{R}}_2$, ${\mathfrak{R}}_3$, and ${\mathfrak{R}}_4$ be IFSs in $X=x_1,x_2,x_3,x_4,x_5$ and are defined as follows:$$\begin{array}{c}\text{(45)}& {\mathfrak{R}}_1=\frac{\mathrm{0.1,0.8}}{x_1},\frac{\mathrm{0.3,0.5}}{x_2},\frac{\mathrm{0.6,0.2}}{x_3},\frac{\mathrm{0.9,0}}{x_4},\frac{\mathrm{1,0}}{x_5}\text{,}{\mathfrak{R}}_2=\frac{\mathrm{0.01,0.96}}{x_1},\frac{\mathrm{0.09,0.75}}{x_2},\frac{\mathrm{0.36,0.36}}{x_3},\frac{\mathrm{0.81,0}}{x_4},\frac{\mathrm{1,0}}{x_5}\text{,}\\ {\mathfrak{R}}_3=\frac{\mathrm{0.96,0.01}}{x_1},\frac{\mathrm{0.75,0.09}}{x_2},\frac{\mathrm{0.36,0.36}}{x_3},\frac{\mathrm{0,0.81}}{x_4},\frac{\mathrm{0,1}}{x_5}\text{,}\\ {\mathfrak{R}}_4=\frac{\mathrm{0,1}}{x_1},\frac{\mathrm{0.01,0.94}}{x_2},\frac{\mathrm{0.13,0.59}}{x_3},\frac{\mathrm{0.66,0}}{x_4},\frac{\mathrm{1,0}}{x_5}\text{.}\end{array}$$

We aim to compute the PCC between the IFAs by removing the effect of IFA ${\mathfrak{R}}_1$. Using the approach in [30], we obtain the following results:$$\begin{array}{c}\text{(46)}& {\partial }_{{\mathfrak{R}}_2{\mathfrak{R}}_2{\mathfrak{R}}_1}=1.000,{\partial }_{{\mathfrak{R}}_2{\mathfrak{R}}_3{\mathfrak{R}}_1}=-1.000,{\partial }_{{\mathfrak{R}}_2{\mathfrak{R}}_4{\mathfrak{R}}_1}=0.983\text{,}{\partial }_{{\mathfrak{R}}_3{\mathfrak{R}}_3{\mathfrak{R}}_1}=1.000,{\partial }_{{\mathfrak{R}}_3{\mathfrak{R}}_4{\mathfrak{R}}_1}=-0.983,{\partial }_{{\mathfrak{R}}_4{\mathfrak{R}}_4{\mathfrak{R}}_1}=1.000\text{.}\end{array}$$

With the novel approach, we obtain the following results:$$\begin{array}{c}\text{(47)}& {\tilde{\partial }}_{{\mathfrak{R}}_2{\mathfrak{R}}_2{\mathfrak{R}}_1}=1.000,{\tilde{\partial }}_{{\mathfrak{R}}_2{\mathfrak{R}}_3{\mathfrak{R}}_1}=0.508,{\tilde{\partial }}_{{\mathfrak{R}}_2{\mathfrak{R}}_4{\mathfrak{R}}_1}=0.975\text{,}{\tilde{\partial }}_{{\mathfrak{R}}_3{\mathfrak{R}}_3{\mathfrak{R}}_1}=1.000,{\tilde{\partial }}_{{\mathfrak{R}}_3{\mathfrak{R}}_4{\mathfrak{R}}_1}=0.672,{\tilde{\partial }}_{{\mathfrak{R}}_4{\mathfrak{R}}_4{\mathfrak{R}}_1}=1.000\text{.}\end{array}$$

The PCC values are compiled in Table 3.

From Table 3, 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 ${\mathfrak{R}}_1$ is removed. The new approach gives reasonable results for the correlations between ${\mathfrak{R}}_2,{\mathfrak{R}}_3$ and ${\mathfrak{R}}_3,{\mathfrak{R}}_4$ when the effect of ${\mathfrak{R}}_1$ is removed except for ${\mathfrak{R}}_2,{\mathfrak{R}}_4$. From the approach in [30], the correlation coefficient between ${\mathfrak{R}}_2,{\mathfrak{R}}_3$ when the effect of ${\mathfrak{R}}_1$ is removed shows that ${\mathfrak{R}}_2$ and ${\mathfrak{R}}_3$ are negatively perfect correlated, which is not true. Similarly, the correlation coefficient between ${\mathfrak{R}}_3,{\mathfrak{R}}_4$ when the effect of ${\mathfrak{R}}_1$ is removed shows that ${\mathfrak{R}}_3$ and ${\mathfrak{R}}_4$ are negatively correlated, which is also not true. However, the new approach shows that positive correlations exist between ${\mathfrak{R}}_2,{\mathfrak{R}}_3$ and ${\mathfrak{R}}_3,{\mathfrak{R}}_4$ when the effect of ${\mathfrak{R}}_1$ 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.

Approaches	${\mathfrak{R}}_2{\mathfrak{R}}_2{\mathfrak{R}}_1$	${\mathfrak{R}}_2{\mathfrak{R}}_3{\mathfrak{R}}_1$	${\mathfrak{R}}_2{\mathfrak{R}}_4{\mathfrak{R}}_1$	${\mathfrak{R}}_3{\mathfrak{R}}_3{\mathfrak{R}}_1$	${\mathfrak{R}}_3{\mathfrak{R}}_4{\mathfrak{R}}_1$	${\mathfrak{R}}_4{\mathfrak{R}}_4{\mathfrak{R}}_1$
Hung [30]	1.000	$-1.000$	0.983	1.000	$-0.983$	1.000
New method	1.000	0.508	0.975	1.000	0.672	1.000

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 IFAs ${\mathfrak{I}}_i$ for $i=\mathrm{1,2,3,4}$ in the feature space $F=f_1,f_2,f_3,f_4,f_5$ as follows:$$\begin{array}{c}\text{(48)}& {\mathfrak{I}}_1=\frac{\mathrm{0.800,0.100}}{f_1},\frac{\mathrm{0.700,0.200}}{f_2},\frac{\mathrm{0.900,0.000}}{f_3},\frac{\mathrm{0.600,0.300}}{f_4},\frac{\mathrm{0.800,0.100}}{f_5}\text{,}{\mathfrak{I}}_2=\frac{\mathrm{0.900,0.100}}{f_1},\frac{\mathrm{0.800,0.100}}{f_2},\frac{\mathrm{0.800,0.100}}{f_3},\frac{\mathrm{0.500,0.300}}{f_4},\frac{\mathrm{0.700,0.200}}{f_5}\text{,}\\ {\mathfrak{I}}_3=\frac{\mathrm{0.500,0.300}}{f_1},\frac{\mathrm{0.500,0.200}}{f_2},\frac{\mathrm{0.900,0.000}}{f_3},\frac{\mathrm{0.500,0.400}}{f_4},\frac{\mathrm{0.700,0.100}}{f_5}\text{,}\\ {\mathfrak{I}}_4=\frac{\mathrm{0.700,0.200}}{f_1},\frac{\mathrm{0.500,0.400}}{f_2},\frac{\mathrm{0.900,0.100}}{f_3},\frac{\mathrm{0.600,0.300}}{f_4},\frac{\mathrm{0.800,0.000}}{f_5}\text{.}\end{array}$$

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 ${\mathfrak{I}}_i$ for $i=\mathrm{1,2,3,4}$.

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 for $n$ 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:$$\begin{array}{}\text{(49)}& \end{array}$$

From the matrix, we see that pattern ${\mathfrak{I}}_1$ can be associated with pattern ${\mathfrak{I}}_4$, pattern ${\mathfrak{I}}_2$ can be associated with pattern ${\mathfrak{I}}_1$, and patterns ${\mathfrak{I}}_3$ and ${\mathfrak{I}}_4$ can be associated with each other. We observe that the matrix is symmetric and ${\tilde{\partial }}_{{\mathfrak{I}}_i{\mathfrak{I}}_{i+1}}={\tilde{\partial }}_{{\mathfrak{I}}_{i+1}{\mathfrak{I}}_i}$. The correlation coefficient results show that the four random IFAs have positive linear relationship with each other with the exceptions of ${\tilde{\partial }}_{{\mathfrak{I}}_1{\mathfrak{I}}_2}$, ${\tilde{\partial }}_{{\mathfrak{I}}_2{\mathfrak{I}}_3}$, and ${\tilde{\partial }}_{{\mathfrak{I}}_2{\mathfrak{I}}_4}$.

To get a precise correlation coefficient among the building patterns, we find the first-order PCC of the patterns ${\mathfrak{I}}_i$. 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:$$\begin{array}{c}\text{(50)}& {\tilde{\partial }}_{{\mathfrak{I}}_1{\mathfrak{I}}_2{\mathfrak{I}}_3}=-0.0895,{\tilde{\partial }}_{{\mathfrak{I}}_1{\mathfrak{I}}_2{\mathfrak{I}}_4}=-0.0616\text{,}{\tilde{\partial }}_{{\mathfrak{I}}_1{\mathfrak{I}}_3{\mathfrak{I}}_2}=0.1822,{\tilde{\partial }}_{{\mathfrak{I}}_1{\mathfrak{I}}_3{\mathfrak{I}}_4}=0.1091\text{,}\\ {\tilde{\partial }}_{{\mathfrak{I}}_1{\mathfrak{I}}_4{\mathfrak{I}}_2}=0.2607,{\tilde{\partial }}_{{\mathfrak{I}}_1{\mathfrak{I}}_4{\mathfrak{I}}_3}=0.2268\text{,}\\ {\tilde{\partial }}_{{\mathfrak{I}}_2{\mathfrak{I}}_3{\mathfrak{I}}_1}=-0.1280,{\tilde{\partial }}_{{\mathfrak{I}}_2{\mathfrak{I}}_3{\mathfrak{I}}_4}=-0.0805\text{,}\\ {\tilde{\partial }}_{{\mathfrak{I}}_2{\mathfrak{I}}_4{\mathfrak{I}}_1}=-0.1849,{\tilde{\partial }}_{{\mathfrak{I}}_2{\mathfrak{I}}_4{\mathfrak{I}}_3}=-0.1690\text{,}\\ {\tilde{\partial }}_{{\mathfrak{I}}_3{\mathfrak{I}}_4{\mathfrak{I}}_1}=0.3176,{\tilde{\partial }}_{{\mathfrak{I}}_3{\mathfrak{I}}_4{\mathfrak{I}}_2}=0.3338\text{.}\end{array}$$

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: pattern ${\mathfrak{I}}_1$ has a negative effect on the correlation between patterns ${\mathfrak{I}}_2$, ${\mathfrak{I}}_3$ and ${\mathfrak{I}}_2,{\mathfrak{I}}_4$ and a positive effect on pattern ${\mathfrak{I}}_3,{\mathfrak{I}}_4$; pattern ${\mathfrak{I}}_2$ has only positive effect on the correlation between patterns ${\mathfrak{I}}_1,{\mathfrak{I}}_3$, ${\mathfrak{I}}_1,{\mathfrak{I}}_4$, and ${\mathfrak{I}}_3,{\mathfrak{I}}_4$; pattern ${\mathfrak{I}}_3$ has a negative effect on the correlation between patterns ${\mathfrak{I}}_1,{\mathfrak{I}}_2$ and ${\mathfrak{I}}_2,{\mathfrak{I}}_4$ and a positive influence on pattern ${\mathfrak{I}}_1,{\mathfrak{I}}_4$; pattern ${\mathfrak{I}}_4$ has a negative effect on the correlation between patterns ${\mathfrak{I}}_1,{\mathfrak{I}}_2$ and ${\mathfrak{I}}_2,{\mathfrak{I}}_3$ and a positive effect on pattern ${\mathfrak{I}}_1,{\mathfrak{I}}_3$.

By removing the effect of another IFA from the first-order partial correlation coefficients, we obtain the following expressions:$$\begin{array}{c}\text{(51)}& {\tilde{\partial }}_{{\mathfrak{I}}_1{\mathfrak{I}}_2{\mathfrak{I}}_3{\mathfrak{I}}_4}=-0.0533,{\tilde{\partial }}_{{\mathfrak{I}}_1{\mathfrak{I}}_3{\mathfrak{I}}_2{\mathfrak{I}}_4}=0.1047\text{,}{\tilde{\partial }}_{{\mathfrak{I}}_1{\mathfrak{I}}_4{\mathfrak{I}}_2{\mathfrak{I}}_3}=0.2157,{\tilde{\partial }}_{{\mathfrak{I}}_2{\mathfrak{I}}_3{\mathfrak{I}}_1{\mathfrak{I}}_4}=-0.0744\text{,}\\ {\tilde{\partial }}_{{\mathfrak{I}}_2{\mathfrak{I}}_{4}P{\mathfrak{I}}_1{\mathfrak{I}}_3}=-0.1534,{\tilde{\partial }}_{{\mathfrak{I}}_3{\mathfrak{I}}_4{\mathfrak{I}}_1{\mathfrak{I}}_2}=0.3016\text{.}\end{array}$$

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: pattern ${\mathfrak{I}}_1$ has a negative effect on the first-order partial correlation among patterns ${\mathfrak{I}}_2,{\mathfrak{I}}_4{\mathfrak{I}}_3$ and ${\mathfrak{I}}_2,{\mathfrak{I}}_3{\mathfrak{I}}_4$ and a positive effect on pattern ${\mathfrak{I}}_3,{\mathfrak{I}}_4{\mathfrak{I}}_2$; pattern ${\mathfrak{I}}_2$ has only positive effect on the first-order partial correlation among patterns ${\mathfrak{I}}_1,{\mathfrak{I}}_3{\mathfrak{I}}_4$, ${\mathfrak{I}}_1,{\mathfrak{I}}_4{\mathfrak{I}}_3$, and ${\mathfrak{I}}_3,{\mathfrak{I}}_4{\mathfrak{I}}_1$; pattern ${\mathfrak{I}}_3$ has a negative effect on the first-order partial correlation among patterns ${\mathfrak{I}}_1,{\mathfrak{I}}_2{\mathfrak{I}}_4$ and ${\mathfrak{I}}_2,{\mathfrak{I}}_4{\mathfrak{I}}_1$ and a positive influence on pattern ${\mathfrak{I}}_1,{\mathfrak{I}}_4{\mathfrak{I}}_2$; pattern ${\mathfrak{I}}_4$ has a negative effect on the first-order partial correlation among patterns ${\mathfrak{I}}_1,{\mathfrak{I}}_2{\mathfrak{I}}_3$ and ${\mathfrak{I}}_2,{\mathfrak{I}}_3{\mathfrak{I}}_1$ and a positive effect on pattern ${\mathfrak{I}}_1,{\mathfrak{I}}_3{\mathfrak{I}}_2$.

We discover that pattern ${\mathfrak{I}}_2$ 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, ${\tilde{\partial }}_{{\mathfrak{I}}_1{\mathfrak{I}}_2{\mathfrak{I}}_3{\mathfrak{I}}_4}$, ${\tilde{\partial }}_{{\mathfrak{I}}_2{\mathfrak{I}}_3{\mathfrak{I}}_1{P}_4}$, and ${\tilde{\partial }}_{{\mathfrak{I}}_2{\mathfrak{I}}_4{\mathfrak{I}}_1{\mathfrak{I}}_3}$, imply that the linear relationships between ${\mathfrak{I}}_1,{\mathfrak{I}}_2$, ${\mathfrak{I}}_2,{\mathfrak{I}}_3$, and ${\mathfrak{I}}_2,{\mathfrak{I}}_4$ are actually negative. The same reason hold for the positive partial correlation coefficients ${\tilde{\partial }}_{{\mathfrak{I}}_1{\mathfrak{I}}_3{\mathfrak{I}}_2{\mathfrak{I}}_4}$, ${\tilde{\partial }}_{{\mathfrak{I}}_1{\mathfrak{I}}_4{\mathfrak{I}}_2{\mathfrak{I}}_3}$, and ${\tilde{\partial }}_{{\mathfrak{I}}_3{\mathfrak{I}}_4{\mathfrak{I}}_1{\mathfrak{I}}_2}$.

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.

Appendix

A. Proof of Theorem 2

The proof of Theorem 2 is as follows.

Proof.

Assume $r=2$ and $s=1$. Then, we have IFAs ${\mathfrak{R}}_1$, ${\mathfrak{R}}_2$, and ${\mathfrak{R}}_3$. We find the linear correlation between ${\mathfrak{R}}_1$ and ${\mathfrak{R}}_2$ by removing the effect of ${\mathfrak{R}}_3$. Because $r=2$ and $s=1$, we have $r+s=3$ and$$\begin{array}{}\text{(A.1)}& \mathcal{N}=\begin{array}{ccc}\mathrm{\Phi }{C}_1,C_1& \mathrm{\Phi }{C}_1,C_2& \mathrm{\Phi }{C}_1,C_3\\ \mathrm{\Phi }{C}_2,C_1& \mathrm{\Phi }{C}_2,C_2& \mathrm{\Phi }{C}_2,C_3\\ \mathrm{\Phi }{C}_3,C_1& \mathrm{\Phi }{C}_3,C_2& \mathrm{\Phi }{C}_3,C_3\end{array}\text{,}\end{array}$$which can be rewritten as follows:$$\begin{array}{}\text{(A.2)}& \mathcal{N}=\begin{array}{ccc}{\mathrm{\Phi }}_{11}& {\mathrm{\Phi }}_{12}& {\mathrm{\Phi }}_{13}\\ {\mathrm{\Phi }}_{21}& {\mathrm{\Phi }}_{22}& {\mathrm{\Phi }}_{23}\\ {\mathrm{\Phi }}_{31}& {\mathrm{\Phi }}_{32}& {\mathrm{\Phi }}_{33}\end{array}\text{,}\end{array}$$for simplicity sake. Partitioning $\mathcal{N}$ into four parts, we obtain the following expression:$$\begin{array}{}\text{(A.3)}& \mathcal{N}=\begin{array}{cc}\begin{array}{cc}{\mathrm{\Phi }}_{11}& {\mathrm{\Phi }}_{12}\\ {\mathrm{\Phi }}_{21}& {\mathrm{\Phi }}_{22}\end{array}& \begin{array}{c}{\mathrm{\Phi }}_{13}\\ {\mathrm{\Phi }}_{23}\end{array}\\ \begin{array}{cc}{\mathrm{\Phi }}_{31}& {\mathrm{\Phi }}_{32}\end{array}& \begin{array}{c}{\mathrm{\Phi }}_{33}\end{array}\end{array}\text{.}\end{array}$$

Thus,$$\begin{array}{c}\text{(A.4)}& {\mathcal{N}}_{11}=\begin{array}{cc}{\mathrm{\Phi }}_{11}& {\mathrm{\Phi }}_{12}\\ {\mathrm{\Phi }}_{21}& {\mathrm{\Phi }}_{22}\end{array},{\mathcal{N}}_{12}=\begin{array}{c}{\mathrm{\Phi }}_{13}\\ {\mathrm{\Phi }}_{23}\end{array}\text{,}{\mathcal{N}}_{21}=\begin{array}{cc}{\mathrm{\Phi }}_{31}& {\mathrm{\Phi }}_{32}\end{array},{\mathcal{N}}_{22}=\begin{array}{c}{\mathrm{\Phi }}_{33}\end{array}\text{,}\end{array}$$where ${\mathcal{N}}_{11}$ stands for the covariance matrix of ${\mathfrak{R}}_1$ and ${\mathfrak{R}}_2$, ${\mathcal{N}}_{12}$ stands for the covariance matrix of ${\mathfrak{R}}_1$ and ${\mathfrak{R}}_3$, ${\mathcal{N}}_{21}$ stands for the covariance matrix of ${\mathfrak{R}}_2$ and ${\mathfrak{R}}_3$, and ${\mathcal{N}}_{22}$ is the variance of ${\mathfrak{R}}_3$.

Hence, the PCC of ${\mathfrak{R}}_1$ and ${\mathfrak{R}}_2$ by muting ${\mathfrak{R}}_3$ is expressed as follows:$$\begin{array}{}\text{(A.5)}& \tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3=\frac{\mathrm{\Phi }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3}{{\mathrm{\Phi }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3\mathrm{\Phi }{\mathfrak{R}}_2,{\mathfrak{R}}_2{\mathfrak{R}}_3}^{1/2}}\text{,}\end{array}$$or$$\begin{array}{}\text{(A.6)}& {\tilde{\partial }}_{123}=\frac{{\mathrm{\Phi }}_{123}}{{{\mathrm{\Phi }}_{113}{\mathrm{\Phi }}_{223}}^{1/2}}\text{,}\end{array}$$in which ${\mathrm{\Phi }}_{123}$ is the element of ${\mathcal{N}}_{112}={\mathcal{N}}_{11}-{\mathcal{N}}_{12}{\mathcal{N}}_{22}^{-1}{\mathcal{N}}_{21}$. But$$\begin{array}{c}\text{(A.7)}& {\mathcal{N}}_{12}{\mathcal{N}}_{22}^{-1}{\mathcal{N}}_{21}=\begin{array}{c}{\mathrm{\Phi }}_{13}\\ {\mathrm{\Phi }}_{23}\end{array}{\begin{array}{c}{\mathrm{\Phi }}_{33}\end{array}}^{-1}\begin{array}{cc}{\mathrm{\Phi }}_{31}& {\mathrm{\Phi }}_{32}\end{array}=\begin{array}{c}\frac{{\mathrm{\Phi }}_{13}}{{\mathrm{\Phi }}_{33}}\\ \frac{{\mathrm{\Phi }}_{23}}{{\mathrm{\Phi }}_{33}}\end{array}\begin{array}{cc}{\mathrm{\Phi }}_{31}& {\mathrm{\Phi }}_{32}\end{array}\\ =\begin{array}{cc}\frac{{\mathrm{\Phi }}_{13}{\mathrm{\Phi }}_{31}}{{\mathrm{\Phi }}_{33}}& \frac{{\mathrm{\Phi }}_{13}{\mathrm{\Phi }}_{32}}{{\mathrm{\Phi }}_{33}}\\ \frac{{\mathrm{\Phi }}_{23}{\mathrm{\Phi }}_{31}}{{\mathrm{\Phi }}_{33}}& \frac{{\mathrm{\Phi }}_{23}{\mathrm{\Phi }}_{32}}{{\mathrm{\Phi }}_{33}}\end{array}\text{.}\end{array}$$

Thus,$$\begin{array}{}\text{(A.8)}& {\mathcal{N}}_{112}=\begin{array}{cc}{\mathrm{\Phi }}_{11}-\frac{{\mathrm{\Phi }}_{13}{\mathrm{\Phi }}_{31}}{{\mathrm{\Phi }}_{33}}& {\mathrm{\Phi }}_{12}-\frac{{\mathrm{\Phi }}_{13}{\mathrm{\Phi }}_{32}}{{\mathrm{\Phi }}_{33}}\\ {\mathrm{\Phi }}_{21}-\frac{{\mathrm{\Phi }}_{23}{\mathrm{\Phi }}_{31}}{{\mathrm{\Phi }}_{33}}& {\mathrm{\Phi }}_{22}-\frac{{\mathrm{\Phi }}_{23}{\mathrm{\Phi }}_{32}}{{\mathrm{\Phi }}_{33}}\end{array}\text{,}\end{array}$$in which$$\begin{array}{c}\text{(A.9)}& {\mathrm{\Phi }}_{113}={\mathrm{\Phi }}_{11}-\frac{{\mathrm{\Phi }}_{13}{\mathrm{\Phi }}_{31}}{{\mathrm{\Phi }}_{33}},{\mathrm{\Phi }}_{123}={\mathrm{\Phi }}_{12}-\frac{{\mathrm{\Phi }}_{13}{\mathrm{\Phi }}_{32}}{{\mathrm{\Phi }}_{33}}\text{,}{\mathrm{\Phi }}_{213}={\mathrm{\Phi }}_{21}-\frac{{\mathrm{\Phi }}_{23}{\mathrm{\Phi }}_{31}}{{\mathrm{\Phi }}_{33}},{\mathrm{\Phi }}_{223}={\mathrm{\Phi }}_{22}-\frac{{\mathrm{\Phi }}_{23}{\mathrm{\Phi }}_{32}}{{\mathrm{\Phi }}_{33}}\text{.}\end{array}$$

So,$$\begin{array}{}\text{(A.10)}& {\tilde{\partial }}_{123}=\frac{{\mathrm{\Phi }}_{123}}{{{\mathrm{\Phi }}_{113}{\mathrm{\Phi }}_{223}}^{1/2}},\end{array}$$becomes$$\begin{array}{c}\text{(A.11)}& {\tilde{\partial }}_{123}=\frac{{\mathrm{\Phi }}_{12}-{\mathrm{\Phi }}_{13}{\mathrm{\Phi }}_{32}/{\mathrm{\Phi }}_{33}}{{{\mathrm{\Phi }}_{11}-{\mathrm{\Phi }}_{13}{\mathrm{\Phi }}_{31}/{\mathrm{\Phi }}_{33}{\mathrm{\Phi }}_{22}-{\mathrm{\Phi }}_{23}{\mathrm{\Phi }}_{32}/{\mathrm{\Phi }}_{33}}^{1/2}}=\frac{1/{{\mathrm{\Phi }}_{11}{\mathrm{\Phi }}_{22}}^{1/2}{\mathrm{\Phi }}_{12}-{\mathrm{\Phi }}_{13}{\mathrm{\Phi }}_{32}/{\mathrm{\Phi }}_{33}}{1/{{\mathrm{\Phi }}_{11}{\mathrm{\Phi }}_{22}}^{1/2}{{\mathrm{\Phi }}_{11}-{\mathrm{\Phi }}_{13}{\mathrm{\Phi }}_{31}/{\mathrm{\Phi }}_{33}{\mathrm{\Phi }}_{22}-{\mathrm{\Phi }}_{23}{\mathrm{\Phi }}_{32}/{\mathrm{\Phi }}_{33}}^{1/2}}\\ =\frac{{\mathrm{\Phi }}_{12}/{{\mathrm{\Phi }}_{11}{\mathrm{\Phi }}_{22}}^{1/2}-{\mathrm{\Phi }}_{13}/{{\mathrm{\Phi }}_{11}{\mathrm{\Phi }}_{33}}^{1/2}{\mathrm{\Phi }}_{23}/{{\mathrm{\Phi }}_{22}{\mathrm{\Phi }}_{33}}^{1/2}}{{{\mathrm{\Phi }}_{11}/{\mathrm{\Phi }}_{11}-{\mathrm{\Phi }}_{13}{\mathrm{\Phi }}_{31}/{\mathrm{\Phi }}_{11}{\mathrm{\Phi }}_{33}{\mathrm{\Phi }}_{22}/{\mathrm{\Phi }}_{22}-{\mathrm{\Phi }}_{23}{\mathrm{\Phi }}_{32}/{\mathrm{\Phi }}_{22}{\mathrm{\Phi }}_{33}}^{1/2}}\\ =\frac{{\mathrm{\Phi }}_{12}/{{\mathrm{\Phi }}_{11}{\mathrm{\Phi }}_{22}}^{1/2}-{\mathrm{\Phi }}_{13}/{{\mathrm{\Phi }}_{11}{\mathrm{\Phi }}_{33}}^{1/2}{\mathrm{\Phi }}_{23}/{{\mathrm{\Phi }}_{22}{\mathrm{\Phi }}_{33}}^{1/2}}{{{\mathrm{\Phi }}_{11}/{\mathrm{\Phi }}_{11}-{\mathrm{\Phi }}_{13}^2/{{\mathrm{\Phi }}_{11}^2{\mathrm{\Phi }}_{33}^2}^{1/2}{\mathrm{\Phi }}_{22}/{\mathrm{\Phi }}_{22}-{\mathrm{\Phi }}_{23}^2/{{\mathrm{\Phi }}_{22}^2{\mathrm{\Phi }}_{33}^2}^{1/2}}^{1/2}}\\ =\frac{{\tilde{\partial }}_{12}-{\tilde{\partial }}_{13}{\tilde{\partial }}_{23}}{{1-{\tilde{\partial }}_{13}^{2}1-{\tilde{\partial }}_{23}^2}^{1/2}}\text{,}\end{array}$$which is equivalently written as follows:$$\begin{array}{}\text{(A.12)}& \tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2{\mathfrak{R}}_3=\frac{\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_2-\tilde{\partial }{\mathfrak{R}}_1,{\mathfrak{R}}_3\tilde{\partial }{\mathfrak{R}}_2,{\mathfrak{R}}_3}{1-{\tilde{\partial }}^2{\mathfrak{R}}_1,{\mathfrak{R}}_{3}1-{\tilde{\partial }}^2{\mathfrak{R}}_2,{\mathfrak{R}}_3{}^{1/2}}\text{.}\end{array}$$

The result follows.

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