1. Introduction
Garg and Kumar [1] discussed a brief review of the existing
correlation coefficient [2-11]. Furthermore, Garg and Kumar [1,
Section 4, Eqn. 13, pp. 8] used the existing expression\(\left(1\right)\) [12] to evaluate the correlation coefficient of
three known patterns,\(A_{1}=\left\{\left\langle x_{1},0.4,0.5\right\rangle,\ \left\langle x_{2},0.7,0.1\right\rangle,\ \left\langle x_{3},0.3,0.3\right\rangle\right\}\),\(A_{2}=\left\{\left\langle x_{1},0.5,0.4\right\rangle,\ \left\langle x_{2},0.7,0.2\right\rangle,\ \left\langle x_{3},0.4,0.3\right\rangle\right\}\)and\(A_{3}=\left\{\left\langle x_{1},0.4,0.5\right\rangle,\ \left\langle x_{2},0.7,0.1\right\rangle,\ \left\langle x_{3},0.4,0.3\right\rangle\right\}\)with the unknown pattern,\(B=\left\{\left\langle x_{1},0.1,0.1\right\rangle,\ \left\langle x_{2},1.0,0.0\right\rangle,\ \left\langle x_{3},0.0,1.0\right\rangle\right\}\)and shown that the obtained correlation coefficient between \(A_{1}\)and \(B\), the obtained correlation coefficient between \(A_{2}\) and\(B\) as well as the obtained correlation coefficient between \(A_{3}\)and \(B\) are equal. Therefore, expression \(\left(1\right)\) [12]
cannot be used to identify a suitable pattern, for the unknown pattern\(B\), from the known patterns \(A_{1},\ A_{2}\), and \(A_{3}\).
\(\text{ZL}\left(A,B\right)=\frac{\sum_{t=1}^{n}\left(u_{A}\left(x_{t}\right)u_{B}\left(x_{t}\right)+v_{A}\left(x_{t}\right)v_{B}\left(x_{t}\right){+\pi}_{A}\left(x_{t}\right)\pi_{B}\left(x_{t}\right)\right)}{\sqrt{\sum_{t=1}^{n}{\left(u_{A}^{2}\left(x_{t}\right)+v_{A}^{2}\left(x_{t}\right)+\pi_{A}^{2}\left(x_{t}\right)\right).}\sum_{t=1}^{n}\left(u_{B}^{2}\left(x_{t}\right)+v_{B}^{2}\left(x_{t}\right)+\pi_{B}^{2}\left(x_{t}\right)\right)}}\)\(\left(1\right)\)
Garg and Kumar [1, Section 4, Eqn. 14, pp. 9] used the existing
expression \(\left(2\right)\) [13] to evaluate the correlation
coefficient between the IFSs,\(A=\left\{\left\langle x_{1},0.1,0.2\right\rangle,\ \left\langle x_{2},0.2,0.1\right\rangle,\ \left\langle x_{3},0.29,0.0\right\rangle\right\}\)and\(B=\left\{\left\langle x_{1},0.1,0.3\right\rangle,\ \left\langle x_{2},0.2,0.2\right\rangle,\ \left\langle x_{3},0.29,0.1\right\rangle\right\}\)and shown that the obtained correlation coefficient between \(A\) and\(B\) is \(1\), which is mathematically incorrect as it indicates that
the IFSs \(A\) and \(B\) are equal. Whereas, it is obvious that both the
IFSs are not equal. Therefore, the existing expression\(\left(2\right)\) [13] cannot be used to obtain correlation
coefficient between \(A\) and \(B\).
\(r\left(A,B\right)=\frac{r_{1}\left(A,B\right)+r_{2}\left(A,B\right)+r_{3}\left(A,B\right)}{3}\)\(\left(2\right)\)
where,
\(r_{1}\left(A,B\right)=\frac{\sum_{t=1}^{n}{\left(u_{A}\left(x_{t}\right)-\overline{u_{A}}\right)\left(u_{B}\left(x_{t}\right)-\overline{u_{B}}\right)}}{\sqrt{\sum_{t=1}^{n}{\left(u_{A}\left(x_{t}\right)-\overline{u_{A}}\right)^{2}.\sum_{t=1}^{n}\left(u_{B}\left(x_{t}\right)-\overline{u_{B}}\right)^{2}}}}\),
\(r_{2}\left(A,B\right)=\frac{\sum_{t=1}^{n}{\left(v_{A}\left(x_{t}\right)-\overline{v_{A}}\right)\left(v_{B}\left(x_{t}\right)-\overline{v_{B}}\right)}}{\sqrt{\sum_{t=1}^{n}{\left(v_{A}\left(x_{t}\right)-\overline{v_{A}}\right)^{2}.\sum_{t=1}^{n}\left(v_{B}\left(x_{t}\right)-\overline{v_{B}}\right)^{2}}}}\),
\(r_{3}\left(A,B\right)=\frac{\sum_{t=1}^{n}{\left(\pi_{A}\left(x_{t}\right)-\overline{\pi_{A}}\right)\left(\pi_{B}\left(x_{t}\right)-\overline{\pi_{B}}\right)}}{\sqrt{\sum_{t=1}^{n}{\left(\pi_{A}\left(x_{t}\right)-\overline{\pi_{A}}\right)^{2}.\sum_{t=1}^{n}\left(\pi_{B}\left(x_{t}\right)-\overline{\pi_{B}}\right)^{2}}}}\),
\(\overline{u_{A}}=\frac{1}{n}\sum_{t=1}^{n}{u_{A}\left(x_{t}\right)}\),\(\overline{u_{B}}=\frac{1}{n}\sum_{t=1}^{n}{u_{B}\left(x_{t}\right)}\),
\(\overline{v_{A}}=\frac{1}{n}\sum_{t=1}^{n}{v_{A}\left(x_{t}\right)}\),\(\overline{v_{B}}=\frac{1}{n}\sum_{t=1}^{n}{v_{B}\left(x_{t}\right)}\),
\(\overline{\pi_{A}}=\frac{1}{n}\sum_{t=1}^{n}{\pi_{A}\left(x_{t}\right)}\),\(\overline{\pi_{B}}=\frac{1}{n}\sum_{t=1}^{n}{\pi_{B}\left(x_{t}\right)}\).
Garg and Kumar [1, Section 4, Eqn. 15, pp. 10] used the existing
expression \(\left(3\right)\) [14] to evaluate the correlation
coefficient between the three known patterns,\(A_{1}=\left\{\left\langle x_{1},0.4,0.5\right\rangle,\ \left\langle x_{2},0.7,0.1\right\rangle,\ \left\langle x_{3},0.3,0.3\right\rangle\right\}\),\(A_{2}=\left\{\left\langle x_{1},0.5,0.4\right\rangle,\ \left\langle x_{2},0.7,0.2\right\rangle,\ \left\langle x_{3},0.4,0.3\right\rangle\right\}\)and\(A_{3}=\left\{\left\langle x_{1},0.4,0.5\right\rangle,\ \left\langle x_{2},0.7,0.1\right\rangle,\ \left\langle x_{3},0.4,0.3\right\rangle\right\}\)with the unknown pattern,\(B=\left\{\left\langle x_{1},0.1,0.1\right\rangle,\ \left\langle x_{2},1.0,0.0\right\rangle,\ \left\langle x_{3},0.0,1.0\right\rangle\right\}\)and shown that the obtained correlation coefficient between \(A_{1}\)and \(B\), the obtained correlation coefficient between \(A_{2}\) and\(B\) as well as the obtained correlation coefficient between \(A_{3}\)and \(B\) are equal. Therefore, expression \(\left(3\right)\) [14]
cannot be used to identify a suitable pattern, for the unknown pattern\(B\), from the known patterns \(A_{1},\ A_{2}\) and \(A_{3}\).
\(Xu_{1}\left(A,B\right)=\frac{\sum_{t=1}^{n}\left(u_{A}\left(x_{t}\right)u_{B}\left(x_{t}\right)+v_{A}\left(x_{t}\right)v_{B}\left(x_{t}\right){+\pi}_{A}\left(x_{t}\right)\pi_{B}\left(x_{t}\right)\right)}{\max\left\{\sum_{t=1}^{n}{\left(u_{A}^{2}\left(x_{t}\right)+v_{A}^{2}\left(x_{t}\right)+\pi_{A}^{2}\left(x_{t}\right)\right)\text{.\ \ }}\sum_{t=1}^{n}\left(u_{B}^{2}\left(x_{t}\right)+v_{B}^{2}\left(x_{t}\right)+\pi_{B}^{2}\left(x_{t}\right)\right)\right\}}\)\(\left(3\right)\)
To overcome this limitation of the existing expressions\(\left(1\right)-\left(3\right)\), Garg and Kumar [1],
firstly, proposed the expression \(\left(4\right)\) [1, Section 3,
Eqn. 6, pp. 4] to transform an intuitionistic fuzzy number (IFN)\(\left\langle\mu_{p}\left(x_{t}\right),\nu_{p}\left(x_{t}\right)\right\rangle\)into a CN\(a_{p}\left(x_{t}\right)+b_{p}\left(x_{t}\right)i+c_{p}\left(x_{t}\right)j\)[15]. Then, using the proposed expression \(\left(4\right)\), Garg
and Kumar [1] proposed expression \(\left(5\right)\) [1, Section
3, Eqn. 9, pp. 5], expression \(\left(6\right)\) [1, Section 3,
Eqn. 10, pp. 7] for evaluating the correlation coefficient and
expression \(\left(7\right)\) [1, Section 3, Eqn. 11, pp. 7],
expression \(\left(8\right)\) [1, Section 3, Eqn. 12, pp. 7], for
evaluating the weighted correlation coefficient between two IFSs\(A=\left\{\left\langle\mu_{1}\left(x_{t}\right),\nu_{1}\left(x_{t}\right)\right\rangle\ \right\}\)and\(B=\left\{\left\langle\mu_{2}\left(x_{t}\right),\nu_{2}\left(x_{t}\right)\right\rangle,\ \right\}\ \ t=1,2,\ldots,n\).
\(a_{p}\left(x_{t}\right)=\mu_{p}\left(x_{t}\right)\left(1-\nu_{p}\left(x_{t}\right)\right)\),\(b_{p}\left(x_{t}\right)=1-\mu_{p}\left(x_{t}\right)\left(1-\nu_{p}\left(x_{t}\right)\right)-\nu_{p}\left(x_{t}\right)\left(1-\mu_{p}\left(x_{t}\right)\right)\)and\(c_{p}\left(x_{t}\right)=\nu_{p}\left(x_{t}\right)\left(1-\mu_{p}\left(x_{t}\right)\right)\)\(\left(4\right)\)
\(K_{1}\left(A,B\right)=\frac{\sum_{t=1}^{n}\left(a_{1}\left(x_{t}\right)a_{2}\left(x_{t}\right)+b_{1}\left(x_{t}\right)b_{2}\left(x_{t}\right){+c}_{1}\left(x_{t}\right)c_{2}\left(x_{t}\right)\right)}{\sqrt{\sum_{t=1}^{n}{\left(a_{1}^{2}\left(x_{t}\right)+b_{1}^{2}\left(x_{t}\right)+c_{1}^{2}\left(x_{t}\right)\right).}\ \sum_{t=1}^{n}\left(a_{2}^{2}\left(x_{t}\right)+b_{2}^{2}\left(x_{t}\right)+c_{2}^{2}\left(x_{t}\right)\right)}}\)\(\left(5\right)\)
\(K_{2}\left(A,B\right)=\frac{\sum_{t=1}^{n}\left(a_{1}\left(x_{t}\right)a_{2}\left(x_{t}\right)+b_{1}\left(x_{t}\right)b_{2}\left(x_{t}\right){+c}_{1}\left(x_{t}\right)c_{2}\left(x_{t}\right)\right)}{\max\left\{\sum_{t=1}^{n}{\left(a_{1}^{2}\left(x_{t}\right)+b_{1}^{2}\left(x_{t}\right)+c_{1}^{2}\left(x_{t}\right)\right),}\ \sum_{t=1}^{n}\left(a_{2}^{2}\left(x_{t}\right)+b_{2}^{2}\left(x_{t}\right)+c_{2}^{2}\left(x_{t}\right)\right)\right\}}\)\(\left(6\right)\)
\(K_{3}\left(A,B\right)=\frac{\sum_{t=1}^{n}{w_{t}\left(a_{1}\left(x_{t}\right)a_{2}\left(x_{t}\right)+b_{1}\left(x_{t}\right)b_{2}\left(x_{t}\right){+c}_{1}\left(x_{t}\right)c_{2}\left(x_{t}\right)\right)}}{\sqrt{\sum_{t=1}^{n}{w_{t}\left(a_{1}^{2}\left(x_{t}\right)+b_{1}^{2}\left(x_{t}\right)+c_{1}^{2}\left(x_{t}\right)\right)\text{.\ \ }}\sum_{t=1}^{n}{w_{t}\left(a_{2}^{2}\left(x_{t}\right)+b_{2}^{2}\left(x_{t}\right)+c_{2}^{2}\left(x_{t}\right)\right)}}}\)\(\left(7\right)\)
\(K_{4}\left(A,B\right)=\frac{\sum_{t=1}^{n}{w_{t}\left(a_{1}\left(x_{t}\right)a_{2}\left(x_{t}\right)+b_{1}\left(x_{t}\right)b_{2}\left(x_{t}\right){+c}_{1}\left(x_{t}\right)c_{2}\left(x_{t}\right)\right)}}{\max\left\{\sum_{t=1}^{n}{w_{t}\left(a_{1}^{2}\left(x_{t}\right)+b_{1}^{2}\left(x_{t}\right)+c_{1}^{2}\left(x_{t}\right)\right),\ \ \ }\sum_{t=1}^{n}{w_{t}\left(a_{2}^{2}\left(x_{t}\right)+b_{2}^{2}\left(x_{t}\right)+c_{2}^{2}\left(x_{t}\right)\right)}\right\}}\)\(\left(8\right)\)
In this note, it is shown that the correlation coefficients\(\left(5\right)-\left(8\right)\), proposed by Garg and Kumar
[1], also fails to identify a suitable classifier. Furthermore, it
is shown that more computational efforts are required to apply the
correlation coefficients \(\left(5\right)-\left(8\right)\),
proposed by Garg and Kumar [1], as compared to the existing
correlation coefficients \(\left(1\right)-\left(3\right)\). In the
actual case, it is inappropriate to apply the correlation coefficient
for identifying a suitable classifier.