2. Counter examples for Garg and Kumar‘s the correlation
coefficient
As discussed in Section 1, Garg and Kumar [1] have shown that the
existing correlation coefficients\(\left(1\right)-\left(3\right)\) fails to identify a suitable
classifier for the unknown pattern\(B=\left\{\left\langle x_{1},0.1,0.1\right\rangle,\ \left\langle x_{2},1.0,0.0\right\rangle,\ \left\langle x_{3},0.0,1.0\right\rangle\right\}\)from 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\(A_{3}=\left\{\left\langle x_{1},0.4,0.5\right\rangle,\ \left\langle x_{2},0.7,0.1\right\rangle,\ \left\langle x_{3},0.4,0.3\right\rangle\right\}\).
Therefore, it is inappropriate to use the existing correlation
coefficients \(\left(1\right)-\left(3\right)\).
On the same direction, in this section two known patterns\(A_{1}=\left\{\left\langle x_{1},0.1,0.4\right\rangle,\ \left\langle x_{2},0.4,0.3\right\rangle,\ \left\langle x_{3},0.25,0.35\right\rangle\right\}\),\(A_{2}=\left\{\left\langle x_{1},0.4,0.1\right\rangle,\ \left\langle x_{2},0.3,0.4\right\rangle,\ \left\langle x_{3},0.35,0.25\right\rangle\right\}\)and an unknown pattern\(B=\left\{\left\langle x_{1},0.3,0.3\right\rangle,\ \left\langle x_{2},0.2,0.2\right\rangle,\ \left\langle x_{3},0.1,0.1\right\rangle\right\}\),
represented by IFSs, are considered. Also, the weights of a relative
importance are considered as \(\left(0.40,\ 0.45,0.15\right)\), and
shown that the correlation coefficients\(\left(5\right)-\left(8\right)\), proposed by Garg and Kumar
[1], also fails to identify that either \(A_{1}\) or \(A_{2}\) is a
suitable classifier for the unknown pattern \(B\).
To apply the correlation coefficients\(\left(5\right)-\left(8\right)\) [1], proposed by Garg and
Kumar [1], firstly, there is a need to transform each element of\(A_{1},A_{2}\), and \(B\) into a CN.
Using the expression \(\left(4\right)\), proposed by Garg and Kumar
[1] for transforming an IFN into a CN, the elements\(\left\langle 0.1,0.4\right\rangle,\)\(\left\langle 0.4,0.3\right\rangle\),\(\left\langle 0.25,0.35\right\rangle\),\(\left\langle 0.4,0.1\right\rangle,\)\(\left\langle 0.3,0.4\right\rangle,\)\(\left\langle 0.35,0.25\right\rangle,\)\(\left\langle 0.3,0.3\right\rangle\),\(\left\langle 0.2,0.2\right\rangle\) and\(\left\langle 0.1,0.1\right\rangle\) can be transformed into its
equivalent CNs \(\left\langle 0.06,0.58,0.36\right\rangle\),\(\left\langle 0.28,0.54,0.18\right\rangle\),\(\left\langle 0.1625,0.575,0.2625\right\rangle\),\(\left\langle 0.36,0.58,0.06\right\rangle\),\(\left\langle 0.18,0.54,0.28\right\rangle\),\(\left\langle 0.2625,0.575,0.1625\right\rangle\),\(\left\langle 0.21,0.58,0.21\right\rangle\),\(\left\langle 0.16,0.68,0.16\right\rangle\) and\(\left\langle 0.09,0.82,0.09\right\rangle\) respectively. Therefore,\(A_{1}\), \(A_{2}\) and \(B\) in terms of CNs can be rewritten as
\(A_{1}=\left\{\left\langle x_{1},0.06,0.58,0.36\right\rangle,\ \left\langle x_{2},0.28,0.54,0.18\right\rangle,\ \left\langle x_{3},0.1625,0.575,0.2625\right\rangle\right\}\),
\(A_{2}=\left\{\left\langle x_{1},0.36,0.58,0.06\right\rangle,\ \left\langle x_{2},0.18,0.54,0.28\right\rangle,\ \left\langle x_{3},0.2625,0.575,0.1625\right\rangle\right\}\),
and,
\(B=\left\{\left\langle x_{1},0.21,0.58,0.21\right\rangle,\ \left\langle x_{2},0.16,0.68,0.16\right\rangle,\ \left\langle x_{3},0.09,0.82,0.09\right\rangle\right\}\).
Now,
- Using the existing expression \(\left(5\right)\), proposed by Garg
and Kumar [1] for evaluating the correlation coefficient between
IFSs,\(K_{1}\left(A_{1},B\right)=0.946359402\) and\(K_{1}\left(A_{2},B\right)=0.946359402\).
Since \(K_{1}\left(A_{1},B\right)=K_{1}\left(A_{2},\ B\right)\)so it is not possible to identify the suitable classifier for the
unknown pattern \(B\) from the known patterns \(A_{1}\) and \(A_{2}\).
Hence, the limitation pointed out by Garg and Kumar [1] in the
existing correlation coefficients\(\left(1\right)-\left(3\right)\), is also occurring in Garg and
Kumar’s expression \(\left(5\right)\) [1].
- Using the existing expression \(\left(6\right)\), proposed by Garg
and Kumar [1] for evaluating the correlation coefficient between
the IFS,
\(K_{2}\left(A_{1},B\right)=0.84530981\) and\(K_{2}\left(A_{2},B\right)=0.84530981.\)Since \(K_{2}\left(A_{1},B\right)=K_{2}\left(A_{2},B\right)\) so
it is not possible to identify the suitable classifier for the unknown
pattern \(B\) from the known patterns \(A_{1}\ \)and \(A_{2}\).
Hence, the limitation pointed out by Garg and Kumar [1] in the
existing correlation coefficients\(\left(1\right)-\left(3\right)\), is also occurring in Garg and
Kumar’s expression \(\left(6\right)\) [1].
Using the existing expression \(\left(7\right)\), proposed by Garg
and Kumar [1] for evaluating the correlation coefficient between
the IFSs,
\(K_{3}\left(A_{1},B\right)=0.951828261\)and\(\text{\ \ }K_{3}\left(A_{2},B\right)=0.951828261\).
Since \(K_{3}\left(A_{1},B\right)=K_{3}\left(A_{2},B\right)\) so
it is not possible to identify the suitable classifier for the unknown
pattern \(B\) from the known patterns \(A_{1}\) and \(A_{2}\).
Hence, the limitation pointed out by Garg and Kumar [1] in the
existing correlation coefficients\(\left(1\right)-\left(3\right)\), is also occurring in Garg and
Kumar’s expression \(\left(7\right)\) [1].
Using the existing expression \(\left(8\right)\), proposed by Garg
and Kumar [1] for evaluating the correlation coefficient between
the IFSs,
\(K_{4}\left(A_{1},B\right)=0.881829449\) and\(K_{4}\left(A_{2},B\right)=0.881829449\).
Since \(K_{4}\left(A_{1},B\right)=K_{4}\left(A_{2},B\right)\) so
it is not possible to identify the suitable classifier for the unknown
pattern \(B\) from the known patterns \(A_{1}\) and \(A_{2}\).
Hence, the limitation pointed out by Garg and Kumar [1] in the
existing correlation coefficients\(\left(1\right)-\left(3\right)\), is also occurring in Garg and
Kumar’s expression \(\left(8\right)\) [1].
3. Advantages of existing correlation coefficients over
Garg and Kumar’s correlation coefficients
It is obvious from Section \(1\) and Section \(2\) that although neither
the existing correlation coefficients\(\left(1\right)-\left(3\right)\) nor the correlation coefficients\(\left(5\right)-\left(8\right)\), proposed by Garg and Kumar
[1], can be used for identifying a suitable classifier. But, to
apply the correlation coefficients\(\left(5\right)-\left(8\right)\), proposed by Garg and Kumar
[1], there is a need to transform each element of the known
patterns, represented by an intuitionistic fuzzy number, into a CN.
While applying the existing correlation coefficients\(\left(1\right)-\left(3\right)\), no such transformation is
required i.e., more computational efforts are required for applying the
correlation coefficients \(\left(5\right)-\left(8\right)\),
proposed by Garg and Kumar [1], as compared to the existing
correlation coefficients \(\left(1\right)-\left(3\right)\).
Therefore, it is better to use the existing correlation coefficients\(\left(1\right)-\left(3\right)\) as compared to Garg and Kumar’s
correlation coefficients \(\left(5\right)-\left(8\right)\)[1].