4. Correlation module

4.1. Spearman correlation coefficient

The Spearman coefficient for the rank values $rgX$ and $rgY$ can be represented by the formula (ref{equ:spearm}), where the vectors $rgX$ and $rgY$ must have the same number of rank values. When rankings have unique preference values that do not repeat, and all variants have different rankings, the Equation (4.2) can be used to calculate the Spearman coefficient.

(4.1)\[\begin{equation} r_{s}=\frac{\operatorname{cov}\left(r g_{X}, r g_{Y}\right)}{\sigma_{r g_{X}} \sigma_{r g_{Y}}} \end{equation}\]

(4.2)\[\begin{equation} r_{s}=1-\frac{6 \cdot \sum_{i=1}^{N}\left(r g_{X_{i}}-r g_{Y_{i}}\right)}{N\left(N^{2}-1\right)} \end{equation}\]

where $N$ denotes the number of rank.

4.2. Weighted Spearman correlation coefficient

The weighted Spearman correlation coefficient ($r_w$) is being used to make a comparison between two rankings. This coefficient has the benefit of considering the highest ranked alternative as the most important. It can be shown by the Equation (4.3).

(4.3)\[\begin{equation} r_{w} = 1 - \frac{6\sum_{i=1}^{N}(x_{i} - y_{i})^{2} ((N - x_{i} + 1) + (N - y_{i} + 1))}{N^{4} + N^{3} - N^{2} - N} \end{equation}\]

where $x$ and $y$ are the rank vectors, and $N$ is their dimension.

4.3. Kendall rank correlation coefficient

The Kendall rank correlation coefficient is used to evaluate two rank vectors, determining the strength of their relationship. Kendall’s rank correlation coefficient values are in the range $[-1, 1]$. An increase in the value of the coefficient indicates an increase in the ranks of both variables, where the value of the coefficient decreases. This means that the rank of one variable is increasing and the rank of the other variable is decreasing. Kendall’s rank correlation coefficient can be represented by the Equation (4.4).

(4.4)\[\begin{equation} \tau_{b} = \frac{ \sum\limits^{N}_{i=1} \sum\limits^{N}_{j=1} x_{ij} y_{ij} }{ \sqrt{ \sum\limits^{N}_{i=1} \sum\limits^{N}_{j=1} x_{ij}^{2} \sum\limits^{N}_{i=1} \sum\limits^{N}_{j=1} y_{ij}^{2} } } \end{equation}\]

where $N$ denotes the number of ranks, $x_{ij}$ and $y_{ij}$ denote the values of the respective rankings $x$ and $y$.

4.4. Ranking similarity coefficient

The ranking similarity coefficient $WS$ is an asymmetric measure created to assess the similarity of the rankings of two vectors $x$ and $y$. The weight of the comparison is determined according to the relevance of the $x$ ranking position. The higher the rank, the higher the significance for the coefficient $WS$. The values of the ranking similarity coefficient are in the range $[0, 1]$ and can be represented by the Equation (4.5).

(4.5)\[\begin{equation} W S=1-\sum_{i=1}^{N} 2^{-x_{i}} \frac{\left|x_{i}-y_{i}\right|}{\max \left(\left|x_{i}-1\right|,\left|x_{i}-N\right|\right)} \end{equation}\]

where $N$ denotes the number of ranks, $x_i$ and $y_i$ denote the $i-th$ ranks of the $x$ and $y$ vectors.

4.5. Pearson’s correlation coefficient

The Pearson correlation coefficient compares two data sets using covariance and standard deviation. Its value ranges from $-$1 to 1. The smaller the Pearson correlation coefficient value, the less correlation between the data, while the more significant the value, the greater the correlation. Equation (4.6) can represent it.

(4.6)\[\begin{equation} r(x, y)=\frac{\sum_{i=1}^{N}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{\sqrt{\sum_{i=1}^{N}\left(x_{i}-\bar{x}\right)^{2}} \sqrt{\sum_{i=1}^{N}\left(y_{i}-\bar{y}\right)^{2}}} \end{equation}\]

where $N$ is the number of samples and $x$ and $y$ are vectors of values.

4.6. Goodman-Kruskal correlation coefficient

The Goodman-Kruskal Gamma Correlation Coefficient is a measure of rank correlation that measures the strength of association from cross-tabulations. This measure is applied to ordinal variables that are either continuous variables or discrete variables. The values of this measure are in the range [-1,1] and can be represented as follows:

\[\begin{equation} G=\frac{N_{s}-N_{d}}{N_{s}+N_{d}} \end{equation}\]

where $N_s$ is the number of compatible pairs and $N_d$ is the number of non-compliant pairs.

4.7. Weighted Similarity Coefficient

Weighted Similarity Coefficient was created because of the difficulty involved in determining the similarity of two criterion weight vectors. For this purpose, the knowledge that the sum of the weights should be equal to one was used, providing a normalized version of this equation. In addition, it was based on the Manhattan distance metric and can be represented as follows:

\[\begin{equation} \label{eq:wsc} WSC = 1 - \frac{d_1(\mathbf{w},\mathbf{v})}{2 \cdot (1 - min(\mathbf{w}))} = 1 - \frac{\sum_{i=1}^N |w_i - v_i|}{2 \cdot (1 - {min}_{i} w_i)} \end{equation}\]

where $w_i$ and $v_i$ are the criterion weights.

However, if we deal with the decision problems with a small number of criteria, such as 2, 3, and 4, it can be observed that the differences between weights values are naturally bigger in this case. That means that the possibility of achieving maximum distance 2 is different for weight vectors of different lengths. Therefore, the better way to normalize the distance is based on the minimum value on one of the weight vectors (4.7).

(4.7)\[\begin{equation} \label{eq:wsc2} \textit{WSC}_2 = 1 - \frac{d_1(\mathbf{w},\mathbf{v})}{2} = 1 - \frac{\sum_{i=1}^N |w_i - v_i|}{2} \end{equation}\]