. 2020 May 12;33(4):971–987. doi: 10.1007/s10278-020-00338-w

Table 2.

Mathematical formulation of different distance metrics

Metric	Mathematical representation
Euclidean	$D_{v} (X_{1}, X_{2}) = \sqrt{\sum_{i = 1}^{N} {(X_{1_{i}} - X_{2_{i}})}^{2}}$
City block	$D_{v} (X_{1}, X_{2}) = \sum_{i = 1}^{N} \| X_{1_{i}} - X_{2_{i}} \|$
Cosine	$D_{v} (X_{1}, X_{2}) = 1 - \frac{\sum_{i = 1}^{N} X_{1_{i}} . X_{2_{i}}}{\sqrt{\sum_{i = 1}^{N} (X_{1_{i}}^{2}} \sqrt{\sum_{i = 1}^{N} (X_{2_{i}}^{2}}}$
Spearman	$D_{v} (X_{1}, X_{2}) = 1 - \frac{(r_{1} - \bar{r_{1}}) {(r_{2} - \bar{r_{2}})}^{'}}{\sqrt{(r_{1} - \bar{r_{1}}) {(r_{1} - \bar{r_{1}})}^{'}} \sqrt{(r_{2} - \bar{r_{2}}) {(r_{2} - \bar{r_{2}})}^{'}}}$
	$\bar{r_{1}} = \frac{1}{N} \sum_{i = 1}^{N} r_{1_{i}}, \bar{r_{2}} = \frac{1}{N} \sum_{i = 1}^{N} r_{2_{i}}$
Correlation	$D_{v} (X_{1}, X_{2}) = 1 - \frac{\sum_{i = 1}^{N} (X_{1_{i}} - \bar{X_{1}}) (X_{2_{i}} - \bar{X_{2}})}{\sqrt{\sum_{i = 1}^{N} {(X_{1_{i}} - \bar{X_{1}})}^{2}} \sqrt{\sum_{i = 1}^{N} {(X_{2_{i}} - \bar{X_{2}})}^{2}}}$
	$\bar{X_{1}} = \frac{1}{N} \sum_{i = 1}^{N} X_{1_{i}}, \bar{X_{2}} = \frac{1}{N} \sum_{i = 1}^{N} X_{2_{i}}$