Table 1 Extended n-ary similarity indices
| Additive indices | ||||
|---|---|---|---|---|
| Label | Type | Notation | Name | Equation |
| eAC | eAC_1 | eACw |
extended Austin-Colwell | \({s}_{eAC\left(1s\_wd\right)}=\frac{2}{\pi }\text{arcsin}\sqrt{\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}}\) |
| eACnw | \({s}_{eAC\left(1s\_d\right)}=\frac{2}{\pi }\text{arcsin}\sqrt{\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}}\) | |||
| eBUB | eBUB_1 | eBUBw |
extended Baroni-Urbani-Buser | \({s}_{eBUB\left(1s\_wd\right)}=\frac{\sqrt{\left[\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right]\left[\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right]}+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\left\{\begin{array}{c}\sqrt{\left[\sum_{1-s}{f}_{s}\left({\Delta }_{n\left(k\right)}\right){C}_{n\left(k\right)}\right]\left[\sum_{0-s}{f}_{s}\left({\Delta }_{n\left(k\right)}\right){C}_{n\left(k\right)}\right]}+\\ \sum_{1-s}{f}_{s}\left({\Delta }_{n\left(k\right)}\right){C}_{n\left(k\right)}+\sum_{d}{f}_{d}\left({\Delta }_{n\left(k\right)}\right){C}_{n\left(k\right)}\end{array}\right\}}\) |
| eBUBnw | \({s}_{eBUB\left(1s\_d\right)}=\frac{\sqrt{\left[\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right]\left[\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right]}+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\left\{\sqrt{\left[\sum_{1-s}{C}_{n\left(k\right)}\right]\left[\sum_{0-s}{C}_{n\left(k\right)}\right]}+\sum_{1-s}{C}_{n\left(k\right)}+\sum_{d}{C}_{n\left(k\right)}\right\}}\) | |||
| eCT1 | eCT1_1 | eCT1w |
extended Consoni-Todeschini (1) | \({s}_{eCT1\left(1s\_wd\right)}=\frac{\text{ln}\left(1+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\text{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}\) |
| eCT1nw | \({s}_{eCT1\left(1s\_d\right)}=\frac{\text{ln}\left(1+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\text{ln}\left(1+\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}\right)}\) | |||
| eCT2 | eCT2_1 | eCT2w |
extended Consoni-Todeschini (2) | \({s}_{eCT2\left(1s\_wd\right)}=\frac{\text{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)-\text{ln}\left(1+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\text{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}\) |
| eCT2nw | \({s}_{eCT2\left(1s\_d\right)}=\frac{\text{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)-\text{ln}\left(1+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\text{ln}\left(1+\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}\right)}\) | |||
| eFai | eFai_1 | eFaiw |
extended Faith | \({s}_{eFai\left(1s\_wd\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+0.5\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) |
| eFainw | \({s}_{eFai\left(1s\_d\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+0.5\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\) | |||
| eGK | eGK_1 | eGKw |
extended Goodman–Kruskal | \({s}_{eGK\left(1s\_wd\right)}=\frac{2\text{min}\left(\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)},\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)-\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\text{min}\left(\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)},\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) |
| eGKnw | \({s}_{eGK\left(1s\_d\right)}=\frac{2\text{min}\left(\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)},\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)-\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\text{min}\left(\sum_{1-s}{C}_{n(k)},\sum_{0-s}{C}_{n(k)}\right)+\sum_{d}{C}_{n(k)}}\) | |||
| eHD | eHD_1 | eHDw |
extended Hawkins-Dotson | \({s}_{eHD\left(1s\_wd\right)}=\frac{1}{2}\left(\begin{array}{c}\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}+\\ \frac{\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\end{array}\right)\) |
| eHDnw | \({s}_{eHD\left(1s\_d\right)}=\frac{1}{2}\left(\begin{array}{c}\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{1-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}+\\ \frac{\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{0-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\end{array}\right)\) | |||
| eRT | eRT_1 | eRTw |
extended Rogers-Tanimoto | \({s}_{eRT\left(1s\_wd\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+2\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) |
| eRTnw | \({s}_{eRT\left(1s\_d\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+2\sum_{d}{C}_{n(k)}}\) | |||
| eRG | eRG_1 | eRGw |
extended Rogot-Goldberg | \({s}_{eRG\left(1s\_wd\right)}=\begin{array}{c}\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}+\\ \frac{\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\end{array}\) |
| eRGnw | \({s}_{eRG\left(1s\_d\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{1-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}+\frac{\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{0-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\) | |||
| eSM | eSM_1 | eSMw |
extended Simple matching, Sokal-Michener | \({s}_{eSM\left(1s\_wd\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) |
| eSMnw | \({s}_{eSM\left(1s\_d\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\) | |||
| eSS2 | eSS2_1 | eSS2w |
extended Sokal-Sneath (2) | \({s}_{eSS2\left(1s\_wd\right)}=\frac{2\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) |
| eSS2nw | \({s}_{eSS2\left(1s\_wd\right)}=\frac{2\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\) | |||
| Asymmetric indices | ||||
|---|---|---|---|---|
| Label | Type | Notation | Name | Equation |
| eCT3 | eCT3_1 | eCT3w |
extended Consoni-Todeschini (3) | \({s}_{eCT3\left(1s\_wd\right)}=\frac{\text{ln}\left(1+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\text{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}\) |
| eCT3nw | \({s}_{eCT3\left(1s\_d\right)}=\frac{\text{ln}\left(1+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\text{ln}\left(1+\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}\right)}\) | |||
| eCT3_0 | eCT30w | \({s}_{eCT3\left(s\_wd\right)}=\frac{\text{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\text{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}\) | ||
| eCT30nw | \({s}_{eCT3\left(s\_d\right)}=\frac{\text{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\text{ln}\left(1+\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}\right)}\) | |||
| eCT4 | eCT4_1 | eCT4w |
extended Consoni-Todeschini (4) | \({s}_{eCT4\left(1s\_wd\right)}=\frac{\text{ln}\left(1+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\text{ln}\left(1+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}\) |
| eCT4nw | \({s}_{eCT4\left(1s\_d\right)}=\frac{\text{ln}\left(1+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\text{ln}\left(1+\sum_{1-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}\right)}\) | |||
| eCT4_0 | eCT40w | \({s}_{eCT4\left(s\_wd\right)}=\frac{\text{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\text{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}\) | ||
| eCT4nw | \({s}_{eCT4\left(s\_d\right)}=\frac{\text{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\text{ln}\left(1+\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}\right)}\) | |||
| eGle | eGle_1 | eGlew |
extended Gleason | \({s}_{eGle\left(1s\_wd\right)}=\frac{2\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) |
| eGlenw | \({s}_{eGle\left(1s\_d\right)}=\frac{2\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{1-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\) | |||
| eGle_0 | eGle0w | \({s}_{eGle\left(s\_wd\right)}=\frac{2\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) | ||
| eGle0nw | \({s}_{eGle\left(s\_d\right)}=\frac{2\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\) | |||
| eJa | eJa_1 | eJaw |
extended Jaccard | \({s}_{eJa\left(1s\_wd\right)}=\frac{3\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{3\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) |
| eJanw | \({s}_{eJa\left(1s\_d\right)}=\frac{3\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{3\sum_{1-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\) | |||
| eJa_0 | eJa0w | \({s}_{eJa\left(s\_wd\right)}=\frac{3\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{3\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) | ||
| eJa0nw | \({s}_{eJa\left(s\_d\right)}=\frac{3\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{3\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\) | |||
| eRR | eRR_1 | eRRw |
extended Russel-Rao | \({s}_{eRR\left(1s\_wd\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) |
| eRRnw | \({s}_{eRR\left(1s\_d\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\) | |||
| eRR_0 | eRR0w | \({s}_{eRR\left(s\_wd\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) | ||
| eRR0nw | \({s}_{eRR\left(s\_d\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\) | |||
| eSS1 | eSS1_0 | eSSw |
extended Sokal-Sneath (1) | \({s}_{eSS1\left(1s\_wd\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+2\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) |
| eSSnw | \({s}_{eSS1\left(1s\_d\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{1-s}{C}_{n(k)}+2\sum_{d}{C}_{n(k)}}\) | |||
| eSS1_1 | eSS0w | \({s}_{eSS1\left(s\_wd\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+2\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) | ||
| eSS0nw | \({s}_{eSS1\left(s\_d\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+2\sum_{d}{C}_{n(k)}}\) | |||
| eJT | eJT_1 | eJTw |
extended Jaccard-Tanimoto | \({s}_{eJT\left(1s\_wd\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) |
| eJTnw | \({s}_{eJT\left(1s\_d\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{1-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\) | |||
| eJT_0 | eJT0w | \({s}_{eJT\left(s\_wd\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\) | ||
| eJT0nw | \({s}_{eJT\left(s\_d\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\) | |||