Table 1 Extended n-ary similarity indices

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

\({s}_{eSS2\left(1s\_wd\right)}=\frac{2\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\)

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

\({s}_{eJT\left(s\_d\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\)