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[![NPM version][npm-image]][npm-url] [](https://github.com/mljs/distance/actions/workflows/nodejs.yml) [![Test coverage][codecov-image]][codecov-url] [![npm download][download-image]][download-url] Distance functions to compare vectors. ## Installation `$ npm i ml-distance` This package is ESM-only. CommonJS consumers need Node.js >= 20.19, >= 22.12, or any 24.x or later to `require()` it, or should migrate to `import`. ## Usage ```js import { distance, similarity } from 'ml-distance'; const p = [0, 1, 2, 3, 4]; const q = [4, 3, 2, 1, 0]; distance.euclidean(p, q); // 6.324555320336759 distance.manhattan(p, q); // 12 similarity.cosine(p, q); // 0.3333333333333333 ``` ## Methods ### Distances - `euclidean(p, q)` Returns the [euclidean distance](http://en.wikipedia.org/wiki/Euclidean_distance#n_dimensions) between vectors p and q $d(p,q)=\sqrt{\sum\limits_{i=1}^{n}(p_i-q_i)^2}$ - `manhattan(p, q)` Returns the [city block distance](http://en.wikipedia.org/wiki/Taxicab_geometry) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{\left|p_i-q_i\right|}$ - `minkowski(p, q, d)` Returns the [Minkowski distance](http://en.wikipedia.org/wiki/Minkowski_distance) between vectors p and q for order d - `chebyshev(p, q)` Returns the [Chebyshev distance](http://en.wikipedia.org/wiki/Chebyshev_distance) between vectors p and q $d(p,q)=\max\limits_i(|p_i-q_i|)$ - `sorensen(p, q)` Returns the [Sørensen distance](http://en.wikipedia.org/wiki/S%C3%B8rensen%E2%80%93Dice_coefficient) between vectors p and q $d(p,q)=\frac{\sum\limits_{i=1}^{n}{\left|p_i-q_i\right|}}{\sum\limits_{i=1}^{n}{p_i+q_i}}$ - `gower(p, q)` Returns the [Gower distance](https://stat.ethz.ch/education/semesters/ss2012/ams/slides/v4.2.pdf) between vectors p and q $d(p,q)=\frac{\sum\limits_{i=1}^{n}{\left|p_i-q_i\right|}}{n}$ - `soergel(p, q)` Returns the [Soergel distance](http://www.orgchm.bas.bg/~vmonev/SimSearch.pdf) between vectors p and q $d(p,q)=\frac{\sum\limits_{i=1}^{n}{\left|p_i-q_i\right|}}{\sum\limits_{i=1}^{n}{max(p_i,q_i)}}$ - `kulczynski(p, q)` Returns the [Kulczynski distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\frac{\sum\limits_{i=1}^{n}{\left|p_i-q_i\right|}}{\sum\limits_{i=1}^{n}{min(p_i,q_i)}}$ - `canberra(p, q)` Returns the [Canberra distance](http://en.wikipedia.org/wiki/Canberra_distance) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}\frac{\left|{p_i-q_i}\right|}{p_i+q_i}$ - `lorentzian(p, q)` Returns the [Lorentzian distance](https://stat.ethz.ch/education/semesters/ss2012/ams/slides/v4.2.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}\ln(\left|{p_i-q_i}\right|+1)$ - `intersection(p, q)` Returns the [Intersection distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=1-\sum\limits_{i=1}^{n}min(p_i,q_i)$ - `waveHedges(p, q)` Returns the [Wave Hedges distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}\left(1-\frac{min(p_i,q_i)}{max(p_i,q_i)}\right)$ - `czekanowski(p, q)` Returns the [Czekanowski distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=1-\frac{2\sum\limits_{i=1}^{n}{min(p_i,q_i)}}{\sum\limits_{i=1}^{n}{p_i+q_i}}$ - `motyka(p, q)` Returns the [Motyka distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=1-\frac{\sum\limits_{i=1}^{n}{min(p_i,q_i)}}{\sum\limits_{i=1}^{n}{p_i+q_i}}$ Note: distance between 2 identical vectors is 0.5 ! - `tanimoto(p, q, [bitVector])` Returns the [Tanimoto distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q, and accepts the bitVector use, see the test case for an example - `innerProduct(p, q)` Returns the [Inner Product similarity](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $s(p,q)=\sum\limits_{i=1}^{n}{p_i\cdot{q_i}}$ - `harmonicMean(p, q)` Returns the [Harmonic mean similarity](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=2\sum\limits_{i=1}^{n}\frac{p_i\cdot{q_i}}{p_i+q_i}$ - `jaccard(p, q)` Returns the [Jaccard distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=1-\frac{\sum\limits_{i=1}^{n}{p_i\cdot{q_i}}}{\sum\limits_{i=1}^{n}{p_i^2}+\sum\limits_{i=1}^{n}{q_i^2}-\sum\limits_{i=1}^{n}{p_i\cdot{q_i}}}$ - `dice(p,q)` Returns the [Dice distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\frac{\sum\limits_{i=1}^{n}{(p_i-q_i)^2}}{\sum\limits_{i=1}^{n}{p_i^2}+\sum\limits_{i=1}^{n}{q_i^2}}$ - `fidelity(p, q)` Returns the [Fidelity similarity](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{\sqrt{p_i\cdot{q_i}}}$ - `bhattacharyya(p, q)` Returns the [Bhattacharyya distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=-\ln\left(\sum\limits_{i=1}^{n}{\sqrt{p_i\cdot{q_i}}}\right)$ - `hellinger(p, q)` Returns the [Hellinger distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=2\cdot\sqrt{1-\sum\limits_{i=1}^{n}{\sqrt{p_i\cdot{q_i}}}}$ - `matusita(p, q)` Returns the [Matusita distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sqrt{2-2\cdot\sum\limits_{i=1}^{n}{\sqrt{p_i\cdot{q_i}}}}$ - `squaredChord(p, q)` Returns the [Squared-chord distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{(\sqrt{p_i}-\sqrt{q_i})^2}$ - `squaredEuclidean(p, q)` Returns the [squared euclidean distance](http://en.wikipedia.org/wiki/Euclidean_distance#Squared_Euclidean_distance) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{(p_i-q_i)^2}$ - `pearson(p, q)` Returns the [Pearson distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{\frac{(p_i-q_i)^2}{q_i}}$ - `neyman(p, q)` Returns the [Neyman distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{\frac{(p_i-q_i)^2}{p_i}}$ - `squared(p, q)` Returns the [Squared distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{\frac{(p_i-q_i)^2}{p_i+q_i}}$ - `probabilisticSymmetric(p, q)` Returns the [Probabilistic Symmetric distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=2\cdot\sum\limits_{i=1}^{n}{\frac{(p_i-q_i)^2}{p_i+q_i}}$ - `divergence(p, q)` Returns the [Divergence distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=2\cdot\sum\limits_{i=1}^{n}{\frac{(p_i-q_i)^2}{(p_i+q_i)^2}}$ - `clark(p, q)` Returns the [Clark distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sqrt{\sum\limits_{i=1}^{n}{\left(\frac{\left|p_i-q_i\right|}{(p_i+q_i)}\right)^2}}$ - `additiveSymmetric(p, q)` Returns the [Additive Symmetric distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{\frac{(p_i-q_i)^2\cdot(p_i+q_i)}{p_i\cdot{q_i}}}$ - `kullbackLeibler(p, q)` Returns the [Kullback-Leibler distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{p_i\cdot\ln\frac{p_i}{q_i}}$ - `jeffreys(p, q)` Returns the [Jeffreys distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{\left((p_i-q_i)\ln\frac{p_i}{q_i}\right)}$ - `kdivergence(p, q)` Returns the [K divergence distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{\left(p_i\cdot\ln\frac{2p_i}{p_i+q_i}\right)}$ - `topsoe(p, q)` Returns the [Topsøe distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{\left(p_i\cdot\ln\frac{2p_i}{p_i+q_i}+q_i\cdot\ln\frac{2q_i}{p_i+q_i}\right)}$ - `jensenShannon(p, q)` Returns the [Jensen-Shannon distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\frac{1}{2}\left[\sum\limits_{i=1}^{n}{p_i\cdot\ln\frac{2p_i}{p_i+q_i}}+\sum\limits_{i=1}^{n}{q_i\cdot\ln\frac{2q_i}{p_i+q_i}}\right]$ - `jensenDifference(p, q)` Returns the [Jensen difference distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{\left[\frac{p_i\ln{p_i}+q_i\ln{q_i}}{2}-\left(\frac{p_i+q_i}{2}\right)\ln\left(\frac{p_i+q_i}{2}\right)\right]}$ - `taneja(p, q)` Returns the [Taneja distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{\left[\frac{p_i+q_i}{2}\ln\left(\frac{p_i+q_i}{2\sqrt{p_i\cdot{q_i}}}\right)\right]}$ - `kumarJohnson(p, q)` Returns the [Kumar-Johnson distance](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $d(p,q)=\sum\limits_{i=1}^{n}{\frac{\left(p_i^2-q_i^2\right)^2}{2(p_i\cdot{q_i})^{3/2}}}$ - `avg(p, q)` Returns the average of city block and Chebyshev distances between vectors p and q $d(p,q)=\frac{\sum\limits_{i=1}^{n}{\left|p_i-q_i\right|}+\max\limits_i(|p_i-q_i|)}{2}$ ### Similarities - `intersection(p, q)` Returns the [Intersection similarity](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q - `czekanowski(p, q)` Returns the [Czekanowski similarity](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q - `motyka(p, q)` Returns the [Motyka similarity](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q - `kulczynski(p, q)` Returns the [Kulczynski similarity](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q - `squaredChord(p, q)` Returns the [Squared-chord similarity](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q - `cosine(p, q)` Returns the [Cosine similarity](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $s(p,q)=\frac{\sum\limits_{i=1}^{n}{p_i\cdot{q_i}}}{\sqrt{\sum\limits_{i=1}^{n}{p_i^2}}\sqrt{\sum\limits_{i=1}^{n}{q_i^2}}}$ - `kumarHassebrook(p, q)` Returns the [Kumar-Hassebrook similarity](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q $s(p,q)=\frac{\sum\limits_{i=1}^{n}{p_i\cdot{q_i}}}{\sum\limits_{i=1}^{n}{p_i^2}+\sum\limits_{i=1}^{n}{q_i^2}-\sum\limits_{i=1}^{n}{p_i\cdot{q_i}}}$ - `pearson(p, q)` Returns the [Pearson correlation](https://en.wikipedia.org/wiki/Pearson_correlation_coefficient) between vectors p and q, i.e. the cosine similarity of the mean-centred vectors - `ruzicka(p, q)` Returns the [Ruzicka similarity](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf), also known as the [weighted Jaccard similarity](https://en.wikipedia.org/wiki/Jaccard_index#Weighted_Jaccard_similarity_and_distance), between vectors p and q. The matching distance is `soergel`. $s(p,q)=\frac{\sum\limits_{i=1}^{n}{min(p_i,q_i)}}{\sum\limits_{i=1}^{n}{max(p_i,q_i)}}$ - `dice(p, q)` Returns the [Dice similarity](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q - `tanimoto(p, q, [bitVector])` Returns the [Tanimoto similarity](http://www.naun.org/main/NAUN/ijmmas/mmmas-49.pdf) between vectors p and q, and accepts the bitVector use, see the test case for an example - `tree(a,b, from, to, [options])` Refer to [ml-tree-similarity](https://github.com/mljs/tree-similarity) ## Contributing A new metric should normally be in its own file in the `src/distances` (or `src/similarities`) directory. There should be a corresponding test file in the neighbouring `__tests__` directory. The metric should be then added in the exports of `src/distances.ts` (or `src/similarities.ts`) with a relatively small but understandable name (use camelCase). It should also be added to this README with either a link to the formula or an inline description. ## Authors - [Michaël Zasso](https://github.com/targos) - [Miguel Angel Asencio Hurtado](https://github.com/maasencioh) ## License [MIT](./LICENSE) [npm-image]: https://img.shields.io/npm/v/ml-distance.svg?style=flat-square [npm-url]: https://npmjs.org/package/ml-distance [codecov-image]: https://img.shields.io/codecov/c/github/mljs/distance.svg [codecov-url]: https://codecov.io/gh/mljs/distance [download-image]: https://img.shields.io/npm/dm/ml-distance.svg?style=flat-square [download-url]: https://npmjs.org/package/ml-distance