`scipy.spatial.distance:jaccard`

source: /scipy/spatial/distance.py :778

Signature

def jaccard ( u , v , w = None )

Summary

Compute the Jaccard dissimilarity between two boolean vectors.

Extended Summary

Given boolean vectors $u \equiv (u_{1}, \dots, u_{n})$ and $v \equiv (v_{1}, \dots, v_{n})$ that are not both zero, their Jaccard dissimilarity is defined as (^[1], p. 26)

d_{jaccard} (u, v) := \frac{c _{10} + c _{01}}{c _{11} + c _{10} + c _{01}}

where

c_{ij} := 1 \leq k \leq n, u_{k} = i, v_{k} = j \sum 1

for $i, j \in {0, 1}$ . If $u$ and $v$ are both zero, their Jaccard dissimilarity is defined to be zero. ^[2]

If a (non-negative) weight vector $w \equiv (w_{1}, \dots, w_{n})$ is supplied, the weighted Jaccard dissimilarity is defined similarly but with $c_{ij}$ replaced by

\tilde{c}_{ij} := 1 \leq k \leq n, u_{k} = i, v_{k} = j \sum w_{k}

Parameters

u : (N,) array_like of bools: Input vector.
v : (N,) array_like of bools: Input vector.
w : (N,) array_like of floats, optional: Weights for each pair of $(u_{k}, v_{k})$ . Default is None, which gives each pair a weight of 1.0.

Returns

jaccard : float: The Jaccard dissimilarity between vectors u and v, optionally weighted by w if supplied.

Notes

The Jaccard dissimilarity satisfies the triangle inequality and is qualified as a metric. ^[2]

The Jaccard index, or Jaccard similarity coefficient, is equal to one minus the Jaccard dissimilarity. ^[3]

The dissimilarity between general (finite) sets may be computed by encoding them as boolean vectors and computing the dissimilarity between the encoded vectors. For example, subsets $A, B$ of ${1, 2, ..., n}$ may be encoded into boolean vectors $u, v$ by setting $u_{k} := 1_{k \in A}$ , $v_{k} := 1_{k \in B}$ for $k = 1, 2, \dots, n$ .

Examples

from scipy.spatial import distance

✓

Non-zero vectors with no matching 1s have dissimilarity of 1.0:

distance.jaccard([1, 0, 0], [0, 1, 0])

✗

Vectors with some matching 1s have dissimilarity less than 1.0:

distance.jaccard([1, 0, 0, 0], [1, 1, 1, 0])

✗

Identical vectors, including zero vectors, have dissimilarity of 0.0:

distance.jaccard([1, 0, 0], [1, 0, 0])
distance.jaccard([0, 0, 0], [0, 0, 0])

✗

The following example computes the dissimilarity from a confusion matrix directly by setting the weight vector to the frequency of True Positive, False Negative, False Positive, and True Negative:

distance.jaccard([1, 1, 0, 0], [1, 0, 1, 0], [31, 41, 59, 26])

✗

Aliases

scipy.spatial.distance.jaccard

Referenced by

This package

release:1.2.0-notes