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Module « scipy.spatial.distance »

Fonction jaccard - module scipy.spatial.distance

Signature de la fonction jaccard

def jaccard(u, v, w=None) 

Description

help(scipy.spatial.distance.jaccard)

Compute the Jaccard dissimilarity between two boolean vectors.

Given boolean vectors :math:`u \equiv (u_1, \cdots, u_n)`
and :math:`v \equiv (v_1, \cdots, v_n)` that are not both zero,
their *Jaccard dissimilarity* is defined as ([1]_, p. 26)

.. math::

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

where

.. math::

   c_{ij} := \sum_{1 \le k \le n, u_k=i, v_k=j} 1

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

If a (non-negative) weight vector :math:`w \equiv (w_1, \cdots, w_n)`
is supplied, the *weighted Jaccard dissimilarity* is defined similarly
but with :math:`c_{ij}` replaced by

.. math::

   \tilde{c}_{ij} := \sum_{1 \le k \le n, u_k=i, v_k=j} 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 :math:`(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 :math:`A,B` of :math:`\{ 1, 2, ..., n \}` may be
encoded into boolean vectors :math:`u, v` by setting
:math:`u_k := 1_{k \in A}`, :math:`v_k := 1_{k \in B}`
for :math:`k = 1,2,\cdots,n`.

.. versionchanged:: 1.2.0
   Previously, if all (positively weighted) elements in `u` and `v` are
   zero, the function would return ``nan``.  This was changed to return
   ``0`` instead.

.. versionchanged:: 1.15.0
   Non-0/1 numeric input used to produce an ad hoc result.  Since 1.15.0,
   numeric input is converted to Boolean before computation.

References
----------
.. [1] Kaufman, L. and Rousseeuw, P. J.  (1990).  "Finding Groups in Data:
       An Introduction to Cluster Analysis."  John Wiley & Sons, Inc.
.. [2] Kosub, S.  (2019).  "A note on the triangle inequality for the
       Jaccard distance."  *Pattern Recognition Letters*, 120:36-38.
.. [3] https://en.wikipedia.org/wiki/Jaccard_index

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])
1.0

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

>>> distance.jaccard([1, 0, 0, 0], [1, 1, 1, 0])
0.6666666666666666

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

>>> distance.jaccard([1, 0, 0], [1, 0, 0])
0.0
>>> distance.jaccard([0, 0, 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])
0.7633587786259542  # (41+59)/(31+41+59)

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