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def pdist(X, metric='euclidean', *, out=None, **kwargs)
Pairwise distances between observations in n-dimensional space.
See Notes for common calling conventions.
Parameters
----------
X : array_like
An m by n array of m original observations in an
n-dimensional space.
metric : str or function, optional
The distance metric to use. The distance function can
be 'braycurtis', 'canberra', 'chebyshev', 'cityblock',
'correlation', 'cosine', 'dice', 'euclidean', 'hamming',
'jaccard', 'jensenshannon', 'kulczynski1',
'mahalanobis', 'matching', 'minkowski', 'rogerstanimoto',
'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath',
'sqeuclidean', 'yule'.
out : ndarray, optional
The output array.
If not None, condensed distance matrix Y is stored in this array.
**kwargs : dict, optional
Extra arguments to `metric`: refer to each metric documentation for a
list of all possible arguments.
Some possible arguments:
p : scalar
The p-norm to apply for Minkowski, weighted and unweighted.
Default: 2.
w : ndarray
The weight vector for metrics that support weights (e.g., Minkowski).
V : ndarray
The variance vector for standardized Euclidean.
Default: var(X, axis=0, ddof=1)
VI : ndarray
The inverse of the covariance matrix for Mahalanobis.
Default: inv(cov(X.T)).T
Returns
-------
Y : ndarray
Returns a condensed distance matrix Y. For each :math:`i` and :math:`j`
(where :math:`i<j<m`),where m is the number of original observations.
The metric ``dist(u=X[i], v=X[j])`` is computed and stored in entry ``m
* i + j - ((i + 2) * (i + 1)) // 2``.
See Also
--------
squareform : converts between condensed distance matrices and
square distance matrices.
Notes
-----
See ``squareform`` for information on how to calculate the index of
this entry or to convert the condensed distance matrix to a
redundant square matrix.
The following are common calling conventions.
1. ``Y = pdist(X, 'euclidean')``
Computes the distance between m points using Euclidean distance
(2-norm) as the distance metric between the points. The points
are arranged as m n-dimensional row vectors in the matrix X.
2. ``Y = pdist(X, 'minkowski', p=2.)``
Computes the distances using the Minkowski distance
:math:`\|u-v\|_p` (:math:`p`-norm) where :math:`p > 0` (note
that this is only a quasi-metric if :math:`0 < p < 1`).
3. ``Y = pdist(X, 'cityblock')``
Computes the city block or Manhattan distance between the
points.
4. ``Y = pdist(X, 'seuclidean', V=None)``
Computes the standardized Euclidean distance. The standardized
Euclidean distance between two n-vectors ``u`` and ``v`` is
.. math::
\sqrt{\sum {(u_i-v_i)^2 / V[x_i]}}
V is the variance vector; V[i] is the variance computed over all
the i'th components of the points. If not passed, it is
automatically computed.
5. ``Y = pdist(X, 'sqeuclidean')``
Computes the squared Euclidean distance :math:`\|u-v\|_2^2` between
the vectors.
6. ``Y = pdist(X, 'cosine')``
Computes the cosine distance between vectors u and v,
.. math::
1 - \frac{u \cdot v}
{{\|u\|}_2 {\|v\|}_2}
where :math:`\|*\|_2` is the 2-norm of its argument ``*``, and
:math:`u \cdot v` is the dot product of ``u`` and ``v``.
7. ``Y = pdist(X, 'correlation')``
Computes the correlation distance between vectors u and v. This is
.. math::
1 - \frac{(u - \bar{u}) \cdot (v - \bar{v})}
{{\|(u - \bar{u})\|}_2 {\|(v - \bar{v})\|}_2}
where :math:`\bar{v}` is the mean of the elements of vector v,
and :math:`x \cdot y` is the dot product of :math:`x` and :math:`y`.
8. ``Y = pdist(X, 'hamming')``
Computes the normalized Hamming distance, or the proportion of
those vector elements between two n-vectors ``u`` and ``v``
which disagree. To save memory, the matrix ``X`` can be of type
boolean.
9. ``Y = pdist(X, 'jaccard')``
Computes the Jaccard distance between the points. Given two
vectors, ``u`` and ``v``, the Jaccard distance is the
proportion of those elements ``u[i]`` and ``v[i]`` that
disagree.
10. ``Y = pdist(X, 'jensenshannon')``
Computes the Jensen-Shannon distance between two probability arrays.
Given two probability vectors, :math:`p` and :math:`q`, the
Jensen-Shannon distance is
.. math::
\sqrt{\frac{D(p \parallel m) + D(q \parallel m)}{2}}
where :math:`m` is the pointwise mean of :math:`p` and :math:`q`
and :math:`D` is the Kullback-Leibler divergence.
11. ``Y = pdist(X, 'chebyshev')``
Computes the Chebyshev distance between the points. The
Chebyshev distance between two n-vectors ``u`` and ``v`` is the
maximum norm-1 distance between their respective elements. More
precisely, the distance is given by
.. math::
d(u,v) = \max_i {|u_i-v_i|}
12. ``Y = pdist(X, 'canberra')``
Computes the Canberra distance between the points. The
Canberra distance between two points ``u`` and ``v`` is
.. math::
d(u,v) = \sum_i \frac{|u_i-v_i|}
{|u_i|+|v_i|}
13. ``Y = pdist(X, 'braycurtis')``
Computes the Bray-Curtis distance between the points. The
Bray-Curtis distance between two points ``u`` and ``v`` is
.. math::
d(u,v) = \frac{\sum_i {|u_i-v_i|}}
{\sum_i {|u_i+v_i|}}
14. ``Y = pdist(X, 'mahalanobis', VI=None)``
Computes the Mahalanobis distance between the points. The
Mahalanobis distance between two points ``u`` and ``v`` is
:math:`\sqrt{(u-v)(1/V)(u-v)^T}` where :math:`(1/V)` (the ``VI``
variable) is the inverse covariance. If ``VI`` is not None,
``VI`` will be used as the inverse covariance matrix.
15. ``Y = pdist(X, 'yule')``
Computes the Yule distance between each pair of boolean
vectors. (see yule function documentation)
16. ``Y = pdist(X, 'matching')``
Synonym for 'hamming'.
17. ``Y = pdist(X, 'dice')``
Computes the Dice distance between each pair of boolean
vectors. (see dice function documentation)
18. ``Y = pdist(X, 'kulczynski1')``
Computes the kulczynski1 distance between each pair of
boolean vectors. (see kulczynski1 function documentation)
.. deprecated:: 1.15.0
This metric is deprecated and will be removed in SciPy 1.17.0.
Replace usage of ``pdist(X, 'kulczynski1')`` with
``1 / pdist(X, 'jaccard') - 1``.
19. ``Y = pdist(X, 'rogerstanimoto')``
Computes the Rogers-Tanimoto distance between each pair of
boolean vectors. (see rogerstanimoto function documentation)
20. ``Y = pdist(X, 'russellrao')``
Computes the Russell-Rao distance between each pair of
boolean vectors. (see russellrao function documentation)
21. ``Y = pdist(X, 'sokalmichener')``
Computes the Sokal-Michener distance between each pair of
boolean vectors. (see sokalmichener function documentation)
.. deprecated:: 1.15.0
This metric is deprecated and will be removed in SciPy 1.17.0.
Replace usage of ``pdist(X, 'sokalmichener')`` with
``pdist(X, 'rogerstanimoto')``.
22. ``Y = pdist(X, 'sokalsneath')``
Computes the Sokal-Sneath distance between each pair of
boolean vectors. (see sokalsneath function documentation)
23. ``Y = pdist(X, 'kulczynski1')``
Computes the Kulczynski 1 distance between each pair of
boolean vectors. (see kulczynski1 function documentation)
24. ``Y = pdist(X, f)``
Computes the distance between all pairs of vectors in X
using the user supplied 2-arity function f. For example,
Euclidean distance between the vectors could be computed
as follows::
dm = pdist(X, lambda u, v: np.sqrt(((u-v)**2).sum()))
Note that you should avoid passing a reference to one of
the distance functions defined in this library. For example,::
dm = pdist(X, sokalsneath)
would calculate the pair-wise distances between the vectors in
X using the Python function sokalsneath. This would result in
sokalsneath being called :math:`{n \choose 2}` times, which
is inefficient. Instead, the optimized C version is more
efficient, and we call it using the following syntax.::
dm = pdist(X, 'sokalsneath')
Examples
--------
>>> import numpy as np
>>> from scipy.spatial.distance import pdist
``x`` is an array of five points in three-dimensional space.
>>> x = np.array([[2, 0, 2], [2, 2, 3], [-2, 4, 5], [0, 1, 9], [2, 2, 4]])
``pdist(x)`` with no additional arguments computes the 10 pairwise
Euclidean distances:
>>> pdist(x)
array([2.23606798, 6.40312424, 7.34846923, 2.82842712, 4.89897949,
6.40312424, 1. , 5.38516481, 4.58257569, 5.47722558])
The following computes the pairwise Minkowski distances with ``p = 3.5``:
>>> pdist(x, metric='minkowski', p=3.5)
array([2.04898923, 5.1154929 , 7.02700737, 2.43802731, 4.19042714,
6.03956994, 1. , 4.45128103, 4.10636143, 5.0619695 ])
The pairwise city block or Manhattan distances:
>>> pdist(x, metric='cityblock')
array([ 3., 11., 10., 4., 8., 9., 1., 9., 7., 8.])
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