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Distance computations (:mod:`scipy.spatial.distance`)
=====================================================
.. sectionauthor:: Damian Eads
Function reference
------------------
Distance matrix computation from a collection of raw observation vectors
stored in a rectangular array.
.. autosummary::
:toctree: generated/
pdist -- pairwise distances between observation vectors.
cdist -- distances between two collections of observation vectors
squareform -- convert distance matrix to a condensed one and vice versa
directed_hausdorff -- directed Hausdorff distance between arrays
Predicates for checking the validity of distance matrices, both
condensed and redundant. Also contained in this module are functions
for computing the number of observations in a distance matrix.
.. autosummary::
:toctree: generated/
is_valid_dm -- checks for a valid distance matrix
is_valid_y -- checks for a valid condensed distance matrix
num_obs_dm -- # of observations in a distance matrix
num_obs_y -- # of observations in a condensed distance matrix
Distance functions between two numeric vectors ``u`` and ``v``. Computing
distances over a large collection of vectors is inefficient for these
functions. Use ``pdist`` for this purpose.
.. autosummary::
:toctree: generated/
braycurtis -- the Bray-Curtis distance.
canberra -- the Canberra distance.
chebyshev -- the Chebyshev distance.
cityblock -- the Manhattan distance.
correlation -- the Correlation distance.
cosine -- the Cosine distance.
euclidean -- the Euclidean distance.
jensenshannon -- the Jensen-Shannon distance.
mahalanobis -- the Mahalanobis distance.
minkowski -- the Minkowski distance.
seuclidean -- the normalized Euclidean distance.
sqeuclidean -- the squared Euclidean distance.
Distance functions between two boolean vectors (representing sets) ``u`` and
``v``. As in the case of numerical vectors, ``pdist`` is more efficient for
computing the distances between all pairs.
.. autosummary::
:toctree: generated/
dice -- the Dice dissimilarity.
hamming -- the Hamming distance.
jaccard -- the Jaccard distance.
kulsinski -- the Kulsinski distance.
kulczynski1 -- the Kulczynski 1 distance.
rogerstanimoto -- the Rogers-Tanimoto dissimilarity.
russellrao -- the Russell-Rao dissimilarity.
sokalmichener -- the Sokal-Michener dissimilarity.
sokalsneath -- the Sokal-Sneath dissimilarity.
yule -- the Yule dissimilarity.
:func:`hamming` also operates over discrete numerical vectors.
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Compute the directed Hausdorff distance between two 2-D arrays.
Distances between pairs are calculated using a Euclidean metric.
Parameters
----------
u : (M,N) array_like
Input array.
v : (O,N) array_like
Input array.
seed : int or None
Local `numpy.random.RandomState` seed. Default is 0, a random
shuffling of u and v that guarantees reproducibility.
Returns
-------
d : double
The directed Hausdorff distance between arrays `u` and `v`,
index_1 : int
index of point contributing to Hausdorff pair in `u`
index_2 : int
index of point contributing to Hausdorff pair in `v`
Raises
------
ValueError
An exception is thrown if `u` and `v` do not have
the same number of columns.
Notes
-----
Uses the early break technique and the random sampling approach
described by [1]_. Although worst-case performance is ``O(m * o)``
(as with the brute force algorithm), this is unlikely in practice
as the input data would have to require the algorithm to explore
every single point interaction, and after the algorithm shuffles
the input points at that. The best case performance is O(m), which
is satisfied by selecting an inner loop distance that is less than
cmax and leads to an early break as often as possible. The authors
have formally shown that the average runtime is closer to O(m).
.. versionadded:: 0.19.0
References
----------
.. [1] A. A. Taha and A. Hanbury, "An efficient algorithm for
calculating the exact Hausdorff distance." IEEE Transactions On
Pattern Analysis And Machine Intelligence, vol. 37 pp. 2153-63,
2015.
See Also
--------
scipy.spatial.procrustes : Another similarity test for two data sets
Examples
--------
Find the directed Hausdorff distance between two 2-D arrays of
coordinates:
>>> from scipy.spatial.distance import directed_hausdorff
>>> import numpy as np
>>> u = np.array([(1.0, 0.0),
... (0.0, 1.0),
... (-1.0, 0.0),
... (0.0, -1.0)])
>>> v = np.array([(2.0, 0.0),
... (0.0, 2.0),
... (-2.0, 0.0),
... (0.0, -4.0)])
>>> directed_hausdorff(u, v)[0]
2.23606797749979
>>> directed_hausdorff(v, u)[0]
3.0
Find the general (symmetric) Hausdorff distance between two 2-D
arrays of coordinates:
>>> max(directed_hausdorff(u, v)[0], directed_hausdorff(v, u)[0])
3.0
Find the indices of the points that generate the Hausdorff distance
(the Hausdorff pair):
>>> directed_hausdorff(v, u)[1:]
(3, 3)
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Compute the Minkowski distance between two 1-D arrays.
The Minkowski distance between 1-D arrays `u` and `v`,
is defined as
.. math::
{\|u-v\|}_p = (\sum{|u_i - v_i|^p})^{1/p}.
\left(\sum{w_i(|(u_i - v_i)|^p)}\right)^{1/p}.
Parameters
----------
u : (N,) array_like
Input array.
v : (N,) array_like
Input array.
p : scalar
The order of the norm of the difference :math:`{\|u-v\|}_p`. Note
that for :math:`0 < p < 1`, the triangle inequality only holds with
an additional multiplicative factor, i.e. it is only a quasi-metric.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
minkowski : double
The Minkowski distance between vectors `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.minkowski([1, 0, 0], [0, 1, 0], 1)
2.0
>>> distance.minkowski([1, 0, 0], [0, 1, 0], 2)
1.4142135623730951
>>> distance.minkowski([1, 0, 0], [0, 1, 0], 3)
1.2599210498948732
>>> distance.minkowski([1, 1, 0], [0, 1, 0], 1)
1.0
>>> distance.minkowski([1, 1, 0], [0, 1, 0], 2)
1.0
>>> distance.minkowski([1, 1, 0], [0, 1, 0], 3)
1.0
r r� Nr' r* )�ord)r� ru rv rJ �sqrt�inf�powerr+ )rW rX r� rY �u_v�root_w�dists r3 r r � s� � �d ����A�����A��A�v�v��3�4�4�4�
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Computes the Euclidean distance between two 1-D arrays.
The Euclidean distance between 1-D arrays `u` and `v`, is defined as
.. math::
{\|u-v\|}_2
\left(\sum{(w_i |(u_i - v_i)|^2)}\right)^{1/2}
Parameters
----------
u : (N,) array_like
Input array.
v : (N,) array_like
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
euclidean : double
The Euclidean distance between vectors `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.euclidean([1, 0, 0], [0, 1, 0])
1.4142135623730951
>>> distance.euclidean([1, 1, 0], [0, 1, 0])
1.0
r* )r� rY )r �rW rX rY s r3 r r � s � �H �Q��Q�!� $� $� $�$r5 c � � d\ }}t | d� � r$t j | j t j � � s t j }t |d� � r$t j |j t j � � s t j }t
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Compute the squared Euclidean distance between two 1-D arrays.
The squared Euclidean distance between `u` and `v` is defined as
.. math::
{\|u-v\|}_2^2
\left(\sum{(w_i |(u_i - v_i)|^2)}\right)
Parameters
----------
u : (N,) array_like
Input array.
v : (N,) array_like
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
sqeuclidean : double
The squared Euclidean distance between vectors `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.sqeuclidean([1, 0, 0], [0, 1, 0])
2.0
>>> distance.sqeuclidean([1, 1, 0], [0, 1, 0])
1.0
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Compute the correlation distance between two 1-D arrays.
The correlation distance between `u` and `v`, is
defined as
.. math::
1 - \frac{(u - \bar{u}) \cdot (v - \bar{v})}
{{\|(u - \bar{u})\|}_2 {\|(v - \bar{v})\|}_2}
where :math:`\bar{u}` is the mean of the elements of `u`
and :math:`x \cdot y` is the dot product of :math:`x` and :math:`y`.
Parameters
----------
u : (N,) array_like
Input array.
v : (N,) array_like
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
centered : bool, optional
If True, `u` and `v` will be centered. Default is True.
Returns
-------
correlation : double
The correlation distance between 1-D array `u` and `v`.
N��weightsrP )r� rv rJ �average�squarer� �abs)
rW rX rY �centered�umu�vmu�uv�uu�vvr� s
r3 r r A s� � �B ����A�����A��}� �a� � ��� ��j��A�&�&�&���j��A�&�&�&��
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�6�$�<�<�r5 c �` � t dt t | ||d�� � d� � � � S )a�
Compute the Cosine distance between 1-D arrays.
The Cosine distance between `u` and `v`, is defined as
.. math::
1 - \frac{u \cdot v}
{\|u\|_2 \|v\|_2}.
where :math:`u \cdot v` is the dot product of :math:`u` and
:math:`v`.
Parameters
----------
u : (N,) array_like
Input array.
v : (N,) array_like
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
cosine : double
The Cosine distance between vectors `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.cosine([1, 0, 0], [0, 1, 0])
1.0
>>> distance.cosine([100, 0, 0], [0, 1, 0])
1.0
>>> distance.cosine([1, 1, 0], [0, 1, 0])
0.29289321881345254
r F)rY r� r� )�max�minr r� s r3 r r s s0 � �V �q�#�k�!�Q�!�e�<�<�<�c�B�B� C� C�Cr5 c �� � t | � � } t |� � }| j |j k rt d� � �| |k }|�t |� � }t j ||�� � S )a�
Compute the Hamming distance between two 1-D arrays.
The Hamming distance between 1-D arrays `u` and `v`, is simply the
proportion of disagreeing components in `u` and `v`. If `u` and `v` are
boolean vectors, the Hamming distance is
.. math::
\frac{c_{01} + c_{10}}{n}
where :math:`c_{ij}` is the number of occurrences of
:math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for
:math:`k < n`.
Parameters
----------
u : (N,) array_like
Input array.
v : (N,) array_like
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
hamming : double
The Hamming distance between vectors `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.hamming([1, 0, 0], [0, 1, 0])
0.66666666666666663
>>> distance.hamming([1, 0, 0], [1, 1, 0])
0.33333333333333331
>>> distance.hamming([1, 0, 0], [2, 0, 0])
0.33333333333333331
>>> distance.hamming([1, 0, 0], [3, 0, 0])
0.33333333333333331
z&The 1d arrays must have equal lengths.Nr� )r� rt ru rv rJ r� )rW rX rY �u_ne_vs r3 r r � sm � �X ����A�����A��w�!�'����A�B�B�B�
�!�V�F��}� �a� � ��
�:�f�a� (� (� (�(r5 c � � t | � � } t |� � }t j | dk |dk � � }t j | |k |� � }|�t |� � }||z }||z }t j |� � � � � }t j |� � � � � }|dk r||z ndS )a�
Compute the Jaccard-Needham dissimilarity between two boolean 1-D arrays.
The Jaccard-Needham dissimilarity between 1-D boolean arrays `u` and `v`,
is defined as
.. math::
\frac{c_{TF} + c_{FT}}
{c_{TT} + c_{FT} + c_{TF}}
where :math:`c_{ij}` is the number of occurrences of
:math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for
:math:`k < n`.
Parameters
----------
u : (N,) array_like, bool
Input array.
v : (N,) array_like, bool
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
jaccard : double
The Jaccard distance between vectors `u` and `v`.
Notes
-----
When both `u` and `v` lead to a `0/0` division i.e. there is no overlap
between the items in the vectors the returned distance is 0. See the
Wikipedia page on the Jaccard index [1]_, and this paper [2]_.
.. versionchanged:: 1.2.0
Previously, when `u` and `v` lead to a `0/0` division, the function
would return NaN. This was changed to return 0 instead.
References
----------
.. [1] https://en.wikipedia.org/wiki/Jaccard_index
.. [2] S. Kosub, "A note on the triangle inequality for the Jaccard
distance", 2016, :arxiv:`1612.02696`
Examples
--------
>>> from scipy.spatial import distance
>>> distance.jaccard([1, 0, 0], [0, 1, 0])
1.0
>>> distance.jaccard([1, 0, 0], [1, 1, 0])
0.5
>>> distance.jaccard([1, 0, 0], [1, 2, 0])
0.5
>>> distance.jaccard([1, 0, 0], [1, 1, 1])
0.66666666666666663
r )r� rJ �
bitwise_or� bitwise_andrv r{ rS )rW rX rY �nonzero�unequal_nonzeror2 �bs r3 r r � s� � �x ����A�����A��m�A��F�A��F�+�+�G��n�a�1�f�w�7�7�O��}� �a� � ���g�+���o�-��
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� �'�+�+�-�-� � �A��1�f�f�A��E�E�!�#r5 z�Kulsinski has been deprecated from scipy.spatial.distance in SciPy 1.9.0 and it will be removed in SciPy 1.11.0. It is superseded by scipy.spatial.distance.kulczynski1.c � � t | � � } t |� � }|� t t | � � � � }n#t |� � }|� � � }t | ||�� � \ }}}}||z |z
|z ||z |z z S )a@
Compute the Kulsinski dissimilarity between two boolean 1-D arrays.
The Kulsinski dissimilarity between two boolean 1-D arrays `u` and `v`,
is defined as
.. math::
\frac{c_{TF} + c_{FT} - c_{TT} + n}
{c_{FT} + c_{TF} + n}
where :math:`c_{ij}` is the number of occurrences of
:math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for
:math:`k < n`.
.. deprecated:: 0.12.0
`kulsinski` has been deprecated from `scipy.spatial.distance` in
SciPy 1.9.0 and it will be removed in SciPy 1.11.0. It is superseded
by `scipy.spatial.distance.kulczynski1`.
Parameters
----------
u : (N,) array_like, bool
Input array.
v : (N,) array_like, bool
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
kulsinski : double
The Kulsinski distance between vectors `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.kulsinski([1, 0, 0], [0, 1, 0])
1.0
>>> distance.kulsinski([1, 0, 0], [1, 1, 0])
0.75
>>> distance.kulsinski([1, 0, 0], [2, 1, 0])
0.33333333333333331
>>> distance.kulsinski([1, 0, 0], [3, 1, 0])
-0.5
N�rY )r� �floatr� rv rS r` )rW rX rY rl r\ r] r^ r_ s r3 r r ! s� � �h ����A�����A��y� �#�a�&�&�M�M��� �a� � ��
�E�E�G�G��0��A��;�;�;��S�#�s�C� �#�I��O�a� �C�#�I��M� 2�2r5 r� c � � t | � � } t |� � }|�t |� � }t | ||�� � \ }}}}|||z z S )aJ
Compute the Kulczynski 1 dissimilarity between two boolean 1-D arrays.
The Kulczynski 1 dissimilarity between two boolean 1-D arrays `u` and `v`
of length ``n``, is defined as
.. math::
\frac{c_{11}}
{c_{01} + c_{10}}
where :math:`c_{ij}` is the number of occurrences of
:math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for
:math:`k \in {0, 1, ..., n-1}`.
Parameters
----------
u : (N,) array_like, bool
Input array.
v : (N,) array_like, bool
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
kulczynski1 : float
The Kulczynski 1 distance between vectors `u` and `v`.
Notes
-----
This measure has a minimum value of 0 and no upper limit.
It is un-defined when there are no non-matches.
.. versionadded:: 1.8.0
References
----------
.. [1] Kulczynski S. et al. Bulletin
International de l'Academie Polonaise des Sciences
et des Lettres, Classe des Sciences Mathematiques
et Naturelles, Serie B (Sciences Naturelles). 1927;
Supplement II: 57-203.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.kulczynski1([1, 0, 0], [0, 1, 0])
0.0
>>> distance.kulczynski1([True, False, False], [True, True, False])
1.0
>>> distance.kulczynski1([True, False, False], [True])
0.5
>>> distance.kulczynski1([1, 0, 0], [3, 1, 0])
-3.0
Nr� )r� rv r` )rW rX rY �_r] r^ r_ s r3 r r a s] � �v ����A�����A��}� �a� � ��.�q�!�q�9�9�9��Q��S�#� �#��)� � r5 c �, � t | � � } t |� � }t |t j �� � }|j d | j d k s | j d |j d k rt d� � �t | |d|z �� � S )a]
Return the standardized Euclidean distance between two 1-D arrays.
The standardized Euclidean distance between `u` and `v`.
Parameters
----------
u : (N,) array_like
Input array.
v : (N,) array_like
Input array.
V : (N,) array_like
`V` is an 1-D array of component variances. It is usually computed
among a larger collection vectors.
Returns
-------
seuclidean : double
The standardized Euclidean distance between vectors `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.seuclidean([1, 0, 0], [0, 1, 0], [0.1, 0.1, 0.1])
4.4721359549995796
>>> distance.seuclidean([1, 0, 0], [0, 1, 0], [1, 0.1, 0.1])
3.3166247903553998
>>> distance.seuclidean([1, 0, 0], [0, 1, 0], [10, 0.1, 0.1])
3.1780497164141406
rH r z7V must be a 1-D array of the same dimension as u and v.r' r� )r� rJ r� rt � TypeErrorr )rW rX r� s r3 r r � s� � �@ ����A�����A���"�*�-�-�-�A��w�q�z�Q�W�Q�Z� � �1�7�1�:�����#;�#;�� &� '� '� '� �Q��Q�q�S� !� !� !�!r5 c � � t | � � } t |� � }t | |z
� � }|�t |� � }||z }|� � � S )aD
Compute the City Block (Manhattan) distance.
Computes the Manhattan distance between two 1-D arrays `u` and `v`,
which is defined as
.. math::
\sum_i {\left| u_i - v_i \right|}.
Parameters
----------
u : (N,) array_like
Input array.
v : (N,) array_like
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
cityblock : double
The City Block (Manhattan) distance between vectors `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.cityblock([1, 0, 0], [0, 1, 0])
2
>>> distance.cityblock([1, 0, 0], [0, 2, 0])
3
>>> distance.cityblock([1, 0, 0], [1, 1, 0])
1
)r� r� rv rS )rW rX rY �l1_diffs r3 r r � sW � �J ����A�����A��!�a�%�j�j�G��}� �a� � ���g�+�� �;�;�=�=�r5 c �� � t | � � } t |� � }t j |� � }| |z
}t j t j ||� � |� � }t j |� � S )a�
Compute the Mahalanobis distance between two 1-D arrays.
The Mahalanobis distance between 1-D arrays `u` and `v`, is defined as
.. math::
\sqrt{ (u-v) V^{-1} (u-v)^T }
where ``V`` is the covariance matrix. Note that the argument `VI`
is the inverse of ``V``.
Parameters
----------
u : (N,) array_like
Input array.
v : (N,) array_like
Input array.
VI : array_like
The inverse of the covariance matrix.
Returns
-------
mahalanobis : double
The Mahalanobis distance between vectors `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> iv = [[1, 0.5, 0.5], [0.5, 1, 0.5], [0.5, 0.5, 1]]
>>> distance.mahalanobis([1, 0, 0], [0, 1, 0], iv)
1.0
>>> distance.mahalanobis([0, 2, 0], [0, 1, 0], iv)
1.0
>>> distance.mahalanobis([2, 0, 0], [0, 1, 0], iv)
1.7320508075688772
)r� rJ r� r� r� )rW rX r� �deltarx s r3 r r � sb � �N ����A�����A� � �r� � �B�
��E�E�
��r�v�e�R� � �%�(�(�A�
�7�1�:�:�r5 c � � t | � � } t |� � }|�Bt |� � }|dk }|� � � |j k r| | } || }t t | |z
� � � � S )a�
Compute the Chebyshev distance.
Computes the Chebyshev distance between two 1-D arrays `u` and `v`,
which is defined as
.. math::
\max_i {|u_i-v_i|}.
Parameters
----------
u : (N,) array_like
Input vector.
v : (N,) array_like
Input vector.
w : (N,) array_like, optional
Unused, as 'max' is a weightless operation. Here for API consistency.
Returns
-------
chebyshev : double
The Chebyshev distance between vectors `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.chebyshev([1, 0, 0], [0, 1, 0])
1
>>> distance.chebyshev([1, 1, 0], [0, 1, 0])
1
Nr )r� rv rS �sizer� r� )rW rX rY �
has_weights r3 r r + sy � �D ����A�����A��}� �a� � ����U�
� �>�>� � �a�f� $� $��*�
�A��*�
�A� �s�1�q�5�z�z�?�?�r5 c �& � t | � � } t |t j �� � }t | |z
� � }t | |z � � }|�t |� � }||z }||z }|� � � |� � � z S )a{
Compute the Bray-Curtis distance between two 1-D arrays.
Bray-Curtis distance is defined as
.. math::
\sum{|u_i-v_i|} / \sum{|u_i+v_i|}
The Bray-Curtis distance is in the range [0, 1] if all coordinates are
positive, and is undefined if the inputs are of length zero.
Parameters
----------
u : (N,) array_like
Input array.
v : (N,) array_like
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
braycurtis : double
The Bray-Curtis distance between 1-D arrays `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.braycurtis([1, 0, 0], [0, 1, 0])
1.0
>>> distance.braycurtis([1, 1, 0], [0, 1, 0])
0.33333333333333331
rH )r� rJ r� r� rv rS )rW rX rY r� �l1_sums r3 r r X s� � �J ����A���"�*�-�-�-�A��!�a�%�j�j�G�
��Q��Z�Z�F��}� �a� � ���g�+���V��� �;�;�=�=�6�:�:�<�<� '�'r5 c �| � t | � � } t |t j �� � }|�t |� � }t j d�� � 5 t | |z
� � }t | � � }t |� � }|||z z }|�||z }t j |� � }ddd� � n # 1 swxY w Y |S )a{
Compute the Canberra distance between two 1-D arrays.
The Canberra distance is defined as
.. math::
d(u,v) = \sum_i \frac{|u_i-v_i|}
{|u_i|+|v_i|}.
Parameters
----------
u : (N,) array_like
Input array.
v : (N,) array_like
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
canberra : double
The Canberra distance between vectors `u` and `v`.
Notes
-----
When `u[i]` and `v[i]` are 0 for given i, then the fraction 0/0 = 0 is
used in the calculation.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.canberra([1, 0, 0], [0, 1, 0])
2.0
>>> distance.canberra([1, 1, 0], [0, 1, 0])
1.0
rH N�ignore)�invalid)r� rJ r� rv �errstater� �nansum)rW rX rY �abs_uv�abs_u�abs_v�ds r3 r r � s� � �P ����A���"�*�-�-�-�A��}� �a� � �� ��X� &� &� &� � ��Q��U�����A�����A���� �e�e�m� $�� �=��A��A� �I�a�L�L��� � � � � � � � � � ���� � � � �
�Hs �AB1�1B5�8B5Fr8 c �� � t j | � � } t j |� � }| t j | |d�� � z } |t j ||d�� � z }| |z dz }t | |� � }t ||� � }t j |||�� � }t j |||�� � } || z }
|�|
t j |� � z }
t j |
dz � � S )a�
Compute the Jensen-Shannon distance (metric) between
two probability arrays. This is the square root
of the Jensen-Shannon divergence.
The Jensen-Shannon distance between two probability
vectors `p` and `q` is defined as,
.. 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.
This routine will normalize `p` and `q` if they don't sum to 1.0.
Parameters
----------
p : (N,) array_like
left probability vector
q : (N,) array_like
right probability vector
base : double, optional
the base of the logarithm used to compute the output
if not given, then the routine uses the default base of
scipy.stats.entropy.
axis : int, optional
Axis along which the Jensen-Shannon distances are computed. The default
is 0.
.. versionadded:: 1.7.0
keepdims : bool, optional
If this is set to `True`, the reduced axes are left in the
result as dimensions with size one. With this option,
the result will broadcast correctly against the input array.
Default is False.
.. versionadded:: 1.7.0
Returns
-------
js : double or ndarray
The Jensen-Shannon distances between `p` and `q` along the `axis`.
Notes
-----
.. versionadded:: 1.2.0
Examples
--------
>>> from scipy.spatial import distance
>>> import numpy as np
>>> distance.jensenshannon([1.0, 0.0, 0.0], [0.0, 1.0, 0.0], 2.0)
1.0
>>> distance.jensenshannon([1.0, 0.0], [0.5, 0.5])
0.46450140402245893
>>> distance.jensenshannon([1.0, 0.0, 0.0], [1.0, 0.0, 0.0])
0.0
>>> a = np.array([[1, 2, 3, 4],
... [5, 6, 7, 8],
... [9, 10, 11, 12]])
>>> b = np.array([[13, 14, 15, 16],
... [17, 18, 19, 20],
... [21, 22, 23, 24]])
>>> distance.jensenshannon(a, b, axis=0)
array([0.1954288, 0.1447697, 0.1138377, 0.0927636])
>>> distance.jensenshannon(a, b, axis=1)
array([0.1402339, 0.0399106, 0.0201815])
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�b�f�T�l�l���
�7�2��8� � � r5 c �� � t | � � } t |� � }|�t |� � }t | ||�� � \ }}}}||z }|dk rdS t d|z ||z |z z � � S )a�
Compute the Yule dissimilarity between two boolean 1-D arrays.
The Yule dissimilarity is defined as
.. math::
\frac{R}{c_{TT} * c_{FF} + \frac{R}{2}}
where :math:`c_{ij}` is the number of occurrences of
:math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for
:math:`k < n` and :math:`R = 2.0 * c_{TF} * c_{FT}`.
Parameters
----------
u : (N,) array_like, bool
Input array.
v : (N,) array_like, bool
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
yule : double
The Yule dissimilarity between vectors `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.yule([1, 0, 0], [0, 1, 0])
2.0
>>> distance.yule([1, 1, 0], [0, 1, 0])
0.0
Nr� r � r� �r� rv r` r� )rW rX rY r\ r] r^ r_ �half_Rs r3 r r s� � �L ����A�����A��}� �a� � ��0��A��;�;�;��S�#�s�C�
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Compute the Dice dissimilarity between two boolean 1-D arrays.
The Dice dissimilarity between `u` and `v`, is
.. math::
\frac{c_{TF} + c_{FT}}
{2c_{TT} + c_{FT} + c_{TF}}
where :math:`c_{ij}` is the number of occurrences of
:math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for
:math:`k < n`.
Parameters
----------
u : (N,) array_like, bool
Input 1-D array.
v : (N,) array_like, bool
Input 1-D array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
dice : double
The Dice dissimilarity between 1-D arrays `u` and `v`.
Notes
-----
This function computes the Dice dissimilarity index. To compute the
Dice similarity index, convert one to the other with similarity =
1 - dissimilarity.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.dice([1, 0, 0], [0, 1, 0])
1.0
>>> distance.dice([1, 0, 0], [1, 1, 0])
0.3333333333333333
>>> distance.dice([1, 0, 0], [2, 0, 0])
-0.3333333333333333
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Compute the Rogers-Tanimoto dissimilarity between two boolean 1-D arrays.
The Rogers-Tanimoto dissimilarity between two boolean 1-D arrays
`u` and `v`, is defined as
.. math::
\frac{R}
{c_{TT} + c_{FF} + R}
where :math:`c_{ij}` is the number of occurrences of
:math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for
:math:`k < n` and :math:`R = 2(c_{TF} + c_{FT})`.
Parameters
----------
u : (N,) array_like, bool
Input array.
v : (N,) array_like, bool
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
rogerstanimoto : double
The Rogers-Tanimoto dissimilarity between vectors
`u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.rogerstanimoto([1, 0, 0], [0, 1, 0])
0.8
>>> distance.rogerstanimoto([1, 0, 0], [1, 1, 0])
0.5
>>> distance.rogerstanimoto([1, 0, 0], [2, 0, 0])
-1.0
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Compute the Russell-Rao dissimilarity between two boolean 1-D arrays.
The Russell-Rao dissimilarity between two boolean 1-D arrays, `u` and
`v`, is defined as
.. math::
\frac{n - c_{TT}}
{n}
where :math:`c_{ij}` is the number of occurrences of
:math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for
:math:`k < n`.
Parameters
----------
u : (N,) array_like, bool
Input array.
v : (N,) array_like, bool
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
russellrao : double
The Russell-Rao dissimilarity between vectors `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.russellrao([1, 0, 0], [0, 1, 0])
1.0
>>> distance.russellrao([1, 0, 0], [1, 1, 0])
0.6666666666666666
>>> distance.russellrao([1, 0, 0], [2, 0, 0])
0.3333333333333333
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Compute the Sokal-Michener dissimilarity between two boolean 1-D arrays.
The Sokal-Michener dissimilarity between boolean 1-D arrays `u` and `v`,
is defined as
.. math::
\frac{R}
{S + R}
where :math:`c_{ij}` is the number of occurrences of
:math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for
:math:`k < n`, :math:`R = 2 * (c_{TF} + c_{FT})` and
:math:`S = c_{FF} + c_{TT}`.
Parameters
----------
u : (N,) array_like, bool
Input array.
v : (N,) array_like, bool
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
sokalmichener : double
The Sokal-Michener dissimilarity between vectors `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.sokalmichener([1, 0, 0], [0, 1, 0])
0.8
>>> distance.sokalmichener([1, 0, 0], [1, 1, 0])
0.5
>>> distance.sokalmichener([1, 0, 0], [2, 0, 0])
-1.0
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Compute the Sokal-Sneath dissimilarity between two boolean 1-D arrays.
The Sokal-Sneath dissimilarity between `u` and `v`,
.. math::
\frac{R}
{c_{TT} + R}
where :math:`c_{ij}` is the number of occurrences of
:math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for
:math:`k < n` and :math:`R = 2(c_{TF} + c_{FT})`.
Parameters
----------
u : (N,) array_like, bool
Input array.
v : (N,) array_like, bool
Input array.
w : (N,) array_like, optional
The weights for each value in `u` and `v`. Default is None,
which gives each value a weight of 1.0
Returns
-------
sokalsneath : double
The Sokal-Sneath dissimilarity between vectors `u` and `v`.
Examples
--------
>>> from scipy.spatial import distance
>>> distance.sokalsneath([1, 0, 0], [0, 1, 0])
1.0
>>> distance.sokalsneath([1, 0, 0], [1, 1, 0])
0.66666666666666663
>>> distance.sokalsneath([1, 0, 0], [2, 1, 0])
0.0
>>> distance.sokalsneath([1, 0, 0], [3, 1, 0])
-2.0
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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'.
**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
out : ndarray.
The output array
If not None, condensed distance matrix Y is stored in this array.
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)
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)
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')
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Convert a vector-form distance vector to a square-form distance
matrix, and vice-versa.
Parameters
----------
X : array_like
Either a condensed or redundant distance matrix.
force : str, optional
As with MATLAB(TM), if force is equal to ``'tovector'`` or
``'tomatrix'``, the input will be treated as a distance matrix or
distance vector respectively.
checks : bool, optional
If set to False, no checks will be made for matrix
symmetry nor zero diagonals. This is useful if it is known that
``X - X.T1`` is small and ``diag(X)`` is close to zero.
These values are ignored any way so they do not disrupt the
squareform transformation.
Returns
-------
Y : ndarray
If a condensed distance matrix is passed, a redundant one is
returned, or if a redundant one is passed, a condensed distance
matrix is returned.
Notes
-----
1. ``v = squareform(X)``
Given a square n-by-n symmetric distance matrix ``X``,
``v = squareform(X)`` returns a ``n * (n-1) / 2``
(i.e. binomial coefficient n choose 2) sized vector `v`
where :math:`v[{n \choose 2} - {n-i \choose 2} + (j-i-1)]`
is the distance between distinct points ``i`` and ``j``.
If ``X`` is non-square or asymmetric, an error is raised.
2. ``X = squareform(v)``
Given a ``n * (n-1) / 2`` sized vector ``v``
for some integer ``n >= 1`` encoding distances as described,
``X = squareform(v)`` returns a n-by-n distance matrix ``X``.
The ``X[i, j]`` and ``X[j, i]`` values are set to
:math:`v[{n \choose 2} - {n-i \choose 2} + (j-i-1)]`
and all diagonal elements are zero.
In SciPy 0.19.0, ``squareform`` stopped casting all input types to
float64, and started returning arrays of the same dtype as the input.
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Return True if input array is a valid distance matrix.
Distance matrices must be 2-dimensional numpy arrays.
They must have a zero-diagonal, and they must be symmetric.
Parameters
----------
D : array_like
The candidate object to test for validity.
tol : float, optional
The distance matrix should be symmetric. `tol` is the maximum
difference between entries ``ij`` and ``ji`` for the distance
metric to be considered symmetric.
throw : bool, optional
An exception is thrown if the distance matrix passed is not valid.
name : str, optional
The name of the variable to checked. This is useful if
throw is set to True so the offending variable can be identified
in the exception message when an exception is thrown.
warning : bool, optional
Instead of throwing an exception, a warning message is
raised.
Returns
-------
valid : bool
True if the variable `D` passed is a valid distance matrix.
Notes
-----
Small numerical differences in `D` and `D.T` and non-zeroness of
the diagonal are ignored if they are within the tolerance specified
by `tol`.
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Return True if the input array is a valid condensed distance matrix.
Condensed distance matrices must be 1-dimensional numpy arrays.
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for some positive integer n.
Parameters
----------
y : array_like
The condensed distance matrix.
warning : bool, optional
Invokes a warning if the variable passed is not a valid
condensed distance matrix. The warning message explains why
the distance matrix is not valid. `name` is used when
referencing the offending variable.
throw : bool, optional
Throws an exception if the variable passed is not a valid
condensed distance matrix.
name : bool, optional
Used when referencing the offending variable in the
warning or exception message.
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Return the number of original observations that correspond to a
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Parameters
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d : array_like
The target distance matrix.
Returns
-------
num_obs_dm : int
The number of observations in the redundant distance matrix.
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Return the number of original observations that correspond to a
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Parameters
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Y : array_like
Condensed distance matrix.
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-------
n : int
The number of observations in the condensed distance matrix `Y`.
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Compute distance between each pair of the two collections of inputs.
See Notes for common calling conventions.
Parameters
----------
XA : array_like
An :math:`m_A` by :math:`n` array of :math:`m_A`
original observations in an :math:`n`-dimensional space.
Inputs are converted to float type.
XB : array_like
An :math:`m_B` by :math:`n` array of :math:`m_B`
original observations in an :math:`n`-dimensional space.
Inputs are converted to float type.
metric : str or callable, optional
The distance metric to use. If a string, 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'.
**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 : array_like
The weight vector for metrics that support weights (e.g., Minkowski).
V : array_like
The variance vector for standardized Euclidean.
Default: var(vstack([XA, XB]), axis=0, ddof=1)
VI : array_like
The inverse of the covariance matrix for Mahalanobis.
Default: inv(cov(vstack([XA, XB].T))).T
out : ndarray
The output array
If not None, the distance matrix Y is stored in this array.
Returns
-------
Y : ndarray
A :math:`m_A` by :math:`m_B` distance matrix is returned.
For each :math:`i` and :math:`j`, the metric
``dist(u=XA[i], v=XB[j])`` is computed and stored in the
:math:`ij` th entry.
Raises
------
ValueError
An exception is thrown if `XA` and `XB` do not have
the same number of columns.
Notes
-----
The following are common calling conventions:
1. ``Y = cdist(XA, XB, 'euclidean')``
Computes the distance between :math:`m` points using
Euclidean distance (2-norm) as the distance metric between the
points. The points are arranged as :math:`m`
:math:`n`-dimensional row vectors in the matrix X.
2. ``Y = cdist(XA, XB, '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 = cdist(XA, XB, 'cityblock')``
Computes the city block or Manhattan distance between the
points.
4. ``Y = cdist(XA, XB, '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 = cdist(XA, XB, 'sqeuclidean')``
Computes the squared Euclidean distance :math:`\|u-v\|_2^2` between
the vectors.
6. ``Y = cdist(XA, XB, '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 :math:`u` and :math:`v`.
7. ``Y = cdist(XA, XB, '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 = cdist(XA, XB, '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 = cdist(XA, XB, '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 where at least one of them is non-zero.
10. ``Y = cdist(XA, XB, '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 = cdist(XA, XB, '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 = cdist(XA, XB, '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 = cdist(XA, XB, '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 = cdist(XA, XB, '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 = cdist(XA, XB, 'yule')``
Computes the Yule distance between the boolean
vectors. (see `yule` function documentation)
16. ``Y = cdist(XA, XB, 'matching')``
Synonym for 'hamming'.
17. ``Y = cdist(XA, XB, 'dice')``
Computes the Dice distance between the boolean vectors. (see
`dice` function documentation)
18. ``Y = cdist(XA, XB, 'kulczynski1')``
Computes the kulczynski distance between the boolean
vectors. (see `kulczynski1` function documentation)
19. ``Y = cdist(XA, XB, 'rogerstanimoto')``
Computes the Rogers-Tanimoto distance between the boolean
vectors. (see `rogerstanimoto` function documentation)
20. ``Y = cdist(XA, XB, 'russellrao')``
Computes the Russell-Rao distance between the boolean
vectors. (see `russellrao` function documentation)
21. ``Y = cdist(XA, XB, 'sokalmichener')``
Computes the Sokal-Michener distance between the boolean
vectors. (see `sokalmichener` function documentation)
22. ``Y = cdist(XA, XB, 'sokalsneath')``
Computes the Sokal-Sneath distance between the vectors. (see
`sokalsneath` function documentation)
23. ``Y = cdist(XA, XB, 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 = cdist(XA, XB, 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 = cdist(XA, XB, 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 = cdist(XA, XB, 'sokalsneath')
Examples
--------
Find the Euclidean distances between four 2-D coordinates:
>>> from scipy.spatial import distance
>>> import numpy as np
>>> coords = [(35.0456, -85.2672),
... (35.1174, -89.9711),
... (35.9728, -83.9422),
... (36.1667, -86.7833)]
>>> distance.cdist(coords, coords, 'euclidean')
array([[ 0. , 4.7044, 1.6172, 1.8856],
[ 4.7044, 0. , 6.0893, 3.3561],
[ 1.6172, 6.0893, 0. , 2.8477],
[ 1.8856, 3.3561, 2.8477, 0. ]])
Find the Manhattan distance from a 3-D point to the corners of the unit
cube:
>>> a = np.array([[0, 0, 0],
... [0, 0, 1],
... [0, 1, 0],
... [0, 1, 1],
... [1, 0, 0],
... [1, 0, 1],
... [1, 1, 0],
... [1, 1, 1]])
>>> b = np.array([[ 0.1, 0.2, 0.4]])
>>> distance.cdist(a, b, 'cityblock')
array([[ 0.7],
[ 0.9],
[ 1.3],
[ 1.5],
[ 1.5],
[ 1.7],
[ 2.1],
[ 2.3]])
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