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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.
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 with M points in N dimensions.
v : (O,N) array_like
Input array with O points in N dimensions.
rng : int or `numpy.random.Generator` or None, optional
Pseudorandom number generator state. Default is 0 so the
shuffling of `u` and `v` is reproducible.
If `rng` is passed by keyword, types other than `numpy.random.Generator` are
passed to `numpy.random.default_rng` to instantiate a ``Generator``.
If `rng` is already a ``Generator`` instance, then the provided instance is
used.
If this argument is passed by position or `seed` is passed by keyword,
legacy behavior for the argument `seed` applies:
- If `seed` is None, a new ``RandomState`` instance is used. The state is
initialized using data from ``/dev/urandom`` (or the Windows analogue)
if available or from the system clock otherwise.
- If `seed` is an int, a new ``RandomState`` instance is used,
seeded with `seed`.
- If `seed` is already a ``Generator`` or ``RandomState`` instance, then
that instance is used.
.. versionchanged:: 1.15.0
As part of the `SPEC-007 <https://scientific-python.org/specs/spec-0007/>`_
transition from use of `numpy.random.RandomState` to
`numpy.random.Generator`, this keyword was changed from `seed` to `rng`.
For an interim period, both keywords will continue to work, although only
one may be specified at a time. After the interim period, function calls
using the `seed` keyword will emit warnings. The behavior of both `seed`
and `rng` are outlined above, but only the `rng` keyword should be used in
new code.
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.
See Also
--------
scipy.spatial.procrustes : Another similarity test for two data sets
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.
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
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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� rV )r �rT rU rV s r0 r r � s � �H �Q��Q�!� $� $� $�$r2 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::
\sum_i{w_i |u_i - v_i|^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
-------
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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d�� � |�&t
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t j ||� � } d| t j |
| z � � z z
} t j | dd� � S )a/
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 of floats
Input array.
.. deprecated:: 1.15.0
Complex `u` is deprecated and will raise an error in SciPy 1.17.0
v : (N,) array_like of floats
Input array.
.. deprecated:: 1.15.0
Complex `v` is deprecated and will raise an error in SciPy 1.17.0
w : (N,) array_like of floats, 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`.
Examples
--------
Find the correlation between two arrays.
>>> from scipy.spatial.distance import correlation
>>> correlation([1, 0, 1], [1, 1, 0])
1.5
Using a weighting array, the correlation can be calculated as:
>>> correlation([1, 0, 1], [1, 1, 0], w=[0.9, 0.1, 0.1])
1.1
If centering is not needed, the correlation can be calculated as:
>>> correlation([1, 0, 1], [1, 1, 0], centered=False)
0.5
zKComplex `u` and `v` are deprecated and will raise an error in SciPy 1.17.0.r( ��
stacklevelNrM � r� )
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�7�4��c� "� "�"r2 c �( � t | ||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 of floats
Input array.
.. deprecated:: 1.15.0
Complex `u` is deprecated and will raise an error in SciPy 1.17.0
v : (N,) array_like of floats
Input array.
.. deprecated:: 1.15.0
Complex `v` is deprecated and will raise an error in SciPy 1.17.0
w : (N,) array_like of floats, 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
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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.Nz/'w' should have the same length as 'u' and 'v'.)r� rq rr rs rP rG r� r8 )rT rU rV �u_ne_vs r0 r r � s� � �X ����A�����A��w�!�'����A�B�B�B�
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������K���v�f�a� � � �
�7�6�?�?�r2 c � � t | � � } t |� � }t j | dk |dk � � }t j | dk |dk � � }|�t |� � }||z }||z }t j |� � � � � }t j |� � � � � }|dk r||z nt j d� � S )a,
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)
r )r� rG � bitwise_xor�
bitwise_orrs r� rP )rT rU rV �unequal�nonzeror/ �bs r0 r r s� � �X ����A�����A��n�Q�!�V�Q�!�V�,�,�G��m�A��F�A��F�+�+�G��}� �a� � ���g�+���g�+��
�
�7�;�;�=�=�!�!�A�
�
�7�;�;�=�=�!�!�A��1�f�f�A��E�E�"�*�Q�-�-�/r2 z�The kulczynski1 metric is deprecated since SciPy 1.15.0 and will be removed in SciPy 1.17.0. Replace usage of 'kulczynski1(u, v)' with '1/jaccard(u, v) - 1'.�rV c � � t | � � } t |� � }|�t |� � }t | ||�� � \ }}}}|||z z S )a�
Compute the Kulczynski 1 dissimilarity between two boolean 1-D arrays.
.. deprecated:: 1.15.0
This function is deprecated and will be removed in SciPy 1.17.0.
Replace usage of ``kulczynski1(u, v)`` with ``1/jaccard(u, v) - 1``.
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� rs r] )rT rU rV �_rZ r[ r\ s r0 r r � s] � �@ ����A�����A��}� �a� � ��.�q�!�q�9�9�9��Q��S�#� �#��)� � r2 c �, � t | � � } t |� � }t |t j �� � }|j d | j d k s | j d |j d k rt d� � �t | |d|z �� � S )ad
Return the standardized Euclidean distance between two 1-D arrays.
The standardized Euclidean distance between two n-vectors `u` and `v` is
.. math::
\sqrt{\sum\limits_i \frac{1}{V_i} \left(u_i-v_i \right)^2}
``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.
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 of 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
rE r z7V must be a 1-D array of the same dimension as u and v.r% r� )r� rG r� rq � TypeErrorr )rT rU r� s r0 r r � s� � �N ����A�����A���"�*�-�-�-�A��w�q�z�Q�W�Q�Z� � �1�7�1�:�����#;�#;�� &� '� '� '� �Q��Q�q�S� !� !� !�!r2 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� �absrs rP )rT rU rV �l1_diffs r0 r r sW � �J ����A�����A��!�a�%�j�j�G��}� �a� � ���g�+�� �;�;�=�=�r2 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� rG r� r� r� )rT rU r} �deltaru s r0 r r G sb � �N ����A�����A� � �r� � �B�
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��r�v�e�R� � �%�(�(�A�
�7�1�:�:�r2 c �� � t | � � } t |� � }|�5t |� � }t |dk t | |z
� � z � � S t t | |z
� � � � S )a�
Compute the Chebyshev distance.
The *Chebyshev distance* between real vectors
:math:`u \equiv (u_1, \cdots, u_n)` and
:math:`v \equiv (v_1, \cdots, v_n)` is defined as [1]_
.. math::
d_\textrm{chebyshev}(u,v) := \max_{1 \le i \le n} |u_i-v_i|
If a (non-negative) weight vector :math:`w \equiv (w_1, \cdots, w_n)`
is supplied, the *weighted Chebyshev distance* is defined to be the
weighted Minkowski distance of infinite order; that is,
.. math::
\begin{align}
d_\textrm{chebyshev}(u,v;w) &:= \lim_{p\rightarrow \infty}
\left( \sum_{i=1}^n w_i | u_i-v_i |^p \right)^\frac{1}{p} \\
&= \max_{1 \le i \le n} 1_{w_i > 0} | u_i - v_i |
\end{align}
Parameters
----------
u : (N,) array_like of floats
Input vector.
v : (N,) array_like of floats
Input vector.
w : (N,) array_like of floats, optional
Weight vector. Default is ``None``, which gives all pairs
:math:`(u_i, v_i)` the same weight ``1.0``.
Returns
-------
chebyshev : float
The Chebyshev distance between vectors `u` and `v`, optionally weighted
by `w`.
References
----------
.. [1] https://en.wikipedia.org/wiki/Chebyshev_distance
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� rs �maxr� r� s r0 r r v sh � �j ����A�����A��}� �a� � ���A��E�S��Q��Z�Z�'�(�(�(� �s�1�q�5�z�z�?�?�r2 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
rE )r� rG r� r� rs rP )rT rU rV r� �l1_sums r0 r r � s� � �J ����A���"�*�-�-�-�A��!�a�%�j�j�G�
��Q��Z�Z�F��}� �a� � ���g�+���V��� �;�;�=�=�6�:�:�<�<� '�'r2 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
rE N�ignore)�invalid)r� rG r� rs �errstater� �nansum)rT rU rV �abs_uv�abs_u�abs_v�ds r0 r r � s� � �P ����A���"�*�-�-�-�A��}� �a� � �� ��X� &� &� &� � ��Q��U�����A�����A���� �e�e�m� $�� �=��A��A� �I�a�L�L��� � � � � � � � � � ���� � � � �
�Hs �AB1�1B5�8B5r5 c �� � t j | � � } t j |� � }| t j | |d�� � z } |t j ||d�� � z }| |z dz }t | |� � }t ||� � }t j |||�� � }t j |||�� � } || z }
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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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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� �r� rs r] �float)rT rU rV rY rZ r[ r\ �half_Rs r0 r r r s� � �L ����A�����A��}� �a� � ��0��A��;�;�;��S�#�s�C�
�3�Y�F�
��{�{��s��S�6�\�S�3�Y��%7�8�9�9�9r2 c �b � t | � � } t |� � }|�t |� � }| j |j cxk r t k r n n|�| |z � � � }n�t j t | j |j � � }| � |� � } |� |� � }|�| |z � � � }n| |z |z � � � }t | ||�� � \ }}t ||z t j d|z |z |z � � z � � S )a�
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
Nr� r� ) r� rs rF rO rP rG rQ rR rS r_ r� �array)rT rU rV r\ rF rZ r[ s r0 r r � s � �^ ����A�����A��}� �a� � ���w�!�'�!�!�!�!�T�!�!�!�!�!�a�i��1�u�k�k�m�m�����s�A�G�Q�W�5�5��
�H�H�U�O�O��
�H�H�U�O�O�� �9��q�5�+�+�-�-�C�C��q�5�1�9�/�/�#�#�C�(��A��3�3�3�J�S�#� �#��)�r�x��c� �C��#�(=� >� >�>� ?� ?�?r2 c �� � t | � � } t |� � }|�t |� � }t | ||�� � \ }}}}t d||z z � � t ||z d||z z z � � z S )aI
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
Nr� r� r� �rT rU rV rY rZ r[ r\ s r0 r r � s� � �T ����A�����A��}� �a� � ��0��A��;�;�;��S�#�s�C� ���c� �"� #� #�e�C�#�I���c� �9J�,K�&L�&L� L�Lr2 c �� � t | � � } t |� � }| j |j cxk r t k r9n n6|�4| |z � � � }t t | � � � � }ns|�4| |z � � � }t t | � � � � }n=t
|� � }| |z |z � � � }|� � � }t ||z
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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
)r� rF rO rP r� r� rs )rT rU rV r\ ri s r0 r r s� � �T ����A�����A��w�!�'�!�!�!�!�T�!�!�!�!�!�a�i��1�u�k�k�m�m�� �#�a�&�&�M�M���
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�E�E�G�G�� ��S��>�>�A� � r2 z�The sokalmichener metric is deprecated since SciPy 1.15.0 and will be removed in SciPy 1.17.0. Replace usage of 'sokalmichener(u, v)' with 'rogerstanimoto(u, v)'.c �� � t | � � } t |� � }|�t |� � }t | ||�� � \ }}}}t d||z z � � t ||z d||z z z � � z S )a
Compute the Sokal-Michener dissimilarity between two boolean 1-D arrays.
.. deprecated:: 1.15.0
This function is deprecated and will be removed in SciPy 1.17.0.
Replace usage of ``sokalmichener(u, v)`` with ``rogerstanimoto(u, v)``.
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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j |d||z z z � � }|� � � st d� � �t d||z z � � |z S )aF
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'.
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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|d dz � � � � � � }||dz
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� � �t j ||f| j � � � }t | � � } t j
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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.
Examples
--------
>>> import numpy as np
>>> from scipy.spatial.distance import pdist, squareform
``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)`` computes the Euclidean distances between each pair of
points in ``x``. The distances are returned in a one-dimensional
array with length ``5*(5 - 1)/2 = 10``.
>>> distvec = pdist(x)
>>> distvec
array([2.23606798, 6.40312424, 7.34846923, 2.82842712, 4.89897949,
6.40312424, 1. , 5.38516481, 4.58257569, 5.47722558])
``squareform(distvec)`` returns the 5x5 distance matrix.
>>> m = squareform(distvec)
>>> m
array([[0. , 2.23606798, 6.40312424, 7.34846923, 2.82842712],
[2.23606798, 0. , 4.89897949, 6.40312424, 1. ],
[6.40312424, 4.89897949, 0. , 5.38516481, 4.58257569],
[7.34846923, 6.40312424, 5.38516481, 0. , 5.47722558],
[2.82842712, 1. , 4.58257569, 5.47722558, 0. ]])
When given a square distance matrix ``m``, ``squareform(m)`` returns
the one-dimensional condensed distance vector associated with the
matrix. In this case, we recover ``distvec``.
>>> squareform(m)
array([2.23606798, 6.40312424, 7.34846923, 2.82842712, 4.89897949,
6.40312424, 1. , 5.38516481, 4.58257569, 5.47722558])
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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`.
Examples
--------
>>> import numpy as np
>>> from scipy.spatial.distance import is_valid_dm
This matrix is a valid distance matrix.
>>> d = np.array([[0.0, 1.1, 1.2, 1.3],
... [1.1, 0.0, 1.0, 1.4],
... [1.2, 1.0, 0.0, 1.5],
... [1.3, 1.4, 1.5, 0.0]])
>>> is_valid_dm(d)
True
In the following examples, the input is not a valid distance matrix.
Not square:
>>> is_valid_dm([[0, 2, 2], [2, 0, 2]])
False
Nonzero diagonal element:
>>> is_valid_dm([[0, 1, 1], [1, 2, 3], [1, 3, 0]])
False
Not symmetric:
>>> is_valid_dm([[0, 1, 3], [2, 0, 1], [3, 1, 0]])
False
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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.
Their length must be a binomial coefficient :math:`{n \choose 2}`
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.
Returns
-------
bool
True if the input array is a valid condensed distance matrix,
False otherwise.
Examples
--------
>>> from scipy.spatial.distance import is_valid_y
This vector is a valid condensed distance matrix. The length is 6,
which corresponds to ``n = 4``, since ``4*(4 - 1)/2`` is 6.
>>> v = [1.0, 1.2, 1.0, 0.5, 1.3, 0.9]
>>> is_valid_y(v)
True
An input vector with length, say, 7, is not a valid condensed distance
matrix.
>>> is_valid_y([1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7])
False
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Return the number of original observations that correspond to a
square, redundant distance matrix.
Parameters
----------
d : array_like
The target distance matrix.
Returns
-------
num_obs_dm : int
The number of observations in the redundant distance matrix.
Examples
--------
Find the number of original observations corresponding
to a square redundant distance matrix d.
>>> from scipy.spatial.distance import num_obs_dm
>>> d = [[0, 100, 200], [100, 0, 150], [200, 150, 0]]
>>> num_obs_dm(d)
3
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Return the number of original observations that correspond to a
condensed distance matrix.
Parameters
----------
Y : array_like
Condensed distance matrix.
Returns
-------
n : int
The number of observations in the condensed distance matrix `Y`.
Examples
--------
Find the number of original observations corresponding to a
condensed distance matrix Y.
>>> from scipy.spatial.distance import num_obs_y
>>> Y = [1, 2, 3.5, 7, 10, 4]
>>> num_obs_y(Y)
4
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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)
.. deprecated:: 1.15.0
This metric is deprecated and will be removed in SciPy 1.17.0.
Replace usage of ``cdist(XA, XB, 'kulczynski1')`` with
``1 / cdist(XA, XB, 'jaccard') - 1``.
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)
.. deprecated:: 1.15.0
This metric is deprecated and will be removed in SciPy 1.17.0.
Replace usage of ``cdist(XA, XB, 'sokalmichener')`` with
``cdist(XA, XB, 'rogerstanimoto')``.
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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