scipy.spatial.distance.cdist — SciPy v1.13.1 Manual (2024)

scipy.spatial.distance.cdist(XA, XB, metric='euclidean', *, out=None, **kwargs)[source]#

Compute distance between each pair of the two collections of inputs.

See Notes for common calling conventions.

Parameters:

XAarray_like

An \(m_A\) by \(n\) array of \(m_A\)original observations in an \(n\)-dimensional space.Inputs are converted to float type.

XBarray_like

An \(m_B\) by \(n\) array of \(m_B\)original observations in an \(n\)-dimensional space.Inputs are converted to float type.

metricstr 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’.

**kwargsdict, optional

Extra arguments to metric: refer to each metric documentation for alist of all possible arguments.

Some possible arguments:

p : scalarThe p-norm to apply for Minkowski, weighted and unweighted.Default: 2.

w : array_likeThe weight vector for metrics that support weights (e.g., Minkowski).

V : array_likeThe variance vector for standardized Euclidean.Default: var(vstack([XA, XB]), axis=0, ddof=1)

VI : array_likeThe inverse of the covariance matrix for Mahalanobis.Default: inv(cov(vstack([XA, XB].T))).T

out : ndarrayThe output arrayIf not None, the distance matrix Y is stored in this array.

Returns:

Yndarray: A \(m_A\) by \(m_B\) distance matrix is returned.For each \(i\) and \(j\), the metricdist(u=XA[i], v=XB[j]) is computed and stored in the\(ij\) th entry.

Raises:

ValueError: An exception is thrown if XA and XB do not havethe same number of columns.

Notes

The following are common calling conventions:

Y = cdist(XA, XB, 'euclidean')
Computes the distance between \(m\) points usingEuclidean distance (2-norm) as the distance metric between thepoints. The points are arranged as \(m\)\(n\)-dimensional row vectors in the matrix X.
Y = cdist(XA, XB, 'minkowski', p=2.)
Computes the distances using the Minkowski distance\(\|u-v\|_p\) (\(p\)-norm) where \(p > 0\) (notethat this is only a quasi-metric if \(0 < p < 1\)).
Y = cdist(XA, XB, 'cityblock')
Computes the city block or Manhattan distance between thepoints.
Y = cdist(XA, XB, 'seuclidean', V=None)
See Also
scipy.cluster.hierarchy.linkage — SciPy v1.13.1 Manual Clustering Custom Data Using the K-Means Algorithm — Python Definitive Guide to K-Means Clustering with Scikit-Learn Custom Distance Function in K-Means Clustering with scikit-learn in Python 3 - DNMTechs - Sharing and Storing Technology Knowledge
Computes the standardized Euclidean distance. The standardizedEuclidean distance between two n-vectors u and v is
\[\sqrt{\sum {(u_i-v_i)^2 / V[x_i]}}.\]
V is the variance vector; V[i] is the variance computed over allthe i’th components of the points. If not passed, it isautomatically computed.
Y = cdist(XA, XB, 'sqeuclidean')
Computes the squared Euclidean distance \(\|u-v\|_2^2\) betweenthe vectors.
Y = cdist(XA, XB, 'cosine')
Computes the cosine distance between vectors u and v,
\[1 - \frac{u \cdot v} {{\|u\|}_2 {\|v\|}_2}\]
where \(\|*\|_2\) is the 2-norm of its argument *, and\(u \cdot v\) is the dot product of \(u\) and \(v\).
Y = cdist(XA, XB, 'correlation')
Computes the correlation distance between vectors u and v. This is
\[1 - \frac{(u - \bar{u}) \cdot (v - \bar{v})} {{\|(u - \bar{u})\|}_2 {\|(v - \bar{v})\|}_2}\]
where \(\bar{v}\) is the mean of the elements of vector v,and \(x \cdot y\) is the dot product of \(x\) and \(y\).
Y = cdist(XA, XB, 'hamming')
Computes the normalized Hamming distance, or the proportion ofthose vector elements between two n-vectors u and vwhich disagree. To save memory, the matrix X can be of typeboolean.
Y = cdist(XA, XB, 'jaccard')
Computes the Jaccard distance between the points. Given twovectors, u and v, the Jaccard distance is theproportion of those elements u[i] and v[i] thatdisagree where at least one of them is non-zero.
Y = cdist(XA, XB, 'jensenshannon')
Computes the Jensen-Shannon distance between two probability arrays.Given two probability vectors, \(p\) and \(q\), theJensen-Shannon distance is
\[\sqrt{\frac{D(p \parallel m) + D(q \parallel m)}{2}}\]
where \(m\) is the pointwise mean of \(p\) and \(q\)and \(D\) is the Kullback-Leibler divergence.
Y = cdist(XA, XB, 'chebyshev')
Computes the Chebyshev distance between the points. TheChebyshev distance between two n-vectors u and v is themaximum norm-1 distance between their respective elements. Moreprecisely, the distance is given by
\[d(u,v) = \max_i {|u_i-v_i|}.\]
Y = cdist(XA, XB, 'canberra')
Computes the Canberra distance between the points. TheCanberra distance between two points u and v is
\[d(u,v) = \sum_i \frac{|u_i-v_i|} {|u_i|+|v_i|}.\]
Y = cdist(XA, XB, 'braycurtis')
Computes the Bray-Curtis distance between the points. TheBray-Curtis distance between two points u and v is
\[d(u,v) = \frac{\sum_i (|u_i-v_i|)} {\sum_i (|u_i+v_i|)}\]
Y = cdist(XA, XB, 'mahalanobis', VI=None)
Computes the Mahalanobis distance between the points. TheMahalanobis distance between two points u and v is\(\sqrt{(u-v)(1/V)(u-v)^T}\) where \((1/V)\) (the VIvariable) is the inverse covariance. If VI is not None,VI will be used as the inverse covariance matrix.
Y = cdist(XA, XB, 'yule')
Computes the Yule distance between the booleanvectors. (see yule function documentation)
Y = cdist(XA, XB, 'matching')
Synonym for ‘hamming’.
Y = cdist(XA, XB, 'dice')
Computes the Dice distance between the boolean vectors. (seedice function documentation)
Y = cdist(XA, XB, 'kulczynski1')
Computes the kulczynski distance between the booleanvectors. (see kulczynski1 function documentation)
Y = cdist(XA, XB, 'rogerstanimoto')
Computes the Rogers-Tanimoto distance between the booleanvectors. (see rogerstanimoto function documentation)
Y = cdist(XA, XB, 'russellrao')
Computes the Russell-Rao distance between the booleanvectors. (see russellrao function documentation)
Y = cdist(XA, XB, 'sokalmichener')
Computes the Sokal-Michener distance between the booleanvectors. (see sokalmichener function documentation)
Y = cdist(XA, XB, 'sokalsneath')
Computes the Sokal-Sneath distance between the vectors. (seesokalsneath function documentation)
Y = cdist(XA, XB, f)
Computes the distance between all pairs of vectors in Xusing the user supplied 2-arity function f. For example,Euclidean distance between the vectors could be computedas follows:
```
dm = cdist(XA, XB, lambda u, v: np.sqrt(((u-v)**2).sum()))
```
Note that you should avoid passing a reference to one ofthe distance functions defined in this library. For example,:
```
dm = cdist(XA, XB, sokalsneath)
```
would calculate the pair-wise distances between the vectors inX using the Python function sokalsneath. This would result insokalsneath being called \({n \choose 2}\) times, whichis inefficient. Instead, the optimized C version is moreefficient, 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 unitcube:

>>> 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]])