scipy.spatial.distance.cdist#

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 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:

Yndarray: A \(m_A\) by \(m_B\) distance matrix is returned. For each \(i\) and \(j\), the metric dist(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 have the same number of columns.

Notes

The following are common calling conventions:

Y = cdist(XA, XB, '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.
Y = cdist(XA, XB, 'minkowski', p=2.)

Computes the distances using the Minkowski distance \(\|u-v\|_p\) (\(p\)-norm) where \(p > 0\) (note that this is only a quasi-metric if \(0 < p < 1\)).
Y = cdist(XA, XB, 'cityblock')

Computes the city block or Manhattan distance between the points.
Y = cdist(XA, XB, 'seuclidean', V=None)

Computes the standardized Euclidean distance. The standardized Euclidean 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 all the i’th components of the points. If not passed, it is automatically computed.
Y = cdist(XA, XB, 'sqeuclidean')

Computes the squared Euclidean distance \(\|u-v\|_2^2\) between the 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 of those vector elements between two n-vectors u and v which disagree. To save memory, the matrix X can be of type boolean.
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.
Y = cdist(XA, XB, 'jensenshannon')

Computes the Jensen-Shannon distance between two probability arrays. Given two probability vectors, \(p\) and \(q\), the Jensen-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. 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

\[d(u,v) = \max_i {|u_i-v_i|}.\]
Y = cdist(XA, XB, 'canberra')

Computes the Canberra distance between the points. The Canberra 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. The Bray-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. The Mahalanobis distance between two points u and v is \(\sqrt{(u-v)(1/V)(u-v)^T}\) where \((1/V)\) (the VI variable) 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 boolean vectors. (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. (see dice function documentation)
Y = cdist(XA, XB, 'kulczynski1')

Computes the kulczynski distance between the boolean vectors. (see kulczynski1 function documentation)
Y = cdist(XA, XB, 'rogerstanimoto')

Computes the Rogers-Tanimoto distance between the boolean vectors. (see rogerstanimoto function documentation)
Y = cdist(XA, XB, 'russellrao')

Computes the Russell-Rao distance between the boolean vectors. (see russellrao function documentation)
Y = cdist(XA, XB, 'sokalmichener')

Computes the Sokal-Michener distance between the boolean vectors. (see sokalmichener function documentation)
Y = cdist(XA, XB, 'sokalsneath')

Computes the Sokal-Sneath distance between the vectors. (see sokalsneath function documentation)
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 \({n \choose 2}\) 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]])