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’, ‘kulsinski’, ‘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
andv
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
andv
which disagree. To save memory, the matrixX
can be of type boolean.Y = cdist(XA, XB, 'jaccard')
Computes the Jaccard distance between the points. Given two vectors,
u
andv
, the Jaccard distance is the proportion of those elementsu[i]
andv[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
andv
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
andv
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
andv
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
andv
is \(\sqrt{(u-v)(1/V)(u-v)^T}\) where \((1/V)\) (theVI
variable) is the inverse covariance. IfVI
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, 'kulsinski')
Computes the Kulsinski distance between the boolean vectors. (see
kulsinski
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 >>> 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]])