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scipy.spatial.distance.

# cdist

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

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

  See Notes for common calling conventions.

  Parameters:
    XA array_like

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

    XB array_like

      An \(m_B\) by \(n\) array of \(m_B\) original observations in an \(n\)-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’, ‘mahalanobis’, ‘matching’, ‘minkowski’, ‘rogerstanimoto’, ‘russellrao’, ‘seuclidean’, ‘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 \(m_A\) 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:

  1. `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.

  2. `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 \(0 < p < 1\)).

  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

     \[\sqrt{\sum {(u_i-v_i)^2 / V[x_i]}}.\]

     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 \(\|u-v\|_2^2\) between the vectors.

  6. `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 \(v\).

  7. `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 \(y\).

  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, \(p\) 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.

  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

      \[d(u,v) = \max_i {|u_i-v_i|}.\]

  12. `Y = cdist(XA, XB, 'canberra')`

      Computes the Canberra distance between the points. The Canberra distance between two points `u` and `v` is

      \[d(u,v) = \sum_i \frac{|u_i-v_i|} {|u_i|+|v_i|}.\]

  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

      \[d(u,v) = \frac{\sum_i (|u_i-v_i|)} {\sum_i (|u_i+v_i|)}\]

  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 \(\sqrt{(u-v)(1/V)(u-v)^T}\) where \((1/V)\) (the `VI` variable) is the inverse covariance. If `VI` is not None, `VI` will be used as the inverse covariance matrix.

  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, 'rogerstanimoto')`

      Computes the Rogers-Tanimoto distance between the boolean vectors. (see `rogerstanimoto` function documentation)

  19. `Y = cdist(XA, XB, 'russellrao')`

      Computes the Russell-Rao distance between the boolean vectors. (see `russellrao` function documentation)

  20. `Y = cdist(XA, XB, 'sokalsneath')`

      Computes the Sokal-Sneath distance between the vectors. (see `sokalsneath` function documentation)

  21. `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]])