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# scipy.spatial.distance.pdist

scipy.spatial.distance.pdist(X, metric='euclidean', *, out=None, **kwargs)

  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 \(i\) and \(j\) (where \(i<j<m\)),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 \(\|u-v\|_p\) (\(p\)-norm) where \(p > 0\) (note that this is only a quasi-metric if \(0 < p < 1\)).

  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

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

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

  6. `Y = pdist(X, '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 = pdist(X, '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 = 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, \(p\) 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 = 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

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

  12. `Y = pdist(X, 'canberra')`

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

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

  13. `Y = pdist(X, 'braycurtis')`

      Computes the Bray-Curtis distance between the points. The Bray-Curtis distance between two points `u` and `v` is

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

  14. `Y = pdist(X, 'mahalanobis', VI=None)`

      Computes the Mahalanobis distance between the points. The Mahalanobis distance between two points `u` and `v` is \(\sqrt{(u-v)(1/V)(u-v)^T}\) 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 = 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)

  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)

  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 \({n \choose 2}\) 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.])