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# scipy.cluster.hierarchy.ward

scipy.cluster.hierarchy.ward(y)

  Perform Ward’s linkage on a condensed distance matrix.

  See `linkage` for more information on the return structure and algorithm.

  The following are common calling conventions:

  1. `Z = ward(y)` Performs Ward’s linkage on the condensed distance matrix `y`.

  2. `Z = ward(X)` Performs Ward’s linkage on the observation matrix `X` using Euclidean distance as the distance metric.

  Parameters:
    y ndarray

      A condensed distance matrix. A condensed distance matrix is a flat array containing the upper triangular of the distance matrix. This is the form that `pdist` returns. Alternatively, a collection of m observation vectors in n dimensions may be passed as an m by n array.

  Returns:
    Z ndarray

      The hierarchical clustering encoded as a linkage matrix. See `linkage` for more information on the return structure and algorithm.

  See also

  `linkage`

    for advanced creation of hierarchical clusterings.

  `scipy.spatial.distance.pdist`

    pairwise distance metrics

  Examples

      >>> from scipy.cluster.hierarchy import ward, fcluster
      >>> from scipy.spatial.distance import pdist

  First, we need a toy dataset to play with:

      x x    x x
      x        x

      x        x
      x x    x x

      >>> X = [[0, 0], [0, 1], [1, 0],
      ...      [0, 4], [0, 3], [1, 4],
      ...      [4, 0], [3, 0], [4, 1],
      ...      [4, 4], [3, 4], [4, 3]]

  Then, we get a condensed distance matrix from this dataset:

      >>> y = pdist(X)

  Finally, we can perform the clustering:

      >>> Z = ward(y)
      >>> Z
      array([[ 0.        ,  1.        ,  1.        ,  2.        ],
             [ 3.        ,  4.        ,  1.        ,  2.        ],
             [ 6.        ,  7.        ,  1.        ,  2.        ],
             [ 9.        , 10.        ,  1.        ,  2.        ],
             [ 2.        , 12.        ,  1.29099445,  3.        ],
             [ 5.        , 13.        ,  1.29099445,  3.        ],
             [ 8.        , 14.        ,  1.29099445,  3.        ],
             [11.        , 15.        ,  1.29099445,  3.        ],
             [16.        , 17.        ,  5.77350269,  6.        ],
             [18.        , 19.        ,  5.77350269,  6.        ],
             [20.        , 21.        ,  8.16496581, 12.        ]])

  The linkage matrix `Z` represents a dendrogram - see `scipy.cluster.hierarchy.linkage` for a detailed explanation of its contents.

  We can use `scipy.cluster.hierarchy.fcluster` to see to which cluster each initial point would belong given a distance threshold:

      >>> fcluster(Z, 0.9, criterion='distance')
      array([ 1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12], dtype=int32)
      >>> fcluster(Z, 1.1, criterion='distance')
      array([1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8], dtype=int32)
      >>> fcluster(Z, 3, criterion='distance')
      array([1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4], dtype=int32)
      >>> fcluster(Z, 9, criterion='distance')
      array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int32)

  Also, `scipy.cluster.hierarchy.dendrogram` can be used to generate a plot of the dendrogram.