# scipy.cluster.hierarchy.leaders

scipy.cluster.hierarchy.leaders(Z, T)

  Return the root nodes in a hierarchical clustering.

  Returns the root nodes in a hierarchical clustering corresponding to a cut defined by a flat cluster assignment vector `T`. See the `fcluster` function for more information on the format of `T`.

  For each flat cluster \(j\) of the \(k\) flat clusters represented in the n-sized flat cluster assignment vector `T`, this function finds the lowest cluster node \(i\) in the linkage tree Z, such that:

  > - leaf descendants belong only to flat cluster j (i.e., `T[p]==j` for all \(p\) in \(S(i)\), where \(S(i)\) is the set of leaf ids of descendant leaf nodes with cluster node \(i\))

  > - there does not exist a leaf that is not a descendant with \(i\) that also belongs to cluster \(j\) (i.e., `T[q]!=j` for all \(q\) not in \(S(i)\)). If this condition is violated, `T` is not a valid cluster assignment vector, and an exception will be thrown.

  Parameters:
    Z ndarray

      The hierarchical clustering encoded as a matrix. See `linkage` for more information.

    T ndarray

      The flat cluster assignment vector.

  Returns:
    L ndarray

      The leader linkage node id’s stored as a k-element 1-D array, where `k` is the number of flat clusters found in `T`.

      `L[j]=i` is the linkage cluster node id that is the leader of flat cluster with id M[j]. If `i < n`, `i` corresponds to an original observation, otherwise it corresponds to a non-singleton cluster.

    M ndarray

      The leader linkage node id’s stored as a k-element 1-D array, where `k` is the number of flat clusters found in `T`. This allows the set of flat cluster ids to be any arbitrary set of `k` integers.

      For example: if `L[3]=2` and `M[3]=8`, the flat cluster with id 8’s leader is linkage node 2.

  See also

  `fcluster`

    for the creation of flat cluster assignments.

  Examples

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

  Given a linkage matrix `Z` - obtained after apply a clustering method to a dataset `X` - and a flat cluster assignment array `T`:

      >>> 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]]

      >>> Z = ward(pdist(X))
      >>> 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.        ]])

      >>> T = fcluster(Z, 3, criterion='distance')
      >>> T
      array([1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4], dtype=int32)

  `scipy.cluster.hierarchy.leaders` returns the indices of the nodes in the dendrogram that are the leaders of each flat cluster:

      >>> L, M = leaders(Z, T)
      >>> L
      array([16, 17, 18, 19], dtype=int32)

  (remember that indices 0-11 point to the 12 data points in `X`, whereas indices 12-22 point to the 11 rows of `Z`)

  `scipy.cluster.hierarchy.leaders` also returns the indices of the flat clusters in `T`:

      >>> M
      array([1, 2, 3, 4], dtype=int32)