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scipy.cluster.hierarchy.

# fcluster

scipy.cluster.hierarchy.fcluster(Z, t, criterion='inconsistent', depth=2, R=None, monocrit=None)

  Form flat clusters from the hierarchical clustering defined by the given linkage matrix.

  Parameters:
    Zndarray

      The hierarchical clustering encoded with the matrix returned by the `linkage` function.

    tscalar
      For criteria ‘inconsistent’, ‘distance’ or ‘monocrit’,

        this is the threshold to apply when forming flat clusters.

      For ‘maxclust’ or ‘maxclust_monocrit’ criteria,

        this would be max number of clusters requested.

    criterionstr, optional

      The criterion to use in forming flat clusters. This can be any of the following values:

      > `inconsistent` :

      >   If a cluster node and all its descendants have an inconsistent value less than or equal to t, then all its leaf descendants belong to the same flat cluster. When no non-singleton cluster meets this criterion, every node is assigned to its own cluster. (Default)

      > `distance` :

      >   Forms flat clusters so that the original observations in each flat cluster have no greater a cophenetic distance than t.

      > `maxclust` :

      >   Finds a minimum threshold `r` so that the cophenetic distance between any two original observations in the same flat cluster is no more than `r` and no more than t flat clusters are formed.

      > `monocrit` :

      >   Forms a flat cluster from a cluster node c with index i when `monocrit[j] <= t`.

      >   For example, to threshold on the maximum mean distance as computed in the inconsistency matrix R with a threshold of 0.8 do:

      >       MR = maxRstat(Z, R, 3)
      >       fcluster(Z, t=0.8, criterion='monocrit', monocrit=MR)

      > `maxclust_monocrit` :

      >   Forms a flat cluster from a non-singleton cluster node `c` when `monocrit[i] <= r` for all cluster indices `i` below and including `c`. `r` is minimized such that no more than `t` flat clusters are formed. monocrit must be monotonic. For example, to minimize the threshold t on maximum inconsistency values so that no more than 3 flat clusters are formed, do:

      >       MI = maxinconsts(Z, R)
      >       fcluster(Z, t=3, criterion='maxclust_monocrit', monocrit=MI)

    depthint, optional

      The maximum depth to perform the inconsistency calculation. It has no meaning for the other criteria. Default is 2.

    Rndarray, optional

      The inconsistency matrix to use for the `'inconsistent'` criterion. This matrix is computed if not provided.

    monocritndarray, optional

      An array of length n-1. monocrit[i] is the statistics upon which non-singleton i is thresholded. The monocrit vector must be monotonic, i.e., given a node c with index i, for all node indices j corresponding to nodes below c, `monocrit[i] >= monocrit[j]`.

  Returns:
    fclusterndarray

      An array of length `n`. `T[i]` is the flat cluster number to which original observation `i` belongs.

  See also

  `linkage`

    for information about hierarchical clustering methods work.

  Notes

  `fcluster` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ⛔
  PyTorch | ✅ | ⛔
  JAX | ⚠️ no JIT | ⛔
  Dask | ⚠️ computes graph | n/a

  See Support for the array API standard for more information.

  Examples

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

  All cluster linkage methods - e.g., `scipy.cluster.hierarchy.ward` generate a linkage matrix `Z` as their output:

      >>> 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.        ]])

  This matrix represents a dendrogram, where the first and second elements are the two clusters merged at each step, the third element is the distance between these clusters, and the fourth element is the size of the new cluster - the number of original data points included.

  `scipy.cluster.hierarchy.fcluster` can be used to flatten the dendrogram, obtaining as a result an assignation of the original data points to single clusters.

  This assignation mostly depends on a distance threshold `t` - the maximum inter-cluster distance allowed:

      >>> fcluster(Z, t=0.9, criterion='distance')
      array([ 1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12], dtype=int32)

      >>> fcluster(Z, t=1.1, criterion='distance')
      array([1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8], dtype=int32)

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

      >>> fcluster(Z, t=9, criterion='distance')
      array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int32)

  In the first case, the threshold `t` is too small to allow any two samples in the data to form a cluster, so 12 different clusters are returned.

  In the second case, the threshold is large enough to allow the first 4 points to be merged with their nearest neighbors. So, here, only 8 clusters are returned.

  The third case, with a much higher threshold, allows for up to 8 data points to be connected - so 4 clusters are returned here.

  Lastly, the threshold of the fourth case is large enough to allow for all data points to be merged together - so a single cluster is returned.