Skip to main content
`Ctrl`+`K`
scipy.cluster.hierarchy.

# average

scipy.cluster.hierarchy.average(y)

  Perform average/UPGMA linkage on a condensed distance matrix.

  Parameters:
    y ndarray

      The upper triangular of the distance matrix. The result of `pdist` is returned in this form.

  Returns:
    Z ndarray

      A linkage matrix containing the hierarchical clustering. See `linkage` for more information on its structure.

  See also

  `linkage`

    for advanced creation of hierarchical clusterings.

  `scipy.spatial.distance.pdist`

    pairwise distance metrics

  Notes

  `average` 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 | ✅ | ⛔
  Dask | ⚠️ merges chunks | n/a

  See Support for the array API standard for more information.

  Examples

      >>> from scipy.cluster.hierarchy import average, 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 = average(y)
      >>> Z
      array([[ 0.        ,  1.        ,  1.        ,  2.        ],
             [ 3.        ,  4.        ,  1.        ,  2.        ],
             [ 6.        ,  7.        ,  1.        ,  2.        ],
             [ 9.        , 10.        ,  1.        ,  2.        ],
             [ 2.        , 12.        ,  1.20710678,  3.        ],
             [ 5.        , 13.        ,  1.20710678,  3.        ],
             [ 8.        , 14.        ,  1.20710678,  3.        ],
             [11.        , 15.        ,  1.20710678,  3.        ],
             [16.        , 17.        ,  3.39675184,  6.        ],
             [18.        , 19.        ,  3.39675184,  6.        ],
             [20.        , 21.        ,  4.09206523, 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, 1.5, criterion='distance')
      array([1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4], dtype=int32)
      >>> fcluster(Z, 4, criterion='distance')
      array([1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2], dtype=int32)
      >>> fcluster(Z, 6, 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.