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scipy.cluster.hierarchy.

# is_monotonic

scipy.cluster.hierarchy.is_monotonic(Z)

  Return True if the linkage passed is monotonic.

  The linkage is monotonic if for every cluster \(s\) and \(t\) joined, the distance between them is no less than the distance between any previously joined clusters.

  Parameters:
    Zndarray

      The linkage matrix to check for monotonicity.

  Returns:
    bbool

      A boolean indicating whether the linkage is monotonic.

  See also

  `linkage`

    for a description of what a linkage matrix is.

  Notes

  `is_monotonic` 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 | ✅ | n/a

  See Support for the array API standard for more information.

  Examples

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

  By definition, some hierarchical clustering algorithms - such as `scipy.cluster.hierarchy.ward` - produce monotonic assignments of samples to clusters; however, this is not always true for other hierarchical methods - e.g. `scipy.cluster.hierarchy.median`.

  Given a linkage matrix `Z` (as the result of a hierarchical clustering method) we can test programmatically whether it has the monotonicity property or not, using `scipy.cluster.hierarchy.is_monotonic`:

      >>> 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.        ]])
      >>> is_monotonic(Z)
      True

      >>> Z = median(pdist(X))
      >>> Z
      array([[ 0.        ,  1.        ,  1.        ,  2.        ],
             [ 3.        ,  4.        ,  1.        ,  2.        ],
             [ 9.        , 10.        ,  1.        ,  2.        ],
             [ 6.        ,  7.        ,  1.        ,  2.        ],
             [ 2.        , 12.        ,  1.11803399,  3.        ],
             [ 5.        , 13.        ,  1.11803399,  3.        ],
             [ 8.        , 15.        ,  1.11803399,  3.        ],
             [11.        , 14.        ,  1.11803399,  3.        ],
             [18.        , 19.        ,  3.        ,  6.        ],
             [16.        , 17.        ,  3.5       ,  6.        ],
             [20.        , 21.        ,  3.25      , 12.        ]])
      >>> is_monotonic(Z)
      False

  Note that this method is equivalent to just verifying that the distances in the third column of the linkage matrix appear in a monotonically increasing order.