# scipy.cluster.hierarchy.is_monotonic¶

scipy.cluster.hierarchy.is_monotonic(Z)[source]

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.

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.