# scipy.cluster.hierarchy.average¶

scipy.cluster.hierarchy.average(y)[source]

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. Z : ndarray A linkage matrix containing the hierarchical clustering. See linkage for more information on its structure.

linkage
for advanced creation of hierarchical clusterings.
scipy.spatial.distance.pdist
pairwise distance metrics

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.

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