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Module « scipy.cluster.hierarchy »

Fonction cophenet - module scipy.cluster.hierarchy

Signature de la fonction cophenet

def cophenet(Z, Y=None) 

Description

help(scipy.cluster.hierarchy.cophenet)

Calculate the cophenetic distances between each observation in
the hierarchical clustering defined by the linkage ``Z``.

Suppose ``p`` and ``q`` are original observations in
disjoint clusters ``s`` and ``t``, respectively and
``s`` and ``t`` are joined by a direct parent cluster
``u``. The cophenetic distance between observations
``i`` and ``j`` is simply the distance between
clusters ``s`` and ``t``.

Parameters
----------
Z : ndarray
    The hierarchical clustering encoded as an array
    (see `linkage` function).
Y : ndarray (optional)
    Calculates the cophenetic correlation coefficient ``c`` of a
    hierarchical clustering defined by the linkage matrix `Z`
    of a set of :math:`n` observations in :math:`m`
    dimensions. `Y` is the condensed distance matrix from which
    `Z` was generated.

Returns
-------
c : ndarray
    The cophentic correlation distance (if ``Y`` is passed).
d : ndarray
    The cophenetic distance matrix in condensed form. The
    :math:`ij` th entry is the cophenetic distance between
    original observations :math:`i` and :math:`j`.

See Also
--------
linkage :
    for a description of what a linkage matrix is.
scipy.spatial.distance.squareform :
    transforming condensed matrices into square ones.

Examples
--------
>>> from scipy.cluster.hierarchy import single, cophenet
>>> from scipy.spatial.distance import pdist, squareform

Given a dataset ``X`` and a linkage matrix ``Z``, the cophenetic distance
between two points of ``X`` is the distance between the largest two
distinct clusters that each of the points:

>>> 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]]

``X`` corresponds to this dataset ::

    x x    x x
    x        x

    x        x
    x x    x x

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

The output of the `scipy.cluster.hierarchy.cophenet` method is
represented in condensed form. We can use
`scipy.spatial.distance.squareform` to see the output as a
regular matrix (where each element ``ij`` denotes the cophenetic distance
between each ``i``, ``j`` pair of points in ``X``):

>>> squareform(cophenet(Z))
array([[0., 1., 1., 2., 2., 2., 2., 2., 2., 2., 2., 2.],
       [1., 0., 1., 2., 2., 2., 2., 2., 2., 2., 2., 2.],
       [1., 1., 0., 2., 2., 2., 2., 2., 2., 2., 2., 2.],
       [2., 2., 2., 0., 1., 1., 2., 2., 2., 2., 2., 2.],
       [2., 2., 2., 1., 0., 1., 2., 2., 2., 2., 2., 2.],
       [2., 2., 2., 1., 1., 0., 2., 2., 2., 2., 2., 2.],
       [2., 2., 2., 2., 2., 2., 0., 1., 1., 2., 2., 2.],
       [2., 2., 2., 2., 2., 2., 1., 0., 1., 2., 2., 2.],
       [2., 2., 2., 2., 2., 2., 1., 1., 0., 2., 2., 2.],
       [2., 2., 2., 2., 2., 2., 2., 2., 2., 0., 1., 1.],
       [2., 2., 2., 2., 2., 2., 2., 2., 2., 1., 0., 1.],
       [2., 2., 2., 2., 2., 2., 2., 2., 2., 1., 1., 0.]])

In this example, the cophenetic distance between points on ``X`` that are
very close (i.e., in the same corner) is 1. For other pairs of points is 2,
because the points will be located in clusters at different
corners - thus, the distance between these clusters will be larger.



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