sklearn.cluster.ward_tree

sklearn.cluster.ward_tree(X, *, connectivity=None, n_clusters=None, return_distance=False)

Ward clustering based on a Feature matrix.

Recursively merges the pair of clusters that minimally increases within-cluster variance.

The inertia matrix uses a Heapq-based representation.

This is the structured version, that takes into account some topological structure between samples.

Read more in the User Guide.

Parameters:

Xarray-like of shape (n_samples, n_features)

Feature matrix representing n_samples samples to be clustered.

connectivity{array-like, sparse matrix}, default=None

Connectivity matrix. Defines for each sample the neighboring samples following a given structure of the data. The matrix is assumed to be symmetric and only the upper triangular half is used. Default is None, i.e, the Ward algorithm is unstructured.

n_clustersint, default=None

n_clusters should be less than n_samples. Stop early the construction of the tree at n_clusters. This is useful to decrease computation time if the number of clusters is not small compared to the number of samples. In this case, the complete tree is not computed, thus the ‘children’ output is of limited use, and the ‘parents’ output should rather be used. This option is valid only when specifying a connectivity matrix.

return_distancebool, default=False

If True, return the distance between the clusters.

Returns:

childrenndarray of shape (n_nodes-1, 2)

The children of each non-leaf node. Values less than n_samples correspond to leaves of the tree which are the original samples. A node i greater than or equal to n_samples is a non-leaf node and has children children_[i - n_samples]. Alternatively at the i-th iteration, children[i][0] and children[i][1] are merged to form node n_samples + i.

n_connected_componentsint

The number of connected components in the graph.

n_leavesint

The number of leaves in the tree.

parentsndarray of shape (n_nodes,) or None

The parent of each node. Only returned when a connectivity matrix is specified, elsewhere ‘None’ is returned.

distancesndarray of shape (n_nodes-1,)

Only returned if return_distance is set to True (for compatibility). The distances between the centers of the nodes. distances[i] corresponds to a weighted Euclidean distance between the nodes children[i, 1] and children[i, 2]. If the nodes refer to leaves of the tree, then distances[i] is their unweighted Euclidean distance. Distances are updated in the following way (from scipy.hierarchy.linkage):

The new entry \(d(u,v)\) is computed as follows,

\[d(u,v) = \sqrt{\frac{|v|+|s|} {T}d(v,s)^2 + \frac{|v|+|t|} {T}d(v,t)^2 - \frac{|v|} {T}d(s,t)^2}\]

where \(u\) is the newly joined cluster consisting of clusters \(s\) and \(t\), \(v\) is an unused cluster in the forest, \(T=|v|+|s|+|t|\), and \(|*|\) is the cardinality of its argument. This is also known as the incremental algorithm.

Examples

>>> import numpy as np
>>> from sklearn.cluster import ward_tree
>>> X = np.array([[1, 2], [1, 4], [1, 0],
...               [4, 2], [4, 4], [4, 0]])
>>> children, n_connected_components, n_leaves, parents = ward_tree(X)
>>> children
array([[0, 1],
       [3, 5],
       [2, 6],
       [4, 7],
       [8, 9]])
>>> n_connected_components
1
>>> n_leaves
6