#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on March 2019
@author: Thomas Bonald <bonald@enst.fr>
"""
import numpy as np
from scipy import sparse
from sknetwork.hierarchy.paris import AggregateGraph
from sknetwork.utils.check import check_format, get_probs, check_square
from sknetwork.utils.check import check_min_size, check_min_nnz
from sknetwork.utils.format import directed2undirected
def _instantiate_vars(adjacency: sparse.csr_matrix, weights: str = 'uniform'):
"""Initialize standard variables for metrics."""
weights_row = get_probs(weights, adjacency)
weights_col = get_probs(weights, adjacency.T)
sym_adjacency = directed2undirected(adjacency)
aggregate_graph = AggregateGraph(weights_row, weights_col, sym_adjacency.data.astype(float),
sym_adjacency.indices, sym_adjacency.indptr)
return aggregate_graph, weights_row, weights_col
def get_sampling_distributions(adjacency: sparse.csr_matrix, dendrogram: np.ndarray, weights: str = 'uniform'):
"""Get sampling distributions over each internal node of the tree.
Parameters
----------
adjacency :
Adjacency matrix of the graph.
dendrogram :
Dendrogram.
weights :
Weights of nodes.
``'degree'`` or ``'uniform'`` (default).
Returns
-------
edge_sampling: np.ndarray
Edge sampling distribution.
node_sampling: np.ndarray
Node sampling distribution.
cluster_weights: np.ndarray
Cluster weights.
"""
n = adjacency.shape[0]
aggregate_graph, weights_row, weights_col = _instantiate_vars(adjacency, weights)
cluster_weight = np.zeros(n-1)
edge_sampling = np.zeros(n-1)
node_sampling = np.zeros(n-1)
for t in range(n - 1):
i = int(dendrogram[t][0])
j = int(dendrogram[t][1])
if j in aggregate_graph.neighbors[i]:
edge_sampling[t] += 2 * aggregate_graph.neighbors[i][j]
node_sampling[t] += aggregate_graph.cluster_out_weights[i] * aggregate_graph.cluster_in_weights[j] + \
aggregate_graph.cluster_out_weights[j] * aggregate_graph.cluster_in_weights[i]
cluster_weight[t] = aggregate_graph.cluster_out_weights[i] + aggregate_graph.cluster_out_weights[j] + \
aggregate_graph.cluster_in_weights[i] + aggregate_graph.cluster_in_weights[j]
for node in {i, j}:
if node < n:
# self-loop
node_sampling[t] += aggregate_graph.cluster_out_weights[node] * aggregate_graph.cluster_in_weights[node]
if node in aggregate_graph.neighbors[node]:
edge_sampling[t] += aggregate_graph.neighbors[node][node]
aggregate_graph.merge(i, j)
return edge_sampling, node_sampling, cluster_weight / 2
[docs]def dasgupta_cost(adjacency: sparse.csr_matrix, dendrogram: np.ndarray, weights: str = 'uniform',
normalized: bool = False) -> float:
"""Dasgupta's cost of a hierarchy.
Expected size (weights = ``'uniform'``) or expected volume (weights = ``'degree'``) of the cluster induced by
random edge sampling (closest ancestor of the two nodes in the hierarchy).
Parameters
----------
adjacency :
Adjacency matrix of the graph.
dendrogram :
Dendrogram.
weights :
Weights of nodes.
``'degree'`` or ``'uniform'`` (default).
normalized :
If ``True``, normalized cost (between 0 and 1).
Returns
-------
cost : float
Cost.
Example
-------
>>> from sknetwork.hierarchy import dasgupta_score, Paris
>>> from sknetwork.data import house
>>> paris = Paris()
>>> adjacency = house()
>>> dendrogram = paris.fit_transform(adjacency)
>>> cost = dasgupta_cost(adjacency, dendrogram)
>>> np.round(cost, 2)
3.33
References
----------
Dasgupta, S. (2016). A cost function for similarity-based hierarchical clustering.
Proceedings of ACM symposium on Theory of Computing.
"""
adjacency = check_format(adjacency)
check_square(adjacency)
n = adjacency.shape[0]
check_min_size(n, 2)
edge_sampling, _, cluster_weight = get_sampling_distributions(adjacency, dendrogram, weights)
cost = edge_sampling.dot(cluster_weight)
if not normalized:
if weights == 'degree':
cost *= adjacency.data.sum()
else:
cost *= n
return cost
[docs]def dasgupta_score(adjacency: sparse.csr_matrix, dendrogram: np.ndarray, weights: str = 'uniform') -> float:
"""Dasgupta's score of a hierarchy (quality metric, between 0 and 1).
Defined as 1 - normalized Dasgupta's cost.
Parameters
----------
adjacency :
Adjacency matrix of the graph.
dendrogram :
Dendrogram.
weights :
Weights of nodes.
``'degree'`` or ``'uniform'`` (default).
Returns
-------
score : float
Score.
Example
-------
>>> from sknetwork.hierarchy import dasgupta_score, Paris
>>> from sknetwork.data import house
>>> paris = Paris()
>>> adjacency = house()
>>> dendrogram = paris.fit_transform(adjacency)
>>> score = dasgupta_score(adjacency, dendrogram)
>>> np.round(score, 2)
0.33
References
----------
Dasgupta, S. (2016). A cost function for similarity-based hierarchical clustering.
Proceedings of ACM symposium on Theory of Computing.
"""
return 1 - dasgupta_cost(adjacency, dendrogram, weights, normalized=True)
[docs]def tree_sampling_divergence(adjacency: sparse.csr_matrix, dendrogram: np.ndarray, weights: str = 'degree',
normalized: bool = True) -> float:
"""Tree sampling divergence of a hierarchy (quality metric).
Parameters
----------
adjacency :
Adjacency matrix of the graph.
dendrogram :
Dendrogram.
weights :
Weights of nodes.
``'degree'`` (default) or ``'uniform'``.
normalized :
If ``True``, normalized score (between 0 and 1).
Returns
-------
score : float
Score.
Example
-------
>>> from sknetwork.hierarchy import tree_sampling_divergence, Paris
>>> from sknetwork.data import house
>>> paris = Paris()
>>> adjacency = house()
>>> dendrogram = paris.fit_transform(adjacency)
>>> score = tree_sampling_divergence(adjacency, dendrogram)
>>> np.round(score, 2)
0.05
References
----------
Charpentier, B. & Bonald, T. (2019).
`Tree Sampling Divergence: An Information-Theoretic Metric for
Hierarchical Graph Clustering.
<https://hal.telecom-paristech.fr/hal-02144394/document>`_
Proceedings of IJCAI.
"""
adjacency = check_format(adjacency)
check_square(adjacency)
check_min_nnz(adjacency.nnz, 1)
adjacency = adjacency.astype(float)
n = adjacency.shape[0]
check_min_size(n, 2)
adjacency.data /= adjacency.data.sum()
edge_sampling, node_sampling, _ = get_sampling_distributions(adjacency, dendrogram, weights)
index = np.where(edge_sampling)[0]
score = edge_sampling[index].dot(np.log(edge_sampling[index] / node_sampling[index]))
if normalized:
weights_row = get_probs(weights, adjacency)
weights_col = get_probs(weights, adjacency.T)
inv_out_weights = sparse.diags(weights_row, shape=(n, n), format='csr')
inv_out_weights.data = 1 / inv_out_weights.data
inv_in_weights = sparse.diags(weights_col, shape=(n, n), format='csr')
inv_in_weights.data = 1 / inv_in_weights.data
sampling_ratio = inv_out_weights.dot(adjacency.dot(inv_in_weights))
inv_out_weights.data = np.ones(len(inv_out_weights.data))
inv_in_weights.data = np.ones(len(inv_in_weights.data))
edge_sampling = inv_out_weights.dot(adjacency.dot(inv_in_weights))
mutual_information = edge_sampling.data.dot(np.log(sampling_ratio.data))
if mutual_information > 0:
score /= mutual_information
return score