def test_attribute_tree_sampling_probability_edge_model(self):
g = hg.get_4_adjacency_graph((3, 3))
edge_weights = np.asarray((0, 6, 2, 6, 0, 0, 5, 4, 5, 3, 2, 2))
tree, altitudes = hg.quasi_flat_zone_hierarchy(g, edge_weights)
res = hg.attribute_tree_sampling_probability(tree,
g,
edge_weights,
model='edge')
ref = np.asarray((0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 3,
26)) / np.sum(edge_weights)
self.assertTrue(np.allclose(ref, res))
def tree_sampling_divergence(tree, edge_weights, leaf_graph):
"""
Tree sampling divergence is an unsupervised measure of the quality of a hierarchical clustering of an
edge weighted graph.
It measures how well the given edge weighted graph can be reconstructed from the tree alone.
It is equal to 0 if and only if the given graph can be fully recovered from the tree.
It is defined as the Kullback-Leibler divergence between the edge sampling model :math:`p` and the independent
(null) sampling model :math:`q` of the nodes of a tree (see :func:`~higra.attribute_tree_sampling_probability`).
The tree sampling divergence on a tree :math:`T` is then
.. math::
TSD(T) = \sum_{x \in T} p(x) \log\\frac{p(x)}{q(x)}
The definition of the tree sampling divergence was proposed in:
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.
:Complexity:
The tree sampling divergence is computed in :math:`\mathcal{O}(N (\log(N) + C^2) + M)` with :math:`N` the number of
nodes in the tree, :math:`M` the number of edges in the leaf graph, and :math:`C` the maximal number of children of
a node in the tree.
:param tree: Input tree
:param edge_weights: Edge weights on the leaf graph (similarities)
:param leaf_graph: Leaf graph of the input tree (deduced from :class:`~higra.CptHierarchy`)
:return: a real number
"""
num_l = tree.num_leaves()
p = hg.attribute_tree_sampling_probability(tree, leaf_graph, edge_weights,
'edge')[num_l:]
q = hg.attribute_tree_sampling_probability(tree, leaf_graph, edge_weights,
'null')[num_l:]
index, = np.where(p)
return np.sum(p[index] * np.log(p[index] / q[index]))
def test_attribute_tree_sampling_probability_null_model(self):
g = hg.get_4_adjacency_graph((3, 3))
edge_weights = np.asarray((0, 6, 2, 6, 0, 0, 5, 4, 5, 3, 2, 2))
tree, altitudes = hg.quasi_flat_zone_hierarchy(g, edge_weights)
res = hg.attribute_tree_sampling_probability(tree,
g,
edge_weights,
model='null')
Z = np.sum(edge_weights)
ref = np.asarray(
(0, 0, 0, 0, 0, 0, 0, 0, 0,
6 * 8, 2 * 7,
11 * 15,
6 * 2 + 6 * 7 + 8 * 2 + 8 * 7,
7 * 9 + 7 * 5 + 9 * 5,
6 * 7 + 6 * 9 + 6 * 5 + 8 * 7 + 8 * 9 + 8 * 5 + 2 * 7 + 2 * 9 + 2 * 5 + 7 * 7 + 7 * 9 + 7 * 5,
6 * 11 + 6 * 15 + 8 * 11 + 8 * 15 + 2 * 11 + 2 * 15 + 11 * 7 + 11 * 7 + 11 * 9 + 11 * 5 + 15 * 7 + 15 * 7 + 15 * 9 + 15 * 5)) / \
(Z * Z)
self.assertTrue(np.allclose(ref, res))