# Copyright (c) Microsoft Corporation and contributors.
# Licensed under the MIT License.
import numpy as np
from scipy.linalg import orthogonal_procrustes
from ..align import OrthogonalProcrustes
from ..embed import AdjacencySpectralEmbed, OmnibusEmbed, select_dimension
from ..simulations import rdpg
from ..utils import import_graph, is_symmetric
from .base import BaseInference
[docs]class LatentPositionTest(BaseInference):
r"""
Two-sample hypothesis test for the problem of determining whether two random
dot product graphs have the same latent positions.
This test assumes that the two input graphs are vertex aligned, that is,
there is a known mapping between vertices in the two graphs and the input graphs
have their vertices sorted in the same order. Currently, the function only
supports undirected graphs.
Read more in the :ref:`tutorials <inference_tutorials>`
Parameters
----------
embedding : string, { 'ase' (default), 'omnibus'}
String describing the embedding method to use:
- 'ase'
Embed each graph separately using adjacency spectral embedding
and use Procrustes to align the embeddings.
- 'omnibus'
Embed all graphs simultaneously using omnibus embedding.
n_components : None (default), or int
Number of embedding dimensions. If None, the optimal embedding
dimensions are found by the Zhu and Godsi algorithm.
test_case : string, {'rotation' (default), 'scalar-rotation', 'diagonal-rotation'}
describes the exact form of the hypothesis to test when using 'ase' or 'lse'
as an embedding method. Ignored if using 'omnibus'. Given two latent positions,
:math:`X_1` and :math:`X_2`, and an orthogonal rotation matrix :math:`R` that
minimizes :math:`||X_1 - X_2 R||_F`:
- 'rotation'
.. math:: H_o: X_1 = X_2 R
- 'scalar-rotation'
.. math:: H_o: X_1 = c X_2 R
where :math:`c` is a scalar, :math:`c > 0`
- 'diagonal-rotation'
.. math:: H_o: X_1 = D X_2 R
where :math:`D` is an arbitrary diagonal matrix
n_bootstraps : int, optional (default 500)
Number of bootstrap simulations to run to generate the null distribution
Attributes
----------
null_distribution_1_, null_distribution_2_ : np.ndarray (n_bootstraps,)
The distribution of T statistics generated under the null, using the first and
and second input graph, respectively. The latent positions of each sample graph
are used independently to sample random dot product graphs, so two null
distributions are generated
sample_T_statistic_ : float
The observed difference between the embedded positions of the two input graphs
after an alignment (the type of alignment depends on ``test_case``)
p_value_1_, p_value_2_ : float
The p value estimated from the null distributions from sample 1 and sample 2.
p_value_ : float
The overall p value from the test; this is the max of ``p_value_1_`` and ``p_value_2_``
See also
--------
graspologic.embed.AdjacencySpectralEmbed
graspologic.embed.OmnibusEmbed
graspologic.embed.selectSVD
References
----------
.. [1] Tang, M., A. Athreya, D. Sussman, V. Lyzinski, Y. Park, Priebe, C.E.
"A Semiparametric Two-Sample Hypothesis Testing Problem for Random Graphs"
Journal of Computational and Graphical Statistics, Vol. 26(2), 2017
"""
def __init__(
self, embedding="ase", n_components=None, n_bootstraps=500, test_case="rotation"
):
if type(embedding) is not str:
raise TypeError("embedding must be str")
if type(n_bootstraps) is not int:
raise TypeError()
if type(test_case) is not str:
raise TypeError()
if n_bootstraps < 1:
raise ValueError(
"{} is invalid number of bootstraps, must be greater than 1".format(
n_bootstraps
)
)
if embedding not in ["ase", "omnibus"]:
raise ValueError("{} is not a valid embedding method.".format(embedding))
if test_case not in ["rotation", "scalar-rotation", "diagonal-rotation"]:
raise ValueError(
"test_case must be one of 'rotation', 'scalar-rotation',"
+ "'diagonal-rotation'"
)
super().__init__(n_components=n_components)
self.embedding = embedding
self.n_bootstraps = n_bootstraps
self.test_case = test_case
# paper uses these always, but could be kwargs eventually. need to test
self.rescale = False
self.loops = False
def _bootstrap(self, X_hat):
t_bootstrap = np.zeros(self.n_bootstraps)
for i in range(self.n_bootstraps):
A1_simulated = rdpg(X_hat, rescale=self.rescale, loops=self.loops)
A2_simulated = rdpg(X_hat, rescale=self.rescale, loops=self.loops)
X1_hat_simulated, X2_hat_simulated = self._embed(
A1_simulated, A2_simulated, check_lcc=False
)
t_bootstrap[i] = self._difference_norm(X1_hat_simulated, X2_hat_simulated)
return t_bootstrap
def _difference_norm(self, X1, X2):
if self.embedding in ["ase"]:
if self.test_case == "rotation":
pass
elif self.test_case == "scalar-rotation":
X1 = X1 / np.linalg.norm(X1, ord="fro")
X2 = X2 / np.linalg.norm(X2, ord="fro")
elif self.test_case == "diagonal-rotation":
normX1 = np.sum(X1 ** 2, axis=1)
normX2 = np.sum(X2 ** 2, axis=1)
normX1[normX1 <= 1e-15] = 1
normX2[normX2 <= 1e-15] = 1
X1 = X1 / np.sqrt(normX1[:, None])
X2 = X2 / np.sqrt(normX2[:, None])
aligner = OrthogonalProcrustes()
X1 = aligner.fit_transform(X1, X2)
return np.linalg.norm(X1 - X2)
def _embed(self, A1, A2, check_lcc=True):
if self.embedding == "ase":
X1_hat = AdjacencySpectralEmbed(
n_components=self.n_components, check_lcc=check_lcc
).fit_transform(A1)
X2_hat = AdjacencySpectralEmbed(
n_components=self.n_components, check_lcc=check_lcc
).fit_transform(A2)
elif self.embedding == "omnibus":
X_hat_compound = OmnibusEmbed(
n_components=self.n_components, check_lcc=check_lcc
).fit_transform((A1, A2))
X1_hat = X_hat_compound[0]
X2_hat = X_hat_compound[1]
return (X1_hat, X2_hat)
[docs] def fit(self, A1, A2):
"""
Fits the test to the two input graphs
Parameters
----------
A1, A2 : nx.Graph, nx.DiGraph, nx.MultiDiGraph, nx.MultiGraph, np.ndarray
The two graphs to run a hypothesis test on.
If np.ndarray, shape must be ``(n_vertices, n_vertices)`` for both graphs,
where ``n_vertices`` is the same for both
Returns
-------
self
"""
A1 = import_graph(A1)
A2 = import_graph(A2)
if not is_symmetric(A1) or not is_symmetric(A2):
raise NotImplementedError() # TODO asymmetric case
if A1.shape != A2.shape:
raise ValueError("Input matrices do not have matching dimensions")
if self.n_components is None:
# get the last elbow from ZG for each and take the maximum
num_dims1 = select_dimension(A1)[0][-1]
num_dims2 = select_dimension(A2)[0][-1]
self.n_components = max(num_dims1, num_dims2)
X_hats = self._embed(A1, A2)
sample_T_statistic = self._difference_norm(X_hats[0], X_hats[1])
null_distribution_1 = self._bootstrap(X_hats[0])
null_distribution_2 = self._bootstrap(X_hats[1])
# uisng exact mc p-values (see, for example, Phipson and Smyth, 2010)
p_value_1 = (
len(null_distribution_1[null_distribution_1 >= sample_T_statistic]) + 1
) / (self.n_bootstraps + 1)
p_value_2 = (
len(null_distribution_2[null_distribution_2 >= sample_T_statistic]) + 1
) / (self.n_bootstraps + 1)
p_value = max(p_value_1, p_value_2)
self.null_distribution_1_ = null_distribution_1
self.null_distribution_2_ = null_distribution_2
self.sample_T_statistic_ = sample_T_statistic
self.p_value_1_ = p_value_1
self.p_value_2_ = p_value_2
self.p_value_ = p_value
return self