#!/usr/bin/env python
from warnings import warn
from itertools import combinations
from joblib import Parallel, delayed
import networkx as nx
from pgmpy.base import PDAG
from pgmpy.estimators import StructureEstimator
from pgmpy.estimators.CITests import chi_square, pearsonr, independence_match
[docs]class PC(StructureEstimator):
def __init__(self, data=None, independencies=None, **kwargs):
"""
Class for constraint-based estimation of DAGs using the PC algorithm
from a given data set. Identifies (conditional) dependencies in data
set using chi_square dependency test and uses the PC algorithm to
estimate a DAG pattern that satisfies the identified dependencies. The
DAG pattern can then be completed to a faithful DAG, if possible.
Parameters
----------
data: pandas DataFrame object
datafame object where each column represents one variable. (If some
values in the data are missing the data cells should be set to
`numpy.NaN`. Note that pandas converts each column containing
`numpy.NaN`s to dtype `float`.)
References
----------
[1] Koller & Friedman, Probabilistic Graphical Models - Principles and Techniques,
2009, Section 18.2
[2] Neapolitan, Learning Bayesian Networks, Section 10.1.2 for the PC algorithm (page 550), http://www.cs.technion.ac.il/~dang/books/Learning%20Bayesian%20Networks(Neapolitan,%20Richard).pdf
"""
super(PC, self).__init__(data=data, independencies=independencies, **kwargs)
[docs] def estimate(
self,
variant="stable",
ci_test="chi_square",
max_cond_vars=5,
return_type="dag",
significance_level=0.01,
n_jobs=-1,
**kwargs,
):
"""
Estimates a DAG/PDAG from the given dataset using the PC algorithm which
is a constraint-based structure learning algorithm[1]. The independencies
in the dataset are identified by doing statistical independece test. This
method returns a DAG/PDAG structure which is faithful to the independencies
implied by the dataset
Parameters
----------
variant: str (one of "orig", "stable", "parallel")
The variant of PC algorithm to run.
"orig": The original PC algorithm. Might not give the same
results in different runs but does less independence
tests compared to stable.
"stable": Gives the same result in every run but does needs to
do more statistical independence tests.
"parallel": Parallel version of PC Stable. Can run on multiple
cores with the same result on each run.
ci_test: str or fun
The statistical test to use for testing conditional independence in
the dataset. If `str` values should be one of:
"independence_match": If using this option, an additional parameter
`independencies` must be specified.
"chi_square": Uses the Chi-Square independence test. This works
only for discrete datasets.
"pearsonr": Uses the pertial correlation based on pearson
correlation coefficient to test independence. This works
only for continuous datasets.
max_cond_vars: int
The maximum number of conditional variables allowed to do the statistical
test with.
return_type: str (one of "dag", "cpdag", "pdag", "skeleton")
The type of structure to return.
If `return_type=pdag` or `return_type=cpdag`: a partially directed structure is returned.
If `return_type=dag`, a fully directed structure is returned if it
is possible to orient all the edges.
If `return_type="skeleton", returns an undirected graph along
with the separating sets.
significance_level: float (default: 0.01)
The statistical tests use this value to compare with the p-value of
the test to decide whether the tested variables are independent or
not. Different tests can treat this parameter differently:
1. Chi-Square: If p-value > significance_level, it assumes that the
independence condition satisfied in the data.
2. pearsonr: If p-value > significance_level, it assumes that the
independence condition satisfied in the data.
Returns
-------
model: DAG-instance, PDAG-instance, or (networkx.UndirectedGraph, dict)
The estimated model structure, can be a partially directed graph (PDAG)
or a fully directed graph (DAG), or (Undirected Graph, separating sets)
depending on the value of `return_type` argument.
References
----------
[1] Original PC: P. Spirtes, C. Glymour, and R. Scheines, Causation,
Prediction, and Search, 2nd ed. Cambridge, MA: MIT Press, 2000.
[2] Stable PC: D. Colombo and M. H. Maathuis, “A modification of the PC algorithm
yielding order-independent skeletons,” ArXiv e-prints, Nov. 2012.
[3] Parallel PC: Le, Thuc, et al. "A fast PC algorithm for high dimensional causal
discovery with multi-core PCs." IEEE/ACM transactions on computational
biology and bioinformatics (2016).
Examples
--------
>>> import pandas as pd
>>> import numpy as np
>>> from pgmpy.estimators import ConstraintBasedEstimator
>>> data = pd.DataFrame(np.random.randint(0, 5, size=(2500, 3)), columns=list('XYZ'))
>>> data['sum'] = data.sum(axis=1)
>>> print(data)
X Y Z sum
0 3 0 1 4
1 1 4 3 8
2 0 0 3 3
3 0 2 3 5
4 2 1 1 4
... .. .. .. ...
2495 2 3 0 5
2496 1 1 2 4
2497 0 4 2 6
2498 0 0 0 0
2499 2 4 0 6
[2500 rows x 4 columns]
>>> c = PC(data)
>>> model = c.estimate()
>>> print(model.edges())
[('Z', 'sum'), ('X', 'sum'), ('Y', 'sum')]
"""
# Step 0: Do checks that the specified parameters are correct, else throw meaningful error.
if variant not in ("orig", "stable", "parallel"):
raise ValueError(
f"variant must be one of: orig, stable, or parallel. Got: {variant}"
)
elif (not callable(ci_test)) and (
ci_test not in ("chi_square", "independence_match", "pearsonr")
):
raise ValueError(
"ci_test must be a callable or one of: chi_square, pearsonr, independence_match"
)
if (ci_test == "independence_match") and (self.independencies is None):
raise ValueError(
"For using independence_match, independencies argument must be specified"
)
elif (ci_test in ("chi_square", "pearsonr")) and (self.data is None):
raise ValueError(
"For using Chi Square or Pearsonr, data arguement must be specified"
)
# Step 1: Run the PC algorithm to build the skeleton and get the separating sets.
skel, separating_sets = self.build_skeleton(
ci_test=ci_test,
max_cond_vars=max_cond_vars,
significance_level=significance_level,
variant=variant,
n_jobs=n_jobs,
**kwargs,
)
if return_type.lower() == "skeleton":
return skel, separating_sets
# Step 2: Orient the edges based on build the PDAG/CPDAG.
pdag = self.skeleton_to_pdag(skel, separating_sets)
# Step 3: Either return the CPDAG or fully orient the edges to build a DAG.
if return_type.lower() in ("pdag", "cpdag"):
return pdag
elif return_type.lower() == "dag":
return pdag.to_dag()
else:
raise ValueError(
f"return_type must be one of: dag, pdag, cpdag, or skeleton. Got: {return_type}"
)
[docs] def build_skeleton(
self,
ci_test="chi_square",
max_cond_vars=5,
significance_level=0.01,
variant="stable",
n_jobs=-1,
**kwargs,
):
"""
Estimates a graph skeleton (UndirectedGraph) from a set of independencies
using (the first part of) the PC algorithm. The independencies can either be
provided as an instance of the `Independencies`-class or by passing a
decision function that decides any conditional independency assertion.
Returns a tuple `(skeleton, separating_sets)`.
If an Independencies-instance is passed, the contained IndependenceAssertions
have to admit a faithful BN representation. This is the case if
they are obtained as a set of d-seperations of some Bayesian network or
if the independence assertions are closed under the semi-graphoid axioms.
Otherwise the procedure may fail to identify the correct structure.
Parameters
----------
Returns
-------
skeleton: UndirectedGraph
An estimate for the undirected graph skeleton of the BN underlying the data.
separating_sets: dict
A dict containing for each pair of not directly connected nodes a
separating set ("witnessing set") of variables that makes then
conditionally independent. (needed for edge orientation procedures)
References
----------
[1] Neapolitan, Learning Bayesian Networks, Section 10.1.2, Algorithm 10.2 (page 550)
http://www.cs.technion.ac.il/~dang/books/Learning%20Bayesian%20Networks(Neapolitan,%20Richard).pdf
[2] Koller & Friedman, Probabilistic Graphical Models - Principles and Techniques, 2009
Section 3.4.2.1 (page 85), Algorithm 3.3
Examples
--------
>>> from pgmpy.estimators import ConstraintBasedEstimator
>>> from pgmpy.models import DAG
>>> from pgmpy.independencies import Independencies
>>> # build skeleton from list of independencies:
... ind = Independencies(['B', 'C'], ['A', ['B', 'C'], 'D'])
>>> # we need to compute closure, otherwise this set of independencies doesn't
... # admit a faithful representation:
... ind = ind.closure()
>>> skel, sep_sets = ConstraintBasedEstimator.build_skeleton("ABCD", ind)
>>> print(skel.edges())
[('A', 'D'), ('B', 'D'), ('C', 'D')]
>>> # build skeleton from d-seperations of DAG:
... model = DAG([('A', 'C'), ('B', 'C'), ('B', 'D'), ('C', 'E')])
>>> skel, sep_sets = ConstraintBasedEstimator.build_skeleton(model.nodes(), model.get_independencies())
>>> print(skel.edges())
[('A', 'C'), ('B', 'C'), ('B', 'D'), ('C', 'E')]
"""
# Initialize initial values and structures.
lim_neighbors = 0
separating_sets = dict()
if ci_test == "chi_square":
ci_test = chi_square
elif ci_test == "pearsonr":
ci_test = pearsonr
elif ci_test == "independence_match":
ci_test = independence_match
elif callable(ci_test):
ci_test = ci_test
else:
raise ValueError("CI test must be this this")
# Step 1: Initialize a fully connected undirected graph
graph = nx.complete_graph(n=self.variables, create_using=nx.Graph)
# Exit condition: 1. If all the nodes in graph has less than `lim_neighbors` neighbors.
# or 2. `lim_neighbors` is greater than `max_conditional_variables`.
while not all(
[len(list(graph.neighbors(var))) < lim_neighbors for var in self.variables]
):
# Step 2: Iterate over the edges and find a conditioning set of
# size `lim_neighbors` which makes u and v independent.
if variant == "orig":
for (u, v) in graph.edges():
for separating_set in combinations(
set(graph.neighbors(u)) - set([v]), lim_neighbors
):
# If a conditioning set exists remove the edge, store the separating set
# and move on to finding conditioning set for next edge.
if ci_test(
u,
v,
separating_set,
data=self.data,
independencies=self.independencies,
significance_level=significance_level,
):
separating_sets[frozenset((u, v))] = separating_set
graph.remove_edge(u, v)
break
elif variant == "stable":
# In case of stable, precompute neighbors as this is the stable algorithm.
neighbors = {node: set(graph[node]) for node in graph.nodes()}
for (u, v) in graph.edges():
for separating_set in combinations(
neighbors[u] - set([v]), lim_neighbors
):
# If a conditioning set exists remove the edge, store the
# separating set and move on to finding conditioning set for next edge.
if ci_test(
u,
v,
separating_set,
data=self.data,
independencies=self.independencies,
significance_level=significance_level,
):
separating_sets[frozenset((u, v))] = separating_set
graph.remove_edge(u, v)
break
elif variant == "parallel":
neighbors = {node: set(graph[node]) for node in graph.nodes()}
def _parallel_fun(u, v):
for separating_set in combinations(
neighbors[u] - set([v]), lim_neighbors
):
if ci_test(
u,
v,
separating_set,
data=self.data,
independencies=self.independencies,
significance_level=significance_level,
):
return (u, v), separating_set
results = Parallel(n_jobs=n_jobs, prefer="threads")(
delayed(_parallel_fun)(u, v) for (u, v) in graph.edges()
)
for result in results:
if result is not None:
(u, v), sep_set = result
graph.remove_edge(u, v)
separating_sets[frozenset((u, v))] = sep_set
else:
raise ValueError(
f"variant must be one of (orig, stable, parallel). Got: {variant}"
)
# Step 3: After iterating over all the edges, expand the search space by increasing the size
# of conditioning set by 1.
if lim_neighbors >= max_cond_vars:
warn("Reached maximum number of allowed conditional variables. Exiting")
lim_neighbors += 1
return graph, separating_sets
[docs] @staticmethod
def skeleton_to_pdag(skeleton, separating_sets):
"""Orients the edges of a graph skeleton based on information from
`separating_sets` to form a DAG pattern (DAG).
Parameters
----------
skeleton: UndirectedGraph
An undirected graph skeleton as e.g. produced by the
estimate_skeleton method.
separating_sets: dict
A dict containing for each pair of not directly connected nodes a
separating set ("witnessing set") of variables that makes then
conditionally independent. (needed for edge orientation)
Returns
-------
pdag: DAG
An estimate for the DAG pattern of the BN underlying the data. The
graph might contain some nodes with both-way edges (X->Y and Y->X).
Any completion by (removing one of the both-way edges for each such
pair) results in a I-equivalent Bayesian network DAG.
References
----------
Neapolitan, Learning Bayesian Networks, Section 10.1.2, Algorithm 10.2 (page 550)
http://www.cs.technion.ac.il/~dang/books/Learning%20Bayesian%20Networks(Neapolitan,%20Richard).pdf
Examples
--------
>>> import pandas as pd
>>> import numpy as np
>>> from pgmpy.estimators import ConstraintBasedEstimator
>>> data = pd.DataFrame(np.random.randint(0, 4, size=(5000, 3)), columns=list('ABD'))
>>> data['C'] = data['A'] - data['B']
>>> data['D'] += data['A']
>>> c = ConstraintBasedEstimator(data)
>>> pdag = c.skeleton_to_pdag(*c.estimate_skeleton())
>>> pdag.edges() # edges: A->C, B->C, A--D (not directed)
[('B', 'C'), ('A', 'C'), ('A', 'D'), ('D', 'A')]
"""
pdag = skeleton.to_directed()
node_pairs = combinations(pdag.nodes(), 2)
# 1) for each X-Z-Y, if Z not in the separating set of X,Y, then orient edges as X->Z<-Y
# (Algorithm 3.4 in Koller & Friedman PGM, page 86)
for X, Y in node_pairs:
if not skeleton.has_edge(X, Y):
for Z in set(skeleton.neighbors(X)) & set(skeleton.neighbors(Y)):
if Z not in separating_sets[frozenset((X, Y))]:
pdag.remove_edges_from([(Z, X), (Z, Y)])
progress = True
while progress: # as long as edges can be oriented (removed)
num_edges = pdag.number_of_edges()
# 2) for each X->Z-Y, orient edges to Z->Y
for X, Y in node_pairs:
for Z in (set(pdag.successors(X)) - set(pdag.predecessors(X))) & (
set(pdag.successors(Y)) & set(pdag.predecessors(Y))
):
pdag.remove(Y, Z)
# 3) for each X-Y with a directed path from X to Y, orient edges to X->Y
for X, Y in node_pairs:
for path in nx.all_simple_paths(pdag, X, Y):
is_directed = True
for src, dst in path:
if pdag.has_edge(dst, src):
is_directed = False
if is_directed:
pdag.remove(Y, X)
break
# 4) for each X-Z-Y with X->W, Y->W, and Z-W, orient edges to Z->W
for X, Y in node_pairs:
for Z in (
set(pdag.successors(X))
& set(pdag.predecessors(X))
& set(pdag.successors(Y))
& set(pdag.predecessors(Y))
):
for W in (
(set(pdag.successors(X)) - set(pdag.predecessors(X)))
& (set(pdag.successors(Y)) - set(pdag.predecessors(Y)))
& (set(pdag.successors(Z)) & set(pdag.predecessors(Z)))
):
pdag.remove(W, Z)
progress = num_edges > pdag.number_of_edges()
# TODO: This is temp fix to get a PDAG object.
edges = set(pdag.edges())
undirected_edges = []
directed_edges = []
for u, v in edges:
if (v, u) in edges:
undirected_edges.append((u, v))
else:
directed_edges.append((u, v))
return PDAG(directed_ebunch=directed_edges, undirected_ebunch=undirected_edges)
