Source code for pgmpy.models.MarkovModel

#!/usr/bin/env python3
import itertools
from collections import defaultdict

import networkx as nx
import numpy as np

from pgmpy.base import UndirectedGraph
from pgmpy.factors.discrete import factor_product, DiscreteFactor
from pgmpy.independencies import Independencies
from pgmpy.extern.six.moves import map, range, zip


[docs]class MarkovModel(UndirectedGraph): """ Base class for markov model. A MarkovModel stores nodes and edges with potentials MarkovModel holds undirected edges. Parameters ---------- data : input graph Data to initialize graph. If data=None (default) an empty graph is created. The data can be an edge list, or any NetworkX graph object. Examples -------- Create an empty Markov Model with no nodes and no edges. >>> from pgmpy.models import MarkovModel >>> G = MarkovModel() G can be grown in several ways. **Nodes:** Add one node at a time: >>> G.add_node('a') Add the nodes from any container (a list, set or tuple or the nodes from another graph). >>> G.add_nodes_from(['a', 'b']) **Edges:** G can also be grown by adding edges. Add one edge, >>> G.add_edge('a', 'b') a list of edges, >>> G.add_edges_from([('a', 'b'), ('b', 'c')]) If some edges connect nodes not yet in the model, the nodes are added automatically. There are no errors when adding nodes or edges that already exist. **Shortcuts:** Many common graph features allow python syntax for speed reporting. >>> 'a' in G # check if node in graph True >>> len(G) # number of nodes in graph 3 Public Methods -------------- add_node('node1') add_nodes_from(['node1', 'node2', ...]) add_edge('node1', 'node2') add_edges_from([('node1', 'node2'),('node3', 'node4')]) """ def __init__(self, ebunch=None): super(MarkovModel, self).__init__() if ebunch: self.add_edges_from(ebunch) self.factors = []
[docs] def add_edge(self, u, v, **kwargs): """ Add an edge between u and v. The nodes u and v will be automatically added if they are not already in the graph Parameters ---------- u,v : nodes Nodes can be any hashable Python object. Examples -------- >>> from pgmpy.models import MarkovModel >>> G = MarkovModel() >>> G.add_nodes_from(['Alice', 'Bob', 'Charles']) >>> G.add_edge('Alice', 'Bob') """ # check that there is no self loop. if u != v: super(MarkovModel, self).add_edge(u, v, **kwargs) else: raise ValueError('Self loops are not allowed')
[docs] def add_factors(self, *factors): """ Associate a factor to the graph. See factors class for the order of potential values Parameters ---------- *factor: pgmpy.factors.factors object A factor object on any subset of the variables of the model which is to be associated with the model. Returns ------- None Examples -------- >>> from pgmpy.models import MarkovModel >>> from pgmpy.factors.discrete import DiscreteFactor >>> student = MarkovModel([('Alice', 'Bob'), ('Bob', 'Charles'), ... ('Charles', 'Debbie'), ('Debbie', 'Alice')]) >>> factor = DiscreteFactor(['Alice', 'Bob'], cardinality=[3, 2], ... values=np.random.rand(6)) >>> student.add_factors(factor) """ for factor in factors: if set(factor.variables) - set(factor.variables).intersection( set(self.nodes())): raise ValueError("Factors defined on variable not in the model", factor) self.factors.append(factor)
[docs] def get_factors(self): """ Returns the factors that have been added till now to the graph Examples -------- >>> from pgmpy.models import MarkovModel >>> from pgmpy.factors.discrete import DiscreteFactor >>> student = MarkovModel([('Alice', 'Bob'), ('Bob', 'Charles')]) >>> factor = DiscreteFactor(['Alice', 'Bob'], cardinality=[2, 2], ... values=np.random.rand(4)) >>> student.add_factors(factor) >>> student.get_factors() """ return self.factors
[docs] def remove_factors(self, *factors): """ Removes the given factors from the added factors. Examples -------- >>> from pgmpy.models import MarkovModel >>> from pgmpy.factors.discrete import DiscreteFactor >>> student = MarkovModel([('Alice', 'Bob'), ('Bob', 'Charles')]) >>> factor = DiscreteFactor(['Alice', 'Bob'], cardinality=[2, 2], ... values=np.random.rand(4)) >>> student.add_factors(factor) >>> student.remove_factors(factor) """ for factor in factors: self.factors.remove(factor)
[docs] def get_cardinality(self, check_cardinality=False): """ Returns a dictionary with the given factors as keys and their respective cardinality as values. Parameters ---------- check_cardinality: boolean, optional If, check_cardinality=True it checks if cardinality information for all the variables is availble or not. If not it raises an error. Examples -------- >>> from pgmpy.models import MarkovModel >>> from pgmpy.factors.discrete import DiscreteFactor >>> student = MarkovModel([('Alice', 'Bob'), ('Bob', 'Charles')]) >>> factor = DiscreteFactor(['Alice', 'Bob'], cardinality=[2, 2], ... values=np.random.rand(4)) >>> student.add_factors(factor) >>> student.get_cardinality() defaultdict(<class 'int'>, {'Bob': 2, 'Alice': 2}) """ cardinalities = defaultdict(int) for factor in self.factors: for variable, cardinality in zip(factor.scope(), factor.cardinality): cardinalities[variable] = cardinality if check_cardinality and len(self.nodes()) != len(cardinalities): raise ValueError('Factors for all the variables not defined') return cardinalities
[docs] def check_model(self): """ Check the model for various errors. This method checks for the following errors - * Checks if the cardinalities of all the variables are consistent across all the factors. * Factors are defined for all the random variables. Returns ------- check: boolean True if all the checks are passed """ cardinalities = self.get_cardinality() for factor in self.factors: for variable, cardinality in zip(factor.scope(), factor.cardinality): if cardinalities[variable] != cardinality: raise ValueError( 'Cardinality of variable {var} not matching among factors'.format(var=variable)) for var1, var2 in itertools.combinations(factor.variables, 2): if var2 not in self.neighbors(var1): raise ValueError("DiscreteFactor inconsistent with the model.") return True
[docs] def to_factor_graph(self): """ Converts the markov model into factor graph. A factor graph contains two types of nodes. One type corresponds to random variables whereas the second type corresponds to factors over these variables. The graph only contains edges between variables and factor nodes. Each factor node is associated with one factor whose scope is the set of variables that are its neighbors. Examples -------- >>> from pgmpy.models import MarkovModel >>> from pgmpy.factors.discrete import DiscreteFactor >>> student = MarkovModel([('Alice', 'Bob'), ('Bob', 'Charles')]) >>> factor1 = DiscreteFactor(['Alice', 'Bob'], [3, 2], np.random.rand(6)) >>> factor2 = DiscreteFactor(['Bob', 'Charles'], [2, 2], np.random.rand(4)) >>> student.add_factors(factor1, factor2) >>> factor_graph = student.to_factor_graph() """ from pgmpy.models import FactorGraph factor_graph = FactorGraph() if not self.factors: raise ValueError('Factors not associated with the random variables.') factor_graph.add_nodes_from(self.nodes()) for factor in self.factors: scope = factor.scope() factor_node = 'phi_' + '_'.join(scope) factor_graph.add_edges_from(itertools.product(scope, [factor_node])) factor_graph.add_factors(factor) return factor_graph
[docs] def triangulate(self, heuristic='H6', order=None, inplace=False): """ Triangulate the graph. If order of deletion is given heuristic algorithm will not be used. Parameters ---------- heuristic: H1 | H2 | H3 | H4 | H5 | H6 The heuristic algorithm to use to decide the deletion order of the variables to compute the triangulated graph. Let X be the set of variables and X(i) denotes the i-th variable. * S(i) - The size of the clique created by deleting the variable. * E(i) - Cardinality of variable X(i). * M(i) - Maximum size of cliques given by X(i) and its adjacent nodes. * C(i) - Sum of size of cliques given by X(i) and its adjacent nodes. The heuristic algorithm decide the deletion order if this way: * H1 - Delete the variable with minimal S(i). * H2 - Delete the variable with minimal S(i)/E(i). * H3 - Delete the variable with minimal S(i) - M(i). * H4 - Delete the variable with minimal S(i) - C(i). * H5 - Delete the variable with minimal S(i)/M(i). * H6 - Delete the variable with minimal S(i)/C(i). order: list, tuple (array-like) The order of deletion of the variables to compute the triagulated graph. If order is given heuristic algorithm will not be used. inplace: True | False if inplace is true then adds the edges to the object from which it is called else returns a new object. Reference --------- http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.3607 Examples -------- >>> from pgmpy.models import MarkovModel >>> from pgmpy.factors.discrete import DiscreteFactor >>> G = MarkovModel() >>> G.add_nodes_from(['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7']) >>> G.add_edges_from([('x1', 'x3'), ('x1', 'x4'), ('x2', 'x4'), ... ('x2', 'x5'), ('x3', 'x6'), ('x4', 'x6'), ... ('x4', 'x7'), ('x5', 'x7')]) >>> phi = [DiscreteFactor(edge, [2, 2], np.random.rand(4)) for edge in G.edges()] >>> G.add_factors(*phi) >>> G_chordal = G.triangulate() """ self.check_model() if self.is_triangulated(): if inplace: return else: return self graph_copy = nx.Graph(self.edges()) edge_set = set() def _find_common_cliques(cliques_list): """ Finds the common cliques among the given set of cliques for corresponding node. """ common = set([tuple(x) for x in cliques_list[0]]) for i in range(1, len(cliques_list)): common = common & set([tuple(x) for x in cliques_list[i]]) return list(common) def _find_size_of_clique(clique, cardinalities): """ Computes the size of a clique. Size of a clique is defined as product of cardinalities of all the nodes present in the clique. """ return list(map(lambda x: np.prod([cardinalities[node] for node in x]), clique)) def _get_cliques_dict(node): """ Returns a dictionary in the form of {node: cliques_formed} of the node along with its neighboring nodes. clique_dict_removed would be containing the cliques created after deletion of the node clique_dict_node would be containing the cliques created before deletion of the node """ graph_working_copy = nx.Graph(graph_copy.edges()) neighbors = graph_working_copy.neighbors(node) graph_working_copy.add_edges_from(itertools.combinations(neighbors, 2)) clique_dict = nx.cliques_containing_node(graph_working_copy, nodes=([node] + neighbors)) graph_working_copy.remove_node(node) clique_dict_removed = nx.cliques_containing_node(graph_working_copy, nodes=neighbors) return clique_dict, clique_dict_removed if not order: order = [] cardinalities = self.get_cardinality() for index in range(self.number_of_nodes()): # S represents the size of clique created by deleting the # node from the graph S = {} # M represents the size of maximum size of cliques given by # the node and its adjacent node M = {} # C represents the sum of size of the cliques created by the # node and its adjacent node C = {} for node in set(graph_copy.nodes()) - set(order): clique_dict, clique_dict_removed = _get_cliques_dict(node) S[node] = _find_size_of_clique( _find_common_cliques(list(clique_dict_removed.values())), cardinalities )[0] common_clique_size = _find_size_of_clique( _find_common_cliques(list(clique_dict.values())), cardinalities ) M[node] = np.max(common_clique_size) C[node] = np.sum(common_clique_size) if heuristic == 'H1': node_to_delete = min(S, key=S.get) elif heuristic == 'H2': S_by_E = {key: S[key] / cardinalities[key] for key in S} node_to_delete = min(S_by_E, key=S_by_E.get) elif heuristic == 'H3': S_minus_M = {key: S[key] - M[key] for key in S} node_to_delete = min(S_minus_M, key=S_minus_M.get) elif heuristic == 'H4': S_minus_C = {key: S[key] - C[key] for key in S} node_to_delete = min(S_minus_C, key=S_minus_C.get) elif heuristic == 'H5': S_by_M = {key: S[key] / M[key] for key in S} node_to_delete = min(S_by_M, key=S_by_M.get) else: S_by_C = {key: S[key] / C[key] for key in S} node_to_delete = min(S_by_C, key=S_by_C.get) order.append(node_to_delete) graph_copy = nx.Graph(self.edges()) for node in order: for edge in itertools.combinations(graph_copy.neighbors(node), 2): graph_copy.add_edge(edge[0], edge[1]) edge_set.add(edge) graph_copy.remove_node(node) if inplace: for edge in edge_set: self.add_edge(edge[0], edge[1]) return self else: graph_copy = MarkovModel(self.edges()) for edge in edge_set: graph_copy.add_edge(edge[0], edge[1]) return graph_copy
[docs] def to_junction_tree(self): """ Creates a junction tree (or clique tree) for a given markov model. For a given markov model (H) a junction tree (G) is a graph 1. where each node in G corresponds to a maximal clique in H 2. each sepset in G separates the variables strictly on one side of the edge to other. Examples -------- >>> from pgmpy.models import MarkovModel >>> from pgmpy.factors.discrete import DiscreteFactor >>> mm = MarkovModel() >>> mm.add_nodes_from(['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7']) >>> mm.add_edges_from([('x1', 'x3'), ('x1', 'x4'), ('x2', 'x4'), ... ('x2', 'x5'), ('x3', 'x6'), ('x4', 'x6'), ... ('x4', 'x7'), ('x5', 'x7')]) >>> phi = [DiscreteFactor(edge, [2, 2], np.random.rand(4)) for edge in mm.edges()] >>> mm.add_factors(*phi) >>> junction_tree = mm.to_junction_tree() """ from pgmpy.models import JunctionTree # Check whether the model is valid or not self.check_model() # Triangulate the graph to make it chordal triangulated_graph = self.triangulate() # Find maximal cliques in the chordal graph cliques = list(map(tuple, nx.find_cliques(triangulated_graph))) # If there is only 1 clique, then the junction tree formed is just a # clique tree with that single clique as the node if len(cliques) == 1: clique_trees = JunctionTree() clique_trees.add_node(cliques[0]) # Else if the number of cliques is more than 1 then create a complete # graph with all the cliques as nodes and weight of the edges being # the length of sepset between two cliques elif len(cliques) >= 2: complete_graph = UndirectedGraph() edges = list(itertools.combinations(cliques, 2)) weights = list(map(lambda x: len(set(x[0]).intersection(set(x[1]))), edges)) for edge, weight in zip(edges, weights): complete_graph.add_edge(*edge, weight=-weight) # Create clique trees by minimum (or maximum) spanning tree method clique_trees = JunctionTree(nx.minimum_spanning_tree(complete_graph).edges()) # Check whether the factors are defined for all the random variables or not all_vars = itertools.chain(*[factor.scope() for factor in self.factors]) if set(all_vars) != set(self.nodes()): ValueError('DiscreteFactor for all the random variables not specified') # Dictionary stating whether the factor is used to create clique # potential or not # If false, then it is not used to create any clique potential is_used = {factor: False for factor in self.factors} for node in clique_trees.nodes(): clique_factors = [] for factor in self.factors: # If the factor is not used in creating any clique potential as # well as has any variable of the given clique in its scope, # then use it in creating clique potential if not is_used[factor] and set(factor.scope()).issubset(node): clique_factors.append(factor) is_used[factor] = True # To compute clique potential, initially set it as unity factor var_card = [self.get_cardinality()[x] for x in node] clique_potential = DiscreteFactor(node, var_card, np.ones(np.product(var_card))) # multiply it with the factors associated with the variables present # in the clique (or node) clique_potential *= factor_product(*clique_factors) clique_trees.add_factors(clique_potential) if not all(is_used.values()): raise ValueError('All the factors were not used to create Junction Tree.' 'Extra factors are defined.') return clique_trees
[docs] def markov_blanket(self, node): """ Returns a markov blanket for a random variable. Markov blanket is the neighboring nodes of the given node. Examples -------- >>> from pgmpy.models import MarkovModel >>> mm = MarkovModel() >>> mm.add_nodes_from(['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7']) >>> mm.add_edges_from([('x1', 'x3'), ('x1', 'x4'), ('x2', 'x4'), ... ('x2', 'x5'), ('x3', 'x6'), ('x4', 'x6'), ... ('x4', 'x7'), ('x5', 'x7')]) >>> mm.markov_blanket('x1') """ return self.neighbors(node)
[docs] def get_local_independencies(self, latex=False): """ Returns all the local independencies present in the markov model. Local independencies are the independence assertion in the form of .. math:: {X \perp W - {X} - MB(X) | MB(X)} where MB is the markov blanket of all the random variables in X Parameters ---------- latex: boolean If latex=True then latex string of the indepedence assertion would be created Examples -------- >>> from pgmpy.models import MarkovModel >>> mm = MarkovModel() >>> mm.add_nodes_from(['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7']) >>> mm.add_edges_from([('x1', 'x3'), ('x1', 'x4'), ('x2', 'x4'), ... ('x2', 'x5'), ('x3', 'x6'), ('x4', 'x6'), ... ('x4', 'x7'), ('x5', 'x7')]) >>> mm.get_local_independecies() """ local_independencies = Independencies() all_vars = set(self.nodes()) for node in self.nodes(): markov_blanket = set(self.markov_blanket(node)) rest = all_vars - set([node]) - markov_blanket try: local_independencies.add_assertions([node, list(rest), list(markov_blanket)]) except ValueError: pass local_independencies.reduce() if latex: return local_independencies.latex_string() else: return local_independencies
[docs] def to_bayesian_model(self): """ Creates a Bayesian Model which is a minimum I-Map for this markov model. The ordering of parents may not remain constant. It would depend on the ordering of variable in the junction tree (which is not constant) all the time. Examples -------- >>> from pgmpy.models import MarkovModel >>> from pgmpy.factors.discrete import DiscreteFactor >>> mm = MarkovModel() >>> mm.add_nodes_from(['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7']) >>> mm.add_edges_from([('x1', 'x3'), ('x1', 'x4'), ('x2', 'x4'), ... ('x2', 'x5'), ('x3', 'x6'), ('x4', 'x6'), ... ('x4', 'x7'), ('x5', 'x7')]) >>> phi = [DiscreteFactor(edge, [2, 2], np.random.rand(4)) for edge in mm.edges()] >>> mm.add_factors(*phi) >>> bm = mm.to_bayesian_model() """ from pgmpy.models import BayesianModel bm = BayesianModel() var_clique_dict = defaultdict(tuple) var_order = [] # Create a junction tree from the markov model. # Creation of clique tree involves triangulation, finding maximal cliques # and creating a tree from these cliques junction_tree = self.to_junction_tree() # create an ordering of the nodes based on the ordering of the clique # in which it appeared first root_node = junction_tree.nodes()[0] bfs_edges = nx.bfs_edges(junction_tree, root_node) for node in root_node: var_clique_dict[node] = root_node var_order.append(node) for edge in bfs_edges: clique_node = edge[1] for node in clique_node: if not var_clique_dict[node]: var_clique_dict[node] = clique_node var_order.append(node) # create a bayesian model by adding edges from parent of node to node as # par(x_i) = (var(c_k) - x_i) \cap {x_1, ..., x_{i-1}} for node_index in range(len(var_order)): node = var_order[node_index] node_parents = (set(var_clique_dict[node]) - set([node])).intersection( set(var_order[:node_index])) bm.add_edges_from([(parent, node) for parent in node_parents]) # TODO : Convert factor into CPDs return bm
[docs] def get_partition_function(self): """ Returns the partition function for a given undirected graph. A partition function is defined as .. math:: \sum_{X}(\prod_{i=1}^{m} \phi_i) where m is the number of factors present in the graph and X are all the random variables present. Examples -------- >>> from pgmpy.models import MarkovModel >>> from pgmpy.factors.discrete import DiscreteFactor >>> G = MarkovModel() >>> G.add_nodes_from(['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7']) >>> G.add_edges_from([('x1', 'x3'), ('x1', 'x4'), ('x2', 'x4'), ... ('x2', 'x5'), ('x3', 'x6'), ('x4', 'x6'), ... ('x4', 'x7'), ('x5', 'x7')]) >>> phi = [DiscreteFactor(edge, [2, 2], np.random.rand(4)) for edge in G.edges()] >>> G.add_factors(*phi) >>> G.get_partition_function() """ self.check_model() factor = self.factors[0] factor = factor_product(factor, *[self.factors[i] for i in range(1, len(self.factors))]) if set(factor.scope()) != set(self.nodes()): raise ValueError('DiscreteFactor for all the random variables not defined.') return np.sum(factor.values)
[docs] def copy(self): """ Returns a copy of this Markov Model. Returns ------- MarkovModel: Copy of this Markov model. Examples ------- >>> from pgmpy.factors.discrete import DiscreteFactor >>> from pgmpy.models import MarkovModel >>> G = MarkovModel() >>> G.add_nodes_from([('a', 'b'), ('b', 'c')]) >>> G.add_edge(('a', 'b'), ('b', 'c')) >>> G_copy = G.copy() >>> G_copy.edges() [(('a', 'b'), ('b', 'c'))] >>> G_copy.nodes() [('a', 'b'), ('b', 'c')] >>> factor = DiscreteFactor([('a', 'b')], cardinality=[3], ... values=np.random.rand(3)) >>> G.add_factors(factor) >>> G.get_factors() [<DiscreteFactor representing phi(('a', 'b'):3) at 0x...>] >>> G_copy.get_factors() [] """ clone_graph = MarkovModel(self.edges()) clone_graph.add_nodes_from(self.nodes()) if self.factors: factors_copy = [factor.copy() for factor in self.factors] clone_graph.add_factors(*factors_copy) return clone_graph