import networkx as nx
import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression
from pgmpy.base import DAG
from pgmpy.factors.continuous import LinearGaussianCPD
from pgmpy.global_vars import logger
from pgmpy.models import BayesianNetwork
[docs]
class LinearGaussianBayesianNetwork(BayesianNetwork):
"""
A Linear Gaussian Bayesian Network is a Bayesian Network, all
of whose variables are continuous, and where all of the CPDs
are linear Gaussians.
An important result is that the linear Gaussian Bayesian Networks
are an alternative representation for the class of multivariate
Gaussian distributions.
"""
[docs]
def add_cpds(self, *cpds):
"""
Add linear Gaussian CPD (Conditional Probability Distribution)
to the Bayesian Network.
Parameters
----------
cpds : instances of LinearGaussianCPD
List of LinearGaussianCPDs which will be associated with the model
Examples
--------
>>> from pgmpy.models import LinearGaussianBayesianNetwork
>>> from pgmpy.factors.continuous import LinearGaussianCPD
>>> model = LinearGaussianBayesianNetwork([('x1', 'x2'), ('x2', 'x3')])
>>> cpd1 = LinearGaussianCPD('x1', [1], 4)
>>> cpd2 = LinearGaussianCPD('x2', [-5, 0.5], 4, ['x1'])
>>> cpd3 = LinearGaussianCPD('x3', [4, -1], 3, ['x2'])
>>> model.add_cpds(cpd1, cpd2, cpd3)
>>> for cpd in model.cpds:
... print(cpd)
P(x1) = N(1; 4)
P(x2| x1) = N(0.5*x1_mu); -5)
P(x3| x2) = N(-1*x2_mu); 4)
"""
for cpd in cpds:
if not isinstance(cpd, LinearGaussianCPD):
raise ValueError("Only LinearGaussianCPD can be added.")
if set(cpd.variables) - set(cpd.variables).intersection(set(self.nodes())):
raise ValueError("CPD defined on variable not in the model", cpd)
for prev_cpd_index in range(len(self.cpds)):
if self.cpds[prev_cpd_index].variable == cpd.variable:
logger.warning(f"Replacing existing CPD for {cpd.variable}")
self.cpds[prev_cpd_index] = cpd
break
else:
self.cpds.append(cpd)
[docs]
def get_cpds(self, node=None):
"""
Returns the cpd of the node. If node is not specified returns all the CPDs
that have been added till now to the graph
Parameter
---------
node: any hashable python object (optional)
The node whose CPD we want. If node not specified returns all the
CPDs added to the model.
Returns
-------
A list of linear Gaussian CPDs.
Examples
--------
>>> from pgmpy.models import LinearGaussianBayesianNetwork
>>> from pgmpy.factors.continuous import LinearGaussianCPD
>>> model = LinearGaussianBayesianNetwork([('x1', 'x2'), ('x2', 'x3')])
>>> cpd1 = LinearGaussianCPD('x1', [1], 4)
>>> cpd2 = LinearGaussianCPD('x2', [-5, 0.5], 4, ['x1'])
>>> cpd3 = LinearGaussianCPD('x3', [4, -1], 3, ['x2'])
>>> model.add_cpds(cpd1, cpd2, cpd3)
>>> model.get_cpds()
"""
return super(LinearGaussianBayesianNetwork, self).get_cpds(node)
[docs]
def remove_cpds(self, *cpds):
"""
Removes the cpds that are provided in the argument.
Parameters
----------
*cpds: LinearGaussianCPD object
A LinearGaussianCPD object on any subset of the variables
of the model which is to be associated with the model.
Examples
--------
>>> from pgmpy.models import LinearGaussianBayesianNetwork
>>> from pgmpy.factors.continuous import LinearGaussianCPD
>>> model = LinearGaussianBayesianNetwork([('x1', 'x2'), ('x2', 'x3')])
>>> cpd1 = LinearGaussianCPD('x1', [1], 4)
>>> cpd2 = LinearGaussianCPD('x2', [-5, 0.5], 4, ['x1'])
>>> cpd3 = LinearGaussianCPD('x3', [4, -1], 3, ['x2'])
>>> model.add_cpds(cpd1, cpd2, cpd3)
>>> for cpd in model.get_cpds():
... print(cpd)
P(x1) = N(1; 4)
P(x2| x1) = N(0.5*x1_mu); -5)
P(x3| x2) = N(-1*x2_mu); 4)
>>> model.remove_cpds(cpd2, cpd3)
>>> for cpd in model.get_cpds():
... print(cpd)
P(x1) = N(1; 4)
"""
return super(LinearGaussianBayesianNetwork, self).remove_cpds(*cpds)
[docs]
def get_random_cpds(self, loc=0, scale=1, inplace=False, seed=None):
"""
Generates random Linear Gaussian CPDs for the model. The coefficients
are sampled from a normal distribution with mean `loc` and standard
deviation `scale`.
Parameters
----------
loc: float
The mean of the normal distribution from which the coefficients are
sampled.
scale: float
The standard deviation of the normal distribution from which the
coefficients are sampled.
inplace: bool (default: False)
If inplace=True, adds the generated LinearGaussianCPDs to `model` itself,
else creates a copy of the model.
seed: int
The seed for the random number generator.
"""
rng = np.random.default_rng(seed=seed)
cpds = []
for var in self.nodes():
parents = self.get_parents(var)
cpds.append(
LinearGaussianCPD(
variable=var,
beta=rng.normal(loc=loc, scale=scale, size=(len(parents) + 1)),
std=abs(rng.normal(loc=loc, scale=scale)),
evidence=parents,
)
)
if inplace:
self.add_cpds(*cpds)
else:
return cpds
[docs]
def to_joint_gaussian(self):
"""
Linear Gaussian Bayesian Networks can be represented using a joint
Gaussian distribution over all the variables. This method gives
the mean and covariance of this equivalent joint gaussian distribution.
Returns
-------
mean, cov: np.ndarray, np.ndarray
The mean and the covariance matrix of the joint gaussian distribution.
Examples
--------
>>> from pgmpy.models import LinearGaussianBayesianNetwork
>>> from pgmpy.factors.continuous import LinearGaussianCPD
>>> model = LinearGaussianBayesianNetwork([('x1', 'x2'), ('x2', 'x3')])
>>> cpd1 = LinearGaussianCPD('x1', [1], 4)
>>> cpd2 = LinearGaussianCPD('x2', [-5, 0.5], 4, ['x1'])
>>> cpd3 = LinearGaussianCPD('x3', [4, -1], 3, ['x2'])
>>> model.add_cpds(cpd1, cpd2, cpd3)
>>> mean, cov = model.to_joint_gaussian()
>>> mean
array([ 1. ], [-4.5], [ 8.5])
>>> cov
array([[ 4., 2., -2.],
[ 2., 5., -5.],
[-2., -5., 8.]])
"""
variables = list(nx.topological_sort(self))
var_to_index = {var: i for i, var in enumerate(variables)}
n_nodes = len(self.nodes())
# Step 1: Compute the mean for each variable.
mean = {}
for var in variables:
cpd = self.get_cpds(node=var)
mean[var] = (
cpd.beta * (np.array([1] + [mean[u] for u in cpd.evidence]))
).sum()
mean = np.array([mean[u] for u in variables])
# Step 2: Populate the adjacency matrix, and variance matrix
B = np.zeros((n_nodes, n_nodes))
omega = np.zeros((n_nodes, n_nodes))
for var in variables:
cpd = self.get_cpds(node=var)
for i, evidence_var in enumerate(cpd.evidence):
B[var_to_index[evidence_var], var_to_index[var]] = cpd.beta[i + 1]
omega[var_to_index[var], var_to_index[var]] = cpd.std
# Step 3: Compute the implied covariance matrix
I = np.eye(n_nodes)
inv = np.linalg.inv((I - B))
implied_cov = inv.T @ omega @ inv
# Round because numerical errors can lead to non-symmetric cov matrix.
return mean.round(decimals=8), implied_cov.round(decimals=8)
[docs]
def simulate(self, n=1000, seed=None):
"""
Simulates data from the given model.
Parameters
----------
n: int
The number of samples to draw from the model.
seed: int (default: None)
Seed for the random number generator.
Returns
-------
pandas.DataFrame: generated samples
A pandas data frame with the generated samples.
Examples
--------
>>> from pgmpy.models import LinearGaussianBayesianNetwork
>>> from pgmpy.factors.continuous import LinearGaussianCPD
>>> model = LinearGaussianBayesianNetwork([('x1', 'x2'), ('x2', 'x3')])
>>> cpd1 = LinearGaussianCPD('x1', [1], 4)
>>> cpd2 = LinearGaussianCPD('x2', [-5, 0.5], 4, ['x1'])
>>> cpd3 = LinearGaussianCPD('x3', [4, -1], 3, ['x2'])
>>> model.add_cpds(cpd1, cpd2, cpd3)
>>> model.simulate(n=500, seed=42)
"""
if len(self.cpds) != len(self.nodes()):
raise ValueError(
"Each node in the model should have a CPD associated with it"
)
mean, cov = self.to_joint_gaussian()
variables = list(nx.topological_sort(self))
rng = np.random.default_rng(seed=seed)
return pd.DataFrame(
rng.multivariate_normal(mean=mean, cov=cov, size=n), columns=variables
)
[docs]
def check_model(self):
"""
Checks the model for various errors. This method checks for the following
error -
* Checks if the CPDs associated with nodes are consistent with their parents.
Returns
-------
check: boolean
True if all the checks pass.
"""
for node in self.nodes():
cpd = self.get_cpds(node=node)
if isinstance(cpd, LinearGaussianCPD):
if set(cpd.evidence) != set(self.get_parents(node)):
raise ValueError(
"CPD associated with %s doesn't have "
"proper parents associated with it." % node
)
return True
[docs]
def get_cardinality(self, node):
"""
Cardinality is not defined for continuous variables.
"""
raise ValueError("Cardinality is not defined for continuous variables.")
[docs]
def fit(self, data, method="mle"):
"""
Estimates the parameters of the model using the given `data`.
Parameters
----------
data: pd.DataFrame
A pandas DataFrame with the data to which to fit the model
structure. All variables must be continuous valued.
Returns
-------
None: The estimated LinearGaussianCPDs are added to the model. They can
be accessed using `model.cpds`.
Examples
--------
>>> import numpy as np
>>> import pandas as pd
>>> from pgmpy.models import LinearGaussianBayesianNetwork
>>> df = pd.DataFrame(np.random.normal(0, 1, (100, 3)), columns=['x1', 'x2', 'x3'])
>>> model = LinearGaussianBayesianNetwork([('x1', 'x2'), ('x2', 'x3')])
>>> model.fit(df)
>>> model.cpds
[<LinearGaussianCPD: P(x1) = N(-0.114; 0.911) at 0x7eb77d30cec0,
<LinearGaussianCPD: P(x2 | x1) = N(0.07*x1 + -0.075; 1.172) at 0x7eb77171fb60,
<LinearGaussianCPD: P(x3 | x2) = N(0.006*x2 + -0.1; 0.922) at 0x7eb6abbdba10]
"""
# Step 1: Check the input
if len(missing_vars := (set(self.nodes()) - set(data.columns))) > 0:
raise ValueError(
f"Following variables are missing in the data: {missing_vars}"
)
# Step 2: Estimate the LinearGaussianCPDs
cpds = []
for node in self.nodes():
parents = self.get_parents(node)
# Step 2.1: If node doesn't have any parents (i.e. root node),
# simply take the mean and variance.
if len(parents) == 0:
cpds.append(
LinearGaussianCPD(
variable=node,
beta=[data.loc[:, node].mean()],
std=data.loc[:, node].var(),
)
)
# Step 2.2: Else, fit a linear regression model and take the coefficients and intercept.
# Compute error variance using predicted values.
else:
lm = LinearRegression().fit(data.loc[:, parents], data.loc[:, node])
error_var = (data.loc[:, node] - lm.predict(data.loc[:, parents])).var()
cpds.append(
LinearGaussianCPD(
variable=node,
beta=np.append([lm.intercept_], lm.coef_),
std=error_var,
evidence=parents,
)
)
# Step 3: Add the estimated CPDs to the model
self.add_cpds(*cpds)
[docs]
def predict(self, data, distribution="joint"):
"""
Predicts the distribution of the missing variable (i.e. missing columns) in the given dataset.
Parameters
----------
data: pandas.DataFrame
The dataframe with missing variable which to predict.
Returns
-------
variables: list
The list of variables on which the returned conditional distribution is defined on.
mu: np.array
The mean array of the conditional joint distribution over the missing variables corresponding to each row of data.
cov: np.array
The covariance of the conditional joint distribution over the missing variables.
Examples
--------
>>>
"""
# Step 0: Check the inputs
missing_vars = list(set(self.nodes()) - set(data.columns))
if len(missing_vars) == 0:
raise ValueError("No missing variables in the data")
# Step 1: Create separate mean and cov matrices for missing and known variables.
mu, cov = self.to_joint_gaussian()
variable_order = list(nx.topological_sort(self))
missing_indexes = [variable_order.index(var) for var in missing_vars]
remain_vars = [var for var in variable_order if var not in missing_vars]
mu_a = mu[missing_indexes]
mu_b = np.delete(mu, missing_indexes)
cov_aa = cov[missing_indexes, missing_indexes]
cov_bb = np.delete(
np.delete(cov, missing_indexes, axis=0), missing_indexes, axis=1
)
cov_ab = np.delete(cov[missing_indexes, :], missing_indexes, axis=1)
# Step 2: Compute the conditional distributions
cov_bb_inv = np.linalg.inv(cov_bb)
mu_cond = (
np.atleast_2d(mu_a)
+ (
cov_ab
[docs]
@ cov_bb_inv
@ (data.loc[:, remain_vars].values - np.atleast_2d(mu_b)).T
).T
)
cov_cond = cov_aa - cov_ab @ cov_bb_inv @ cov_ab.T
# Step 3: Return values
return ([variable_order[i] for i in missing_indexes], mu_cond, cov_cond)
def to_markov_model(self):
"""
For now, to_markov_model method has not been implemented for LinearGaussianBayesianNetwork.
"""
raise NotImplementedError(
"to_markov_model method has not been implemented for LinearGaussianBayesianNetwork."
)
[docs]
def is_imap(self, JPD):
"""
For now, is_imap method has not been implemented for LinearGaussianBayesianNetwork.
"""
raise NotImplementedError(
"is_imap method has not been implemented for LinearGaussianBayesianNetwork."
)
[docs]
@staticmethod
def get_random(
n_nodes=5,
edge_prob=0.5,
node_names=None,
latents=False,
loc=0,
scale=1,
seed=None,
):
"""
Returns a randomly generated Linear Gaussian Bayesian Network on `n_nodes` variables
with edge probabiliy of `edge_prob` between variables.
Parameters
----------
n_nodes: int
The number of nodes in the randomly generated DAG.
edge_prob: float
The probability of edge between any two nodes in the topologically
sorted DAG.
node_names: list (default: None)
A list of variables names to use in the random graph.
If None, the node names are integer values starting from 0.
latents: bool (default: False)
If True, also creates latent variables.
loc: float
The mean of the normal distribution from which the coefficients are
sampled.
scale: float
The standard deviation of the normal distribution from which the
coefficients are sampled.
seed: int
The seed for the random number generator.
Returns
-------
Random DAG: pgmpy.base.DAG
The randomly generated DAG.
Examples
--------
>>> from pgmpy.models import LinearGaussianBayesianNetwork
>>> model = LinearGaussianBayesianNetwork.get_random(n_nodes=5)
>>> model.nodes()
NodeView((0, 3, 1, 2, 4))
>>> model.edges()
OutEdgeView([(0, 3), (3, 4), (1, 3), (2, 4)])
>>> model.cpds
[<LinearGaussianCPD: P(0) = N(1.764; 1.613) at 0x2732f41aae0,
<LinearGaussianCPD: P(3 | 0, 1) = N(-0.721*0 + -0.079*1 + 0.943; 0.12) at 0x2732f16db20,
<LinearGaussianCPD: P(1) = N(-0.534; 0.208) at 0x2732f320b30,
<LinearGaussianCPD: P(2) = N(-0.023; 0.166) at 0x2732d8d5f40,
<LinearGaussianCPD: P(4 | 2, 3) = N(-0.24*2 + -0.907*3 + 0.625; 0.48) at 0x2737fecdaf0]
"""
if node_names is None:
node_names = list(range(n_nodes))
dag = DAG.get_random(
n_nodes=n_nodes, edge_prob=edge_prob, node_names=node_names, latents=latents
)
lgbn_model = LinearGaussianBayesianNetwork(dag.edges(), latents=dag.latents)
lgbn_model.add_nodes_from(dag.nodes())
cpds = lgbn_model.get_random_cpds(loc=loc, scale=scale, seed=seed)
lgbn_model.add_cpds(*cpds)
return lgbn_model
