CAM-UV

Model

This method CAM-UV (Causal Additive Models with Unobserved Variables) assumes an extension of the basic CAM model [1] to include unobserved variables. This method makes the following assumptions:

  1. The effects of direct causes on a variable form generalized additive models (GAMs).
  2. The causal structures form directed acyclic graphs (DAGs).

CAM-UV allows the existence of unobserved variables. It outputs a causal graph where a undirected edge indicates the pair of variables that have an unobserved causal path (UCP) or an unobserved backdoor path (UBP), and a directed edge indicates the causal direction of a pair of variables that do not have an UCP or UBP.

Definition of UCPs ans UBPs: As shown in the below figure, a causal path from \(x_j\) to \(x_i\) is called an UCP if it ends with the directed edge connecting \(x_i\) and its unobserved direct cause. A backdoor path between \(x_i\) and \(x_j\) is called an UBP if it starts with the edge connecting \(x_i\) and its unobserved direct cause, and ends with the edge connecting \(x_j\) and its unobserved direct cause.

../_images/camuvexmpl.png

References

[1]P. Bühlmann, J. Peters, and J. Ernest. CAM: Causal additive models, high-dimensional order search and penalized regression. Annals of Statistics, 42(6):2526–2556, 2014.
[2]T. N. Maeda and S. Shimizu. Causal additive models with unobserved variables. In Proc. 27th Conference on Uncertainty in Artificial Intelligence (UAI2021), PMLR 161:97-106, 2021.

Import and settings

In this example, we need to import numpy in addition to lingam.

import numpy as np
import random
import lingam

Test data

First, we generate a causal structure with 2 unobserved variables (y6 and y7) and 6 observed variables (x0–x5) as shown in the below figure.

../_images/datageneration_CAMUV.png 
def get_noise(n):
    noise = ((np.random.rand(1, n)-0.5)*5).reshape(n)
    mean = get_random_constant(0.0,2.0)
    noise += mean
    return noise



def causal_func(cause):
    a = get_random_constant(-5.0,5.0)
    b = get_random_constant(-1.0,1.0)
    c = int(random.uniform(2,3))
    return ((cause+a)**(c))+b


def get_random_constant(s,b):
    constant = random.uniform(-1.0, 1.0)
    if constant>0:
        constant = random.uniform(s, b)
    else:
        constant = random.uniform(-b, -s)
    return constant


def create_data(n):
    causal_pairs = [[0,1],[0,3],[2,4]]
    intermediate_pairs = [[2,5]]
    confounder_pairs = [[3,4]]

    n_variables = 6

    data = np.zeros((n, n_variables)) # observed data
    confounders = np.zeros((n, len(confounder_pairs))) # data of unobserced common causes

    # Adding external effects
    for i in range(n_variables):
        data[:,i] = get_noise(n)
    for i in range(len(confounder_pairs)):
        confounders[:,i] = get_noise(n)
        confounders[:,i] = confounders[:,i] / np.std(confounders[:,i])

    # Adding the effects of unobserved common causes
    for i, cpair in enumerate(confounder_pairs):
        cpair = list(cpair)
        cpair.sort()
        data[:,cpair[0]] += causal_func(confounders[:,i])
        data[:,cpair[1]] += causal_func(confounders[:,i])

    for i1 in range(n_variables)[0:n_variables]:
        data[:,i1] = data[:,i1] / np.std(data[:,i1])
        for i2 in range(n_variables)[i1+1:n_variables+1]:
            # Adding direct effects between observed variables
            if [i1, i2] in causal_pairs:
                data[:,i2] += causal_func(data[:,i1])
            # Adding undirected effects between observed variables mediated through unobserved variables
            if [i1, i2] in intermediate_pairs:
                interm = causal_func(data[:,i1])+get_noise(n)
                interm = interm / np.std(interm)
                data[:,i2] += causal_func(interm)

    return data


sample_size = 2000
X = create_data(sample_size)

Causal Discovery

To run causal discovery, we create a CAMUV object and call the fit method.

model = lingam.CAMUV()
model.fit(X)

Using the adjacency_matrix_ properties, we can see the adjacency matrix as a result of the causal discovery. When the value of a variable pair is np.nan, the variables have a UCP or UBP.

model.adjacency_matrix_

array([[ 0.,  0.,  0.,  0.,  0.,  0.],
       [ 1.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0., nan],
       [ 1.,  0.,  0.,  0., nan,  0.],
       [ 0.,  0.,  1., nan,  0.,  0.],
       [ 0.,  0., nan,  0.,  0.,  0.]])