jan-glx / cam

Causal Additive Model (CAM)

Causal Additive Model (CAM)

And R package to infer the causal DAG under the assumtion of an additive model.

The code takes an n x p data matrix and fits a Causal Additive Model (CAM) for estimating the causal structure of the underlying process. The output is a p x p adjacency matrix (a one in entry (i,j) indicates an edge from i to j).

Details of the algorithm can be found in: P. Bühlmann, J. Peters, J. Ernest: "CAM: Causal Additive Models, high-dimensional order search and penalized regression", Annals of Statistics 42:2526-2556, 2014.

Changes

• adds bootstrap based hypothesis test for total causal effects (or at least for smaller causal order)
• adds functions to simulate related models
• adds some documentation & unit-tests
• simplifies the code base & speeds up model fits (using data.table)

Installation

# install.packages("remotes")
remotes::install_github("jan-glx/CAM")

Usage

library(CAM)

simple model fit

# set-up data

set.seed(1)
n <- 5000
eps1 <- rnorm(n)
eps2 <- rnorm(n)
eps3 <- rnorm(n)
eps4 <- rnorm(n)

x2 <- 0.5 * eps2
x1 <- 0.9 * sign(x2) * (abs(x2) ^ (0.5)) + 0.5 * eps1
x3 <- 0.8 * x2 ^ 2 + 0.5 * eps3
x4 <- -0.9 * sin(x3) - abs(x1) + 0.5 * eps4

X <- cbind(x1, x2, x3, x4)

trueDAG <- edges2adj(i = c(3, 2, 2, 1),
                     j = c(4, 3, 1, 4)) 
## x4 <- x3 <- x2 -> x1 
##  ^               /
##   \_____________/
## adjacency matrix:
## 0 0 0 1
## 1 0 1 0
## 0 0 0 1
## 0 0 0 0

# fit CAM

fit1 <- CAM(X)
fit1$Adj # fitted DAG
#>       [,1]  [,2]  [,3]  [,4]
#> [1,] FALSE FALSE FALSE  TRUE
#> [2,]  TRUE FALSE  TRUE  TRUE
#> [3,]  TRUE FALSE FALSE  TRUE
#> [4,] FALSE FALSE FALSE FALSE

# check if causal ordering of inferred causal DAG is compatible with true causal DAG
areAllCausalOrdersCompatible(fit1$Adj, trueDAG)
#> [1] TRUE

Bootstrap-based hypothesis test

# set-up example DAG

p=3
trueDAG <- matrix(FALSE, ncol=p, nrow=p)
trueDAG[matrix(c(1, 2,
                 3, 3), ncol=2)] <- TRUE
## 1 -> 3 <- 2
trueDAG
#>       [,1]  [,2]  [,3]
#> [1,] FALSE FALSE  TRUE
#> [2,] FALSE FALSE  TRUE
#> [3,] FALSE FALSE FALSE

# simulate example data

sem_object <- random_additive_polynomial_SEM(trueDAG, seed_ = 5)
sem_object <- rescale_sem_object(sem_object, seed_ = 4)
X <- simulate_additive_SEM(sem_object, n = 400, seed_ = 3)
pairs(X)

# perform tests

# Test H0: 2 <~/~> 1 (TRUE)
bootstrap.cam(X, matrix(c(2, 1), ncol=2), B=100, method = "two-sided")$pvalue 
#> [1] 0.86

# Test H0: 2 <~/~> 3 (FALSE)
bootstrap.cam(X, matrix(c(2, 3), ncol=2), B=100, method = "two-sided")$pvalue 
#> [1] 0

Authors

This basically the package from CRAN by J. Peters, J. Ernest with some minor changes by @jan-glx and @nignatiadis.

cam's Issues

do not compute score matrix entries that are disallowed by fixedOrders

Brute Force Optimization

To avoid local minima and as sanity check, allow ability to brute force optimize over all possible causal orderings. This can be improved via a branch and bound procedure (use score matrix bound)