causalXtreme

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An R package for causal discovery in heavy-tailed models

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# causalXtreme

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The goal of causalXtreme is to provide an interface to perform causal
discovery in linear structural equation models (SEM) with heavy-tailed
noise. For more details see Gnecco et al. (2019,
https://arxiv.org/abs/1908.05097).

## Installation

You can install the development version from
[GitHub](https://github.com/) with:

``` r
# install.packages("devtools")
devtools::install_github("nicolagnecco/causalXtreme")
```

## Example

Let us generate 500 observations from a SEM of two Student-t variables,
$X_1$ and $X_2$, with 1.5 degrees of freedom (i.e., heavy-tailed). When
the function `simulate_data` is called with the default values, it
returns a list containing:

- the simulated `dataset` represented as a matrix of size $n \times p$.
  Here, $n = 500$ is the number of observations and $p = 2$ is the
  number of variables,
- the underlying directed acyclic graph (DAG) represented as an
  adjacency matrix `dag` of size $p\times p$.

``` r
library(causalXtreme)

# basic example code
set.seed(1)
sem <- simulate_data(n = 500, p = 2, prob_connect = 0.5,
                     distr = "student_t", tail_index = 1.5)
```

At this point, we can look at the randomly simulated DAG.

``` r
sem$dag
#>      [,1] [,2]
#> [1,]    0    1
#> [2,]    0    0
```

We interpret the adjacency matrix as follows. Loosely speaking, we say
that variable $X_i$ causes variable $X_j$ if the entry $(i, j)$ of the
adjacency matrix is equal to 1. We see that the first variable $X_1$
causes the second variable $X_2$, since the entry $(1, 2)$ of the matrix
`sem$dag` is equal to 1. We can plot the simulated dataset.

``` r
plot(sem$dataset, pch = 20,
     xlab = "X1", ylab = "X2")
```

At this point, we can estimate the causal direction between $X_1$ and
$X_2$ by computing the *causal tail coefficients* $\Gamma_{12}$ and
$\Gamma_{21}$ (see Gnecco et al. 2019, Definition 1).

``` r
X1 <- sem$dataset[, 1]
X2 <- sem$dataset[, 2]

# gamma_12
causal_tail_coeff(X1, X2)
#> [1] 0.9523333

# gamma_21
causal_tail_coeff(X2, X1)
#> [1] 0.4816667
```

We see that the coefficient $\Gamma_{12} \approx 1$ (entry $(1, 2)$ of
the matrix) and $\Gamma_{21} < 1$ (entry $(2, 1)$ of the matrix). This
is evidence for a causal relationship from $X_1$ to $X_2$.

We can also run the *extremal ancestral search* (EASE) algorithm, based
on the causal tail coefficients (see Gnecco et al. 2019, sec. 3.1). The
algorithm estimates from the data a *causal order* of the DAG.

``` r
ease(dat = sem$dataset)
#> [1] 1 2
```

In this case, we can see that the estimated causal order is correct,
since $X_1$ (the cause) is placed before $X_2$ (the effect).

## References

Gnecco, Nicola, Nicolai Meinshausen, Jonas Peters, and Sebastian
Engelke. 2019. “Causal Discovery in Heavy-Tailed Models.” *arXiv
Preprint arXiv:1908.05097*.

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