# LiM¶

## Model¶

Linear Mixed (LiM) causal discovery algorithm [1] extends LiNGAM to handle the mixed data that consists of both continuous and discrete variables. The estimation is performed by first globally optimizing the log-likelihood function on the joint distribution of data with the acyclicity constraint, and then applying a local combinatorial search to output a causal graph.

This method is based on the LiM model as shown below,

i) As for the continuous variable, its value assigned to each of \(x_i\) is a linear function of its parent variables denoted by \(x_{\mathrm{pa}(i)}\) plus a non-Gaussian error term \(e_i\), that is,

where the error terms \(e_i\) are continuous random variables with non-Gaussian densities, and the error variables \(e_i\) are independent of each other. The coefficients \(b_{ij}\) and intercepts \(c_i\) are constants.

ii) As for the discrete variable, its value equals 1 if the linear function of its parent variables \(x_{\mathrm{pa}(i)}\) plus a Logistic error term \(e_i\) is larger than 0, otherwise, its value equals 0. That is,

where the error terms \(e_i\) follow the Logistic distribution, while the other notations are identical to those in continuous variables.

This method makes the following assumptions.

- Continous variables and binary variables.
- Linearity
- Acyclicity
- No hidden common causes
- Baselines are the same when predicting one binary variable from the other for every pair of binary variables.

References

[1] Y. Zeng, S. Shimizu, H. Matsui, F. Sun. Causal discovery for linear mixed data. In Proc. First Conference on Causal Learning and Reasoning (CLeaR2022). PMLR 177, pp. 994-1009, 2022.

## Import and settings¶

In this example, we need to import `numpy`

, and `random`

,
in addition to `lingam`

.

```
import numpy as np
import random
import lingam
import lingam.utils as ut
print([np.__version__, lingam.__version__])
```

```
['1.20.3', '1.6.0']
```

## Test data¶

First, we generate a causal structure with 2 variables, where one of them is randomly set to be a discrete variable.

```
ut.set_random_seed(1)
n_samples, n_features, n_edges, graph_type, sem_type = 1000, 2, 1, 'ER', 'mixed_random_i_dis'
B_true = ut.simulate_dag(n_features, n_edges, graph_type)
W_true = ut.simulate_parameter(B_true) # row to column
no_dis = np.random.randint(1, n_features) # number of discrete vars.
print('There are %d discrete variable(s).' % (no_dis))
nodes = [iii for iii in range(n_features)]
dis_var = random.sample(nodes, no_dis) # randomly select no_dis discrete variables
dis_con = np.full((1, n_features), np.inf)
for iii in range(n_features):
if iii in dis_var:
dis_con[0, iii] = 0 # 1:continuous; 0:discrete
else:
dis_con[0, iii] = 1
X = ut.simulate_linear_mixed_sem(W_true, n_samples, sem_type, dis_con)
print('The true adjacency matrix is:\n', W_true)
```

```
There are 1 discrete variable(s).
The true adjacency matrix is:
[[0. 0. ]
[1.3082251 0. ]]
```

## Causal Discovery for linear mixed data¶

To run causal discovery, we create a `LiM`

object and call the `fit`

method.

```
model = lingam.LiM()
model.fit(X, dis_con)
```

```
<lingam.lim.LiM at 0x174d475f850>
```

Using the `_adjacency_matrix`

properties, we can see the estimated adjacency matrix between mixed variables.

```
print('The estimated adjacency matrix is:\n', model._adjacency_matrix)
```

```
The estimated adjacency matrix is:
[[ 0. , 0. ],
[-1.09938457, 0. ]]
```