Bootstrap

Import and settings

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

import numpy as np
import pandas as pd
import graphviz
import lingam
from lingam.utils import print_causal_directions, print_dagc, make_dot

print([np.__version__, pd.__version__, graphviz.__version__, lingam.__version__])

np.set_printoptions(precision=3, suppress=True)
np.random.seed(0)

['1.16.2', '0.24.2', '0.11.1', '1.5.4']

Test data

We create test data consisting of 6 variables.

x3 = np.random.uniform(size=1000)
x0 = 3.0*x3 + np.random.uniform(size=1000)
x2 = 6.0*x3 + np.random.uniform(size=1000)
x1 = 3.0*x0 + 2.0*x2 + np.random.uniform(size=1000)
x5 = 4.0*x0 + np.random.uniform(size=1000)
x4 = 8.0*x0 - 1.0*x2 + np.random.uniform(size=1000)
X = pd.DataFrame(np.array([x0, x1, x2, x3, x4, x5]).T ,columns=['x0', 'x1', 'x2', 'x3', 'x4', 'x5'])
X.head()
x0 x1 x2 x3 x4 x5
0 2.239321 15.340724 4.104399 0.548814 14.176947 9.249925
1 2.155632 16.630954 4.767220 0.715189 12.775458 9.189045
2 2.284116 15.910406 4.139736 0.602763 14.201794 9.273880
3 2.343420 14.921457 3.519820 0.544883 15.580067 9.723392
4 1.314940 11.055176 3.146972 0.423655 7.604743 5.312976 

m = np.array([[0.0, 0.0, 0.0, 3.0, 0.0, 0.0],
              [3.0, 0.0, 2.0, 0.0, 0.0, 0.0],
              [0.0, 0.0, 0.0, 6.0, 0.0, 0.0],
              [0.0, 0.0, 0.0, 0.0, 0.0, 0.0],
              [8.0, 0.0,-1.0, 0.0, 0.0, 0.0],
              [4.0, 0.0, 0.0, 0.0, 0.0, 0.0]])

make_dot(m)
../_images/bootstrap_dag.svg

Bootstrapping

We call bootstrap() method instead of fit(). Here, the second argument specifies the number of bootstrap sampling.

model = lingam.DirectLiNGAM()
result = model.bootstrap(X, n_sampling=100)

Causal Directions

Since BootstrapResult object is returned, we can get the ranking of the causal directions extracted by get_causal_direction_counts() method. In the following sample code, n_directions option is limited to the causal directions of the top 8 rankings, and min_causal_effect option is limited to causal directions with a coefficient of 0.01 or more.

cdc = result.get_causal_direction_counts(n_directions=8, min_causal_effect=0.01, split_by_causal_effect_sign=True)

We can check the result by utility function.

print_causal_directions(cdc, 100)

x1 <--- x0 (b>0) (100.0%)
x1 <--- x2 (b>0) (100.0%)
x5 <--- x0 (b>0) (100.0%)
x0 <--- x3 (b>0) (99.0%)
x4 <--- x0 (b>0) (98.0%)
x2 <--- x3 (b>0) (96.0%)
x4 <--- x2 (b<0) (94.0%)
x4 <--- x5 (b>0) (20.0%)

Directed Acyclic Graphs

Also, using the get_directed_acyclic_graph_counts() method, we can get the ranking of the DAGs extracted. In the following sample code, n_dags option is limited to the dags of the top 3 rankings, and min_causal_effect option is limited to causal directions with a coefficient of 0.01 or more.

dagc = result.get_directed_acyclic_graph_counts(n_dags=3, min_causal_effect=0.01, split_by_causal_effect_sign=True)

We can check the result by utility function.

print_dagc(dagc, 100)

DAG[0]: 54.0%
    x0 <--- x3 (b>0)
    x1 <--- x0 (b>0)
    x1 <--- x2 (b>0)
    x2 <--- x3 (b>0)
    x4 <--- x0 (b>0)
    x4 <--- x2 (b<0)
    x5 <--- x0 (b>0)
DAG[1]: 16.0%
    x0 <--- x3 (b>0)
    x1 <--- x0 (b>0)
    x1 <--- x2 (b>0)
    x2 <--- x3 (b>0)
    x4 <--- x0 (b>0)
    x4 <--- x2 (b<0)
    x4 <--- x5 (b>0)
    x5 <--- x0 (b>0)
DAG[2]: 7.0%
    x0 <--- x3 (b>0)
    x1 <--- x0 (b>0)
    x1 <--- x2 (b>0)
    x1 <--- x3 (b>0)
    x2 <--- x3 (b>0)
    x4 <--- x0 (b>0)
    x4 <--- x2 (b<0)
    x5 <--- x0 (b>0)

Probability

Using the get_probabilities() method, we can get the probability of bootstrapping.

prob = result.get_probabilities(min_causal_effect=0.01)
print(prob)

[[0.   0.   0.1  0.99 0.02 0.  ]
 [1.   0.   1.   0.1  0.   0.05]
 [0.   0.   0.   0.96 0.   0.  ]
 [0.   0.   0.04 0.   0.   0.  ]
 [0.98 0.   0.94 0.02 0.   0.2 ]
 [1.   0.   0.   0.   0.   0.  ]]

Total Causal Effects

Using the get_total_causal_effects() method, we can get the list of total causal effect. The total causal effects we can get are dictionary type variable. We can display the list nicely by assigning it to pandas.DataFrame. Also, we have replaced the variable index with a label below.

causal_effects = result.get_total_causal_effects(min_causal_effect=0.01)

# Assign to pandas.DataFrame for pretty display
df = pd.DataFrame(causal_effects)
labels = [f'x{i}' for i in range(X.shape[1])]
df['from'] = df['from'].apply(lambda x : labels[x])
df['to'] = df['to'].apply(lambda x : labels[x])
df
from to effect probability
0 x3 x0 3.006190 1.00
1 x0 x1 3.004868 1.00
2 x2 x1 2.092102 1.00
3 x3 x1 20.931938 1.00
4 x0 x5 3.982892 1.00
5 x3 x5 12.024250 1.00
6 x2 x4 -0.887620 1.00
7 x3 x4 18.077244 1.00
8 x0 x4 7.993145 0.98
9 x3 x2 5.970163 0.96
10 x5 x1 0.011708 0.79
11 x2 x5 0.024284 0.72
12 x0 x2 0.014228 0.70
13 x5 x4 0.015170 0.66
14 x2 x0 0.015480 0.30
15 x1 x5 0.021215 0.21
16 x4 x1 -0.004251 0.04
17 x2 x3 0.163050 0.04
18 x4 x0 0.122301 0.02
19 x4 x5 0.009574 0.02

We can easily perform sorting operations with pandas.DataFrame.

df.sort_values('effect', ascending=False).head()
from to effect probability
3 x3 x1 20.931938 1.00
7 x3 x4 18.077244 1.00
5 x3 x5 12.024250 1.00
8 x0 x4 7.993145 0.98
9 x3 x2 5.970163 0.96 

df.sort_values('probability', ascending=True).head()
from to effect probability
19 x4 x5 0.009574 0.02
18 x4 x0 0.122301 0.02
17 x2 x3 0.163050 0.04
16 x4 x1 -0.004251 0.04
15 x1 x5 0.021215 0.21

And with pandas.DataFrame, we can easily filter by keywords. The following code extracts the causal direction towards x1.

df[df['to']=='x1'].head()
from to effect probability
1 x0 x1 3.004868 1.00
2 x2 x1 2.092102 1.00
3 x3 x1 20.931938 1.00
10 x5 x1 0.011708 0.79
16 x4 x1 -0.004251 0.04

Because it holds the raw data of the total causal effect (the original data for calculating the median), it is possible to draw a histogram of the values of the causal effect, as shown below.

import matplotlib.pyplot as plt
import seaborn as sns
sns.set()
%matplotlib inline

from_index = 3 # index of x3
to_index = 0 # index of x0
plt.hist(result.total_effects_[:, to_index, from_index])
../_images/bootstrap_hist.png

Bootstrap Probability of Path

Using the get_paths() method, we can explore all paths from any variable to any variable and calculate the bootstrap probability for each path. The path will be output as an array of variable indices. For example, the array [3, 0, 1] shows the path from variable X3 through variable X0 to variable X1.

from_index = 3 # index of x3
to_index = 1 # index of x0

pd.DataFrame(result.get_paths(from_index, to_index))
path effect probability
0 [3, 0, 1] 8.644765 0.99
1 [3, 2, 1] 11.653517 0.96
2 [3, 1] 0.102936 0.10
3 [3, 2, 0, 1] 1.007787 0.08
4 [3, 0, 5, 1] 0.889629 0.05
5 [3, 4, 0, 1] 8.419805 0.02
6 [3, 2, 4, 0, 1] -3.802326 0.01