Regional Effects (unknown black-box function)

This tutorial use the same dataset with the previous tutorial, but instead of explaining the known (synthetic) predictive function, we fit a neural network on the data and explain the neural network. This is a more realistic scenario, since in real-world applications we do not know the underlying function and we only have access to the data. We advise the reader to first read the previous tutorial.

import numpy as np
import effector
import keras
import tensorflow as tf

np.random.seed(12345)
tf.random.set_seed(12345)

Simulation example

Data Generating Distribution

We will generate $N=1000$ examples with $D=3$ features, which are in the uncorrelated setting all uniformly distributed as follows:

Feature	Description	Distribution
$x_1$	Uniformly distributed between $-1$ and $1$	$x_1 \sim \mathcal{U}(-1,1)$
$x_2$	Uniformly distributed between $-1$ and $1$	$x_2 \sim \mathcal{U}(-1,1)$
$x_3$	Uniformly distributed between $-1$ and $1$	$x_3 \sim \mathcal{U}(-1,1)$

For the correlated setting, we keep the distributional assumptions for $x_2$ and $x_3$, but define $x_1$ such that it is highly correlated with $x_3$ by:
$x_1 = x_3 + \delta$ with $\delta \sim \mathcal{N}(0,0.0625)$.

def generate_dataset_uncorrelated(N):
    x1 = np.random.uniform(-1, 1, size=N)
    x2 = np.random.uniform(-1, 1, size=N)
    x3 = np.random.uniform(-1, 1, size=N)
    return np.stack((x1, x2, x3), axis=-1)

def generate_dataset_correlated(N):
    x3 = np.random.uniform(-1, 1, size=N)
    x2 = np.random.uniform(-1, 1, size=N)
    x1 = x3 + np.random.normal(loc = np.zeros_like(x3), scale = 0.25)
    return np.stack((x1, x2, x3), axis=-1)

# generate the dataset for the uncorrelated and correlated setting
N = 1000
X_uncor_train = generate_dataset_uncorrelated(N)
X_uncor_test = generate_dataset_uncorrelated(10000)
X_cor_train = generate_dataset_correlated(N)
X_cor_test = generate_dataset_correlated(10000)

Black-box function

We will use the following linear model with a subgroup-specific interaction term: $$ y = 3x_1I_{x_3>0} - 3x_1I_{x_3\leq0} + x_3$$

On a global level, there is a high heterogeneity for the features $x_1$ and $x_3$ due to their interaction with each other. However, this heterogeneity vanishes to 0 if the feature space is separated into subregions:

| Feature | Region | Average Effect | Heterogeneity | |---------|-------------|----------------|---------------| | $x_1$ | $x_3>0$ | $3x_1$ | 0 | | $x_1$ | $x_3\leq 0$ | $-3x_1$ | 0 | | $x_2$ | all | 0 | 0 | | $x_3$ | $x_3>0$ | $x_3$ | 0 | | $x_3$ | $x_3\leq 0$ | $x_3$ | 0 |

def generate_target(X):
    f = np.where(X[:,2] > 0, 3*X[:,0] + X[:,2], -3*X[:,0] + X[:,2])
    epsilon = np.random.normal(loc = np.zeros_like(X[:,0]), scale = 0.1)
    Y = f + epsilon
    return(Y)

# generate target for uncorrelated and correlated setting
Y_uncor_train = generate_target(X_uncor_train)
Y_uncor_test = generate_target(X_uncor_test)
Y_cor_train = generate_target(X_cor_train)
Y_cor_test = generate_target(X_cor_test)

Fit a Neural Network

We create a two-layer feedforward Neural Network, a weight decay of 0.01 for 100 epochs. We train two instances of this NN, one on the uncorrelated and one on the correlated setting. In both cases, the NN achieves a Mean Squared Error of about $0.17$ units.

# Train - Evaluate - Explain a neural network
model_uncor = keras.Sequential([
    keras.layers.Dense(10, activation="relu", input_shape=(3,)),
    keras.layers.Dense(10, activation="relu", input_shape=(3,)),
    keras.layers.Dense(1)
])

optimizer = keras.optimizers.Adam(learning_rate=0.01)
model_uncor.compile(optimizer=optimizer, loss="mse")
model_uncor.fit(X_uncor_train, Y_uncor_train, epochs=100)
model_uncor.evaluate(X_uncor_test, Y_uncor_test)

Epoch 1/100


/Users/dimitriskyriakopoulos/Documents/ath/Effector/Code/eff-env/lib/python3.10/site-packages/keras/src/layers/core/dense.py:93: UserWarning: Do not pass an `input_shape`/`input_dim` argument to a layer. When using Sequential models, prefer using an `Input(shape)` object as the first layer in the model instead.
  super().__init__(activity_regularizer=activity_regularizer, **kwargs)


[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 3.0840   
Epoch 2/100
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Epoch 3/100
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Epoch 4/100
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Epoch 5/100
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Epoch 6/100
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Epoch 7/100
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Epoch 8/100
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Epoch 9/100
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Epoch 10/100
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Epoch 11/100
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Epoch 12/100
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Epoch 13/100
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Epoch 14/100
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Epoch 15/100
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Epoch 16/100
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Epoch 17/100
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Epoch 18/100
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Epoch 19/100
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Epoch 20/100
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Epoch 21/100
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Epoch 22/100
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Epoch 23/100
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Epoch 24/100
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Epoch 25/100
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Epoch 26/100
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Epoch 27/100
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Epoch 28/100
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Epoch 29/100
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Epoch 30/100
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Epoch 31/100
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Epoch 32/100
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Epoch 33/100
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Epoch 34/100
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Epoch 35/100
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Epoch 36/100
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Epoch 37/100
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Epoch 38/100
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Epoch 39/100
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Epoch 40/100
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Epoch 41/100
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Epoch 42/100
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Epoch 43/100
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Epoch 44/100
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Epoch 45/100
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Epoch 46/100
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Epoch 47/100
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Epoch 48/100
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Epoch 49/100
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Epoch 50/100
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Epoch 51/100
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Epoch 52/100
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Epoch 53/100
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Epoch 54/100
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Epoch 55/100
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Epoch 56/100
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Epoch 57/100
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Epoch 58/100
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Epoch 59/100
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Epoch 60/100
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Epoch 61/100
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Epoch 62/100
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Epoch 63/100
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Epoch 64/100
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Epoch 65/100
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Epoch 66/100
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Epoch 67/100
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Epoch 68/100
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Epoch 69/100
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Epoch 70/100
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Epoch 71/100
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Epoch 72/100
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Epoch 73/100
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Epoch 74/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 867us/step - loss: 0.0535
Epoch 75/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 879us/step - loss: 0.0525
Epoch 76/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 853us/step - loss: 0.0530
Epoch 77/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 858us/step - loss: 0.0538
Epoch 78/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 867us/step - loss: 0.0532
Epoch 79/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 864us/step - loss: 0.0530
Epoch 80/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 869us/step - loss: 0.0536
Epoch 81/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 848us/step - loss: 0.0534
Epoch 82/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0533 
Epoch 83/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 893us/step - loss: 0.0530
Epoch 84/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 889us/step - loss: 0.0521
Epoch 85/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 877us/step - loss: 0.0537
Epoch 86/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 896us/step - loss: 0.0537
Epoch 87/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 893us/step - loss: 0.0514
Epoch 88/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 886us/step - loss: 0.0525
Epoch 89/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 898us/step - loss: 0.0531
Epoch 90/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 875us/step - loss: 0.0515
Epoch 91/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 884us/step - loss: 0.0514
Epoch 92/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 874us/step - loss: 0.0507
Epoch 93/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 2ms/step - loss: 0.0518
Epoch 94/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 891us/step - loss: 0.0518
Epoch 95/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 881us/step - loss: 0.0507
Epoch 96/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 915us/step - loss: 0.0510
Epoch 97/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 885us/step - loss: 0.0502
Epoch 98/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 897us/step - loss: 0.0511
Epoch 99/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 852us/step - loss: 0.0502
Epoch 100/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 848us/step - loss: 0.0508
[1m313/313[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 440us/step - loss: 0.0729





0.06930544227361679

model_cor = keras.Sequential([
    keras.layers.Dense(10, activation="relu", input_shape=(3,)),
    keras.layers.Dense(10, activation="relu", input_shape=(3,)),
    keras.layers.Dense(1)
])

optimizer = keras.optimizers.Adam(learning_rate=0.01)
model_cor.compile(optimizer=optimizer, loss="mse")
model_cor.fit(X_cor_train, Y_cor_train, epochs=100)
model_cor.evaluate(X_cor_test, Y_cor_test)

Epoch 1/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 3.7841   
Epoch 2/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 961us/step - loss: 0.7420
Epoch 3/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 937us/step - loss: 0.2709
Epoch 4/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 865us/step - loss: 0.1657
Epoch 5/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 869us/step - loss: 0.1346
Epoch 6/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 856us/step - loss: 0.1218
Epoch 7/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 864us/step - loss: 0.1147
Epoch 8/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 838us/step - loss: 0.1112
Epoch 9/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 877us/step - loss: 0.1089
Epoch 10/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 842us/step - loss: 0.1071
Epoch 11/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 859us/step - loss: 0.1050
Epoch 12/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 853us/step - loss: 0.1031
Epoch 13/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 857us/step - loss: 0.0995
Epoch 14/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 855us/step - loss: 0.0954
Epoch 15/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 850us/step - loss: 0.0904
Epoch 16/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 864us/step - loss: 0.0852
Epoch 17/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 869us/step - loss: 0.0805
Epoch 18/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 833us/step - loss: 0.0753
Epoch 19/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 854us/step - loss: 0.0675
Epoch 20/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 846us/step - loss: 0.0602
Epoch 21/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 862us/step - loss: 0.0560
Epoch 22/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 850us/step - loss: 0.0536
Epoch 23/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0503 
Epoch 24/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 874us/step - loss: 0.0484
Epoch 25/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 878us/step - loss: 0.0464
Epoch 26/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 845us/step - loss: 0.0440
Epoch 27/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 824us/step - loss: 0.0422
Epoch 28/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 855us/step - loss: 0.0410
Epoch 29/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 855us/step - loss: 0.0391
Epoch 30/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 847us/step - loss: 0.0377
Epoch 31/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 851us/step - loss: 0.0367
Epoch 32/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 853us/step - loss: 0.0350
Epoch 33/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 858us/step - loss: 0.0341
Epoch 34/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 854us/step - loss: 0.0323
Epoch 35/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 832us/step - loss: 0.0306
Epoch 36/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 855us/step - loss: 0.0300
Epoch 37/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 856us/step - loss: 0.0286
Epoch 38/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 847us/step - loss: 0.0274
Epoch 39/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 858us/step - loss: 0.0274
Epoch 40/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 862us/step - loss: 0.0259
Epoch 41/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 856us/step - loss: 0.0253
Epoch 42/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 857us/step - loss: 0.0250
Epoch 43/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 834us/step - loss: 0.0240
Epoch 44/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 859us/step - loss: 0.0240
Epoch 45/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 849us/step - loss: 0.0230
Epoch 46/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 843us/step - loss: 0.0233
Epoch 47/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 855us/step - loss: 0.0232
Epoch 48/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 869us/step - loss: 0.0229
Epoch 49/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 858us/step - loss: 0.0217
Epoch 50/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 864us/step - loss: 0.0225
Epoch 51/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 857us/step - loss: 0.0210
Epoch 52/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 856us/step - loss: 0.0221
Epoch 53/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 864us/step - loss: 0.0212
Epoch 54/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 853us/step - loss: 0.0220
Epoch 55/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 847us/step - loss: 0.0202
Epoch 56/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 860us/step - loss: 0.0211
Epoch 57/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 859us/step - loss: 0.0207
Epoch 58/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 2ms/step - loss: 0.0193
Epoch 59/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 855us/step - loss: 0.0206
Epoch 60/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 877us/step - loss: 0.0194
Epoch 61/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 852us/step - loss: 0.0204
Epoch 62/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 846us/step - loss: 0.0190
Epoch 63/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 854us/step - loss: 0.0196
Epoch 64/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 901us/step - loss: 0.0192
Epoch 65/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 870us/step - loss: 0.0193
Epoch 66/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 845us/step - loss: 0.0187
Epoch 67/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 844us/step - loss: 0.0188
Epoch 68/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 857us/step - loss: 0.0185
Epoch 69/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 851us/step - loss: 0.0189
Epoch 70/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 853us/step - loss: 0.0183
Epoch 71/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 856us/step - loss: 0.0187
Epoch 72/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 848us/step - loss: 0.0183
Epoch 73/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 840us/step - loss: 0.0184
Epoch 74/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 862us/step - loss: 0.0184
Epoch 75/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 861us/step - loss: 0.0180
Epoch 76/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 861us/step - loss: 0.0182
Epoch 77/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 854us/step - loss: 0.0179
Epoch 78/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 852us/step - loss: 0.0177
Epoch 79/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 846us/step - loss: 0.0178
Epoch 80/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 860us/step - loss: 0.0172
Epoch 81/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 848us/step - loss: 0.0186
Epoch 82/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 845us/step - loss: 0.0172
Epoch 83/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 842us/step - loss: 0.0184
Epoch 84/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 855us/step - loss: 0.0171
Epoch 85/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 849us/step - loss: 0.0172
Epoch 86/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 840us/step - loss: 0.0176
Epoch 87/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 830us/step - loss: 0.0174
Epoch 88/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 845us/step - loss: 0.0176
Epoch 89/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 832us/step - loss: 0.0173
Epoch 90/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 844us/step - loss: 0.0170
Epoch 91/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 839us/step - loss: 0.0173
Epoch 92/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 860us/step - loss: 0.0172
Epoch 93/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 864us/step - loss: 0.0174
Epoch 94/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 836us/step - loss: 0.0170
Epoch 95/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 841us/step - loss: 0.0174
Epoch 96/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 860us/step - loss: 0.0167
Epoch 97/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 848us/step - loss: 0.0173
Epoch 98/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 849us/step - loss: 0.0165
Epoch 99/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 857us/step - loss: 0.0171
Epoch 100/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 861us/step - loss: 0.0167
[1m313/313[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 433us/step - loss: 0.0285





0.0264066681265831

PDP

Uncorrelated setting

Global PDP

pdp = effector.PDP(data=X_uncor_train, model=model_uncor, feature_names=['x1','x2','x3'], target_name="Y")
pdp.plot(feature=0, centering=True, show_avg_output=False, heterogeneity="ice", y_limits=[-5, 5])
pdp.plot(feature=1, centering=True, show_avg_output=False, heterogeneity="ice", y_limits=[-5, 5])
pdp.plot(feature=2, centering=True, show_avg_output=False, heterogeneity="ice", y_limits=[-5, 5])

Regional PDP

regional_pdp = effector.RegionalPDP(data=X_uncor_train, model=model_uncor, feature_names=['x1','x2','x3'], axis_limits=np.array([[-1,1],[-1,1],[-1,1]]).T)
space_partitioner = effector.space_partitioning.Best(min_heterogeneity_decrease_pcg=0.3, numerical_features_grid_size=10)
regional_pdp.fit(features="all", space_partitioner=space_partitioner)

100%|██████████| 3/3 [00:00<00:00, 67.34it/s]

regional_pdp.summary(features=0)

Feature 0 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x1 🔹 [id: 0 | heter: 3.15 | inst: 1000 | w: 1.00]
    x3 ≤ 0.00 🔹 [id: 1 | heter: 0.05 | inst: 500 | w: 0.50]
    x3 > 0.00 🔹 [id: 2 | heter: 0.07 | inst: 500 | w: 0.50]
--------------------------------------------------
Feature 0 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 3.15
    Level 1🔹heter: 0.06 | 🔻3.09 (98.09%)

regional_pdp.plot(feature=0, node_idx=1, heterogeneity="ice", y_limits=[-5, 5])
regional_pdp.plot(feature=0, node_idx=2, heterogeneity="ice", y_limits=[-5, 5])

regional_pdp.summary(features=1)

Feature 1 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x2 🔹 [id: 0 | heter: 0.00 | inst: 1000 | w: 1.00]
--------------------------------------------------
Feature 1 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.00

regional_pdp.summary(features=2)

Feature 2 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x3 🔹 [id: 0 | heter: 2.95 | inst: 1000 | w: 1.00]
    x1 ≤ 0.00 🔹 [id: 1 | heter: 0.73 | inst: 494 | w: 0.49]
        x1 ≤ -0.60 🔹 [id: 2 | heter: 0.13 | inst: 196 | w: 0.20]
        x1 > -0.60 🔹 [id: 3 | heter: 0.27 | inst: 298 | w: 0.30]
    x1 > 0.00 🔹 [id: 4 | heter: 0.71 | inst: 506 | w: 0.51]
        x1 ≤ 0.40 🔹 [id: 5 | heter: 0.13 | inst: 199 | w: 0.20]
        x1 > 0.40 🔹 [id: 6 | heter: 0.26 | inst: 307 | w: 0.31]
--------------------------------------------------
Feature 2 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 2.95
    Level 1🔹heter: 0.72 | 🔻2.22 (75.50%)
        Level 2🔹heter: 0.21 | 🔻0.51 (70.90%)

regional_pdp.plot(feature=2, node_idx=1, heterogeneity="ice", centering=True, y_limits=[-5, 5])
regional_pdp.plot(feature=2, node_idx=4, heterogeneity="ice", centering=True, y_limits=[-5, 5])

Conclusion

For the Global PDP:

the average effect of $x_1$ is $0$ with some heterogeneity implied by the interaction with $x_1$. The heterogeneity is expressed with two opposite lines; $-3x_1$ when $x_1 \leq 0$ and $3x_1$ when $x_1 >0$
the average effect of $x_2$ to be $0$ without heterogeneity
the average effect of $x_3$ to be $x_3$ with some heterogeneity due to the interaction with $x_1$. The heterogeneity is expressed with a discontinuity around $x_3=0$, with either a positive or a negative offset depending on the value of $x_1^i$

For the Regional PDP:

For $x_1$, the algorithm finds two regions, one for $x_3 \leq 0$ and one for $x_3 > 0$
when $x_3>0$ the effect is $3x_1$
when $x_3 \leq 0$, the effect is $-3x_1$
For $x_2$ the algorithm does not find any subregion
For $x_3$, there is a change in the offset:
when $x_1>0$ the line is $x_3 - 3x_1^i$ in the first half and $x_3 + 3x_1^i$ later
when $x_1<0$ the line is $x_3 + 3x_1^i$ in the first half and $x_3 - 3x_1^i$ later

Correlated setting

Global PDP

pdp = effector.PDP(data=X_cor_train, model=model_cor, feature_names=['x1','x2','x3'], target_name="Y")
pdp.plot(feature=0, centering=True, show_avg_output=False, heterogeneity="ice", y_limits=[-5, 5])
pdp.plot(feature=1, centering=True, show_avg_output=False, heterogeneity="ice", y_limits=[-5, 5])
pdp.plot(feature=2, centering=True, show_avg_output=False, heterogeneity="ice", y_limits=[-5, 5])

Regional-PDP

regional_pdp = effector.RegionalPDP(data=X_cor_train, model=model_cor, feature_names=['x1','x2','x3'], axis_limits=np.array([[-1,1],[-1,1],[-1,1]]).T)
space_partitioner = effector.space_partitioning.Best(min_heterogeneity_decrease_pcg=0.4, numerical_features_grid_size=10)
regional_pdp.fit(features="all", space_partitioner=space_partitioner)

100%|██████████| 3/3 [00:00<00:00, 89.17it/s]

regional_pdp.summary(features=0)

Feature 0 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x1 🔹 [id: 0 | heter: 2.82 | inst: 900 | w: 1.00]
    x3 ≤ 0.00 🔹 [id: 1 | heter: 0.28 | inst: 436 | w: 0.48]
        x3 ≤ -0.20 🔹 [id: 2 | heter: 0.08 | inst: 337 | w: 0.37]
        x3 > -0.20 🔹 [id: 3 | heter: 0.22 | inst: 99 | w: 0.11]
    x3 > 0.00 🔹 [id: 4 | heter: 0.13 | inst: 464 | w: 0.52]
        x3 ≤ 0.20 🔹 [id: 5 | heter: 0.13 | inst: 115 | w: 0.13]
        x3 > 0.20 🔹 [id: 6 | heter: 0.04 | inst: 349 | w: 0.39]
--------------------------------------------------
Feature 0 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 2.82
    Level 1🔹heter: 0.20 | 🔻2.62 (92.84%)
        Level 2🔹heter: 0.09 | 🔻0.12 (57.09%)

regional_pdp.plot(feature=0, node_idx=1, heterogeneity="ice", centering=True, y_limits=[-5, 5])
regional_pdp.plot(feature=0, node_idx=4, heterogeneity="ice", centering=True, y_limits=[-5, 5])

regional_pdp.summary(features=1)

Feature 1 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x2 🔹 [id: 0 | heter: 0.00 | inst: 900 | w: 1.00]
--------------------------------------------------
Feature 1 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.00

regional_pdp.summary(features=2)

Feature 2 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x3 🔹 [id: 0 | heter: 2.48 | inst: 900 | w: 1.00]
    x1 ≤ 0.00 🔹 [id: 1 | heter: 0.62 | inst: 463 | w: 0.51]
        x1 ≤ -0.40 🔹 [id: 2 | heter: 0.21 | inst: 246 | w: 0.27]
        x1 > -0.40 🔹 [id: 3 | heter: 0.11 | inst: 217 | w: 0.24]
    x1 > 0.00 🔹 [id: 4 | heter: 0.76 | inst: 437 | w: 0.49]
        x1 ≤ 0.40 🔹 [id: 5 | heter: 0.16 | inst: 189 | w: 0.21]
        x1 > 0.40 🔹 [id: 6 | heter: 0.21 | inst: 248 | w: 0.28]
--------------------------------------------------
Feature 2 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 2.48
    Level 1🔹heter: 0.69 | 🔻1.79 (72.25%)
        Level 2🔹heter: 0.18 | 🔻0.51 (74.47%)

regional_pdp.plot(feature=2, node_idx=1, heterogeneity="ice", centering=True, y_limits=[-5, 5])
regional_pdp.plot(feature=2, node_idx=4, heterogeneity="ice", centering=True, y_limits=[-5, 5])

Conclusion

(RH)ALE

def model_uncor_jac(x):
    x_tensor = tf.convert_to_tensor(x, dtype=tf.float32)
    with tf.GradientTape() as t:
        t.watch(x_tensor)
        pred = model_uncor(x_tensor)
        grads = t.gradient(pred, x_tensor)
    return grads.numpy()

def model_cor_jac(x):
    x_tensor = tf.convert_to_tensor(x, dtype=tf.float32)
    with tf.GradientTape() as t:
        t.watch(x_tensor)
        pred = model_cor(x_tensor)
        grads = t.gradient(pred, x_tensor)
    return grads.numpy()

Uncorrelated setting

Global RHALE

rhale = effector.RHALE(data=X_uncor_train, model=model_uncor, model_jac=model_uncor_jac, feature_names=['x1','x2','x3'], target_name="Y")

binning_method = effector.axis_partitioning.Fixed(10, min_points_per_bin=0)
rhale.fit(features="all", binning_method=binning_method, centering=True)

rhale.plot(feature=0, centering=True, heterogeneity="std", show_avg_output=False, y_limits=[-5, 5], dy_limits=[-5, 5])
rhale.plot(feature=1, centering=True, heterogeneity="std", show_avg_output=False, y_limits=[-5, 5], dy_limits=[-5, 5])
rhale.plot(feature=2, centering=True, heterogeneity="std", show_avg_output=False, y_limits=[-5, 5], dy_limits=[-5, 5])

Regional RHALE

regional_rhale = effector.RegionalRHALE(
    data=X_uncor_train, 
    model=model_uncor, 
    model_jac= model_uncor_jac, 
    feature_names=['x1', 'x2', 'x3'],
    axis_limits=np.array([[-1, 1], [-1, 1], [-1, 1]]).T) 

binning_method = effector.axis_partitioning.Fixed(11, min_points_per_bin=0)
space_partitioner = effector.space_partitioning.Best(min_heterogeneity_decrease_pcg=0.6, numerical_features_grid_size=10)
regional_rhale.fit(
    features="all",
    binning_method=binning_method,
    space_partitioner=space_partitioner
)

100%|██████████| 3/3 [00:00<00:00, 53.67it/s]

regional_rhale.summary(features=0)

Feature 0 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x1 🔹 [id: 0 | heter: 8.82 | inst: 1000 | w: 1.00]
    x3 ≤ 0.00 🔹 [id: 1 | heter: 0.30 | inst: 500 | w: 0.50]
    x3 > 0.00 🔹 [id: 2 | heter: 0.45 | inst: 500 | w: 0.50]
--------------------------------------------------
Feature 0 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 8.82
    Level 1🔹heter: 0.37 | 🔻8.45 (95.76%)

regional_rhale.plot(feature=0, node_idx=1, heterogeneity="std", centering=True, y_limits=[-5, 5])
regional_rhale.plot(feature=0, node_idx=2, heterogeneity="std", centering=True, y_limits=[-5, 5])

regional_rhale.summary(features=1)

Feature 1 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x2 🔹 [id: 0 | heter: 0.02 | inst: 1000 | w: 1.00]
--------------------------------------------------
Feature 1 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.02

regional_rhale.summary(features=2)

Feature 2 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x3 🔹 [id: 0 | heter: 55.57 | inst: 1000 | w: 1.00]
--------------------------------------------------
Feature 2 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 55.57

Conclusion

Correlated setting

Global RHALE

rhale = effector.RHALE(data=X_cor_train, model=model_cor, model_jac=model_cor_jac, feature_names=['x1','x2','x3'], target_name="Y")

binning_method = effector.axis_partitioning.Fixed(10, min_points_per_bin=0)
rhale.fit(features="all", binning_method=binning_method, centering=True)

rhale.plot(feature=0, centering=True, heterogeneity="std", show_avg_output=False, y_limits=[-5, 5], dy_limits=[-5, 5])
rhale.plot(feature=1, centering=True, heterogeneity="std", show_avg_output=False, y_limits=[-5, 5], dy_limits=[-5, 5])
rhale.plot(feature=2, centering=True, heterogeneity="std", show_avg_output=False, y_limits=[-5, 5], dy_limits=[-5, 5])

Regional RHALE

regional_rhale = effector.RegionalRHALE(
    data=X_cor_train, 
    model=model_cor, 
    model_jac= model_cor_jac, 
    feature_names=['x1', 'x2', 'x3'],
    axis_limits=np.array([[-1, 1], [-1, 1], [-1, 1]]).T) 

binning_method = effector.axis_partitioning.Fixed(11, min_points_per_bin=0)
space_partitioner = effector.space_partitioning.Best(min_heterogeneity_decrease_pcg=0.6, numerical_features_grid_size=10)
regional_rhale.fit(
    features="all",
    space_partitioner=space_partitioner,
)

100%|██████████| 3/3 [00:00<00:00, 21.03it/s]

regional_rhale.summary(features=0)

Feature 0 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x1 🔹 [id: 0 | heter: 2.13 | inst: 900 | w: 1.00]
--------------------------------------------------
Feature 0 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 2.13

regional_rhale.summary(features=1)

Feature 1 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x2 🔹 [id: 0 | heter: 0.03 | inst: 900 | w: 1.00]
--------------------------------------------------
Feature 1 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.03

regional_rhale.summary(features=2)

Feature 2 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x3 🔹 [id: 0 | heter: 14.00 | inst: 900 | w: 1.00]
--------------------------------------------------
Feature 2 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 14.00

Conclusion

SHAP DP

Uncorrelated setting

Global SHAP DP

shap = effector.ShapDP(data=X_uncor_train, model=model_uncor, feature_names=['x1', 'x2', 'x3'], target_name="Y")
binning_method = effector.axis_partitioning.Fixed(nof_bins=5, min_points_per_bin=0)
shap.fit(features="all", binning_method=binning_method, centering=True)
shap.plot(feature=0, centering=True, heterogeneity="shap_values", show_avg_output=False, y_limits=[-3, 3])
shap.plot(feature=1, centering=True, heterogeneity="shap_values", show_avg_output=False, y_limits=[-3, 3])
shap.plot(feature=2, centering=True, heterogeneity="shap_values", show_avg_output=False, y_limits=[-3, 3])

Regional SHAP-DP

regional_shap = effector.RegionalShapDP(
    data=X_uncor_train,
    model=model_uncor,
    feature_names=['x1', 'x2', 'x3'],
    axis_limits=np.array([[-1, 1], [-1, 1], [-1, 1]]).T)

space_partitioner = effector.space_partitioning.Best(min_heterogeneity_decrease_pcg=0.6, numerical_features_grid_size=10)
regional_shap.fit(
    features="all",
    space_partitioner=space_partitioner,
    binning_method = effector.axis_partitioning.Fixed(nof_bins=5, min_points_per_bin=0)
)

100%|██████████| 3/3 [00:02<00:00,  1.04it/s]

regional_shap.summary(0)

Feature 0 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x1 🔹 [id: 0 | heter: 0.84 | inst: 1000 | w: 1.00]
    x3 ≤ 0.00 🔹 [id: 1 | heter: 0.04 | inst: 500 | w: 0.50]
    x3 > 0.00 🔹 [id: 2 | heter: 0.03 | inst: 500 | w: 0.50]
--------------------------------------------------
Feature 0 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.84
    Level 1🔹heter: 0.04 | 🔻0.80 (95.39%)

regional_shap.plot(feature=0, node_idx=1, heterogeneity="std", centering=True, y_limits=[-5, 5])
regional_shap.plot(feature=0, node_idx=2, heterogeneity="std", centering=True, y_limits=[-5, 5])

regional_shap.summary(features=1)

Feature 1 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x2 🔹 [id: 0 | heter: 0.00 | inst: 1000 | w: 1.00]
--------------------------------------------------
Feature 1 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.00

regional_shap.summary(features=2)

Feature 2 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x3 🔹 [id: 0 | heter: 0.79 | inst: 1000 | w: 1.00]
    x1 ≤ 0.00 🔹 [id: 1 | heter: 0.25 | inst: 494 | w: 0.49]
    x1 > 0.00 🔹 [id: 2 | heter: 0.37 | inst: 506 | w: 0.51]
--------------------------------------------------
Feature 2 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.79
    Level 1🔹heter: 0.31 | 🔻0.48 (60.80%)

Conclusion

Correlated setting

Global SHAP-DP

shap = effector.ShapDP(data=X_cor_train, model=model_cor, feature_names=['x1', 'x2', 'x3'], target_name="Y")
binning_method = effector.axis_partitioning.Fixed(nof_bins=5, min_points_per_bin=0)
shap.fit(features="all", binning_method=binning_method, centering=True)
shap.plot(feature=0, centering=True, heterogeneity="shap_values", show_avg_output=False, y_limits=[-3, 3])
shap.plot(feature=1, centering=True, heterogeneity="shap_values", show_avg_output=False, y_limits=[-3, 3])
shap.plot(feature=2, centering=True, heterogeneity="shap_values", show_avg_output=False, y_limits=[-3, 3])

Regional SHAP

regional_shap = effector.RegionalShapDP(
    data=X_cor_train,
    model=model_cor,
    feature_names=['x1', 'x2', 'x3'],
    axis_limits=np.array([[-1, 1], [-1, 1], [-1, 1]]).T)

space_partitioner = effector.space_partitioning.Best(min_heterogeneity_decrease_pcg=0.6, numerical_features_grid_size=10)
regional_shap.fit(
    features="all",
    space_partitioner=space_partitioner,
    binning_method = effector.axis_partitioning.Fixed(nof_bins=5, min_points_per_bin=0)
)

100%|██████████| 3/3 [00:02<00:00,  1.17it/s]

regional_shap.summary(0)
regional_shap.summary(1)
regional_shap.summary(2)

Feature 0 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x1 🔹 [id: 0 | heter: 0.08 | inst: 900 | w: 1.00]
--------------------------------------------------
Feature 0 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.08




Feature 1 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x2 🔹 [id: 0 | heter: 0.00 | inst: 900 | w: 1.00]
--------------------------------------------------
Feature 1 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.00




Feature 2 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x3 🔹 [id: 0 | heter: 0.11 | inst: 900 | w: 1.00]
--------------------------------------------------
Feature 2 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.11

Feature	Description	Distribution
\(x_1\)	Uniformly distributed between \(-1\) and \(1\)	\(x_1 \sim \mathcal{U}(-1,1)\)
\(x_2\)	Uniformly distributed between \(-1\) and \(1\)	\(x_2 \sim \mathcal{U}(-1,1)\)
\(x_3\)	Uniformly distributed between \(-1\) and \(1\)	\(x_3 \sim \mathcal{U}(-1,1)\)