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)

2025-02-26 11:26:23.961349: I tensorflow/core/platform/cpu_feature_guard.cc:210] This TensorFlow binary is optimized to use available CPU instructions in performance-critical operations.
To enable the following instructions: AVX2 FMA, in other operations, rebuild TensorFlow with the appropriate compiler flags.

Simulation example

Data Generating Distribution

We will generate $N=500$ 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


/home/givasile/miniconda3/envs/effector-dev/lib/python3.10/site-packages/keras/src/layers/core/dense.py:87: 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)


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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 955us/step - loss: 0.0434
Epoch 75/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 934us/step - loss: 0.0454
Epoch 76/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 910us/step - loss: 0.0434
Epoch 77/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0439
Epoch 78/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0431
Epoch 79/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 917us/step - loss: 0.0434
Epoch 80/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 899us/step - loss: 0.0431
Epoch 81/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0425
Epoch 82/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 935us/step - loss: 0.0423
Epoch 83/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0416
Epoch 84/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0420
Epoch 85/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0412
Epoch 86/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0416
Epoch 87/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0406
Epoch 88/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0404
Epoch 89/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 986us/step - loss: 0.0414
Epoch 90/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0401
Epoch 91/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 983us/step - loss: 0.0391
Epoch 92/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0405
Epoch 93/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 927us/step - loss: 0.0394
Epoch 94/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 858us/step - loss: 0.0396
Epoch 95/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 847us/step - loss: 0.0383
Epoch 96/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 895us/step - loss: 0.0384
Epoch 97/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 840us/step - loss: 0.0381
Epoch 98/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 896us/step - loss: 0.0386
Epoch 99/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 875us/step - loss: 0.0378
Epoch 100/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 938us/step - loss: 0.0391
[1m313/313[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 745us/step - loss: 0.0631





0.0614858940243721

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 [1m1s[0m 959us/step - loss: 2.1464
Epoch 2/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 989us/step - loss: 0.4442
Epoch 3/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 915us/step - loss: 0.2055
Epoch 4/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.1394
Epoch 5/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 978us/step - loss: 0.1068
Epoch 6/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0945
Epoch 7/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 771us/step - loss: 0.0892
Epoch 8/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 929us/step - loss: 0.0851
Epoch 9/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 971us/step - loss: 0.0818
Epoch 10/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 863us/step - loss: 0.0795
Epoch 11/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 908us/step - loss: 0.0759
Epoch 12/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 869us/step - loss: 0.0731
Epoch 13/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 959us/step - loss: 0.0707
Epoch 14/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 964us/step - loss: 0.0684
Epoch 15/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 870us/step - loss: 0.0667
Epoch 16/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0645
Epoch 17/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 916us/step - loss: 0.0624
Epoch 18/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 852us/step - loss: 0.0587
Epoch 19/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 800us/step - loss: 0.0563
Epoch 20/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0526
Epoch 21/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 821us/step - loss: 0.0513
Epoch 22/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 798us/step - loss: 0.0498
Epoch 23/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 793us/step - loss: 0.0481
Epoch 24/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 766us/step - loss: 0.0457
Epoch 25/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 816us/step - loss: 0.0445
Epoch 26/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 865us/step - loss: 0.0424
Epoch 27/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 968us/step - loss: 0.0416
Epoch 28/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0382
Epoch 29/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 972us/step - loss: 0.0372
Epoch 30/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 929us/step - loss: 0.0372
Epoch 31/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 806us/step - loss: 0.0358
Epoch 32/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 913us/step - loss: 0.0340
Epoch 33/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 953us/step - loss: 0.0330
Epoch 34/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 925us/step - loss: 0.0313
Epoch 35/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 861us/step - loss: 0.0308
Epoch 36/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 898us/step - loss: 0.0298
Epoch 37/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 827us/step - loss: 0.0290
Epoch 38/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 993us/step - loss: 0.0283
Epoch 39/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 803us/step - loss: 0.0282
Epoch 40/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 933us/step - loss: 0.0269
Epoch 41/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 822us/step - loss: 0.0263
Epoch 42/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 841us/step - loss: 0.0274
Epoch 43/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 836us/step - loss: 0.0272
Epoch 44/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 810us/step - loss: 0.0259
Epoch 45/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 764us/step - loss: 0.0245
Epoch 46/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 866us/step - loss: 0.0252
Epoch 47/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 798us/step - loss: 0.0248
Epoch 48/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 808us/step - loss: 0.0246
Epoch 49/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 865us/step - loss: 0.0242
Epoch 50/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 881us/step - loss: 0.0231
Epoch 51/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 739us/step - loss: 0.0227
Epoch 52/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 784us/step - loss: 0.0218
Epoch 53/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 846us/step - loss: 0.0227
Epoch 54/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 777us/step - loss: 0.0228
Epoch 55/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 767us/step - loss: 0.0227
Epoch 56/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 778us/step - loss: 0.0215
Epoch 57/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 808us/step - loss: 0.0216
Epoch 58/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 831us/step - loss: 0.0227
Epoch 59/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0212
Epoch 60/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 768us/step - loss: 0.0213
Epoch 61/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 830us/step - loss: 0.0206
Epoch 62/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 862us/step - loss: 0.0203
Epoch 63/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 868us/step - loss: 0.0201
Epoch 64/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 795us/step - loss: 0.0202
Epoch 65/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 930us/step - loss: 0.0205
Epoch 66/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 989us/step - loss: 0.0200
Epoch 67/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 806us/step - loss: 0.0205
Epoch 68/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 856us/step - loss: 0.0202
Epoch 69/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 797us/step - loss: 0.0199
Epoch 70/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 763us/step - loss: 0.0191
Epoch 71/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 801us/step - loss: 0.0194
Epoch 72/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 876us/step - loss: 0.0198
Epoch 73/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 805us/step - loss: 0.0197
Epoch 74/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 813us/step - loss: 0.0191
Epoch 75/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 794us/step - loss: 0.0190
Epoch 76/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 784us/step - loss: 0.0199
Epoch 77/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 766us/step - loss: 0.0200
Epoch 78/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 772us/step - loss: 0.0182
Epoch 79/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 817us/step - loss: 0.0193
Epoch 80/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 985us/step - loss: 0.0193
Epoch 81/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 992us/step - loss: 0.0186
Epoch 82/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 816us/step - loss: 0.0183
Epoch 83/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 812us/step - loss: 0.0190
Epoch 84/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 816us/step - loss: 0.0194
Epoch 85/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 849us/step - loss: 0.0182
Epoch 86/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 804us/step - loss: 0.0183
Epoch 87/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 813us/step - loss: 0.0187
Epoch 88/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 799us/step - loss: 0.0191
Epoch 89/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 802us/step - loss: 0.0178
Epoch 90/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 840us/step - loss: 0.0178
Epoch 91/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 848us/step - loss: 0.0184
Epoch 92/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0176
Epoch 93/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 915us/step - loss: 0.0185
Epoch 94/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 1ms/step - loss: 0.0177
Epoch 95/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 955us/step - loss: 0.0179
Epoch 96/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 790us/step - loss: 0.0179
Epoch 97/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 760us/step - loss: 0.0178
Epoch 98/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 762us/step - loss: 0.0177
Epoch 99/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 777us/step - loss: 0.0178
Epoch 100/100
[1m32/32[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 769us/step - loss: 0.0178
[1m313/313[0m [32m━━━━━━━━━━━━━━━━━━━━[0m[37m[0m [1m0s[0m 627us/step - loss: 0.0250





0.023643650114536285

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, 39.83it/s]

regional_pdp.summary(features=0)

Feature 0 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x1 🔹 [id: 0 | heter: 3.38 | inst: 1000 | w: 1.00]
    x3 ≤ 0.00 🔹 [id: 1 | heter: 0.10 | inst: 500 | w: 0.50]
    x3 > 0.00 🔹 [id: 2 | heter: 0.16 | inst: 500 | w: 0.50]
--------------------------------------------------
Feature 0 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 3.38
    Level 1🔹heter: 0.13 | 🔻3.26 (96.20%)

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: 3.22 | inst: 1000 | w: 1.00]
--------------------------------------------------
Feature 1 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 3.22

regional_pdp.summary(features=2)

Feature 2 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x3 🔹 [id: 0 | heter: 2.90 | inst: 1000 | w: 1.00]
    x1 ≤ 0.00 🔹 [id: 1 | heter: 0.71 | inst: 494 | w: 0.49]
    x1 > 0.00 🔹 [id: 2 | heter: 0.71 | inst: 506 | w: 0.51]
--------------------------------------------------
Feature 2 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 2.90
    Level 1🔹heter: 0.71 | 🔻2.19 (75.41%)

regional_pdp.plot(feature=2, node_idx=1, heterogeneity="ice", centering=True, y_limits=[-5, 5])
regional_pdp.plot(feature=2, node_idx=2, 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, 54.75it/s]

regional_pdp.summary(features=0)

Feature 0 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x1 🔹 [id: 0 | heter: 3.25 | inst: 900 | w: 1.00]
    x3 ≤ 0.00 🔹 [id: 1 | heter: 0.17 | inst: 436 | w: 0.48]
    x3 > 0.00 🔹 [id: 2 | heter: 0.20 | inst: 464 | w: 0.52]
--------------------------------------------------
Feature 0 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 3.25
    Level 1🔹heter: 0.18 | 🔻3.07 (94.40%)

regional_pdp.plot(feature=0, node_idx=1, heterogeneity="ice", centering=True, y_limits=[-5, 5])
regional_pdp.plot(feature=0, node_idx=2, 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: 1.22 | inst: 900 | w: 1.00]
    x1 ≤ 0.40 🔹 [id: 1 | heter: 0.57 | inst: 652 | w: 0.72]
    x1 > 0.40 🔹 [id: 2 | heter: 0.55 | inst: 248 | w: 0.28]
--------------------------------------------------
Feature 1 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 1.22
    Level 1🔹heter: 0.56 | 🔻0.66 (54.17%)

regional_pdp.summary(features=2)

Feature 2 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x3 🔹 [id: 0 | heter: 2.44 | inst: 900 | w: 1.00]
    x1 ≤ 0.00 🔹 [id: 1 | heter: 0.74 | inst: 463 | w: 0.51]
    x1 > 0.00 🔹 [id: 2 | heter: 0.81 | inst: 437 | w: 0.49]
--------------------------------------------------
Feature 2 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 2.44
    Level 1🔹heter: 0.77 | 🔻1.67 (68.34%)

regional_pdp.plot(feature=2, node_idx=1, heterogeneity="ice", centering=True, y_limits=[-5, 5])
regional_pdp.plot(feature=2, node_idx=2, 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, 17.88it/s]

regional_rhale.summary(features=0)

Feature 0 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x1 🔹 [id: 0 | heter: 8.64 | inst: 1000 | w: 1.00]
    x3 ≤ 0.00 🔹 [id: 1 | heter: 0.26 | inst: 500 | w: 0.50]
    x3 > 0.00 🔹 [id: 2 | heter: 0.47 | inst: 500 | w: 0.50]
--------------------------------------------------
Feature 0 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 8.64
    Level 1🔹heter: 0.37 | 🔻8.28 (95.77%)

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: 0.03 | inst: 1000 | 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: 73.33 | inst: 1000 | w: 1.00]
--------------------------------------------------
Feature 2 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 73.33

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,  5.06it/s]

regional_rhale.summary(features=0)

Feature 0 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x1 🔹 [id: 0 | heter: 2.50 | inst: 900 | w: 1.00]
--------------------------------------------------
Feature 0 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 2.50

regional_rhale.summary(features=1)

Feature 1 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x2 🔹 [id: 0 | heter: 0.01 | inst: 900 | w: 1.00]
--------------------------------------------------
Feature 1 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.01

regional_rhale.summary(features=2)

Feature 2 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x3 🔹 [id: 0 | heter: 13.57 | inst: 900 | w: 1.00]
--------------------------------------------------
Feature 2 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 13.57

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 [00:04<00:00,  1.59s/it]

regional_shap.summary(0)

Feature 0 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x1 🔹 [id: 0 | heter: 0.83 | 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: 0.83
    Level 1🔹heter: 0.03 | 🔻0.79 (95.81%)

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: 0.77 | inst: 1000 | w: 1.00]
    x1 ≤ 0.00 🔹 [id: 1 | heter: 0.25 | inst: 494 | w: 0.49]
    x1 > 0.00 🔹 [id: 2 | heter: 0.34 | inst: 506 | w: 0.51]
--------------------------------------------------
Feature 2 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.77
    Level 1🔹heter: 0.30 | 🔻0.47 (60.94%)

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 [00:04<00:00,  1.40s/it]

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

Feature 0 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
x1 🔹 [id: 0 | heter: 0.06 | inst: 900 | w: 1.00]
--------------------------------------------------
Feature 0 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.06




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: 0.14 | inst: 900 | w: 1.00]
--------------------------------------------------
Feature 2 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.14