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)


32/32 ━━━━━━━━━━━━━━━━━━━━ 1s 3ms/step - loss: 3.1056
Epoch 2/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 1.0435
Epoch 3/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 3ms/step - loss: 0.4445
Epoch 4/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.3043
Epoch 5/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.2314
Epoch 6/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.1921
Epoch 7/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.1674
Epoch 8/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.1468
Epoch 9/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.1294 
Epoch 10/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.1176
Epoch 11/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 910us/step - loss: 0.1098
Epoch 12/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.1042
Epoch 13/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.1007 
Epoch 14/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0952
Epoch 15/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0922
Epoch 16/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0873
Epoch 17/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 959us/step - loss: 0.0801
Epoch 18/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0775
Epoch 19/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0741
Epoch 20/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0710
Epoch 21/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0689
Epoch 22/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0675
Epoch 23/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0663
Epoch 24/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0656
Epoch 25/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 3ms/step - loss: 0.0647
Epoch 26/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0633
Epoch 27/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0629
Epoch 28/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0632
Epoch 29/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0624 
Epoch 30/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0613
Epoch 31/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0599
Epoch 32/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0576
Epoch 33/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0567
Epoch 34/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0584
Epoch 35/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 3ms/step - loss: 0.0565
Epoch 36/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0562
Epoch 37/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0570
Epoch 38/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0557
Epoch 39/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0570
Epoch 40/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0532 
Epoch 41/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0558
Epoch 42/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0554
Epoch 43/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 3ms/step - loss: 0.0547
Epoch 44/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0543 
Epoch 45/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0539
Epoch 46/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0554
Epoch 47/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0523
Epoch 48/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0518
Epoch 49/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0519
Epoch 50/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0517
Epoch 51/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0509
Epoch 52/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 2ms/step - loss: 0.0522
Epoch 53/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0505
Epoch 54/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0526
Epoch 55/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0535
Epoch 56/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0526
Epoch 57/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0505
Epoch 58/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0503
Epoch 59/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 956us/step - loss: 0.0497
Epoch 60/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0491
Epoch 61/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0497
Epoch 62/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0496
Epoch 63/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0482
Epoch 64/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0465
Epoch 65/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 959us/step - loss: 0.0473
Epoch 66/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0476
Epoch 67/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0461
Epoch 68/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 909us/step - loss: 0.0462
Epoch 69/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 839us/step - loss: 0.0459
Epoch 70/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 857us/step - loss: 0.0453
Epoch 71/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 999us/step - loss: 0.0448
Epoch 72/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 963us/step - loss: 0.0435
Epoch 73/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 997us/step - loss: 0.0452
Epoch 74/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 955us/step - loss: 0.0434
Epoch 75/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 934us/step - loss: 0.0454
Epoch 76/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 910us/step - loss: 0.0434
Epoch 77/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0439
Epoch 78/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0431
Epoch 79/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 917us/step - loss: 0.0434
Epoch 80/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 899us/step - loss: 0.0431
Epoch 81/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0425
Epoch 82/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 935us/step - loss: 0.0423
Epoch 83/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0416
Epoch 84/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0420
Epoch 85/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0412
Epoch 86/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0416
Epoch 87/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0406
Epoch 88/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0404
Epoch 89/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 986us/step - loss: 0.0414
Epoch 90/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0401
Epoch 91/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 983us/step - loss: 0.0391
Epoch 92/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0405
Epoch 93/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 927us/step - loss: 0.0394
Epoch 94/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 858us/step - loss: 0.0396
Epoch 95/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 847us/step - loss: 0.0383
Epoch 96/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 895us/step - loss: 0.0384
Epoch 97/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 840us/step - loss: 0.0381
Epoch 98/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 896us/step - loss: 0.0386
Epoch 99/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 875us/step - loss: 0.0378
Epoch 100/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 938us/step - loss: 0.0391
313/313 ━━━━━━━━━━━━━━━━━━━━ 0s 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
32/32 ━━━━━━━━━━━━━━━━━━━━ 1s 959us/step - loss: 2.1464
Epoch 2/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 989us/step - loss: 0.4442
Epoch 3/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 915us/step - loss: 0.2055
Epoch 4/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.1394
Epoch 5/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 978us/step - loss: 0.1068
Epoch 6/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0945
Epoch 7/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 771us/step - loss: 0.0892
Epoch 8/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 929us/step - loss: 0.0851
Epoch 9/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 971us/step - loss: 0.0818
Epoch 10/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 863us/step - loss: 0.0795
Epoch 11/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 908us/step - loss: 0.0759
Epoch 12/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 869us/step - loss: 0.0731
Epoch 13/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 959us/step - loss: 0.0707
Epoch 14/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 964us/step - loss: 0.0684
Epoch 15/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 870us/step - loss: 0.0667
Epoch 16/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0645
Epoch 17/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 916us/step - loss: 0.0624
Epoch 18/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 852us/step - loss: 0.0587
Epoch 19/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 800us/step - loss: 0.0563
Epoch 20/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0526
Epoch 21/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 821us/step - loss: 0.0513
Epoch 22/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 798us/step - loss: 0.0498
Epoch 23/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 793us/step - loss: 0.0481
Epoch 24/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 766us/step - loss: 0.0457
Epoch 25/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 816us/step - loss: 0.0445
Epoch 26/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 865us/step - loss: 0.0424
Epoch 27/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 968us/step - loss: 0.0416
Epoch 28/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0382
Epoch 29/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 972us/step - loss: 0.0372
Epoch 30/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 929us/step - loss: 0.0372
Epoch 31/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 806us/step - loss: 0.0358
Epoch 32/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 913us/step - loss: 0.0340
Epoch 33/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 953us/step - loss: 0.0330
Epoch 34/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 925us/step - loss: 0.0313
Epoch 35/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 861us/step - loss: 0.0308
Epoch 36/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 898us/step - loss: 0.0298
Epoch 37/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 827us/step - loss: 0.0290
Epoch 38/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 993us/step - loss: 0.0283
Epoch 39/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 803us/step - loss: 0.0282
Epoch 40/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 933us/step - loss: 0.0269
Epoch 41/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 822us/step - loss: 0.0263
Epoch 42/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 841us/step - loss: 0.0274
Epoch 43/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 836us/step - loss: 0.0272
Epoch 44/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 810us/step - loss: 0.0259
Epoch 45/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 764us/step - loss: 0.0245
Epoch 46/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 866us/step - loss: 0.0252
Epoch 47/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 798us/step - loss: 0.0248
Epoch 48/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 808us/step - loss: 0.0246
Epoch 49/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 865us/step - loss: 0.0242
Epoch 50/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 881us/step - loss: 0.0231
Epoch 51/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 739us/step - loss: 0.0227
Epoch 52/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 784us/step - loss: 0.0218
Epoch 53/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 846us/step - loss: 0.0227
Epoch 54/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 777us/step - loss: 0.0228
Epoch 55/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 767us/step - loss: 0.0227
Epoch 56/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 778us/step - loss: 0.0215
Epoch 57/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 808us/step - loss: 0.0216
Epoch 58/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 831us/step - loss: 0.0227
Epoch 59/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0212
Epoch 60/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 768us/step - loss: 0.0213
Epoch 61/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 830us/step - loss: 0.0206
Epoch 62/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 862us/step - loss: 0.0203
Epoch 63/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 868us/step - loss: 0.0201
Epoch 64/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 795us/step - loss: 0.0202
Epoch 65/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 930us/step - loss: 0.0205
Epoch 66/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 989us/step - loss: 0.0200
Epoch 67/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 806us/step - loss: 0.0205
Epoch 68/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 856us/step - loss: 0.0202
Epoch 69/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 797us/step - loss: 0.0199
Epoch 70/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 763us/step - loss: 0.0191
Epoch 71/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 801us/step - loss: 0.0194
Epoch 72/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 876us/step - loss: 0.0198
Epoch 73/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 805us/step - loss: 0.0197
Epoch 74/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 813us/step - loss: 0.0191
Epoch 75/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 794us/step - loss: 0.0190
Epoch 76/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 784us/step - loss: 0.0199
Epoch 77/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 766us/step - loss: 0.0200
Epoch 78/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 772us/step - loss: 0.0182
Epoch 79/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 817us/step - loss: 0.0193
Epoch 80/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 985us/step - loss: 0.0193
Epoch 81/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 992us/step - loss: 0.0186
Epoch 82/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 816us/step - loss: 0.0183
Epoch 83/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 812us/step - loss: 0.0190
Epoch 84/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 816us/step - loss: 0.0194
Epoch 85/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 849us/step - loss: 0.0182
Epoch 86/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 804us/step - loss: 0.0183
Epoch 87/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 813us/step - loss: 0.0187
Epoch 88/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 799us/step - loss: 0.0191
Epoch 89/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 802us/step - loss: 0.0178
Epoch 90/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 840us/step - loss: 0.0178
Epoch 91/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 848us/step - loss: 0.0184
Epoch 92/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0176
Epoch 93/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 915us/step - loss: 0.0185
Epoch 94/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0177
Epoch 95/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 955us/step - loss: 0.0179
Epoch 96/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 790us/step - loss: 0.0179
Epoch 97/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 760us/step - loss: 0.0178
Epoch 98/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 762us/step - loss: 0.0177
Epoch 99/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 777us/step - loss: 0.0178
Epoch 100/100
32/32 ━━━━━━━━━━━━━━━━━━━━ 0s 769us/step - loss: 0.0178
313/313 ━━━━━━━━━━━━━━━━━━━━ 0s 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

Conclusion