Model Interpretation: PDP vs ALE

In this notebook, we compare Partial Dependence Plots (PDP) and Accumulated Local Effects (ALE).

The Problem with PDP

PDPs work by marginalizing over the distribution of the features. If features \(X_1\) and \(X_2\) are highly correlated, the PDP will calculate predictions for points that are impossible (e.g. a 100 \(\mathrm{m}^2\) apartment with 10 bedrooms). This leads to extrapolation bias.

The ALE Solution

ALE plots, proposed by Apley and Zhu[1], calculate the model’s behavior based on conditional distributions. They only look at how the prediction changes when a feature varies locally, given the values of other features.

Creating a Synthetic Correlated Dataset

To easily witness this phenomenon, we will generate a synthetic dataset containing a deliberate statistical trap:

1. We generate two features, \(X_1\) and \(X_2\), which are strongly correlated.

2. We define the target variable \(y\) to depend only on \(X_1\) via a quadratic function:

\[y = X_1^2 + \varepsilon\]

The Goal: Because \(y\) has no direct connection to \(X_2\), an ideal interpretation method must return a completely flat line at zero for \(X_2\).

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

n_samples = 5000
generator = np.random.default_rng(92)

X1 = generator.uniform(-1, 1, n_samples)
X2 = X1 + generator.normal(0, 0.1, n_samples)  # Strong correlation with X1

y = X1**2 + generator.normal(0, 0.05, n_samples)

X = pd.DataFrame({"X1": X1, "X2": X2})

plt.figure(figsize=(6, 4))
plt.scatter(X["X1"], X["X2"], c=y, cmap="viridis", alpha=0.4, s=15)
cbar = plt.colorbar()
cbar.set_label("Target Value (y)", rotation=270, labelpad=15)
plt.title("Strongly Correlated Features")
plt.xlabel("X1")
plt.ylabel("X2")
plt.show()

Training the model

We train a RandomForestRegressor from scikit-learn. Because \(X_1\) and \(X_2\) are similar, the model might use both to predict \(y\), especially when we set max_features=’sqrt’.

from sklearn.ensemble import RandomForestRegressor
from sklearn.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(
    X, y, test_size=0.3, random_state=92
)

model = RandomForestRegressor(max_features="sqrt", random_state=92)
model.fit(X_train, y_train)

r2_test_score = model.score(X_test, y_test)
print(f"R² score on the test set: {r2_test_score:.2f}")

R² score on the test set: 0.97

Comparison: PDP vs ALE for Feature \(X_2\)

Now, we plot the interpretation curves for \(X_2\) using both methods to observe how they process the correlation.

from hidimstat.visualization import ALE, PDP

# 1. Partial Dependence Plot (PDP)
pdp = PDP(model, feature_names=X.columns)
pdp_axes = pdp.plot(X_test, features=1)

# 2. ALE Plot
ale = ALE(model, feature_names=X.columns)
ale_axes = ale.plot(
    X_test, features=1, grid_resolution=100, confidence_level=0
)

mean_pred = model.predict(X_test).mean()
pdp_ymin, pdp_ymax = pdp_axes[1].get_ylim()
ale_axes[1].set_ylim(pdp_ymin - mean_pred, pdp_ymax - mean_pred)
ale_axes[1].figure.axes[2].set_ylim(pdp_ymin, pdp_ymax)

plt.show()

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Conclusion

• The PDP shows a prominent U-shaped parabola for \(X_2\) because of extrapolation. An analyst looking at this plot could falsely conclude that increasing or decreasing \(X_2\) directly increases the target variable. This is an error since it is actually the result of out-of-distribution artifacts.

• ALE shows a mostly flat line (near zero), isolating the unique contribution of \(X_2\) by blocking the shadow effect of \(X_1\). This isolates the direct non-impact, which is ideal for understanding pure mechanisms, though it leaves out indirect operational levers.

References

[1]

Daniel W. Apley and Jingyu Zhu. Visualizing the effects of predictor variables in black box supervised learning models. Journal of the Royal Statistical Society Series B: Statistical Methodology, 82(4):1059–1086, 2020.

Estimated memory usage: 250 MB