Permutation Importance with Multicollinear or Correlated Features

In this example, we compute the permutation importance on the Wisconsin breast cancer dataset using permutation_importance. The RandomForestClassifier can easily get about 97% accuracy on a test dataset. Because this dataset contains multicollinear features, the permutation importance will show that none of the features are important. One approach to handling multicollinearity is by performing hierarchical clustering on the features’ Spearman rank-order correlations, picking a threshold, and keeping a single feature from each cluster.

from collections import defaultdict

import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import spearmanr
from scipy.cluster import hierarchy
from scipy.spatial.distance import squareform

from sklearn.datasets import load_breast_cancer
from sklearn.ensemble import RandomForestClassifier
from sklearn.inspection import permutation_importance
from sklearn.model_selection import train_test_split

Random Forest Feature Importance on Breast Cancer Data

First, we train a random forest on the breast cancer dataset and evaluate its accuracy on a test set:

data = load_breast_cancer()
X, y =,
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)

clf = RandomForestClassifier(n_estimators=100, random_state=42), y_train)
print("Accuracy on test data: {:.2f}".format(clf.score(X_test, y_test)))
Accuracy on test data: 0.97

Next, we plot the tree based feature importance and the permutation importance. The permutation importance plot shows that permuting a feature drops the accuracy by at most 0.012, which would suggest that none of the features are important. This is in contradiction with the high test accuracy computed above: some feature must be important. The permutation importance is calculated on the training set to show how much the model relies on each feature during training.

result = permutation_importance(clf, X_train, y_train, n_repeats=10, random_state=42)
perm_sorted_idx = result.importances_mean.argsort()

tree_importance_sorted_idx = np.argsort(clf.feature_importances_)
tree_indices = np.arange(0, len(clf.feature_importances_)) + 0.5

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 8))
ax1.barh(tree_indices, clf.feature_importances_[tree_importance_sorted_idx], height=0.7)
ax1.set_ylim((0, len(clf.feature_importances_)))
plot permutation importance multicollinear

Handling Multicollinear Features

When features are collinear, permutating one feature will have little effect on the models performance because it can get the same information from a correlated feature. One way to handle multicollinear features is by performing hierarchical clustering on the Spearman rank-order correlations, picking a threshold, and keeping a single feature from each cluster. First, we plot a heatmap of the correlated features:

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 8))
corr = spearmanr(X).correlation

# Ensure the correlation matrix is symmetric
corr = (corr + corr.T) / 2
np.fill_diagonal(corr, 1)

# We convert the correlation matrix to a distance matrix before performing
# hierarchical clustering using Ward's linkage.
distance_matrix = 1 - np.abs(corr)
dist_linkage = hierarchy.ward(squareform(distance_matrix))
dendro = hierarchy.dendrogram(
    dist_linkage, labels=data.feature_names.tolist(), ax=ax1, leaf_rotation=90
dendro_idx = np.arange(0, len(dendro["ivl"]))

ax2.imshow(corr[dendro["leaves"], :][:, dendro["leaves"]])
ax2.set_xticklabels(dendro["ivl"], rotation="vertical")
plot permutation importance multicollinear

Next, we manually pick a threshold by visual inspection of the dendrogram to group our features into clusters and choose a feature from each cluster to keep, select those features from our dataset, and train a new random forest. The test accuracy of the new random forest did not change much compared to the random forest trained on the complete dataset.

cluster_ids = hierarchy.fcluster(dist_linkage, 1, criterion="distance")
cluster_id_to_feature_ids = defaultdict(list)
for idx, cluster_id in enumerate(cluster_ids):
selected_features = [v[0] for v in cluster_id_to_feature_ids.values()]

X_train_sel = X_train[:, selected_features]
X_test_sel = X_test[:, selected_features]

clf_sel = RandomForestClassifier(n_estimators=100, random_state=42), y_train)
    "Accuracy on test data with features removed: {:.2f}".format(
        clf_sel.score(X_test_sel, y_test)
Accuracy on test data with features removed: 0.97

Total running time of the script: ( 0 minutes 4.341 seconds)

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