IsolationForest example

An example using IsolationForest for anomaly detection.

The Isolation Forest is an ensemble of “Isolation Trees” that “isolate” observations by recursive random partitioning, which can be represented by a tree structure. The number of splittings required to isolate a sample is lower for outliers and higher for inliers.

In the present example we demo two ways to visualize the decision boundary of an Isolation Forest trained on a toy dataset.

Data generation

We generate two clusters (each one containing n_samples) by randomly sampling the standard normal distribution as returned by numpy.random.randn. One of them is spherical and the other one is slightly deformed.

For consistency with the IsolationForest notation, the inliers (i.e. the gaussian clusters) are assigned a ground truth label 1 whereas the outliers (created with numpy.random.uniform) are assigned the label -1.

import numpy as np
from sklearn.model_selection import train_test_split

n_samples, n_outliers = 120, 40
rng = np.random.RandomState(0)
covariance = np.array([[0.5, -0.1], [0.7, 0.4]])
cluster_1 = 0.4 * rng.randn(n_samples, 2) @ covariance + np.array([2, 2])  # general
cluster_2 = 0.3 * rng.randn(n_samples, 2) + np.array([-2, -2])  # spherical
outliers = rng.uniform(low=-4, high=4, size=(n_outliers, 2))

X = np.concatenate([cluster_1, cluster_2, outliers])
y = np.concatenate(
    [np.ones((2 * n_samples), dtype=int), -np.ones((n_outliers), dtype=int)]
)

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

We can visualize the resulting clusters:

import matplotlib.pyplot as plt

scatter = plt.scatter(X[:, 0], X[:, 1], c=y, s=20, edgecolor="k")
handles, labels = scatter.legend_elements()
plt.axis("square")
plt.legend(handles=handles, labels=["outliers", "inliers"], title="true class")
plt.title("Gaussian inliers with \nuniformly distributed outliers")
plt.show()
Gaussian inliers with  uniformly distributed outliers

Training of the model

from sklearn.ensemble import IsolationForest

clf = IsolationForest(max_samples=100, random_state=0)
clf.fit(X_train)
IsolationForest(max_samples=100, random_state=0)
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Plot discrete decision boundary

We use the class DecisionBoundaryDisplay to visualize a discrete decision boundary. The background color represents whether a sample in that given area is predicted to be an outlier or not. The scatter plot displays the true labels.

import matplotlib.pyplot as plt
from sklearn.inspection import DecisionBoundaryDisplay

disp = DecisionBoundaryDisplay.from_estimator(
    clf,
    X,
    response_method="predict",
    alpha=0.5,
)
disp.ax_.scatter(X[:, 0], X[:, 1], c=y, s=20, edgecolor="k")
disp.ax_.set_title("Binary decision boundary \nof IsolationForest")
plt.axis("square")
plt.legend(handles=handles, labels=["outliers", "inliers"], title="true class")
plt.show()
Binary decision boundary  of IsolationForest

Plot path length decision boundary

By setting the response_method="decision_function", the background of the DecisionBoundaryDisplay represents the measure of normality of an observation. Such score is given by the path length averaged over a forest of random trees, which itself is given by the depth of the leaf (or equivalently the number of splits) required to isolate a given sample.

When a forest of random trees collectively produce short path lengths for isolating some particular samples, they are highly likely to be anomalies and the measure of normality is close to 0. Similarly, large paths correspond to values close to 1 and are more likely to be inliers.

disp = DecisionBoundaryDisplay.from_estimator(
    clf,
    X,
    response_method="decision_function",
    alpha=0.5,
)
disp.ax_.scatter(X[:, 0], X[:, 1], c=y, s=20, edgecolor="k")
disp.ax_.set_title("Path length decision boundary \nof IsolationForest")
plt.axis("square")
plt.legend(handles=handles, labels=["outliers", "inliers"], title="true class")
plt.colorbar(disp.ax_.collections[1])
plt.show()
Path length decision boundary  of IsolationForest

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