Extended Isolation Forest with feature sensitivities

TL;DR

This work extends isolation-based anomaly detection by introducing the Anisotropic Isolation Forest (AIF), which samples hyperplane normals from an anisotropic distribution $N(0,A)$ to impose direction-dependent sensitivity across features. It formalizes directional-sensitivity measures, including alpha(n)=sqrt(n^T A n) and tau(B), and shows how mixtures of Gaussians can yield richer anisotropy patterns. Empirical results on synthetic data and real-world data (e.g., Diabetes) demonstrate that AIF can emphasize deviations in task-relevant directions, producing directional anomaly scores and contour patterns that align with the specified sensitivities. The approach provides a practical mechanism to tailor anomaly detection to domain-specific requirements while preserving EIF compatibility and scalability.

Abstract

Compared to theoretical frameworks that assume equal sensitivity to deviations in all features of data, the theory of anomaly detection allowing for variable sensitivity across features is less developed. To the best of our knowledge, this issue has not yet been addressed in the context of isolation-based methods, and this paper represents the first attempt to do so. This paper introduces an Extended Isolation Forest with feature sensitivities, which we refer to as the Anisotropic Isolation Forest (AIF). In contrast to the standard EIF, the AIF enables anomaly detection with controllable sensitivity to deviations in different features or directions in the feature space. The paper also introduces novel measures of directional sensitivity, which allow quantification of AIF's sensitivity in different directions in the feature space. These measures enable adjustment of the AIF's sensitivity to task-specific requirements. We demonstrate the performance of the algorithm by applying it to synthetic and real-world datasets. The results show that the AIF enables anomaly detection that focuses on directions in the feature space where deviations from typical behavior are more important.

Figures (15)

• Figure 1: Partitions of data.
• Figure 2: Distributions and sampled normal vectors.
• Figure 3: Planes and their normal vectors.
• Figure 4: Anomaly scores.
• Figure 5: Anomaly score maps.

Theorems & Definitions (3)

• Example 1
• Example 2
• Example 3