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Kernel-Based Anomaly Detection Using Generalized Hyperbolic Processes

Pauline Bourigault, Danilo P. Mandic

TL;DR

The paper tackles anomaly detection in data exhibiting non-Gaussian characteristics such as heavy tails and skewness. It introduces a Generalized Hyperbolic (GH) kernel to replace traditional Gaussian kernels, enabling kernel-based methods like One-Class SVM (OCSVM) and Kernel Density Estimation (KDE) to better model complex distributions. Key contributions include deriving a PSD GH kernel via convolution of GH PDFs, proving consistency, and integrating the kernel into both OCSVM and GH-KDE with theoretical guarantees. Empirical results on synthetic data and real datasets (KDDCup99 and ForestCover) demonstrate improved detection performance for skewed and heavy-tailed distributions, highlighting the practical impact for robust anomaly detection in non-Gaussian settings.

Abstract

We present a novel approach to anomaly detection by integrating Generalized Hyperbolic (GH) processes into kernel-based methods. The GH distribution, known for its flexibility in modeling skewness, heavy tails, and kurtosis, helps to capture complex patterns in data that deviate from Gaussian assumptions. We propose a GH-based kernel function and utilize it within Kernel Density Estimation (KDE) and One-Class Support Vector Machines (OCSVM) to develop anomaly detection frameworks. Theoretical results confirmed the positive semi-definiteness and consistency of the GH-based kernel, ensuring its suitability for machine learning applications. Empirical evaluation on synthetic and real-world datasets showed that our method improves detection performance in scenarios involving heavy-tailed and asymmetric or imbalanced distributions. https://github.com/paulinebourigault/GHKernelAnomalyDetect

Kernel-Based Anomaly Detection Using Generalized Hyperbolic Processes

TL;DR

The paper tackles anomaly detection in data exhibiting non-Gaussian characteristics such as heavy tails and skewness. It introduces a Generalized Hyperbolic (GH) kernel to replace traditional Gaussian kernels, enabling kernel-based methods like One-Class SVM (OCSVM) and Kernel Density Estimation (KDE) to better model complex distributions. Key contributions include deriving a PSD GH kernel via convolution of GH PDFs, proving consistency, and integrating the kernel into both OCSVM and GH-KDE with theoretical guarantees. Empirical results on synthetic data and real datasets (KDDCup99 and ForestCover) demonstrate improved detection performance for skewed and heavy-tailed distributions, highlighting the practical impact for robust anomaly detection in non-Gaussian settings.

Abstract

We present a novel approach to anomaly detection by integrating Generalized Hyperbolic (GH) processes into kernel-based methods. The GH distribution, known for its flexibility in modeling skewness, heavy tails, and kurtosis, helps to capture complex patterns in data that deviate from Gaussian assumptions. We propose a GH-based kernel function and utilize it within Kernel Density Estimation (KDE) and One-Class Support Vector Machines (OCSVM) to develop anomaly detection frameworks. Theoretical results confirmed the positive semi-definiteness and consistency of the GH-based kernel, ensuring its suitability for machine learning applications. Empirical evaluation on synthetic and real-world datasets showed that our method improves detection performance in scenarios involving heavy-tailed and asymmetric or imbalanced distributions. https://github.com/paulinebourigault/GHKernelAnomalyDetect
Paper Structure (7 sections, 5 theorems, 22 equations, 2 figures, 2 tables)

This paper contains 7 sections, 5 theorems, 22 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. GH Kernel-Based Anomaly Detection
  4. Kernel Derivation from Generalized Hyperbolic Processes
  5. Integration of GH Kernel with Support Vector Machines
  6. Kernel-Based Density Estimation using GH Processes
  7. Empirical Validation

Key Result

Lemma 1

The GH kernel $K_{\text{GH}}(x,y)$ is positive semi-definite (PSD) to satisfy Mercer's condition for appropriately chosen parameters $\lambda$, $\alpha$, $\beta$, $\delta$, $\mu$.

Figures (2)

  • Figure 1: Decision boundaries for anomaly detection using various OCSVM kernels: standard and GH kernels. Blue circles denote normal data points, while red triangles represent anomalies. The black line represents the method's decision boundary. It visually illustrates the kernel's ability to distinguish between normal and anomalous data points. See Table \ref{['tab:anomaly_detection_synthetic']} and Fig. \ref{['fig:enter-label']} for details.
  • Figure 2: Histograms of decision function values for the OCSVM kernels from Fig. \ref{['fig:decbound']} and Tab. \ref{['tab:anomaly_detection_synthetic']}, highlighting the separation between normal data (blue) and anomalies (red). The dashed black line represents the decision boundary at 0.

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof