Authors:
(1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France;
(2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France;
(3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.2. The Signature Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature B. K-SIF and SIF Algorithms C. Additional Numerical Experiments A. Additional Information About the Signature In this section, we provide additional information regarding the signature. Firstly, the formula for the coordinate signature is provided, followed by some further information about the truncated signature and the link of the signature with moments. We then introduce references and further discussion for the computation of the signature, the linear-closed form utilised and, lastly, a review of the inner product of tensors required for the signature method is presented. Computation of the Signature. The signature is usually computed on the linear paths reconstructed from observations (Lyons and McLeod, 2022; Fermanian, 2021) thanks to the two properties listed below. First, the signature benefits from a closed form when computed on linear paths. Second, a concatenation formula has been provided by Chen (1958), where the coordinate signature of two concatenate segments is given by a tensor product of the specific coordinate signatures on each segment. These two properties are described formally below. In practice, the signature is usually computed as follows. First, a linear path is reconstructed by interpolation from the observed data. In a second step, the signature of each segment of the linear path are calculated separately thanks to the closed-form (2). Then the signature of the entire path is computed recursively using the Chen’s identity property. Definition of Inner Product of Tensors for Signature Method. Following Chevyrev and Oberhauser (2022) and Lee and Oberhauser (2023), we introduce the inner product of the tensors required to compute the truncated signature and its kernel. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Authors: Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Background & Preliminaries 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.1. Functional Isolation Forest 2.2. The Signature Method 2.2. The Signature Method 3. Signature Isolation Forest Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature A. Additional Information About the Signature B. K-SIF and SIF Algorithms B. K-SIF and SIF Algorithms C. Additional Numerical Experiments C. Additional Numerical Experiments A. Additional Information About the Signature In this section, we provide additional information regarding the signature. Firstly, the formula for the coordinate signature is provided, followed by some further information about the truncated signature and the link of the signature with moments. We then introduce references and further discussion for the computation of the signature, the linear-closed form utilised and, lastly, a review of the inner product of tensors required for the signature method is presented. Computation of the Signature. The signature is usually computed on the linear paths reconstructed from observations (Lyons and McLeod, 2022; Fermanian, 2021) thanks to the two properties listed below. First, the signature benefits from a closed form when computed on linear paths. Second, a concatenation formula has been provided by Chen (1958), where the coordinate signature of two concatenate segments is given by a tensor product of the specific coordinate signatures on each segment. These two properties are described formally below. Computation of the Signature. In practice, the signature is usually computed as follows. First, a linear path is reconstructed by interpolation from the observed data. In a second step, the signature of each segment of the linear path are calculated separately thanks to the closed-form (2). Then the signature of the entire path is computed recursively using the Chen’s identity property. Definition of Inner Product of Tensors for Signature Method. Following Chevyrev and Oberhauser (2022) and Lee and Oberhauser (2023), we introduce the inner product of the tensors required to compute the truncated signature and its kernel. Definition of Inner Product of Tensors for Signature Method. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Part of HackerNoon's growing list of open-source research papers, promoting free access to academic material.

Unlocking the Power of Signatures in Anomaly Detection

About Author

Comments

TOPICS

THIS ARTICLE WAS FEATURED IN

Related Stories

Untitled Story

A Data-centric Approach to Class-specific Bias in Image Data Augmentation: Abstract and Intro

What is the Signature Isolation Forest?

How Functional Isolation Forest Detects Anomalies

Unveiling Path Signatures: A Key to Geometric Insights in Data

Redefining Anomaly Detection with Signature Isolation Forests

A Data-centric Approach to Class-specific Bias in Image Data Augmentation: Abstract and Intro

What is the Signature Isolation Forest?

How Functional Isolation Forest Detects Anomalies

Unveiling Path Signatures: A Key to Geometric Insights in Data

Redefining Anomaly Detection with Signature Isolation Forests

Light-Mode

Classic

Newspaper

Dark-Mode

Neon Noir

Minty

HN StartUps