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Open AccessArticle

Functional Kernel Density Estimation: Point and Fourier Approaches to Time Series Anomaly Detection

1
Department of Mathematics, University of California, Los Angeles, CA 90024, USA
2
Global Aviation Data Management, International Air Transport Association (IATA), Montréal, QC H2Y 1C6, Canada
3
Department of Decision Sciences, HEC Montréal, Montréal, QC H2Y 1C6, Canada
*
Author to whom correspondence should be addressed.
Entropy 2020, 22(12), 1363; https://doi.org/10.3390/e22121363
Received: 16 November 2020 / Accepted: 27 November 2020 / Published: 30 November 2020
(This article belongs to the Special Issue Time Series Modelling)
We present an unsupervised method to detect anomalous time series among a collection of time series. To do so, we extend traditional Kernel Density Estimation for estimating probability distributions in Euclidean space to Hilbert spaces. The estimated probability densities we derive can be obtained formally through treating each series as a point in a Hilbert space, placing a kernel at those points, and summing the kernels (a “point approach”), or through using Kernel Density Estimation to approximate the distributions of Fourier mode coefficients to infer a probability density (a “Fourier approach”). We refer to these approaches as Functional Kernel Density Estimation for Anomaly Detection as they both yield functionals that can score a time series for how anomalous it is. Both methods naturally handle missing data and apply to a variety of settings, performing well when compared with an outlyingness score derived from a boxplot method for functional data, with a Principal Component Analysis approach for functional data, and with the Functional Isolation Forest method. We illustrate the use of the proposed methods with aviation safety report data from the International Air Transport Association (IATA). View Full-Text
Keywords: time series; anomaly detection; unsupervised learning; kernel density estimation; missing data time series; anomaly detection; unsupervised learning; kernel density estimation; missing data
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MDPI and ACS Style

Lindstrom, M.R.; Jung, H.; Larocque, D. Functional Kernel Density Estimation: Point and Fourier Approaches to Time Series Anomaly Detection. Entropy 2020, 22, 1363. https://doi.org/10.3390/e22121363

AMA Style

Lindstrom MR, Jung H, Larocque D. Functional Kernel Density Estimation: Point and Fourier Approaches to Time Series Anomaly Detection. Entropy. 2020; 22(12):1363. https://doi.org/10.3390/e22121363

Chicago/Turabian Style

Lindstrom, Michael R.; Jung, Hyuntae; Larocque, Denis. 2020. "Functional Kernel Density Estimation: Point and Fourier Approaches to Time Series Anomaly Detection" Entropy 22, no. 12: 1363. https://doi.org/10.3390/e22121363

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