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

Area of Science:

• Statistics
• Machine Learning
• Data Science

Background:

• Time series analysis is crucial for understanding sequential data.
• Detecting anomalous time series is challenging, especially with missing data.
• Existing methods may not adequately handle the complexity of functional data.

Purpose of the Study:

• To develop an unsupervised method for detecting anomalous time series.
• To extend Kernel Density Estimation to Hilbert spaces for functional data.
• To provide a robust anomaly detection technique applicable to various settings.

Main Methods:

• Functional Kernel Density Estimation (FKDE) was developed, offering two approaches: a 'point approach' and a 'Fourier approach'.
• Both FKDE methods estimate probability densities in Hilbert spaces.
• The methods are designed to naturally handle missing data in time series.

Main Results:

• The proposed FKDE methods provide functionals to score time series for anomaly levels.
• Performance was evaluated against established methods like boxplot, Principal Component Analysis (PCA), and Functional Isolation Forest.
• FKDE demonstrated strong performance in anomaly detection compared to baseline methods.

Conclusions:

• Functional Kernel Density Estimation is an effective unsupervised method for anomalous time series detection.
• The approach is versatile, handling missing data and applicable across diverse datasets.
• The method shows promise for applications such as aviation safety analysis.