A Measure of Directional Outlyingness With Applications to Image Data and Video

ABSTRACT

Functional data analysis covers a wide range of data types. They all have in common that the observed objects are functions of a univariate argument (e.g., time or wavelength) or a multivariate argument (say, a spatial position). These functions take on values which can in turn be univariate (such as the absorbance level) or multivariate (such as the red/green/blue color levels of an image). In practice it is important to be able to detect outliers in such data. For this purpose we introduce a new measure of outlyingness that we compute at each gridpoint of the functions’ domain. The proposed directional outlyingness (DO) measure accounts for skewness in the data and only requires $\mathcal{O}\left(n\right)$ computation time per direction. We derive the influence function of the DO and compute a cutoff for outlier detection. The resulting heatmap and functional outlier map reflect local and global outlyingness of a function. To illustrate the performance of the method on real data it is applied to spectra, MRI images, and video surveillance data.

KEYWORDS:

• Functional data
• Influence function
• Outlier detection
• Robustness
• Skewness

Acknowledgments

This research has been supported by projects of Internal Funds KU Leuven. The authors are grateful for interesting discussions with Pieter Segaert.

Funding

Onderzoeksraad, KU Leuven.