A Geometric Perspective on Functional Outlier Detection

Abstract

We consider functional outlier detection from a geometric perspective, specifically: for functional datasets drawn from a functional manifold, which is defined by the data’s modes of variation in shape, translation, and phase. Based on this manifold, we developed a conceptualization of functional outlier detection that is more widely applicable and realistic than previously proposed taxonomies. Our theoretical and experimental analyses demonstrated several important advantages of this perspective: it considerably improves theoretical understanding and allows describing and analyzing complex functional outlier scenarios consistently and in full generality, by differentiating between structurally anomalous outlier data that are off-manifold and distributionally outlying data that are on-manifold, but at its margins. This improves the practical feasibility of functional outlier detection: we show that simple manifold-learning methods can be used to reliably infer and visualize the geometric structure of functional datasets. We also show that standard outlier-detection methods requiring tabular data inputs can be applied to functional data very successfully by simply using their vector-valued representations learned from manifold learning methods as the input features. Our experiments on synthetic and real datasets demonstrated that this approach leads to outlier detection performances at least on par with existing functional-data-specific methods in a large variety of settings, without the highly specialized, complex methodology and narrow domain of application these methods often entail.