We consider functional outlier detection from a geometric perspective, specifically: for functional data sets drawn from a functional manifold which is defined by the data's modes of variation in amplitude and phase. Based on this manifold, we develop a conceptualization of functional outlier detection that is more widely applicable and realistic than previously proposed. Our theoretical and experimental analyses demonstrate several important advantages of this perspective: It considerably improves theoretical understanding and allows to describe and analyse 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 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 data sets. 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 input features. Our experiments on synthetic and real data sets demonstrate 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.
翻译:我们从几何角度考虑外星功能探测,具体地说,我们认为,从数据振幅和阶段变化模式所定义的功能多元体定义的功能数据集,是功能外星探测的功能外星探测,我们从这个多元体出发,对功能外星探测进行概念化的构想,比以前的提议更加广泛适用和现实。我们的理论和实验分析表明,这一角度有一些重要的优点:它通过区分结构异常体外和分布的外星数据,将结构异常体数据分为不同部分和分布外的外星数据,这些数据由数据分布上的不同模式决定。这提高了功能外星探测的实际可行性:我们表明,可以使用简单的多式学习方法,可靠地推断和直观功能性数据集的几何结构。 我们还表明,要求表格数据投入的标准外星探测方法可以非常成功地应用于功能性数据,只需使用从多重学习方法中学到的矢量值表作为输入特征。我们在合成和真实数据上进行的实验表明,这种方法至少可以在现有功能外外星点探测出性探测出,而往往需要非常狭窄的域方法。