Research Group Fabian Scheipl

Multivariate functional data can be intrinsically multivariate like movement trajectories in 2D or complementary such as precipitation, temperature and wind speeds over time at a given weather station. We propose a multivariate functional additive mixed model (multiFAMM) and show its application to both data situations using examples from sports science (movement trajectories of snooker players) and phonetic science (acoustic signals and articulation of consonants). The approach includes linear and nonlinear covariate effects and models the dependency structure between the dimensions of the responses using multivariate functional principal component analysis. Multivariate functional random intercepts capture both the auto-correlation within a given function and cross-correlations between the multivariate functional dimensions. They also allow us to model between-function correlations as induced by, for example, repeated measurements or crossed study designs. Modelling the dependency structure between the dimensions can generate additional insight into the properties of the multivariate functional process, improves the estimation of random effects, and yields corrected confidence bands for covariate effects. Extensive simulation studies indicate that a multivariate modelling approach is more parsimonious than fitting independent univariate models to the data while maintaining or improving model fit.

Outlier or anomaly detection is an important task in data analysis. We discuss the problem from a geometrical perspective and provide a framework which exploits the metric structure of a data set. Our approach rests on the manifold assumption, that is, that the observed, nominally high-dimensional data lie on a much lower dimensional manifold and that this intrinsic structure can be inferred with manifold learning methods. We show that exploiting this structure significantly improves the detection of outlying observations in high dimensional data. We also suggest a novel, mathematically precise and widely applicable distinction between distributional and structural outliers based on the geometry and topology of the data manifold that clarifies conceptual ambiguities prevalent throughout the literature. Our experiments focus on functional data as one class of structured high-dimensional data, but the framework we propose is completely general and we include image and graph data applications. Our results show that the outlier structure of high-dimensional and non-tabular data can be detected and visualized using manifold learning methods and quantified using standard outlier scoring methods applied to the manifold embedding vectors.

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.

In recent years, manifold methods have moved into focus as tools for dimension reduction. Assuming that the high-dimensional data actually lie on or close to a low-dimensional nonlinear manifold, these methods have shown convincing results in several settings. This manifold assumption is often reasonable for functional data, i.e., data representing continuously observed functions, as well. However, the performance of manifold methods recently proposed for tabular or image data has not been systematically assessed in the case of functional data yet. Moreover, it is unclear how to evaluate the quality of learned embeddings that do not yield invertible mappings, since the reconstruction error cannot be used as a performance measure for such representations. In this work, we describe and investigate the specific challenges for nonlinear dimension reduction posed by the functional data setting. The contributions of the paper are three-fold: First of all, we define a theoretical framework which allows to systematically assess specific challenges that arise in the functional data context, transfer several nonlinear dimension reduction methods for tabular and image data to functional data, and show that manifold methods can be used successfully in this setting. Secondly, we subject performance assessment and tuning strategies to a thorough and systematic evaluation based on several different functional data settings and point out some previously undescribed weaknesses and pitfalls which can jeopardize reliable judgment of embedding quality. Thirdly, we propose a nuanced approach to make trustworthy decisions for or against competing nonconforming embeddings more objectively.

Time series classification problems have drawn increasing attention in the machine learning and statistical community. Closely related is the field of functional data analysis (FDA): it refers to the range of problems that deal with the analysis of data that is continuously indexed over some domain. While often employing different methods, both fields strive to answer similar questions, a common example being classification or regression problems with functional covariates. We study methods from functional data analysis, such as functional generalized additive models, as well as functionality to concatenate (functional-) feature extraction or basis representations with traditional machine learning algorithms like support vector machines or classification trees. In order to assess the methods and implementations, we run a benchmark on a wide variety of representative (time series) data sets, with in-depth analysis of empirical results, and strive to provide a reference ranking for which method(s) to use for non-expert practitioners. Additionally, we provide a software framework in R for functional data analysis for supervised learning, including machine learning and more linear approaches from statistics. This allows convenient access, and in connection with the machine-learning toolbox mlr, those methods can now also be tuned and benchmarked.

Functional data typically contain amplitude and phase variation. In many data situations, phase variation is treated as a nuisance effect and is removed during preprocessing, although it may contain valuable information. In this note, we focus on joint principal component analysis (PCA) of amplitude and phase variation. As the space of warping functions has a complex geometric structure, one key element of the analysis is transforming the warping functions to urn:x-wiley:sta4:media:sta4220:sta4220-math-0001. We present different transformation approaches and show how they fit into a general class of transformations. This allows us to compare their strengths and limitations. In the context of PCA, our results offer arguments in favour of the centred log-ratio transformation. We further embed two existing approaches from the literature for joint PCA of amplitude and phase variation into the framework of multivariate functional PCA, where we study the properties of the estimators based on an appropriate metric. The approach is illustrated through an application from seismology.