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Projects
Current third-party funded projects
DFG Project (2025-2028): From dense to sparse functional data: Optimal rates of convergence, asymptotic inference and functional time series
Functional data analysis (FDA) refers to situations where the data involve repeated observations of functions or images. Two central notions in FDA are replication and regularization. Replication makes use of information across the n repeated observations of the functions, while regularization takes advantage of information within functions, which in practice are only observed at discrete points in time or space. In a synchronous design the discrete observation points are at equal, deterministic locations for each replication, which is the typical scenario for machine recorded functional data which are most prevalent in applications.
Depending on the number of observations within functions, one distinguishes between densely and sparsely observed functional data. For synchronous design, our previous results for the mean function and the covariance kernel show that in the dense case, inference can be done at the parametric root - n - rate as if the processes were continuously observed without errors. The sparse case is dominated by discretization errors, while a phase transition occurs in between driven by the additional observational noise. This phase transition only comes fully to light when the supremum norm is used as error measure. The supremum norm is also of substantial practical interest since it corresponds to the visualization of the estimation error, and forms the basis for the construction of uniform confidence bands.
In this project our first goal is to complete the results described above by providing optimal rates in the supremum norm and central limit theorems (CLTs) in the space of continuous functions under synchronous design for estimating derivatives of the mean function and the covariance kernel and for the principle component basis functions. Our second overall objective is to use the results and insights for inference methods. These shall include the construction of confidence bands, inference on the number of design points required and inference on the smoothness of the paths as well as CLTs and inference for transfer learning in functional estimation problems. Finally, while estimation and testing problems with functional time series frequently occur in applications and have been intensely investigated most studies proceed under the idealized assumption that processes are fully observed without errors. Therefore, the third overall objective is to extend our results to discretely and synchronously sampled functional time series.