Gradual changes in functional time series

Patrick Bastian; Holger Dette

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arXiv · 2407.07996

Gradual changes in functional time series

Patrick Bastian·Holger Dette

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Abstract

We consider the problem of detecting gradual changes in the sequence of mean functions from a not necessarily stationary functional time series. Our approach is based on the maximum deviation (calculated over a given time interval) between a benchmark function and the mean functions at different time points. We speak of a gradual change of size $Δ$, if this quantity exceeds a given threshold $Δ>0$. For example, the benchmark function could represent an average of yearly temperature curves from the pre-industrial time, and we are interested in the question if the yearly temperature curves afterwards deviate from the pre-industrial average by more than $Δ=1.5$ degrees Celsius, where the deviations are measured with respect to the sup-norm. Using Gaussian approximations for high-dimensional data we develop a test for hypotheses of this type and estimators for the time where a deviation of size larger than $Δ$ appears for the first time. We prove the validity of our approach and illustrate the new methods by a simulation study and a data example, where we analyze yearly temperature curves at different stations in Australia.

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Patrick Bastian, Holger Dette. 2025-01-10. Gradual changes in functional time series. https://arxiv.org/abs/2407.07996

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