Gradual changes in functional time series

TL;DR

This paper introduces a statistical method to detect and estimate the timing of gradual changes in the mean functions of functional time series, with applications to climate data analysis.

Contribution

It develops a Gaussian approximation-based test and estimators for identifying when mean functions deviate beyond a threshold, addressing a gap in analyzing non-stationary functional data.

Findings

The proposed method effectively detects gradual changes in simulated data.

Application to Australian temperature data demonstrates practical utility.

The approach is validated through theoretical proofs and simulation studies.

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 $\Delta $, if this quantity exceeds a given threshold $\Delta>0$. 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 $\Delta =1.5$ 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 $\Delta$ 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.