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.

翻译：本文研究从非平稳功能时间序列中检测均值函数序列渐变的问题。我们的方法基于基准函数与不同时间点均值函数之间（在给定时间区间上计算的）最大偏差。若该偏差超过给定阈值Δ>0，则称存在幅度为Δ的渐变。例如，基准函数可代表前工业化时期年平均温度曲线的均值，我们关注此后年份的温度曲线是否与前工业化均值偏离超过Δ=1.5摄氏度（偏差采用上确界范数度量）。借助高维数据的高斯近似方法，我们发展了针对此类假设的检验方法，以及首次出现幅度超过Δ的偏差的时间估计量。我们证明了所提方法的有效性，并通过模拟研究和澳大利亚多站点年平均温度曲线的实际数据分析展示了新方法的应用。