Gradual changes in functional time series
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912182491414528 |
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| author | Bastian, Patrick Dette, Holger |
| author_facet | Bastian, Patrick Dette, Holger |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_07996 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gradual changes in functional time series Bastian, Patrick Dette, Holger Statistics Theory Methodology 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. |
| title | Gradual changes in functional time series |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2407.07996 |