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Within the framework of functional data analysis, we develop principal component analysis for periodically correlated time series of functions. We define the components of the above analysis including periodic, operator--valued filters, score processes and the inversion formulas. We show that these objects are defined via convergent series under a simple condition requiring summability of the Hilbert--Schmidt norms of the filter coefficients, and that they possess optimality properties. We explain how the Hilbert space theory reduces to an approximate finite--dimensional setting which is implemented in a custom build \verb|R| package. A data example and a simulation study show that the new methodology is superior to existing tools if the functional time series exhibit periodic characteristics.
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Testing for periodicity in functional time series
We derive several tests for the presence of a periodic component in a time series of functions. We consider both the traditional setting in which the periodic functional signal is contaminated by functional white noise, and a more general setting of a weakly dependent contaminating process. Several forms of the periodic component are considered. Our tests are motivated by the likelihood principle and fall into two broad categories, which we term multivariate and fully functional. Generally, for the functional series that motivate this research, the fully functional tests exhibit a superior balance of size and power. Asymptotic null distributions of all tests are derived and their consistency is established. Their finite sample performance is examined and compared by numerical studies and application to pollution data.
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- Award ID(s):
- 1737795
- PAR ID:
- 10097817
- Date Published:
- Journal Name:
- Annals of statistics
- Volume:
- 46
- ISSN:
- 0090-5364
- Page Range / eLocation ID:
- 2960 - 2984
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/DOI: 10.1214/17-AOS1645
