Spectral estimation for locally stationary time series with missing observations |
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Authors: | Marina I Knight Matthew A Nunes Guy P Nason |
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Institution: | 1.NHS Blood and Transplant,Bristol,UK;2.Department of Mathematics and Statistics, Fylde College,Lancaster University,Lancaster,UK;3.School of Mathematics,University of Bristol,Bristol,UK |
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Abstract: | Time series arising in practice often have an inherently irregular sampling structure or missing values, that can arise for
example due to a faulty measuring device or complex time-dependent nature. Spectral decomposition of time series is a traditionally
useful tool for data variability analysis. However, existing methods for spectral estimation often assume a regularly-sampled
time series, or require modifications to cope with irregular or ‘gappy’ data. Additionally, many techniques also assume that
the time series are stationary, which in the majority of cases is demonstrably not appropriate. This article addresses the
topic of spectral estimation of a non-stationary time series sampled with missing data. The time series is modelled as a locally
stationary wavelet process in the sense introduced by Nason et al. (J. R. Stat. Soc. B 62(2):271–292, 2000) and its realization is assumed to feature missing observations. Our work proposes an estimator (the periodogram) for the
process wavelet spectrum, which copes with the missing data whilst relaxing the strong assumption of stationarity. At the
centre of our construction are second generation wavelets built by means of the lifting scheme (Sweldens, Wavelet Applications
in Signal and Image Processing III, Proc. SPIE, vol. 2569, pp. 68–79, 1995), designed to cope with irregular data. We investigate the theoretical properties of our proposed periodogram, and show that
it can be smoothed to produce a bias-corrected spectral estimate by adopting a penalized least squares criterion. We demonstrate
our method with real data and simulated examples. |
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