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Spectral estimation for locally stationary time series with missing observationsby Marina I. Knight, Matthew A. Nunes and Guy P. Nason
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 timedependent nature. Spectral decomposition of
time series is a traditionally useful tool for data variability
analysis. However, existing methods for spectral estimation
often assume a regularlysampled 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 nonstationary 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 (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, 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 biascorrected spectral estimate by adopting a penalized
least squares criterion. We demonstrate our method with real
data and simulated examples.
