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Wavelet variances and covariances (sometime also known as wavelet spectra
and co-spectra) are a form of analysis of variance that find extensive use
in the study of time series arising from atmospheric science, economics,
hydrology, oceanography, soil science and other areas of geophysics.
Specifically, as precursor to a more formal statistical analysis, wavelet
variances and covariances are often applied to time series to explore
their structure and behavior, to identify important scales, to assess long
memory parameters, to detect inhomogeneity and to estimate time-varying
spectral densities. In this talk, we focus on estimation and inference of
wavelet variances and covariances when the observed time series is
.gappy., i.e., sampled at regular intervals, but certain observations are
missing. In particular, we propose a method of estimation that extends
the usual estimation procedure for the non-gappy data, investigate
statistical properties and discuss large sample theory. We then show how
our approach to this problem opens up new avenues in advancing statistical
calculations for wavelet-based principal components, clustering and
classification of multivariate time series that contain missing values.
Finally, we consider an application of our method to NOAA's tropical sea
level barometric pressure data.
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