Trend Estimation Using Wavelets
Peter Craigmile, Don Percival and Peter Guttorp
A common problem in the
analysis of environmental time series is how to deal with a possible
trend component, which is usually thought of as large scale (or low
frequency) variations or patterns in the series that might be best
modelled separately from the rest of the series. Trend is often
confounded with low frequency stochastic fluctuations, particularly in
the case of models such as fractionally differenced processes (FDPs),
which can account for long memory dependence (slowly decaying
autocorrelation) and can be extended to encompass nonstationary
processes exhibiting quite significant low frequency components. We
assume a model of polynomial trend plus fractionally differenced noise
and apply the discrete wavelet transform (DWT) to separate a time series
into pieces that can be used to estimate both the FDP parameters and the
trend. The estimation of the FDP parameters is based on an approximation
maximum likelihood approach that is made possible by the fact that the
DWT decorrelates FDPs approximately. Once the FDP parameters have been
estimated, we can then test for a nonzero trend. Our methodology has
been applied to an air quality time series. Slides on the subject can be
found at: http://www.stat.washington.edu/pfc/talks/talks.html 

