Research Summary

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 auto-correlation) and can be extended to encompass non-stationary 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 non-zero trend. Our methodology has been applied to an air quality time series. Slides on the subject can be found at: