Nov 30

10:30 am

## Wavelet Based Estimation for Trend Contaminated Long Memory Processes

### Peter Craigmile

Final Exam

University of Washington - Department of Statistics

Advisors: Peter Guttorp and Donald B. Percival

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 (FD) processes, which can account for long memory dependence (slowly decaying autocorrelation) and can be extended to encompass nonstationary processes exhibiting quite significant low frequency components.

In this dissertation 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 FD process parameters and the trend. The estimation of the FD parameters is based on an approximative maximum likelihood approach that is made possible by the fact that the DWT decorrelates FD processes approximately. We consider the decorrelation in some detail, examining the between and within scale wavelet correlations separately. As we increase the length of the wavelet filter, we decorrelate better between different wavelet scales, while a white noise or autoregressive process provides a good approximation to the stationary covariance structure within scales.

Once the FD parameters have been estimated, we can then calculate confidence intervals for an estimate of the trend as well as test for a nonzero (or non constant) trend. To do this we must consider the effects of the choice of the wavelet filter and of estimating the parameters of the error process. We find that the variability of the trend estimate, except for the Haar wavelet filter, is dependent on the shape of the wavelet filter chosen, and increases with the degree of autocorrelation. We can obtain the percentage points of the distribution of the test for trend statistic based on a Monte Carlo method, for which we need to be able to simulate FD processes. We prove that the Davies--Harte algorithm for simulating stationary processes can be applied for this purpose.

We demonstrate our methodology by applying it to several environmental time series, including a popular 150 year northern hemisphere temperature series. We show evidence of a significant upward trend in the latter years, highlighting the fact that the variability of the series was higher in the years preceding 1875.