The estimation of spatial and temporal trends is a particularly important problem for properly modeling space-time processes. For example, if significant trends exist in a space-time process and are just ignored, estimates of spatial covariance can be badly biased. Separation of space-time data into trend and random components is difficult mainly because the notion of trend is somewhat subjective, and hence detrending has often been done using ad hoc techniques.
The last ten years has seen considerable work in using wavelets to
perform a multiresolution analysis of a regularly sampled time series. The
result of such an analysis is to decompose a time series into
new
time series such that (i) the summation of the
new series is equal to
, (ii)
of the new series represent variations in
over
different time scales (usually proportional to
) and
(iii) the
new series is a low resolution approximation to
(related to scales proportional to
and longer). One way to quantify
the trend in a time series is to equate it with this low resolution
approximation.
More recently, Sweldens and Schröder (1995) have introduced a scheme called ``lifting'' that effectively generalizes the notion of wavelets to domains other than regularly sampled time series. In particular, the lifting scheme can be used to perform a multiresolution analysis for (i) space-time data collected over an arbitrary two dimensional spatial grid at arbitrary times or (ii) spatial data collected at a finite number of locations over a sphere.