Wavelet analysis of temporal data

This thesis considers the application of wavelets to problems involving multiple series
of temporal data. Wavelets have proven to be highly effective at extracting frequency
information from data. Their multi-scale nature enables the efficient description of both
transient and long-term signals. Furthermore, only a small number of wavelet coefficients
are needed to describe complicated signals and the wavelet transform is computationally
efficient.

In problems where frequency properties are known to be important, it is proposed that
a modelling approach which attempts to explain the response in terms of a multi-scale
wavelet representation of the explanatory series will be an improvement on standard
regression techniques. The problem with classical regression is that differing frequency
characteristics are not exploited and make the estimates of the model parameters less
stable. The proposed modelling method is presented with application to examples from
seismology and tomography.

In the first part of the thesis, we investigate the use of the non-decimated wavelet transform
in the modelling of data produced from a simulated seismology study. The fact that elastic
waves travel with different velocities in different rock types is exploited and wavelet
models are proposed to avoid the complication of predictions being unstable to small
changes in the input data, that is an inverse problem.

The second part of the thesis uses the non-decimated wavelet transform to model electrical
tomographic data, with the aim of process control. In industrial applications of electrical
tomography, multiple voltages are recorded between electrodes attached to the boundary
of, for example, a pipe. The usual first step of the analysis is then to reconstruct the
conductivity distribution within the pipe. The most commonly used approaches again lead
to inverse problems, and wavelet models are again used here to overcome this difficulty.
We conclude by developing the non-decimated multi-wavelet transform for use in the

modelling processes and investigate the improvements over scalar wavelets.