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.