Applications of continuous wavelet methods in statistical modelling

The wavelet approach is an efficient time-frequency analysis tool to investigate stochastic data both in terms of time and frequency. Wavelet methods provide a decomposition of a signal using a wavelet function which is localised in both time and frequency. This thesis focuses on wavelet methods using the Haar wavelet function to develop a new approach to statistical modelling and data analysis.

We apply a regression model to the wavelet coefficients to classify the state of a gas-fraction for an engineering tomography dataset. In the previous research of Aykroyd et al. (2016), the model based on the wavelet coefficients, from the discrete resolution levels, classified the tomography data well. However, the model is fitted on a limited number of wavelet resolution levels of the Discrete Wavelet Transform (DWT). We expand the scale set to the continuous domain via the continuous wavelet transform (CWT) to see the effectiveness of flexible wavelet scale selection for similar tomography data analysis.

Apart from the modelling using the CWT, the locally stationary wavelet (LSW) method is introduced in Nason et al. (2000) as a model to investigate non-stationary process with wavelet functions. We will give an overview of the standard discrete LSW process and suggest an extension of the continuous LSW (CLSW) process to estimate frequency features. The standard LSW process is built on discrete resolution levels, but we will extend the LSW process to the continuous wavelet scales. However, due to the redundancy of the CWT, the estimation of the evolutionary wavelet spectrum (EWS) does not show spectral densities matching true frequency characteristics. To cope with the problem, we apply the idea of the orthogonal matching pursuit (OMP) algorithm to select the best subset of continuous wavelet scales explaining data the most. We illustrate the modified CLSW process using the reflected doppler data and real tomography data and show the spectral estimate of them. Based on the improvement of spectral estimation, we fit a classification model on the estimate of EWS from the modified CLSW process using the real tomography data.