Wavelets in time series

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The analysis of Point process data is essential in a range of application areas where data occur in time and/or space. Various statistical models have been proposed to analyse such data, from basic Poisson processes to more complex self-exciting processes, such as the Hawkes process. The project aims to extend the locally stationary wavelet-based methodology to estimate time-varying parameter functions for a locally stationary Hawkes process. To estimate the time-varying background intensity,a window of the locally stationary Hawkes process that obeys the necessary first and second-order properties must be considered by establishing an appropriate differential process. It is assumed that a well-defined kernel that obeys square integrability is chosen. By appropriately choosing a kernel for a given set of changes in the background intensity, we propose to adopt a wavelet packet approach to efficiently estimate all the background intensity values at once. This project will develop the locally stationary Hawkes process method alongside the locally stationary wavelet, which will allow for the necessary multiscale resolution to be made to estimate the full time-varying background intensity of a locally stationary Hawkes process. The research will develop novel modelling and analysis techniques, as well as providing a practically useful method for estimating time-varying parameters for a Hawkes process which will have an impact in many application areas with a non-standard intensity of the underlying point process.The proposal clearly contributes to the statistics and applied probability research area, with potential application areas over several strategic themes including Mathematical Sciences, Engineering, Manufacturing the Future, and Physical Sciences.