Wavelets in time series

在数据发生在时间和/或空间上的一系列应用领域中，对点过程数据的分析是必不可少的。人们提出了各种统计模型来分析这些数据，从基本的泊松过程到更复杂的自激过程，如霍克斯过程。该项目旨在扩展基于局部平稳小波的方法，以估计局部平稳Hawkes过程的时变参数函数。为了估计时变背景强度，必须通过建立适当的微分过程来考虑局部平稳Hawkes过程的窗口，该窗口服从必要的一阶和二阶性质。假设选择了一个定义良好且服从平方可积性的核。通过对给定的一组背景强度变化选择合适的核，我们提出采用小波包方法一次有效地估计所有的背景强度值。该项目将发展局部平稳Hawkes过程方法和局部平稳小波，这将允许必要的多尺度分辨率来估计局部平稳Hawkes过程的全时变背景强度。该研究将开发新的建模和分析技术，并提供一种实用的方法来估计霍克斯过程的时变参数，这将对许多具有非标准强度的基础点过程的应用领域产生影响。该提案显然有助于统计和应用概率研究领域，潜在的应用领域涉及几个战略主题，包括数学科学、工程、未来制造业和物理科学。

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.