Wavelets in time series

点过程数据的分析在数据在时间和/或空间中出现的一系列应用领域中是必不可少的。已经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。该项目的目的是扩展局部平稳小波为基础的方法来估计局部平稳霍克斯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服从必要的一阶和二阶性质的局部平稳Hawkes过程的窗口。它是假设一个定义良好的内核，服从平方可积性选择。通过适当地选择一个内核为一组给定的背景强度的变化，我们建议采用小波包的方法来有效地估计所有的背景强度值一次。该项目将开发局部平稳霍克斯过程方法以及局部平稳小波，这将允许进行必要的多尺度分辨率，以估计局部平稳霍克斯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，用于估计Hawkes过程的时变参数，该过程将在许多应用领域产生影响，这些应用领域具有潜在的点过程的非标准强度。该提案显然有助于统计和应用概率研究领域，在几个战略主题中具有潜在的应用领域，包括数学科学，工程，制造未来和物理科学。

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.