局域平稳时间序列的统计推断与预测

Locally stationary models have been widely applied to various fields including engineering science, environmental science, economic and financial studies. On one hand, their assumption of local stationarity is able to well characterize the nonstationary pattern of many large data sets due to their time-varying data generating mechanism. On the other hand, the class of locally stationary models maintains many advantages of stationary models such as model analyzability and interpretability. In the literature, the researches on locally stationary series mainly focus on topics such as regression analysis, hypothesis testing, covariance estimation and Gaussian approximation. However, there are only very few researches on the inference and forecasting of the locally stationary series. The proposal aims to propose a general method for inferencing and forecasting locally stationary series via two steps. The first is to assess all the possible abrupt change pints of the locally stationary series. The second is to estimate the high dimensional covariance of the series after removing abrupt change points. The success of the proposed research will lead to a general prediction approach which is needed for the establishment of the general framework of analyzing the locally stationary data. Furthermore, the method proposed in the proposal will be applied to real data sets that are important to the country and the society, for instance, the environmental pollution data sets, the economic and financial data sets. In this way, the proposed research projects will make contribution to society.

局域平稳时间序列模型是一类广泛应用于各行各业，包括工程学，环境科学，经济金融等领域的时间序列模型。该模型的局域平稳性能很好的刻画因为时变数据产生机制而产生的非平稳性，从而避免大数据时代平稳性假设常常失效的问题。该模型同时也保留了一系列平稳模型的优点，比如可分析性和可解释性。然而，国际国内对该模型的研究主要集中于回归分析，假设检验，协方差估计，高斯逼近等方面，而关于模型的统计学推断以及预测等鲜少涉及。本项目旨在通过1）检测以及估计序列的突然变点 和2）估计突然变点被移除后的序列的高维协方差矩阵来构建预测一般性局域平稳序列的办法，从而弥补学术界在相关预测领域的研究空白，并为建立系统性分析局域平稳序列的框架奠定基础。同时本项目也将通过统计推断以及预测分析实际时间序列，包括环境污染数据，经济金融数据等对国计民生有重要意义的数据对实践提供指导，为祖国建设尽力。

本项目研究了局域平稳时间序列的的统计推断与预测。在变点检验以及推断方面，主要研究了突然变点的检验问题和估计问题。前者通过积分为0的核函数构造统计量进行检验，后者通过在统计量超过理论阈值附近利用CUSUM方法进行估计。在预测方面，提出了利用局部线性回归去除局域平稳序列时变均值，用残差估计大维协方差矩阵，并利用该协方差矩阵的估计进行预测的方法。本项目进一步研究了一系列局域平稳时间序列的推断问题，包括两个局域平稳序列之间的时变期望与时变分位数的比较问题，局域平稳序列非参数回归的长记忆检验问题，局域平稳函数型时间序列的同时置信曲面的构造问题。本项目提出了解决这些问题的统计方法，推导了相应统计量的理论性质，并用蒙特卡洛方法检验他们的有效性。这些方法还被应用实际数据分析中，包括金融数据，流行病学数据，环境数据等。本项目进一步研究了受时间影响的网络数据的建模，将时间序列的研究方法应用到网络数据中，提出了新的受时间影响的网络数据的分析框架和方法。本项目的理论结果可以被广泛运用于局域平稳序列的统计性质分析中，所提出的方法可以用于金融、环境、健康等领域。