Robust estimation and inference for high-dimensional time series

时间序列数据是在金融、经济、医学和社会科学等广泛领域收集的。数据集的示例包括金融中的股票价格数据和神经科学中的EEG数据，这些数据记录了大脑的电活动。对于时间序列数据，建模技术的存在提供了解释过程行为的方法，对观测数据进行推理陈述，并预测未来的观测结果。此外，多变量时间序列分析方法通过联合分析多个时间序列来利用时间序列之间可能的相互依赖性。除了时间序列数据集的流行之外，现代数据集通常包括与观测数量相关的大量变量。因此，有一个多变量的时间序列方法，工作高维data.My项目的目的是robustify现有的方法建模高维时间序列数据，确保模型估计器的性能在宽松的假设。特别是，我们将在向量自回归（VAR）模型的估计中利用重尾稳健协方差估计。其目的是，这将为在较弱假设下拟合稀疏VAR模型的正则化方法提供理论保证，与通常应用于数据的高斯假设相反。在VAR模型的框架下，可以进一步研究数据的问题，例如，检测数据中的组结构或数据中的变点。该项目的下一步可能需要开发有助于解决这些问题的技术，并在允许数据重尾的假设下提供理论保证。这项研究的潜在影响是，当前工具的性能将得到改善，并与更广泛的数据集一起工作。这可能会导致预测和理解时间序列数据的随机结构的能力的改进。这些技术可以用于前面列出的领域。这个项目正在开发统计方法，因此自然属于EPSRC统计和应用概率研究领域的福尔斯。

Time series data are collected in a wide range of areas such as finance, economics, medicine and social sciences. Examples of datasets include, stock price data in finance, and EEG data in Neuroscience, which records electrical activity of the brain. For time series data, modelling techniques exist which provide ways to explain the behaviour of the process, make inferential statements about the observed data, and to forecast future observations. Furthermore, multivariate time series analysis methods make use of the possible interdependences between time series by analysing several time series jointly. In addition to the prevalence of time series datasets, modern datasets often include a large number of variables relative to the number of observations. Therefore, there is a demand for multivariate time series methods that work for high-dimensional data.My project aims to robustify existing methodology for modelling high-dimensional time series data, ensuring the performance of model estimators under relaxed assumptions. In particular, we will be looking to utilise heavy-tailed robust covariance estimators within the estimation of vector autoregressive (VAR) models. With the aim that this will then provide theoretical guarantees for regularised methods for fitting sparse VAR models under weaker assumptions, in contrast to the Gaussian assumptions typically applied to the data. The robust estimation procedure could then be used within downstream methods which require estimation of coefficients of a VAR process.Under the framework of VAR models further questions about the data can be asked, for example, detecting group structures in the data, or change points in the data. Next steps in the project could entail developing techniques that help solve these questions, with theoretical guarantees under assumptions that allow for heavy-tailedness in the data. The potential impact of this research is that the performance of current tools will be improved, and work with a broader range of datasets. This could then lead to improvements in forecasting and ability to understanding the stochastic structure of time series data. These techniques could be used in the fields listed previously.This project is developing statistical methodology, and therefore naturally falls under the EPSRC Statistics and applied probability research area.