Modelling Vast Time Series: Sparsity and Segmentation

在这个现代信息时代，大量或大量时间序列数据的可用性给时间序列分析带来了机遇和挑战。对高维时间序列建模和预测的需求源于各种实际问题，如经济、社会和自然现象（如天气）的小组研究、金融市场分析和通信工程。我们提出了两种新的方法来分析高维时间序列数据，当维度等于甚至大于观测时间序列的长度时。第一种方法是用稀疏向量自回归模型（VAR）拟合数据。对于某些应用，当组件是有序的，我们将进一步探讨由于带结构的稀疏性。注意，我们在VAR模型中直接对系数矩阵施加稀疏性或带状。因此，相关的推理方法和理论与大协方差矩阵的估计不同。我们的第二种方法是通过转换进行分割。我们寻求一个同步的线性变换，这样变换后的时间序列被分成几个子向量，这些子向量是同时和序列不相关的。因此，它们可以分别建模。我们的建议面临两方面的挑战：首先，我们需要开发用于识别稀疏结构和拟合大维度稀疏VAR模型的统计推断方法和相关理论。设p表示时间序列的维数。我们的目标是将模型参数的数量从p的平方阶减少到p阶，并开发log(p)= o(n)时有效的推理方法。其次，我们需要识别线性变换来识别潜在的分割结构，即当存在块对角自协方差结构时。高维数据分析(即“大数据”)是过去十年统计学中最具活力的研究领域之一。迄今为止，大多数工作都集中在具有大量候选回归量的线性回归上（即所谓的“大p小n”范式）。另一个研究方向是大协方差矩阵的推理。虽然打着类似的旗号，但提案中解决的问题是不同的，因为我们处理的是高维时间序列，我们需要估计非正半确定的大变换或系数矩阵。我们的目标是简单有效的推理方法，使它们可以在普通pc上实现，以处理数千维的数据。

In this modern information age the availability of large or vast time series data brings opportunities with challenges to time series analysts. The demand for modelling and forecasting high-dimensional time series arises from various practical problems such as panel study of economic, social and natural phenomena (such as weather), financial market analysis and communications engineering. We propose two new approaches for analyzing high-dimensional time series data when the dimension is as large as, or even greater than, the length of observed time series. The first approach is to fit the data with sparse vector auto-regressive models (VAR). For some applications when the components are ordered, we will further explore the sparsity due to a band structure. Note that we impose sparsity or banding directly on the coefficient matrices in VAR models. Hence, the relevant inference methods and the associated theory are different from those for the estimation of large covariance matrices. Our second approach is segmentation via transformation. We seek for a contemporaneous linear transformation such that the transformed time series is divided into several sub-vectors, and those sub-vectors are both contemporaneously and serially uncorrelated. Therefore, they can be modelled separately.The challenges of our proposal are two-fold: First we need to develop the statistical inference methods and the associated theory for identifying the sparse structure and for fitting sparse VAR models with large dimensions. Let p denote the dimension of the time series. We aim to reduce the number of model parameters from the order of the square of p to the order of p, and to develop the valid inference methods when log(p)= o(n). Secondly, we need to identify the linear transformation to identify the latent segmentation structure, i.e. the block-diagonal autocovariance structure when such a structure exists.High-dimensional data analysis (i.e. 'big data') is one of the most vibrant research areas in statistics in the last decade. Most work to date concentrates on linear regression with a large number of candidate regressors (i.e. the so-called 'large p small n' paradigm). Another stream of the research is on the inference of large covariance matrices. Though bearing a similar banner, the problems addressed in the proposal are different, as we deal with high-dimensional time series and we need to estimate large transformation or coefficient matrices that are not positive semi-definite. We aim for simple and effective inference methods so that they can be implemented with ordinary PCs for the data of dimensions in the order of thousands.