Modelling Vast Time Series: Sparsity and Segmentation

在现代信息时代，大量或海量时间序列数据的可获得性给时间序列分析人员带来了机遇和挑战。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，当其维度等于甚至大于观察时间序列的长度时。第一种方法是用稀疏向量自回归模型(VAR)对数据进行拟合。对于某些应用，当组件被排序时，我们将进一步探索由于带结构而产生的稀疏性。请注意，在VAR模型中，我们直接对系数矩阵施加稀疏性或带状。因此，相关的推断方法和相关理论不同于大型协方差矩阵的估计。我们的第二种方法是通过变换进行分割。我们寻求一种同时线性变换，使得变换后的时间序列被分成几个子矢量，而这些子矢量同时和序列是不相关的。首先，我们需要发展识别稀疏结构和拟合高维稀疏VAR模型的统计推断方法和相关理论。设p表示时间序列的维度。我们的目标是将模型参数的数量从p的平方降到p的数量级，并发展出当log(P)=o(N)时的有效推理方法。其次，我们需要确定线性变换来识别潜在的分段结构，即存在块对角自协方差结构时的这种结构。高维数据分析(即大数据)是近十年来统计学中最活跃的研究领域之一。到目前为止，大多数工作都集中在具有大量候选回归变量的线性回归(即所谓的“大p小n”范式)。另一个研究方向是关于大协方差矩阵的推断。虽然打着类似的旗号，但提案中所涉及的问题是不同的，因为我们处理的是高维时间序列，并且我们需要估计不是半正定的大变换或系数矩阵。我们的目标是寻求简单有效的推理方法，以便在普通PC上实现对数千维数据的推理。

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.