基于序贯变换的高维矩阵值时间序列分割
Segmenting High-dimensional Matrix-valued Time Series via Sequential Transformations
摘要：
矩阵值时间序列建模是一个有趣而重要的研究课题。在本文中，我们将Chang et al.（2017）的方法扩展到矩阵值时间序列。对于任何给定的$p\times q$矩阵值时间序列，我们寻找线性变换将矩阵分割成许多小的子矩阵，对于这些子矩阵中的每一个，它们都与其他子矩阵同时或连续地不相关，因此它们可以被单独分析，这将大大减少在建模方面要估计的参数的数量。为了克服识别问题，我们提出了一个两步和更结构化的过程，分别分割行和列。当$\max(p,q)$相对于样本量$n$较大时，我们假设变换矩阵是稀疏的，并对（自动）协方差阵使用阈值估计。我们还提出了一种分块阈值方法来分离变换后的矩阵值数据的列（或行）。建立了固定和发散$\max(p,q)$的渐近性质。与独立数据的主成分分析（PCA）不同，我们不能保证所需的线性变换存在。当它不，所提出的方法提供了一个近似的分割，这可能是有用的预测。仿真和实际数据的例子说明了所提出的方法。我们还提出了一个序列变换算法来分割高阶张量值时间序列。
英文摘要：
Modeling matrix-valued time series is an interesting and important research topic. In this paper, we extend the method of Chang et al. (2017) to matrix-valued time series. For any given $p\times q$ matrix-valued time series, we look for linear transformations to segment the matrix into many small sub-matrices for which each of them are uncorrelated with the others both contemporaneously and serially, thus they can be analyzed separately, which will greatly reduce the number of parameters to be estimated in terms of modeling. To overcome the identification issue, we propose a two-step and more structured procedure to segment the rows and columns separately. When $\max(p,q)$ is large in relation to the sample size $n$, we assume the transformation matrices are sparse and use threshold estimators for the (auto)covariance matrices. We also propose a block-wisely thresholding method to separate the columns (or rows) of the transformed matrix-valued data. The asymptotic properties are established for both fixed and diverging $\max(p,q)$. Unlike principal component analysis (PCA) for independent data, we cannot guarantee that the required linear transformation exists. When it does not, the proposed method provides an approximate segmentation, which may be useful for forecasting. The proposed method is illustrated with both simulated and real data examples. We also propose a sequential transformation algorithm to segment higher-order tensor-valued time series.
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