PRINCIPAL COMPONENT ANALYSIS FOR SECOND-ORDER STATIONARY VECTOR TIME SERIES
PRINCIPAL COMPONENT ANALYSIS FOR SECOND-ORDER STATIONARY VECTOR TIME SERIES
复制标题
二阶平稳向量时间序列的主成分分析
DOI：
10.1214/17-aos1613
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发表时间：
2018-10-01
影响因子：
4.5
通讯作者：
Yao, Qiwei
中科院分区：
文献类型：
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作者：
Chang, Jinyuan;Guo, Bin;Yao, Qiwei
我们将主成分分析(PCA)推广到二阶平稳向量时间序列，即寻求p元时间序列的同时线性变换，使得变换后的序列被分割成几个低维子序列，并且这些子序列同时和序列彼此不相关。因此，对于线性动态结构而言，这些低维序列可以单独进行分析。从技术上讲，它归结为对正定矩阵的特征分析。当$p$较大时，需要额外的步骤来执行基于多个测试的最大互相关或FDR方面的置换。当样本量n$趋于无穷大时，建立了固定$p$和发散$p$的渐近理论。用模拟和真实数据集进行的数值实验表明，该方法是分析多时间序列数据的一个有效的初始步骤，从而大大降低了高维线性动态结构的建模和预测的维度。与独立数据的主成分分析不同，不能保证存在所需的线性变换。当不是这样时，所提出的方法提供了一个近似的分段，这导致了例如在预测未来价值方面的优势。该方法还可以适用于分割多个波动过程。
We extend the principal component analysis (PCA) to second-order stationary vector time series in the sense that we seek for a contemporaneous linear transformation for a $p$-variate time series such that the transformed series is segmented into several lower-dimensional subseries, and those subseries are uncorrelated with each other both contemporaneously and serially. Therefore those lower-dimensional series can be analysed separately as far as the linear dynamic structure is concerned. Technically it boils down to an eigenanalysis for a positive definite matrix. When $p$ is large, an additional step is required to perform a permutation in terms of either maximum cross-correlations or FDR based on multiple tests. The asymptotic theory is established for both fixed $p$ and diverging $p$ when the sample size $n$ tends to infinity. Numerical experiments with both simulated and real data sets indicate that the proposed method is an effective initial step in analysing multiple time series data, which leads to substantial dimension reduction in modelling and forecasting high-dimensional linear dynamical structures. Unlike PCA for independent data, there is no guarantee that the required linear transformation exists. When it does not, the proposed method provides an approximate segmentation which leads to the advantages in, for example, forecasting for future values. The method can also be adapted to segment multiple volatility processes.