论文标题
多元非组织时间序列的逆协方差运算符
Inverse covariance operators of multivariate nonstationary time series
论文作者
论文摘要
对于多元固定时间序列，许多重要属性，例如部分相关，图形模型和自回归表示形式在其光谱密度矩阵的倒数中编码。对于非组织时间序列而言，情况并非如此，其中相关信息在于与多元时间序列相关的无限尺寸尺寸协方差矩阵运算符。这需要研究多元非组织时间序列的协方差及其与逆的关系。我们表明，如果以一定速率以一定速率衰减的无限尺寸协方差矩阵衰减，则速率（最高因子）将转移到逆协方差矩阵的行/列。这用于获得时间序列的非平稳性自回旋表示形式，以及自动回归无限表示参数与相应有限自回旋投影的参数之间的baxter型绑定。上述结果为局部固定时间序列的后续分析奠定了基础。特别是，我们表明，协方差矩阵上的平滑度转移到（i）逆协方差（ii）矢量自回旋表示的参数以及（iii）部分协方差的参数。所有结果的设置方式是使所涉及的常数仅取决于协方差矩阵的特征值，并且可以应用于具有不变特征值的高维度。
For multivariate stationary time series many important properties, such as partial correlation, graphical models and autoregressive representations are encoded in the inverse of its spectral density matrix. This is not true for nonstationary time series, where the pertinent information lies in the inverse infinite dimensional covariance matrix operator associated with the multivariate time series. This necessitates the study of the covariance of a multivariate nonstationary time series and its relationship to its inverse. We show that if the rows/columns of the infinite dimensional covariance matrix decay at a certain rate then the rate (up to a factor) transfers to the rows/columns of the inverse covariance matrix. This is used to obtain a nonstationary autoregressive representation of the time series and a Baxter-type bound between the parameters of the autoregressive infinite representation and the corresponding finite autoregressive projection. The aforementioned results lay the foundation for the subsequent analysis of locally stationary time series. In particular, we show that smoothness properties on the covariance matrix transfer to (i) the inverse covariance (ii) the parameters of the vector autoregressive representation and (iii) the partial covariances. All results are set up in such a way that the constants involved depend only on the eigenvalue of the covariance matrix and can be applied in the high-dimensional settings with non-diverging eigenvalues.
