论文标题
比率一致估计的远程依赖性toeplitz协方差与矩阵数据的应用
Ratio-consistent estimation for long range dependent Toeplitz covariance with application to matrix data whitening
论文作者
论文摘要
我们考虑来自具有可分离的协方差函数的数据矩阵$ x：= c_n^{1/2} zr_m^{1/2} $，具有可分离的协方差函数，其中$ c_n $是$ n \ times n \ times n $ n $ n $ semi-definite semi-definite matrix，$ z $ z $ z $ n \ a $ n \ a $ a $ n $ n $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ y mmix and $ mmix and $ mmix and $ mmix mm y m。 m $ toeplitz矩阵。在长期依赖性（LRD）的假设下，我们重新检查了两个toeplitzififized估计量的一致性$ \ hat r_m $（无偏见）和$ \ hat r_m^b $（偏见）$ r_m $的$ r_m $，这些$ r_m $与$ r_m $相处于$ r_m $时，在$ r_m $中是$ r_m $的短范围依赖范围（s s srd）。但是，在LRD情况下，一些模拟表明，两个估计器的规范一致性一般都不达到。取而代之的是，为无偏的估算器$ \ hat r_m $建立了较弱的{\ it比率一致性}，并为有偏见的估计值$ \ hat r_m^b $建立了进一步的弱{\ IT比率lsd一致性}。主要结果导致原始数据矩阵$ x $的一致美白过程，该过程进一步应用于两个现实世界问题，一个是一个信号检测问题，另一个是太空协方差$ c_n $上的PCA，以实现降低降噪和数据压缩。
We consider a data matrix $X:=C_N^{1/2}ZR_M^{1/2}$ from a multivariate stationary process with a separable covariance function, where $C_N$ is a $N\times N$ positive semi-definite matrix, $Z$ a $N\times M$ random matrix of uncorrelated standardized white noise, and $R_M$ a $M\times M$ Toeplitz matrix. Under the assumption of long range dependence (LRD), we re-examine the consistency of two toeplitzifized estimators $\hat R_M$ (unbiased) and $\hat R_M^b$ (biased) for $R_M$, which are known to be norm consistent with $R_M$ when the process is short range dependent (SRD). However in the LRD case, some simulations suggest that the norm consistency does not hold in general for both estimators. Instead, a weaker {\it ratio consistency} is established for the unbiased estimator $\hat R_M$, and a further weaker {\it ratio LSD consistency} is established for the biased estimator $\hat R_M^b$. The main result leads to a consistent whitening procedure on the original data matrix $X$, which is further applied to two real world questions, one is a signal detection problem, and the other is PCA on the space covariance $C_N$ to achieve a noise reduction and data compression.
