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.
Fixed-Threshold One-Bit Toeplitz Covariance Estimation under Sparse-Ruler Sampling
School of Computer and Artificial Intelligence, Chaohu University（巢湖大学计算机与人工智能学院）；School of Electronic and Optical Engineering, Nanjing University of Science and Technology（南京理工大学电子与光学工程学院）
Robust Kronecker Product PCA for Spatio-Temporal Covariance Estimation
Kronecker PCA涉及使用空间与时间Kronecker乘积分解来估计时空协方差。在这项工作中，被认为是一个稀疏的校正因子，这对应于一个模型的协方差作为一个低（分离）秩和稀疏矩阵的克罗内克产品的总和。这种稀疏校正扩展了[Greenewald et al 2013，2014]的对角校正Kronecker PCA，以允许协方差阵中任何地方的稀疏非结构化“离群值”，例如，由不适合Kronecker模型的变量或相关性引起，或由传感器噪声或传感器故障等来源引起。我们引入了一种稳健的基于PCA的算法来估计该模型下的协方差，扩展了[Greenewald et al 2014，Tsiligkarnovel et al 2013]的重排核范数惩罚LS Kronecker PCA方法。还提供了一个扩展Toeplitz时间因素，产生一个参数减少时间平稳的测量建模。高维MSE性能界限给出了这些扩展。最后，建议的扩展KronPCA的酵母细胞周期实验的模拟和真实数据进行评估。这确立了鲁棒Kronecker PCA在生物和其他应用中的实际效用。
