Hermitian正定矩阵曲线的内禀小波回归
Intrinsic wavelet regression for curves of Hermitian positive definite matrices
摘要：
介绍了Hermitian正定矩阵的非欧空间曲线的内禀小波变换和小波估计方法，并将其应用于多元平稳时间序列的Fourier谱估计。重点研究了具有仿射不变黎曼度量的正定矩阵空间中的内禀平均插值小波变换，并对Hermitian正定矩阵的内禀光滑曲线导出了线性小波阈值的收敛速度.在多变量傅立叶谱估计中，当时间序列的基发生变化时，固有小波阈值是等变的，而非线性小波阈值能够捕捉到谱密度矩阵在频率上的局部特征，始终保证正定估计。有限样本性能的内在小波阈值评估通过模拟数据，并比较几个基准估计在黎曼流形。通过检查在学习实验期间记录的试验复制的脑信号时间序列的多变量谱，提供了进一步的说明。
英文摘要：
Intrinsic wavelet transforms and wavelet estimation methods are introduced for curves in the non-Euclidean space of Hermitian positive definite matrices, with in mind the application to Fourier spectral estimation of multivariate stationary time series. The main focus is on intrinsic average-interpolation wavelet transforms in the space of positive definite matrices equipped with an affine-invariant Riemannian metric, and convergence rates of linear wavelet thresholding are derived for intrinsically smooth curves of Hermitian positive definite matrices. In the context of multivariate Fourier spectral estimation, intrinsic wavelet thresholding is equivariant under a change of basis of the time series, and nonlinear wavelet thresholding is able to capture localized features in the spectral density matrix across frequency, always guaranteeing positive definite estimates. The finite-sample performance of intrinsic wavelet thresholding is assessed by means of simulated data and compared to several benchmark estimators in the Riemannian manifold. Further illustrations are provided by examining the multivariate spectra of trial-replicated brain signal time series recorded during a learning experiment.
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