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linear algebra – Spectra of spatial and temporal covariance matrices Answer

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linear algebra – Spectra of spatial and temporal covariance matrices

@Carlo’s respond could be very insightful from a physics perspective, thanks quite a bit for the instructing and erudition right here. My respond is from a extra ML and statistic perspective as @Ed Smith requested.

But within the OP, I cerebrate the formulation isn’t correct, your spatial and temporal
covariance matrices would not have to participate the identical clique of eccentric values, noticing that the $B_{st} = sum_{i} x_i(s)x_i(t)$ entails too spatial indices $s$ whereas $A_{ij}$ relies upon solely on temporal indices $t$.

Shared eigenvalues and shared eigenvectors/eigenspaces will not be the identical, the latter is a mighty stronger function in common.

(1) Shared eigenvalues
As identified by @Carlo, “when the empirical spatial and temporal covariance matrices share the same positive eigenvalues, then you know that your data is self-averaging in both space and time.” In addition, if shared eccentric values are detected, the joint inference of spatial temporal succession energy breathe workable (generally known as “Spatial-Temporal Spectrum Sensing” [Do et.al.], the place a Fourier foundation is used). Another necessary sample is that, since your eigenvalues are the identical, so would the ordering of eigenvalues. This extra issue could breathe helpful in detecting change-point by way of SVD when evolutionary dynamics exists [Townsend&Gong]. However, as a consequence of numerical points for sizable matrices or outliers in observations, typically optimistic eigenvalues (since we’re regarding covariance matrices) could breathe very immediate to zero; then shared eigenvalues to rectify numerical or detect outliers.

(2) Shared eigenspaces A extra necessary function could breathe the eigenspaces spatial and temporal covariance spans, for the reason that eigenvectors delineate the spatial and the temporal modes [Greenewald&Hero], shared eigenspaces could permit borrowing info from spatial information to appraise temporal parameters and vice versa. Besides, we could discover a good foundation spanned by eigenvectors (or chosen spectrum foundation) that approximates each spatial and temporal dependence properly. This is in keeping with Carlo’s respond above.

If there are shared eigenvectors for each matrices, one other unaffected thought would breathe to assemble effectual approximations when the mannequin is fitted to sizable datasets love spatial datasets [Genton]. Effective approximations of inescapable fashions love Gaussian processes and Gaussian random fields are of central necessary in machine erudition and spatial statistics [Bauer et.al.], particularly for the situations that require environment friendly suits to sizable datasets. When you employ approximation methods for SVD in sizable spatial-temporal matrices, shared spatial-temporal spectra could permit a greater approximation [Bogaardt et.al.]. It too permits exquisite filtering in picture processing (apart from arresting processing).

Reference
(by date, not essentially previous 7 years.)

[Genton] Separable approximations of space-time covariance matrices, 2007.

[Do et.al.] Joint Spatial-Temporal Spectrum Sensing for Cognitive Radio Networks, 2010.

[Greenewald&Hero]Robust Kronecker Product PCA for Spatio-Temporal Covariance Estimation, 2015.

[Bauer et.al.]Understanding Probabilistic Sparse Gaussian Process Approximations, 2016.

[Baranger et.al.] Adaptive Spatiotemporal SVD Clutter Filtering for Ultrafast Doppler Imaging Using Similarity of Spatial Singular Vectors, 2018.

[Townsend&Gong] Detection and evaluation of spatiotemporal patterns in mind exercise, 2018.

[Bogaardt et.al.] Dataset Reduction Techniques to Speed Up SVD Analyses on Big Geo-Datasets, 2019.

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