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Separable approximations of space-time covariance matrices

Marc G. Genton
2007 Environmetrics  
In this paper, we discuss separable approximations of nonseparable space-time covariance matrices.  ...  The vertical dashed line is atβ = 0.681 We have proposed a general methodology for computing separable approximations of space-time covariance matrices and illustrated that it results in small differences  ...  ACKNOWLEDGEMENTS The author thanks the editor and a referee for suggestions that improved this paper, as well as Professor Charles Van Loan for various discussions on Kronecker products.  ... 
doi:10.1002/env.854 fatcat:5hqvxpzw7jazhpnkuynn7sgrim

Low separation rank covariance estimation using Kronecker product expansions

Theodoros Tsiligkaridis, Alfred O. Hero
2013 2013 IEEE International Symposium on Information Theory  
approximation of covariance matrices [1] .  ...  For covariance matrices of low separation rank, our results establish that PRLS has significantly faster convergence than the standard sample covariance matrix (SCM) estimator.  ...  Future work will be to bound the approximation error term as a function of the factor dimensions p and q for different classes of covariance matrices.  ... 
doi:10.1109/isit.2013.6620417 dblp:conf/isit/TsiligkaridisH13 fatcat:ogioltw3nzhube7255m2h7ed4m

Kronecker Sum Decompositions of Space-Time Data [article]

Kristjan Greenewald, Theodoros Tsiligkaridis, Alfred O Hero III
2013 arXiv   pre-print
To allow a smooth tradeoff between the reduction in the number of parameters (to reduce estimation variance) and the accuracy of the covariance approximation (affecting estimation bias), we introduce a  ...  In this paper we consider the use of the space vs. time Kronecker product decomposition in the estimation of covariance matrices for spatio-temporal data.  ...  ACKNOWLEDGEMENTS This research was partially supported by ARO under grant W911NF-11-1-0391 and by AFRL under grant FA8650-07-D-1220-0006. The CMU data was obtained from mocap.cs.cmu.edu.  ... 
arXiv:1307.7306v2 fatcat:2bk7q4yfcrf3ljoxlqzpqcmneq

Covariance Estimation in High Dimensions Via Kronecker Product Expansions

Theodoros Tsiligkaridis, Alfred O. Hero
2013 IEEE Transactions on Signal Processing  
low rank approximation of covariance matrices.  ...  We show that a class of block Toeplitz covariance matrices is approximatable by low separation rank and give bounds on the minimal separation rank r that ensures a given level of bias.  ...  We show next that the class of block-Toeplitz covariance matrices have bounded approximation error if the separation rank scales like log(max(p, q)).  ... 
doi:10.1109/tsp.2013.2279355 fatcat:xnrnyh4hhne2vbz6pqs7plkyka

Detection of Anomalous Crowd Behavior Using Spatio-Temporal Multiresolution Model and Kronecker Sum Decompositions [article]

Kristjan Greenewald, Alfred Hero
2014 arXiv   pre-print
and greater applicability to spatio-temporal covariance matrices.  ...  The first method considered is the representation of the covariance as a sum of Kronecker products as in (Greenewald et al 2013), which is found to be an accurate approximation in this setting.  ...  Sum of Kronecker Products for Nonrectangular Grids While with enough terms any covariance can be represented as a sum of Kronecker products, the separation rank is significantly lower for those matrices  ... 
arXiv:1401.3291v2 fatcat:siacgmcqtffl3bn4nnwlieez4a

An algebraic approach to ground-volume decomposition from multi-baseline PolInSAR data

Stefano Tebaldini
2009 2009 IEEE International Geoscience and Remote Sensing Symposium  
This paper focuses on the analysis of multiple mechanisms of distributed scattering, as occurring in forested and vegetated areas, on the basis of multi-polarimetric and multi-baseline (MPMB) Synthetic  ...  Two basic assumptions will be retained: i) statistical independence of the different scattering mechanisms (i.e.; ground, volume, ground-trunk scattering); ii) independence of volumetric and temporal coherence  ...  In this case it has been observed that the data covariance matrix may be well approximated as the sum of just two Kronecker products (K = 2).  ... 
doi:10.1109/igarss.2009.5417936 dblp:conf/igarss/Tebaldini09 fatcat:hithmxsgu5g4rglum76avqqpli

Improving computational efficiency in large linear inverse problems: an example from carbon dioxide flux estimation

V. Yadav, A. M. Michalak
2013 Geoscientific Model Development  
The first algorithm can be used to efficiently multiply two matrices, as long as one can be expressed as a Kronecker product of two smaller matrices, a condition that is typical when multiplying a sensitivity  ...  matrix by a covariance matrix in the solution of inverse problems.  ...  This material is based upon work supported by the National Aeronautics and Space Administration under Grant No. NNX12AB90G and the National Science Foundation under Grant No. 1047871. Edited by: L.  ... 
doi:10.5194/gmd-6-583-2013 fatcat:c46rnus3q5gntoeanyanjhsr3q

Exploitation of Kronecker Structure in Gaussian Process Regression for Efficient Biomedical Signal Processing

Jannik Prüßmann, Jan Graßhoff, Philipp Rostalski
2021 Current Directions in Biomedical Engineering  
The methods treated in this work cover GP inference and hyperparameter optimization, exploiting the Kronecker structure of covariance matrices.  ...  To solve regression and source separation problems, two different approaches are presented.  ...  Due to the separability and the definition on a grid, 𝑘((𝑠, 𝑡), (𝑠 ′ , 𝑡 ′ ); θ) is a tensor product kernel and, as mentioned by Saatçi [4] , the covariance matrices can be written as a Kronecker  ... 
doi:10.1515/cdbme-2021-2073 fatcat:z26klcbhwrcepbcxzasfu7z7bm

Kronecker Product Linear Exponent AR(1) Correlation Structures for Multivariate Repeated Measures

Sean L. Simpson, Lloyd J. Edwards, Martin A. Styner, Keith E. Muller, Xi-Nian Zuo
2014 PLoS ONE  
and temporal dependence).  ...  Excellent analytic and numerical properties make the Kronecker product LEAR model a valuable addition to the suite of parsimonious correlation structures for multivariate repeated measures data.  ...  The authors thank the editor and referees for their comments that considerably improved the paper.  ... 
doi:10.1371/journal.pone.0088864 pmid:24586419 pmcid:PMC3931642 fatcat:yesasv4nzrdftddteedekrqyca

Kronecker product linear exponent AR(1) correlation structures and separability tests for multivariate repeated measures [article]

Sean L. Simpson, Lloyd J. Edwards, Martin A. Styner, Keith E. Muller
2012 arXiv   pre-print
We also provide a scientifically informed approach to assessing the adequacy of a Kronecker product LEAR model and a general unstructured Kronecker product model.  ...  Longitudinal medical imaging data of caudate morphology in schizophrenia illustrates the appeal of the Kronecker product LEAR correlation structure.  ...  A covariance matrix is if and only if it can be written separable as , where and are factor specific covariance matrices (e.g. the covariance D > H > H oe OE Thus, as expected, there is no  ... 
arXiv:1010.4471v2 fatcat:2vhkil3h2jgqppgxgsvdhy6zqa

Core Shrinkage Covariance Estimation for Matrix-variate Data [article]

Peter Hoff, Andrew McCormack, Anru R. Zhang
2022 arXiv   pre-print
We show how the set of covariance matrices may be uniquely parametrized in terms of the set of separable covariance matrices and a complementary set of "core" covariance matrices, where the core of a separable  ...  This parametrization defines a Kronecker-core decomposition of a covariance matrix.  ...  The first of these is based on an approximation of an arbitrary positive definite covariance operator by a sum of separable matrices.  ... 
arXiv:2207.12484v1 fatcat:6trgk3dmhnhini5gfcoof7q2gy

Efficient Gaussian Process Inference for Short-Scale Spatio-Temporal Modeling

Jaakko Luttinen, Alexander Ilin
2012 Journal of machine learning research  
Our model uses a sum of separable, compactly supported covariance functions, which yields a full covariance matrix represented in terms of small sparse matrices operating either on the spatial or temporal  ...  In the experimental part, the proposed algorithm is compared to the standard approach using the sparse Cholesky decomposition and it is shown to be much faster and computationally feasible for 100-1000  ...  The use of separable covariance functions have the advantage that the resulting covariance matrix is represented in terms of the Kronecker product of smaller covariance matrices acting only in the spatial  ... 
dblp:journals/jmlr/LuttinenI12 fatcat:v4w5axa7xvacdigg5ej2bmv6iy

Contents [chapter]

2018 Tensor Numerical Methods in Scientific Computing  
applications: electrostatic potential of large biomolecules | 318 5.5.6 Scattered data modeling and tensor approximation of large covariance matrices | 319 X | Contents 5.6 Tensor methods for  ...  of function related tensors | 119 3.5 Matrices in canonical and Tucker tensor formats | 124 3.5.1 Canonical and Tucker matrix (operator) formats | 124 3.5.2 The Kronecker product of matrices revisited  ... 
doi:10.1515/9783110365917-toc fatcat:lllluokyk5dobjcr5xtf2pwvbi

An Adjusted Likelihood Ratio Test for Separability in Unbalanced Multivariate Repeated Measures Data [article]

Sean L. Simpson
2011 arXiv   pre-print
We propose an adjusted likelihood ratio test of two-factor separability (Kronecker product structure) for unbalanced multivariate repeated measures data.  ...  ., temporal and spatial dimensions). However, the test can be easily generalized to factor specific matrices of any structure.  ...  The author also thanks the editor, associate editor, and referee for their comments that considerably improved the paper.  ... 
arXiv:1101.3231v2 fatcat:gw27q2ssajhq3enrueiab74s7i

Power and complex envelope correlation for modeling measured indoor MIMO channels: a beamforming evaluation

J. Wallace, H. Ozcelik, M. Herdin, E. Bonek, M. Jensen
2003 2003 IEEE 58th Vehicular Technology Conference. VTC 2003-Fall (IEEE Cat. No.03CH37484)  
Several models may result depending on how the covariance matrix is specified, i.e. power or complex envelope correlation and full or separable (Kronecker) correlation.  ...  The clustering model is able to match capacity pdfs, and resulting simulated joint beamformer spectra look more realistic than those generated with conventional separable correlation functions.  ...  either full or separable (Kronecker) covariance.  ... 
doi:10.1109/vetecf.2003.1285040 fatcat:fr7ik7itxvf2teyy63sbd2hlsa
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