Sparse Approximations of Fractional Matérn Fields

@article{Roininen2014SparseAO,
  title={Sparse Approximations of Fractional Mat{\'e}rn Fields},
  author={Lassi Roininen and Sari Lasanen and Mikko Orisp{\"a}{\"a} and Simo S{\"a}rkk{\"a}},
  journal={Scandinavian Journal of Statistics},
  year={2014},
  volume={45},
  pages={194 - 216}
}
We consider fast lattice approximation methods for a solution of a certain stochastic non‐local pseudodifferential operator equation. This equation defines a Matérn class random field. We approximate the pseudodifferential operator with truncated Taylor expansion, spectral domain error functional minimization and rounding approximations. This allows us to construct Gaussian Markov random field approximations. We construct lattice approximations with finite‐difference methods. We show that the… 
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9 Citations
Background Citations
5
Methods Citations
1

9 Citations

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New spectral transformation based methods for spectral composition method and a spectral preconditioning method for downscale the computational complexity of Gaussian process regression in the number of data points from cubic to linear are introduced.
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New spectral transformation based methods for spectral composition method and a spectral preconditioning method for downscale the computational complexity of the regression in the number of data points from cubic to linear are introduced.
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This work proposes the use of a new EKI-based framework for ERT which estimates a resistivity model and its uncertainty at a modest computational cost and highlights its readiness and applicability to similar problems in geophysics.
Gaussian Markov Random Field Priors in Ionospheric 3-D Multi-Instrument Tomography
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Gaussian Markov random fields (GMRFs) are used for constructing the prior electron density distribution and the use of GMRF introduces sparsity to the linear system, making the problem computationally feasible.
Applications of Bayesian computational statistics and modeling to large-scale geoscientific problems
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The particular geoscientific problems considered are finding the spatio-temporal distribution of atmospheric carbon dioxide based on sparse remote sensing data, quantifying uncertainties in modeling methane emissions from boreal wetlands, analyzing and quantifying the effect of climate change on growing season in the boreal region, and using statistical methods to calibrate a terrestrial ecosystem model.
Stochastic Differential Equation Methods for Spatio-Temporal Gaussian Process Regression

References

SHOWING 1-10 OF 35 REFERENCES
Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion
TLDR
This thesis proves the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and gives conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian.
Constructing continuous stationary covariances as limits of the second-order stochastic difference equations
In Bayesian statistical inverse problems the a priori probability distributions are often given as stochastic difference equations. We derive a certain class of stochastic partial difference…
Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography
We study flexible and proper smoothness priors for Bayesian statistical inverse problems by using Whittle-Matern Gaussian random fields. We review earlier results on finite-difference…
A comparison between Markov approximations and other methods for large spatial data sets
An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach
TLDR
It is shown that, using an approximate stochastic weak solution to (linear) stochastically partial differential equations, some Gaussian fields in the Matérn class can provide an explicit link, for any triangulation of , between GFs and GMRFs, formulated as a basis function representation.
Covariance Tapering for Interpolation of Large Spatial Datasets
TLDR
It is shown that tapering the correct covariance matrix with an appropriate compactly supported positive definite function reduces the computational burden significantly and still leads to an asymptotically optimal mean squared error.
Infinite-dimensional Bayesian filtering for detection of quasiperiodic phenomena in spatiotemporal data.
  • A. Solin, S. Särkkä
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2013
TLDR
A spatiotemporal resonator model and an inference method for detection and estimation of nearly periodic temporal phenomena in spatiotmporal data are introduced and applied to weather prediction and to physiological noise elimination in functional magnetic resonance imaging brain data.
Non-Gaussian statistical inverse problems. Part I: Posterior distributions
One approach to noisy inverse problems is to use Bayesian methods. In this work, the statistical inverse problem of estimating the probability distribution of an infinite-dimensional unknown given…
Discretized fractional calculus
TLDR
It is shown that the $\alpha $th power of a pth order linear multistep method gives a p third order convolution quadrature for the approximation of $I^\alpha $.
State space regularization in the nonstationary inverse problem for diffuse optical tomography
In this paper, we present a regularization method in the nonstationary inverse problem for diffuse optical tomography (DOT). The regularization is based on a choosing time evolution process such that…
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