Constructing continuous stationary covariances as limits of the
      second-order stochastic difference equations

Roininen, Lassi; Piiroinen, Petteri; Lehtinen, Markku

doi:10.3934/ipi.2013.7.611

Cited by 13 publications

(16 citation statements)

References 10 publications

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“…Here we follow Roininen et al [] to build the matrix L of precision matrix

{boldL}^{T} boldL = {normalΣ}_{pr}^{- 1}

in so that the resulting GMRF covariance matrix approximates a continuous Gaussian‐shaped covariance function. We first define and parametrize the distribution we are aiming at and then show how the corresponding matrix L is formed.…”

Section: Gaussian Markov Random Field Priormentioning

confidence: 99%

“…Then, according to Roininen et al [], we can approximate the continuous spatial differential operator equations as a system of stochastic partial difference equations

({, \begin{matrix} X_{j, k} - \bar{μ} = \sqrt{\frac{α l_{1} l_{2}}{c_{0} h_{1} h_{2}}} W_{j, k}^{(0, 0)} \\ X_{j, k} - X_{j - 1, k} = \sqrt{\frac{α h_{1} l_{2}}{c_{1} l_{1} h_{2}}} W_{j, k}^{(1, 1)} \\ X_{j, k} - X_{j, k - 1} = \sqrt{\frac{α h_{1} l_{2}}{c_{1} l_{1} h_{2}}} W_{j, k}^{(1, 2)} \\ \frac{l_{1}^{2}}{h_{1}^{2}} (X_{j + 1, k} - 2 X_{j, k} + X_{j - 1, k}) + \\ \frac{l_{2}^{2}}{h_{2}^{2}} (X_{j, k + 1} - 2 X_{j, k} + X_{j, k - 1}) = \sqrt{\frac{α l_{1} l_{2}}{c_{2} h}} \end{matrix})

…”

Section: Gaussian Markov Random Field Priormentioning

confidence: 99%

“…Here we follow Roininen et al [, ] to utilize similar smoothness constraints as in Markkanen et al [] but parametrized in a more advanced manner so that the given constraints form a Gaussian Markov random field (GMRF). The GMRFs are implemented with sparse matrices providing computational efficiency, yet the approach provides a known and controllable covariance structure.…”

Section: Introductionmentioning

confidence: 99%

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Ionospheric tomography in Bayesian framework with Gaussian Markov random field priors

et al. 2015

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We present a novel ionospheric tomography reconstruction method. The method is based on Bayesian inference with the use of Gaussian Markov random field priors. We construct the priors as a system of stochastic partial differential equations. Numerical approximations of these equations can be represented with linear systems with sparse matrices, therefore providing computational efficiency. The method enables an interpretable scheme to build the prior distribution based on physical and empirical information on the structure of the ionosphere. We show through synthetic test cases in a two‐dimensional setup of latitude‐altitude slices how this method can be applied to satellite‐based ionospheric tomography and how information about the structure of the ionosphere can be implemented in the prior. The technique is capable of being easily extended to multifrequency tomographic analysis or used for the inclusion of other data sets of ionospheric electron density, such as ground‐based observations by radars or ionosondes.

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“…Here we follow Roininen et al [] to build the matrix L of precision matrix

{boldL}^{T} boldL = {normalΣ}_{pr}^{- 1}

Section: Gaussian Markov Random Field Priormentioning

confidence: 99%

“…Then, according to Roininen et al [], we can approximate the continuous spatial differential operator equations as a system of stochastic partial difference equations

({, \begin{matrix} X_{j, k} - \bar{μ} = \sqrt{\frac{α l_{1} l_{2}}{c_{0} h_{1} h_{2}}} W_{j, k}^{(0, 0)} \\ X_{j, k} - X_{j - 1, k} = \sqrt{\frac{α h_{1} l_{2}}{c_{1} l_{1} h_{2}}} W_{j, k}^{(1, 1)} \\ X_{j, k} - X_{j, k - 1} = \sqrt{\frac{α h_{1} l_{2}}{c_{1} l_{1} h_{2}}} W_{j, k}^{(1, 2)} \\ \frac{l_{1}^{2}}{h_{1}^{2}} (X_{j + 1, k} - 2 X_{j, k} + X_{j - 1, k}) + \\ \frac{l_{2}^{2}}{h_{2}^{2}} (X_{j, k + 1} - 2 X_{j, k} + X_{j, k - 1}) = \sqrt{\frac{α l_{1} l_{2}}{c_{2} h}} \end{matrix})

…”

Section: Gaussian Markov Random Field Priormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ionospheric tomography in Bayesian framework with Gaussian Markov random field priors

et al. 2015

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“…Our main motivation for studying fractional Matérn fields is in applying them as prior distributions in Bayesian statistical inverse problems (Kaipio & Somersalo, ). In our earlier studies, we have considered Gaussian Markov random fields within the framework of Bayesian statistical inverse problems (Roininen et al, ; Roininen et al, ) and applied the methodology to an electrical impedance tomography problem (Roininen et al, ). Studies of very high‐dimensional prior distributions arising from spatially sampled values of random fields in Bayesian inversion are reported by Lasanen () and Stuart ().…”

Section: Introductionmentioning

confidence: 99%

Sparse Approximations of Fractional Matérn Fields

Roininen

Lasanen

Orispää

et al. 2017

Scandinavian J Statistics

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We consider a fast approximation method for a solution of a certain stochastic non-local pseudodifferential equation. This equation defines a Matérn class random field. The approximation method is based on the spectral compactness of the solution. We approximate the pseudodifferential operator with a Taylor expansion. By truncating the expansion, we can construct an approximation with Gaussian Markov random fields. We show that the solution of the truncated version can be constructed with an over-determined system of stochastic matrix equations with sparse matrices. We solve the system of equations with a sparse Cholesky decomposition. We consider the convergence of the discrete approximation of the solution to the continuous one. Finally numerical examples are given.

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“…With the given discretization, the methodology provides inhomogeneous GMRF priors, which take into account the discretization of the unknown, hence providing a discretization-invariant reconstruction method (for references on discretization invariance, see [22]). From the practical point of view, this means that the posterior distributions and reconstructions on different computational meshes are essentially the same, given dense enough mesh.…”

Section: Gaussian Markov Random Field Priormentioning

confidence: 99%

Bayesian approach to ionospheric imaging with Gaussian Markov random field priors

Norberg¹

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Ionosphere is the partly ionised layer of Earth's atmosphere caused by solar radiation and particle precipitation. The ionisation can start from 60 km and extend up to 1000 km altitude. Often the interest in ionosphere is in the quantity and distribution of the free electrons. The electron density is related to the ionospheric refractive index and thus sufficiently high densities affect the electromagnetic waves propagating in the ionised medium. This is the reason for HF radio signals being able to reflect from the ionosphere allowing broadcast over the horizon, but also an error source in satellite positioning systems. The ionospheric electron density can be studied e.g. with specific radars and satellite in situ measurements. These instruments can provide very precise observations, however, typically only in the vicinity of the instrument. To make observations in regional and global scales, due to the volume of the domain and price of the aforementioned instruments, indirect satellite measurements and imaging methods are required. Mathematically ionospheric imaging suffers from two main complications. First, due to very sparse and limited measurement geometry between satellites and receivers, it is an ill-posed inverse problem. The measurements do not have enough information to reconstruct the electron density and thus additional information is required in some form. Second, to obtain sufficient resolution, the resulting numerical model can become computationally infeasible. In this thesis, the Bayesian statistical background for the ionospheric imaging is presented. The Bayesian approach provides a natural way to account for different sources of information with corresponding uncertainties and to update the estimated ionospheric state as new information becomes available. Most importantly, the Gaussian Markov Random Field (GMRF) priors are introduced for the application of ionospheric imaging. The GMRF approach makes the Bayesian approach computationally feasible by sparse prior precision matrices. The Bayesian method is indeed practicable and many of the widely used methods in ionospheric imaging revert back to the Bayesian approach. Unfortunately, the approach cannot escape the inherent lack of information provided by the measurement setup , and similarly to other approaches, it is highly dependent on the additional subjective information required to solve the problem. It is here shown that the use of GMRF provides a genuine improvement for the task as this subjective information can be understood and described probabilistically in a meaningful and physically interpretative way while keeping the computational costs low. Publishing unit Space Research and Observation Technologies Classification (UDC) Keywords 519.676, 551.510.413.5 List of publications This thesis consists of an introductory part and the following original publications:

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Constructing continuous stationary covariances as limits of the second-order stochastic difference equations

Cited by 13 publications

References 10 publications

Ionospheric tomography in Bayesian framework with Gaussian Markov random field priors

Ionospheric tomography in Bayesian framework with Gaussian Markov random field priors

Sparse Approximations of Fractional Matérn Fields

Bayesian approach to ionospheric imaging with Gaussian Markov random field priors

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