Precise estimation of covariance parameters in least-squares collocation by restricted maximum likelihood

Jarmołowski, Wojciech; Bakuła, Mieczysław

doi:10.1007/s11200-013-1213-z

Cited by 10 publications

(11 citation statements)

References 33 publications

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“…Therefore, a decision was taken to control the estimation of the parameters by scoring from the beginning. The mentioned problems are numerically illustrated in the previous work (Jarmołowski and Bakuła 2014), however, this example presents successful scoring, consistent with NLLF values. Nevertheless, the improvement of scoring convergence should be investigated in the future works.…”

Section: Numerical Experimentssupporting

confidence: 73%

“…4) to that from CV results (Jarmołowski 2013). Even if this is not true at all, the examples prove that small changes of these parameters can affect the estimation of δn by REML only negligibly (Jarmołowski and Bakuła 2014). Fixing of C 0 and CL enables a more clear view on δn and makes REML process more effective in the estimation of δn.…”

Section: Initial Data Assessmentmentioning

confidence: 86%

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Least squares collocation with uncorrelated heterogeneous noise estimated by restricted maximum likelihood

Jarmołowski

2015

J Geod

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The article describes the estimation of a priori error associated with heterogeneous, non-correlated noise within one dataset. The errors are estimated by restricted maximum likelihood (REML). The solution is composed of a cross-validation technique named leave-one-out (LOO) and REML estimation of a priori noise for different groups obtained by LOO. A numerical test is the main part of this case study and it presents two options. In the first one, the whole data is split into two subsets using LOO, by finding potentially outlying data. Then a priori errors are estimated in groups for the better and worse subset, where the latter includes the mentioned outlying data. The second option was to select data from the neighborhood of each point and estimate two a priori errors by REML, one for the selected point and one for the surrounding group of data. Both ideas have been validated with the use of LOO performed only in points of the better subset from the first kind of test. The use of homogeneous noise in the two example sets leads to LOO standard deviations equal 1.83 and 1.54 mGal, respectively. The group estimation generates only small improvement at the level of 0.1 mGal, which can also be reached after the removal of worse points. The pointwise REML solution, however, provides LOO standard deviations that are at least 20 % smaller than statistics from the homogeneous noise application.

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Section: Numerical Experimentssupporting

confidence: 73%

Section: Initial Data Assessmentmentioning

confidence: 86%

Least squares collocation with uncorrelated heterogeneous noise estimated by restricted maximum likelihood

Jarmołowski

2015

J Geod

Self Cite

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“…Thus, another aspect of covariance function estimation is the numerical implementation of the estimation procedure. Visual strategies versus least squares versus Maximum Likelihood, point cloud versus representative empirical covariance sequences and non robust versus robust estimators are various implementation possibilities discussed in the literature (see e.g., [11,14,18,47,51,56]). To summarize, the general challenges of covariance function fitting are to find an appropriate set of linear independent base functions, i.e., the type of covariance function, and the nonlinear nature of the set of chosen base functions together with the common problem of outlier detection and finding good initial values for the estimation process.…”

Section: Least Squares Collocationmentioning

confidence: 99%

A Generic Approach to Covariance Function Estimation Using ARMA-Models

et al. 2020

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Covariance function modeling is an essential part of stochastic methodology. Many processes in geodetic applications have rather complex, often oscillating covariance functions, where it is difficult to find corresponding analytical functions for modeling. This paper aims to give the methodological foundations for an advanced covariance modeling and elaborates a set of generic base functions which can be used for flexible covariance modeling. In particular, we provide a straightforward procedure and guidelines for a generic approach to the fitting of oscillating covariance functions to an empirical sequence of covariances. The underlying methodology is developed based on the well known properties of autoregressive processes in time series. The surprising simplicity of the proposed covariance model is that it corresponds to a finite sum of covariance functions of second-order Gauss-Markov (SOGM) processes. Furthermore, the great benefit is that the method is automated to a great extent and directly results in the appropriate model. A manual decision for a set of components is not required. Notably, the numerical method can be easily extended to ARMA-processes, which results in the same linear system of equations. Although the underlying mathematical methodology is extensively complex, the results can be obtained from a simple and straightforward numerical method.

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“…(10). Sometimes the preferred form of the method is restricted maximum likelihood (REML), but it is usually required for biased data (Searle et al, 1992;Jarmoáowski and Bakuáa, 2014). Since the data in the current test is residual after detrending, we decided to apply ML with no projection matrix in the ML equation.…”

Section: Estimation Of Covariance Parametersmentioning

confidence: 99%

“…4). We have assumed that its inaccuracy resulting from the nature of the fitting can have only a few orders smaller impact on ɷn and therefore NLLF values can properly indicate the parameters that are most efficient for LSC solution (Jarmoáowski and Bakuáa, 2014). The example plots of NLLF values for one along-track and one cross-track from the Hellas Montes data are shown in Fig.…”

Section: Numerical Test Using Along-track and Cross-track Profilesmentioning

confidence: 99%

A Study on Along-Track and Cross-Track Noise of Altimetry Data by Maximum Likelihood: Mars Orbiter Laser Altimetry (Mola) Example

Jarmołowski

Łukasiak

2015

Artificial Satellites

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The work investigates the spatial correlation of the data collected along orbital tracks of Mars Orbiter Laser Altimeter (MOLA) with a special focus on the noise variance problem in the covariance matrix. The problem of different correlation parameters in along-track and crosstrack directions of orbital or profile data is still under discussion in relation to Least Squares Collocation (LSC). Different spacing in along-track and transverse directions and anisotropy problem are frequently considered in the context of this kind of data. Therefore the problem is analyzed in this work, using MOLA data samples. The analysis in this paper is focused on a priori errors that correspond to the white noise present in the data and is performed by maximum likelihood (ML) estimation in two, perpendicular directions. Additionally, correlation lengths of assumed planar covariance model are determined by ML and by fitting it into the empirical covariance function (ECF). All estimates considered together confirm substantial influence of different data resolution in along-track and transverse directions on the covariance parameters.

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Precise estimation of covariance parameters in least-squares collocation by restricted maximum likelihood

Cited by 10 publications

References 33 publications

Least squares collocation with uncorrelated heterogeneous noise estimated by restricted maximum likelihood

Least squares collocation with uncorrelated heterogeneous noise estimated by restricted maximum likelihood

A Generic Approach to Covariance Function Estimation Using ARMA-Models

A Study on Along-Track and Cross-Track Noise of Altimetry Data by Maximum Likelihood: Mars Orbiter Laser Altimetry (Mola) Example

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