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

Abstract

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.

Least‐squares collocation modelling of regional ionospheric TEC for accelerating real‐time single‐frequency PPP convergence

“…X ′ can be solved by (7) as: In (7), a key issue is to obtain the covariance matrix D X ′ X . The Hirvonen covariance function has been commonly applied to obtain the covariance matrix in the gravity filed [26, 31, 32]. The Hirvonen covariance function is based on the fact that the correlation between two gravity points is inversely proportional to their geographical distance.…”

Geodesy and Cartography

“…Least squares collocation is used frequently for filtering of geodetic and geophysical data or for investigating the ratio between signal and noise. Jarmołowski (2015) studied the estimation of a priori errors associated with non-correlated noise within one dataset. The author proposed a method that comprises cross-validation based on leave-one-out technique and restricted maximum likelihood estimation of a priori noise for different groups of observations.…”

Multibeam Bathymetric Data Quality Control based on Robust Least Square Collocation of Improved Multi - quadric Function

“…The least square collocation is a method that determines estimation values of the random and non-random parameters in accordance with observation data. This method was applied to study the earth form and gravity field at first [16][17]. Subsequently, the method was widely used in many fields, such as geodetic survey and interpolation problem of spatial random field in physical geography [18][19].…”