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Taking correlations in GPS least squares adjustments into account with a diagonal covariance matrix / Gaël Kermarrec in Journal of geodesy, vol 90 n° 9 (September 2016)
[article]
Titre : Taking correlations in GPS least squares adjustments into account with a diagonal covariance matrix Type de document : Article/Communication Auteurs : Gaël Kermarrec, Auteur ; Steffen Schön, Auteur Année de publication : 2016 Article en page(s) : pp 793 – 805 Note générale : bibliographie Langues : Anglais (eng) Descripteur : [Termes IGN] compensation par moindres carrés
[Termes IGN] corrélation
[Termes IGN] données GPS
[Termes IGN] estimateur
[Termes IGN] matrice de covariance
[Termes IGN] matrice diagonale
[Termes IGN] pondération
[Termes IGN] positionnement différentiel
[Termes IGN] positionnement par GPS
[Termes IGN] régression
[Termes IGN] série temporelle
[Vedettes matières IGN] Traitement de données GNSSRésumé : (auteur) Based on the results of Luati and Proietti (Ann Inst Stat Math 63:673–686, 2011) on an equivalence for a certain class of polynomial regressions between the diagonally weighted least squares (DWLS) and the generalized least squares (GLS) estimator, an alternative way to take correlations into account thanks to a diagonal covariance matrix is presented. The equivalent covariance matrix is much easier to compute than a diagonalization of the covariance matrix via eigenvalue decomposition which also implies a change of the least squares equations. This condensed matrix, for use in the least squares adjustment, can be seen as a diagonal or reduced version of the original matrix, its elements being simply the sums of the rows elements of the weighting matrix. The least squares results obtained with the equivalent diagonal matrices and those given by the fully populated covariance matrix are mathematically strictly equivalent for the mean estimator in terms of estimate and its a priori cofactor matrix. It is shown that this equivalence can be empirically extended to further classes of design matrices such as those used in GPS positioning (single point positioning, precise point positioning or relative positioning with double differences). Applying this new model to simulated time series of correlated observations, a significant reduction of the coordinate differences compared with the solutions computed with the commonly used diagonal elevation-dependent model was reached for the GPS relative positioning with double differences, single point positioning as well as precise point positioning cases. The estimate differences between the equivalent and classical model with fully populated covariance matrix were below the mm for all simulated GPS cases and below the sub-mm for the relative positioning with double differences. These results were confirmed by analyzing real data. Consequently, the equivalent diagonal covariance matrices, compared with the often used elevation-dependent diagonal covariance matrix is appropriate to take correlations in GPS least squares adjustment into account, yielding more accurate cofactor matrices of the unknown. Numéro de notice : A2016-654 Affiliation des auteurs : non IGN Thématique : MATHEMATIQUE/POSITIONNEMENT Nature : Article nature-HAL : ArtAvecCL-RevueIntern DOI : 10.1007/s00190-016-0911-z En ligne : http://dx.doi.org/10.1007/s00190-016-0911-z Format de la ressource électronique : URL article Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=81856
in Journal of geodesy > vol 90 n° 9 (September 2016) . - pp 793 – 805[article]On estimation of the diagonal elements of a sparse precision matrix / Samuel Balmand in Electronic Journal of Statistics, vol 10 n° 1 (January 2016)
[article]
Titre : On estimation of the diagonal elements of a sparse precision matrix Type de document : Article/Communication Auteurs : Samuel Balmand , Auteur ; Arnak Dalalyan, Auteur Année de publication : 2016 Article en page(s) : pp 1551 - 1579 Note générale : Bibliographie Langues : Anglais (eng) Descripteur : [Vedettes matières IGN] Statistiques
[Termes IGN] calcul matriciel
[Termes IGN] estimateur
[Termes IGN] matrice creuse
[Termes IGN] matrice de covariance
[Termes IGN] matrice diagonale
[Termes IGN] méthode du maximum de vraisemblance (estimation)
[Termes IGN] régression linéaire
[Termes IGN] résiduRésumé : (Auteur) In this paper, we present several estimators of the diagonal elements of the inverse of the covariance matrix, called precision matrix, of a sample of independent and identically distributed random vectors. The main focus is on the case of high dimensional vectors having a sparse precision matrix. It is now well understood that when the underlying distribution is Gaussian, the columns of the precision matrix can be estimated independently form one another by solving linear regression problems under sparsity constraints. This approach leads to a computationally efficient strategy for estimating the precision matrix that starts by estimating the regression vectors, then estimates the diagonal entries of the precision matrix and, in a final step, combines these estimators for getting estimators of the off-diagonal entries. While the step of estimating the regression vector has been intensively studied over the past decade, the problem of deriving statistically accurate estimators of the diagonal entries has received much less attention. The goal of the present paper is to fill this gap by presenting four estimators —that seem the most natural ones— of the diagonal entries of the precision matrix and then performing a comprehensive empirical evaluation of these estimators. The estimators under consideration are the residual variance, the relaxed maximum likelihood, the symmetry-enforced maximum likelihood and the penalized maximum likelihood. We show, both theoretically and empirically, that when the aforementioned regression vectors are estimated without error, the symmetry-enforced maximum likelihood estimator has the smallest estimation error. However, in a more realistic setting when the regression vector is estimated by a sparsity-favoring computationally efficient method, the qualities of the estimators become relatively comparable with a slight advantage for the residual variance estimator. Numéro de notice : A2016--107 Affiliation des auteurs : LASTIG MATIS+Ext (2012-2019) Thématique : MATHEMATIQUE Nature : Article nature-HAL : ArtAvecCL-RevueIntern DOI : 10.1214/16-EJS1148 Date de publication en ligne : 31/05/2016 En ligne : http://dx.doi.org/10.1214/16-EJS1148 Format de la ressource électronique : URL Article Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=84707
in Electronic Journal of Statistics > vol 10 n° 1 (January 2016) . - pp 1551 - 1579[article]Documents numériques
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