{"title":"A new approach of multi-dimensional correlation as a separability measure of multiple outliers in GNSS applications","authors":"A. Almagbile","doi":"10.1515/jag-2023-0106","DOIUrl":null,"url":null,"abstract":"\n <jats:p>Detecting and identifying outliers/failures in GNSS measurements has garnered significant attention among researchers aiming to enhance the quality of GNSS positioning and navigation. This study delves into the analysis of the separability of multiple outliers when single, double, and triple outliers occur in single-point positioning (SPP) measurements. To achieve this, a novel method includes introducing a multi-dimensional correlation coefficient among test statistics. This coefficient functions as a measure of outliers separability and, in turn, assesses the possible impact of outliers on other measurements. This multi-dimensional correlation approach is based on a nested correlation (<jats:inline-formula id=\"j_jag-2023-0106_ineq_001\">\n <jats:alternatives>\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n <m:msubsup>\n <m:mrow>\n <m:mi>ρ</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mtext>nested</m:mtext>\n </m:mrow>\n <m:mrow>\n <m:mi>θ</m:mi>\n <m:mo>,</m:mo>\n <m:mi>φ</m:mi>\n </m:mrow>\n </m:msubsup>\n </m:math>\n <jats:tex-math>${\\rho }_{\\text{nested}}^{\\theta ,\\varphi }$</jats:tex-math>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jag-2023-0106_ineq_001.png\" />\n </jats:alternatives>\n </jats:inline-formula>) that explains the variations in test statistic values with and without common measurements in two pairs/combinations. The performance of the <jats:inline-formula id=\"j_jag-2023-0106_ineq_002\">\n <jats:alternatives>\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n <m:msubsup>\n <m:mrow>\n <m:mi>ρ</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mtext>nested</m:mtext>\n </m:mrow>\n <m:mrow>\n <m:mi>θ</m:mi>\n <m:mo>,</m:mo>\n <m:mi>φ</m:mi>\n </m:mrow>\n </m:msubsup>\n </m:math>\n <jats:tex-math>${\\rho }_{\\text{nested}}^{\\theta ,\\varphi }$</jats:tex-math>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jag-2023-0106_ineq_002.png\" />\n </jats:alternatives>\n </jats:inline-formula> is then compared with other two existing methods of multi-dimensional correlation namely the maximum (<jats:inline-formula id=\"j_jag-2023-0106_ineq_003\">\n <jats:alternatives>\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n <m:msubsup>\n <m:mrow>\n <m:mi>ρ</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>max</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>θ</m:mi>\n <m:mo>,</m:mo>\n <m:mi>φ</m:mi>\n </m:mrow>\n </m:msubsup>\n </m:math>\n <jats:tex-math>${\\rho }_{\\mathrm{max}}^{\\theta ,\\varphi }$</jats:tex-math>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jag-2023-0106_ineq_003.png\" />\n </jats:alternatives>\n </jats:inline-formula>) and global (<jats:inline-formula id=\"j_jag-2023-0106_ineq_004\">\n <jats:alternatives>\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n <m:msubsup>\n <m:mrow>\n <m:mi>ρ</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mtext>Global</m:mtext>\n </m:mrow>\n <m:mrow>\n <m:mi>θ</m:mi>\n <m:mo>,</m:mo>\n <m:mi>φ</m:mi>\n </m:mrow>\n </m:msubsup>\n </m:math>\n <jats:tex-math>${\\rho }_{\\text{Global}}^{\\theta ,\\varphi }$</jats:tex-math>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jag-2023-0106_ineq_004.png\" />\n </jats:alternatives>\n </jats:inline-formula>) correlation. The results show that under the presence of two outliers and with and without common measurements in two pairs, the <jats:inline-formula id=\"j_jag-2023-0106_ineq_005\">\n <jats:alternatives>\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n <m:msubsup>\n <m:mrow>\n <m:mi>ρ</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mtext>nested</m:mtext>\n </m:mrow>\n <m:mrow>\n <m:mi>θ</m:mi>\n </m:mrow>\n </m:msubsup>\n </m:math>\n <jats:tex-math>${\\rho }_{\\text{nested}}^{\\theta }$</jats:tex-math>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jag-2023-0106_ineq_005.png\" />\n </jats:alternatives>\n </jats:inline-formula> outperforms the, <jats:inline-formula id=\"j_jag-2023-0106_ineq_006\">\n <jats:alternatives>\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n <m:msubsup>\n <m:mrow>\n <m:mi>ρ</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>max</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>θ</m:mi>\n </m:mrow>\n </m:msubsup>\n </m:math>\n <jats:tex-math>${\\rho }_{\\mathrm{max}}^{\\theta }$</jats:tex-math>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jag-2023-0106_ineq_006.png\" />\n </jats:alternatives>\n </jats:inline-formula> and <jats:inline-formula id=\"j_jag-2023-0106_ineq_007\">\n <jats:alternatives>\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n <m:msubsup>\n <m:mrow>\n <m:mi>ρ</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mtext>Global</m:mtext>\n </m:mrow>\n <m:mrow>\n <m:mi>θ</m:mi>\n </m:mrow>\n </m:msubsup>\n </m:math>\n <jats:tex-math>${\\rho }_{\\text{Global}}^{\\theta }$</jats:tex-math>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jag-2023-0106_ineq_007.png\" />\n </jats:alternatives>\n </jats:inline-formula> exhibiting a determination coefficient (<jats:italic>R</jats:italic>\n <jats:sup>2</jats:sup>) value of approximately 0.95 and 0.62 respectively. The results furthermore reveal that for three outliers and with one, two, and noncommon measurements intersecting between two combinations, the values of <jats:italic>R</jats:italic>\n <jats:sup>2</jats:sup> are 0.62, 0.96, and 0.34. respectively. This means that the <jats:inline-formula id=\"j_jag-2023-0106_ineq_008\">\n <jats:alternatives>\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n <m:msubsup>\n <m:mrow>\n <m:mi>ρ</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mtext>nested</m:mtext>\n </m:mrow>\n <m:mrow>\n <m:mi>θ</m:mi>\n <m:mo>,</m:mo>\n <m:mi>φ</m:mi>\n </m:mrow>\n </m:msubsup>\n </m:math>\n <jats:tex-math>${\\rho }_{\\text{nested}}^{\\theta ,\\varphi }$</jats:tex-math>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":"345 10","pages":""},"PeriodicalIF":16.4000,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/jag-2023-0106","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
Detecting and identifying outliers/failures in GNSS measurements has garnered significant attention among researchers aiming to enhance the quality of GNSS positioning and navigation. This study delves into the analysis of the separability of multiple outliers when single, double, and triple outliers occur in single-point positioning (SPP) measurements. To achieve this, a novel method includes introducing a multi-dimensional correlation coefficient among test statistics. This coefficient functions as a measure of outliers separability and, in turn, assesses the possible impact of outliers on other measurements. This multi-dimensional correlation approach is based on a nested correlation (ρnestedθ,φ${\rho }_{\text{nested}}^{\theta ,\varphi }$) that explains the variations in test statistic values with and without common measurements in two pairs/combinations. The performance of the ρnestedθ,φ${\rho }_{\text{nested}}^{\theta ,\varphi }$ is then compared with other two existing methods of multi-dimensional correlation namely the maximum (ρmaxθ,φ${\rho }_{\mathrm{max}}^{\theta ,\varphi }$) and global (ρGlobalθ,φ${\rho }_{\text{Global}}^{\theta ,\varphi }$) correlation. The results show that under the presence of two outliers and with and without common measurements in two pairs, the ρnestedθ${\rho }_{\text{nested}}^{\theta }$ outperforms the, ρmaxθ${\rho }_{\mathrm{max}}^{\theta }$ and ρGlobalθ${\rho }_{\text{Global}}^{\theta }$ exhibiting a determination coefficient (R2) value of approximately 0.95 and 0.62 respectively. The results furthermore reveal that for three outliers and with one, two, and noncommon measurements intersecting between two combinations, the values of R2 are 0.62, 0.96, and 0.34. respectively. This means that the ρnestedθ,φ${\rho }_{\text{nested}}^{\theta ,\varphi }$
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
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