Fast computation of the eigensystem of genomic similarity matrices

Abstract The computation of a similarity measure for genomic data is a standard tool in computational genetics. The principal components of such matrices are routinely used to correct for biases due to confounding by population stratification, for instance in linear regressions. However, the calcula...
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Autor*in:	Georg Hahn [verfasserIn] Sharon M. Lutz [verfasserIn] Julian Hecker [verfasserIn] Dmitry Prokopenko [verfasserIn] Michael H. Cho [verfasserIn] Edwin K. Silverman [verfasserIn] Scott T. Weiss [verfasserIn] Christoph Lange [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2024

Schlagwörter:	Covariance matrix Fast SVD Genomic relationship matrix Jaccard matrix Principal components Weighted Jaccard matrix

Übergeordnetes Werk:	In: BMC Bioinformatics - BMC, 2003, 25(2024), 1, Seite 20
Übergeordnetes Werk:	volume:25 ; year:2024 ; number:1 ; pages:20

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc

DOI / URN:	10.1186/s12859-024-05650-8

Katalog-ID:	DOAJ092204848

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520			\|a Abstract The computation of a similarity measure for genomic data is a standard tool in computational genetics. The principal components of such matrices are routinely used to correct for biases due to confounding by population stratification, for instance in linear regressions. However, the calculation of both a similarity matrix and its singular value decomposition (SVD) are computationally intensive. The contribution of this article is threefold. First, we demonstrate that the calculation of three matrices (called the covariance matrix, the weighted Jaccard matrix, and the genomic relationship matrix) can be reformulated in a unified way which allows for the application of a randomized SVD algorithm, which is faster than the traditional computation. The fast SVD algorithm we present is adapted from an existing randomized SVD algorithm and ensures that all computations are carried out in sparse matrix algebra. The algorithm only assumes that row-wise and column-wise subtraction and multiplication of a vector with a sparse matrix is available, an operation that is efficiently implemented in common sparse matrix packages. An exception is the so-called Jaccard matrix, which does not have a structure applicable for the fast SVD algorithm. Second, an approximate Jaccard matrix is introduced to which the fast SVD computation is applicable. Third, we establish guaranteed theoretical bounds on the accuracy (in $$L_2$$ L 2 norm and angle) between the principal components of the Jaccard matrix and the ones of our proposed approximation, thus putting the proposed Jaccard approximation on a solid mathematical foundation, and derive the theoretical runtime of our algorithm. We illustrate that the approximation error is low in practice and empirically verify the theoretical runtime scalings on both simulated data and data of the 1000 Genome Project.
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allfields	10.1186/s12859-024-05650-8 doi (DE-627)DOAJ092204848 (DE-599)DOAJ9e85b3a4726d4c48be715654b1c4bd13 DE-627 ger DE-627 rakwb eng R858-859.7 QH301-705.5 Georg Hahn verfasserin aut Fast computation of the eigensystem of genomic similarity matrices 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract The computation of a similarity measure for genomic data is a standard tool in computational genetics. The principal components of such matrices are routinely used to correct for biases due to confounding by population stratification, for instance in linear regressions. However, the calculation of both a similarity matrix and its singular value decomposition (SVD) are computationally intensive. The contribution of this article is threefold. First, we demonstrate that the calculation of three matrices (called the covariance matrix, the weighted Jaccard matrix, and the genomic relationship matrix) can be reformulated in a unified way which allows for the application of a randomized SVD algorithm, which is faster than the traditional computation. The fast SVD algorithm we present is adapted from an existing randomized SVD algorithm and ensures that all computations are carried out in sparse matrix algebra. The algorithm only assumes that row-wise and column-wise subtraction and multiplication of a vector with a sparse matrix is available, an operation that is efficiently implemented in common sparse matrix packages. An exception is the so-called Jaccard matrix, which does not have a structure applicable for the fast SVD algorithm. Second, an approximate Jaccard matrix is introduced to which the fast SVD computation is applicable. Third, we establish guaranteed theoretical bounds on the accuracy (in $$L_2$$ L 2 norm and angle) between the principal components of the Jaccard matrix and the ones of our proposed approximation, thus putting the proposed Jaccard approximation on a solid mathematical foundation, and derive the theoretical runtime of our algorithm. We illustrate that the approximation error is low in practice and empirically verify the theoretical runtime scalings on both simulated data and data of the 1000 Genome Project. Covariance matrix Fast SVD Genomic relationship matrix Jaccard matrix Principal components Weighted Jaccard matrix Computer applications to medicine. Medical informatics Biology (General) Sharon M. Lutz verfasserin aut Julian Hecker verfasserin aut Dmitry Prokopenko verfasserin aut Michael H. Cho verfasserin aut Edwin K. Silverman verfasserin aut Scott T. Weiss verfasserin aut Christoph Lange verfasserin aut In BMC Bioinformatics BMC, 2003 25(2024), 1, Seite 20 (DE-627)326644814 (DE-600)2041484-5 14712105 nnns volume:25 year:2024 number:1 pages:20 https://doi.org/10.1186/s12859-024-05650-8 kostenfrei https://doaj.org/article/9e85b3a4726d4c48be715654b1c4bd13 kostenfrei https://doi.org/10.1186/s12859-024-05650-8 kostenfrei https://doaj.org/toc/1471-2105 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2031 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2061 GBV_ILN_2111 GBV_ILN_2113 GBV_ILN_2190 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 25 2024 1 20
spelling	10.1186/s12859-024-05650-8 doi (DE-627)DOAJ092204848 (DE-599)DOAJ9e85b3a4726d4c48be715654b1c4bd13 DE-627 ger DE-627 rakwb eng R858-859.7 QH301-705.5 Georg Hahn verfasserin aut Fast computation of the eigensystem of genomic similarity matrices 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract The computation of a similarity measure for genomic data is a standard tool in computational genetics. The principal components of such matrices are routinely used to correct for biases due to confounding by population stratification, for instance in linear regressions. However, the calculation of both a similarity matrix and its singular value decomposition (SVD) are computationally intensive. The contribution of this article is threefold. First, we demonstrate that the calculation of three matrices (called the covariance matrix, the weighted Jaccard matrix, and the genomic relationship matrix) can be reformulated in a unified way which allows for the application of a randomized SVD algorithm, which is faster than the traditional computation. The fast SVD algorithm we present is adapted from an existing randomized SVD algorithm and ensures that all computations are carried out in sparse matrix algebra. The algorithm only assumes that row-wise and column-wise subtraction and multiplication of a vector with a sparse matrix is available, an operation that is efficiently implemented in common sparse matrix packages. An exception is the so-called Jaccard matrix, which does not have a structure applicable for the fast SVD algorithm. Second, an approximate Jaccard matrix is introduced to which the fast SVD computation is applicable. Third, we establish guaranteed theoretical bounds on the accuracy (in $$L_2$$ L 2 norm and angle) between the principal components of the Jaccard matrix and the ones of our proposed approximation, thus putting the proposed Jaccard approximation on a solid mathematical foundation, and derive the theoretical runtime of our algorithm. We illustrate that the approximation error is low in practice and empirically verify the theoretical runtime scalings on both simulated data and data of the 1000 Genome Project. Covariance matrix Fast SVD Genomic relationship matrix Jaccard matrix Principal components Weighted Jaccard matrix Computer applications to medicine. Medical informatics Biology (General) Sharon M. Lutz verfasserin aut Julian Hecker verfasserin aut Dmitry Prokopenko verfasserin aut Michael H. Cho verfasserin aut Edwin K. Silverman verfasserin aut Scott T. Weiss verfasserin aut Christoph Lange verfasserin aut In BMC Bioinformatics BMC, 2003 25(2024), 1, Seite 20 (DE-627)326644814 (DE-600)2041484-5 14712105 nnns volume:25 year:2024 number:1 pages:20 https://doi.org/10.1186/s12859-024-05650-8 kostenfrei https://doaj.org/article/9e85b3a4726d4c48be715654b1c4bd13 kostenfrei https://doi.org/10.1186/s12859-024-05650-8 kostenfrei https://doaj.org/toc/1471-2105 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2031 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2061 GBV_ILN_2111 GBV_ILN_2113 GBV_ILN_2190 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 25 2024 1 20
allfields_unstemmed	10.1186/s12859-024-05650-8 doi (DE-627)DOAJ092204848 (DE-599)DOAJ9e85b3a4726d4c48be715654b1c4bd13 DE-627 ger DE-627 rakwb eng R858-859.7 QH301-705.5 Georg Hahn verfasserin aut Fast computation of the eigensystem of genomic similarity matrices 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract The computation of a similarity measure for genomic data is a standard tool in computational genetics. The principal components of such matrices are routinely used to correct for biases due to confounding by population stratification, for instance in linear regressions. However, the calculation of both a similarity matrix and its singular value decomposition (SVD) are computationally intensive. The contribution of this article is threefold. First, we demonstrate that the calculation of three matrices (called the covariance matrix, the weighted Jaccard matrix, and the genomic relationship matrix) can be reformulated in a unified way which allows for the application of a randomized SVD algorithm, which is faster than the traditional computation. The fast SVD algorithm we present is adapted from an existing randomized SVD algorithm and ensures that all computations are carried out in sparse matrix algebra. The algorithm only assumes that row-wise and column-wise subtraction and multiplication of a vector with a sparse matrix is available, an operation that is efficiently implemented in common sparse matrix packages. An exception is the so-called Jaccard matrix, which does not have a structure applicable for the fast SVD algorithm. Second, an approximate Jaccard matrix is introduced to which the fast SVD computation is applicable. Third, we establish guaranteed theoretical bounds on the accuracy (in $$L_2$$ L 2 norm and angle) between the principal components of the Jaccard matrix and the ones of our proposed approximation, thus putting the proposed Jaccard approximation on a solid mathematical foundation, and derive the theoretical runtime of our algorithm. We illustrate that the approximation error is low in practice and empirically verify the theoretical runtime scalings on both simulated data and data of the 1000 Genome Project. Covariance matrix Fast SVD Genomic relationship matrix Jaccard matrix Principal components Weighted Jaccard matrix Computer applications to medicine. Medical informatics Biology (General) Sharon M. Lutz verfasserin aut Julian Hecker verfasserin aut Dmitry Prokopenko verfasserin aut Michael H. Cho verfasserin aut Edwin K. Silverman verfasserin aut Scott T. Weiss verfasserin aut Christoph Lange verfasserin aut In BMC Bioinformatics BMC, 2003 25(2024), 1, Seite 20 (DE-627)326644814 (DE-600)2041484-5 14712105 nnns volume:25 year:2024 number:1 pages:20 https://doi.org/10.1186/s12859-024-05650-8 kostenfrei https://doaj.org/article/9e85b3a4726d4c48be715654b1c4bd13 kostenfrei https://doi.org/10.1186/s12859-024-05650-8 kostenfrei https://doaj.org/toc/1471-2105 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2031 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2061 GBV_ILN_2111 GBV_ILN_2113 GBV_ILN_2190 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 25 2024 1 20
allfieldsGer	10.1186/s12859-024-05650-8 doi (DE-627)DOAJ092204848 (DE-599)DOAJ9e85b3a4726d4c48be715654b1c4bd13 DE-627 ger DE-627 rakwb eng R858-859.7 QH301-705.5 Georg Hahn verfasserin aut Fast computation of the eigensystem of genomic similarity matrices 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract The computation of a similarity measure for genomic data is a standard tool in computational genetics. The principal components of such matrices are routinely used to correct for biases due to confounding by population stratification, for instance in linear regressions. However, the calculation of both a similarity matrix and its singular value decomposition (SVD) are computationally intensive. The contribution of this article is threefold. First, we demonstrate that the calculation of three matrices (called the covariance matrix, the weighted Jaccard matrix, and the genomic relationship matrix) can be reformulated in a unified way which allows for the application of a randomized SVD algorithm, which is faster than the traditional computation. The fast SVD algorithm we present is adapted from an existing randomized SVD algorithm and ensures that all computations are carried out in sparse matrix algebra. The algorithm only assumes that row-wise and column-wise subtraction and multiplication of a vector with a sparse matrix is available, an operation that is efficiently implemented in common sparse matrix packages. An exception is the so-called Jaccard matrix, which does not have a structure applicable for the fast SVD algorithm. Second, an approximate Jaccard matrix is introduced to which the fast SVD computation is applicable. Third, we establish guaranteed theoretical bounds on the accuracy (in $$L_2$$ L 2 norm and angle) between the principal components of the Jaccard matrix and the ones of our proposed approximation, thus putting the proposed Jaccard approximation on a solid mathematical foundation, and derive the theoretical runtime of our algorithm. We illustrate that the approximation error is low in practice and empirically verify the theoretical runtime scalings on both simulated data and data of the 1000 Genome Project. Covariance matrix Fast SVD Genomic relationship matrix Jaccard matrix Principal components Weighted Jaccard matrix Computer applications to medicine. Medical informatics Biology (General) Sharon M. Lutz verfasserin aut Julian Hecker verfasserin aut Dmitry Prokopenko verfasserin aut Michael H. Cho verfasserin aut Edwin K. Silverman verfasserin aut Scott T. Weiss verfasserin aut Christoph Lange verfasserin aut In BMC Bioinformatics BMC, 2003 25(2024), 1, Seite 20 (DE-627)326644814 (DE-600)2041484-5 14712105 nnns volume:25 year:2024 number:1 pages:20 https://doi.org/10.1186/s12859-024-05650-8 kostenfrei https://doaj.org/article/9e85b3a4726d4c48be715654b1c4bd13 kostenfrei https://doi.org/10.1186/s12859-024-05650-8 kostenfrei https://doaj.org/toc/1471-2105 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2031 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2061 GBV_ILN_2111 GBV_ILN_2113 GBV_ILN_2190 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 25 2024 1 20
allfieldsSound	10.1186/s12859-024-05650-8 doi (DE-627)DOAJ092204848 (DE-599)DOAJ9e85b3a4726d4c48be715654b1c4bd13 DE-627 ger DE-627 rakwb eng R858-859.7 QH301-705.5 Georg Hahn verfasserin aut Fast computation of the eigensystem of genomic similarity matrices 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract The computation of a similarity measure for genomic data is a standard tool in computational genetics. The principal components of such matrices are routinely used to correct for biases due to confounding by population stratification, for instance in linear regressions. However, the calculation of both a similarity matrix and its singular value decomposition (SVD) are computationally intensive. The contribution of this article is threefold. First, we demonstrate that the calculation of three matrices (called the covariance matrix, the weighted Jaccard matrix, and the genomic relationship matrix) can be reformulated in a unified way which allows for the application of a randomized SVD algorithm, which is faster than the traditional computation. The fast SVD algorithm we present is adapted from an existing randomized SVD algorithm and ensures that all computations are carried out in sparse matrix algebra. The algorithm only assumes that row-wise and column-wise subtraction and multiplication of a vector with a sparse matrix is available, an operation that is efficiently implemented in common sparse matrix packages. An exception is the so-called Jaccard matrix, which does not have a structure applicable for the fast SVD algorithm. Second, an approximate Jaccard matrix is introduced to which the fast SVD computation is applicable. Third, we establish guaranteed theoretical bounds on the accuracy (in $$L_2$$ L 2 norm and angle) between the principal components of the Jaccard matrix and the ones of our proposed approximation, thus putting the proposed Jaccard approximation on a solid mathematical foundation, and derive the theoretical runtime of our algorithm. We illustrate that the approximation error is low in practice and empirically verify the theoretical runtime scalings on both simulated data and data of the 1000 Genome Project. Covariance matrix Fast SVD Genomic relationship matrix Jaccard matrix Principal components Weighted Jaccard matrix Computer applications to medicine. Medical informatics Biology (General) Sharon M. Lutz verfasserin aut Julian Hecker verfasserin aut Dmitry Prokopenko verfasserin aut Michael H. Cho verfasserin aut Edwin K. Silverman verfasserin aut Scott T. Weiss verfasserin aut Christoph Lange verfasserin aut In BMC Bioinformatics BMC, 2003 25(2024), 1, Seite 20 (DE-627)326644814 (DE-600)2041484-5 14712105 nnns volume:25 year:2024 number:1 pages:20 https://doi.org/10.1186/s12859-024-05650-8 kostenfrei https://doaj.org/article/9e85b3a4726d4c48be715654b1c4bd13 kostenfrei https://doi.org/10.1186/s12859-024-05650-8 kostenfrei https://doaj.org/toc/1471-2105 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2031 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2061 GBV_ILN_2111 GBV_ILN_2113 GBV_ILN_2190 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 25 2024 1 20
language	English
source	In BMC Bioinformatics 25(2024), 1, Seite 20 volume:25 year:2024 number:1 pages:20
sourceStr	In BMC Bioinformatics 25(2024), 1, Seite 20 volume:25 year:2024 number:1 pages:20
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Covariance matrix Fast SVD Genomic relationship matrix Jaccard matrix Principal components Weighted Jaccard matrix Computer applications to medicine. Medical informatics Biology (General)
isfreeaccess_bool	true
container_title	BMC Bioinformatics
authorswithroles_txt_mv	Georg Hahn @@aut@@ Sharon M. Lutz @@aut@@ Julian Hecker @@aut@@ Dmitry Prokopenko @@aut@@ Michael H. Cho @@aut@@ Edwin K. Silverman @@aut@@ Scott T. Weiss @@aut@@ Christoph Lange @@aut@@
publishDateDaySort_date	2024-01-01T00:00:00Z
hierarchy_top_id	326644814
id	DOAJ092204848
language_de	englisch
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callnumber-first	R - Medicine
author	Georg Hahn
spellingShingle	Georg Hahn misc R858-859.7 misc QH301-705.5 misc Covariance matrix misc Fast SVD misc Genomic relationship matrix misc Jaccard matrix misc Principal components misc Weighted Jaccard matrix misc Computer applications to medicine. Medical informatics misc Biology (General) Fast computation of the eigensystem of genomic similarity matrices
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topic_title	R858-859.7 QH301-705.5 Fast computation of the eigensystem of genomic similarity matrices Covariance matrix Fast SVD Genomic relationship matrix Jaccard matrix Principal components Weighted Jaccard matrix
topic	misc R858-859.7 misc QH301-705.5 misc Covariance matrix misc Fast SVD misc Genomic relationship matrix misc Jaccard matrix misc Principal components misc Weighted Jaccard matrix misc Computer applications to medicine. Medical informatics misc Biology (General)
topic_unstemmed	misc R858-859.7 misc QH301-705.5 misc Covariance matrix misc Fast SVD misc Genomic relationship matrix misc Jaccard matrix misc Principal components misc Weighted Jaccard matrix misc Computer applications to medicine. Medical informatics misc Biology (General)
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title	Fast computation of the eigensystem of genomic similarity matrices
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title_full	Fast computation of the eigensystem of genomic similarity matrices
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title_sort	fast computation of the eigensystem of genomic similarity matrices
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title_auth	Fast computation of the eigensystem of genomic similarity matrices
abstract	Abstract The computation of a similarity measure for genomic data is a standard tool in computational genetics. The principal components of such matrices are routinely used to correct for biases due to confounding by population stratification, for instance in linear regressions. However, the calculation of both a similarity matrix and its singular value decomposition (SVD) are computationally intensive. The contribution of this article is threefold. First, we demonstrate that the calculation of three matrices (called the covariance matrix, the weighted Jaccard matrix, and the genomic relationship matrix) can be reformulated in a unified way which allows for the application of a randomized SVD algorithm, which is faster than the traditional computation. The fast SVD algorithm we present is adapted from an existing randomized SVD algorithm and ensures that all computations are carried out in sparse matrix algebra. The algorithm only assumes that row-wise and column-wise subtraction and multiplication of a vector with a sparse matrix is available, an operation that is efficiently implemented in common sparse matrix packages. An exception is the so-called Jaccard matrix, which does not have a structure applicable for the fast SVD algorithm. Second, an approximate Jaccard matrix is introduced to which the fast SVD computation is applicable. Third, we establish guaranteed theoretical bounds on the accuracy (in $$L_2$$ L 2 norm and angle) between the principal components of the Jaccard matrix and the ones of our proposed approximation, thus putting the proposed Jaccard approximation on a solid mathematical foundation, and derive the theoretical runtime of our algorithm. We illustrate that the approximation error is low in practice and empirically verify the theoretical runtime scalings on both simulated data and data of the 1000 Genome Project.
abstractGer	Abstract The computation of a similarity measure for genomic data is a standard tool in computational genetics. The principal components of such matrices are routinely used to correct for biases due to confounding by population stratification, for instance in linear regressions. However, the calculation of both a similarity matrix and its singular value decomposition (SVD) are computationally intensive. The contribution of this article is threefold. First, we demonstrate that the calculation of three matrices (called the covariance matrix, the weighted Jaccard matrix, and the genomic relationship matrix) can be reformulated in a unified way which allows for the application of a randomized SVD algorithm, which is faster than the traditional computation. The fast SVD algorithm we present is adapted from an existing randomized SVD algorithm and ensures that all computations are carried out in sparse matrix algebra. The algorithm only assumes that row-wise and column-wise subtraction and multiplication of a vector with a sparse matrix is available, an operation that is efficiently implemented in common sparse matrix packages. An exception is the so-called Jaccard matrix, which does not have a structure applicable for the fast SVD algorithm. Second, an approximate Jaccard matrix is introduced to which the fast SVD computation is applicable. Third, we establish guaranteed theoretical bounds on the accuracy (in $$L_2$$ L 2 norm and angle) between the principal components of the Jaccard matrix and the ones of our proposed approximation, thus putting the proposed Jaccard approximation on a solid mathematical foundation, and derive the theoretical runtime of our algorithm. We illustrate that the approximation error is low in practice and empirically verify the theoretical runtime scalings on both simulated data and data of the 1000 Genome Project.
abstract_unstemmed	Abstract The computation of a similarity measure for genomic data is a standard tool in computational genetics. The principal components of such matrices are routinely used to correct for biases due to confounding by population stratification, for instance in linear regressions. However, the calculation of both a similarity matrix and its singular value decomposition (SVD) are computationally intensive. The contribution of this article is threefold. First, we demonstrate that the calculation of three matrices (called the covariance matrix, the weighted Jaccard matrix, and the genomic relationship matrix) can be reformulated in a unified way which allows for the application of a randomized SVD algorithm, which is faster than the traditional computation. The fast SVD algorithm we present is adapted from an existing randomized SVD algorithm and ensures that all computations are carried out in sparse matrix algebra. The algorithm only assumes that row-wise and column-wise subtraction and multiplication of a vector with a sparse matrix is available, an operation that is efficiently implemented in common sparse matrix packages. An exception is the so-called Jaccard matrix, which does not have a structure applicable for the fast SVD algorithm. Second, an approximate Jaccard matrix is introduced to which the fast SVD computation is applicable. Third, we establish guaranteed theoretical bounds on the accuracy (in $$L_2$$ L 2 norm and angle) between the principal components of the Jaccard matrix and the ones of our proposed approximation, thus putting the proposed Jaccard approximation on a solid mathematical foundation, and derive the theoretical runtime of our algorithm. We illustrate that the approximation error is low in practice and empirically verify the theoretical runtime scalings on both simulated data and data of the 1000 Genome Project.
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title_short	Fast computation of the eigensystem of genomic similarity matrices
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Fast computation of the eigensystem of genomic similarity matrices

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