Inversion of a part of the numerator relationship matrix using pedigree information

Background In recent theoretical developments, the information available (e.g. genotypes) divides the original population into two groups: animals with this information (selected animals) and animals without this information (excluded animals). These developments require inversion of the part of the pedigree-based numerator relationship matrix that describes the genetic covariance between selected animals (A22). Our main objective was to propose and evaluate methodology that takes advantage of any potential sparsity in the inverse of A22 in order to reduce the computing time required for its inversion. This potential sparsity is brought out by searching the pedigree for dependencies between the selected animals. Jointly, we expected distant ancestors to provide relationship ties that increase the density of matrix A22 but that their effect on might be minor. This hypothesis was also tested. Methods The inverse of A22 can be computed from the inverse of the triangular factor (T-1) obtained by Cholesky root-free decomposition of A22. We propose an algorithm that sets up the sparsity pattern of T-1 using pedigree information. This algorithm provides positions of the elements of T-1 worth to be computed (i.e. different from zero). A recursive computation of is then achieved with or without information on the sparsity pattern and time required for each computation was recorded. For three numbers of selected animals (4000; 8000 and 12 000), A22 was computed using different pedigree extractions and the closeness of the resulting to the inverse computed using the fully extracted pedigree was measured by an appropriate norm. Results The use of prior information on the sparsity of T-1 decreased the computing time for inversion by a factor of 1.73 on average. Computational issues and practical uses of the different algorithms were discussed. Cases involving more than 12 000 selected animals were considered. Inclusion of 10 generations was determined to be sufficient when computing A22. Conclusions Depending on the size and structure of the selected sub-population, gains in time to compute are possible and these gains may increase as the number of selected animals increases. Given the sequential nature of most computational steps, the proposed algorithm can benefit from optimization and may be convenient for genomic evaluations. Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Faux, Pierre [verfasserIn] Gengler, Nicolas

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2013

Schlagwörter:	Genomic Prediction Inversion Algorithm Sparsity Pattern Sparse Vector Genomic Relationship Matrix

Anmerkung:	© Faux and Gengler; licensee BioMed Central Ltd. 2013

Übergeordnetes Werk:	Enthalten in: Genetics, selection, evolution - London : BioMed Central, 1989, 45(2013), 1 vom: 06. Dez.
Übergeordnetes Werk:	volume:45 ; year:2013 ; number:1 ; day:06 ; month:12

Links:	Volltext

DOI / URN:	10.1186/1297-9686-45-45

Katalog-ID:	SPR026807386

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520			\|a Background In recent theoretical developments, the information available (e.g. genotypes) divides the original population into two groups: animals with this information (selected animals) and animals without this information (excluded animals). These developments require inversion of the part of the pedigree-based numerator relationship matrix that describes the genetic covariance between selected animals (A22). Our main objective was to propose and evaluate methodology that takes advantage of any potential sparsity in the inverse of A22 in order to reduce the computing time required for its inversion. This potential sparsity is brought out by searching the pedigree for dependencies between the selected animals. Jointly, we expected distant ancestors to provide relationship ties that increase the density of matrix A22 but that their effect on might be minor. This hypothesis was also tested. Methods The inverse of A22 can be computed from the inverse of the triangular factor (T-1) obtained by Cholesky root-free decomposition of A22. We propose an algorithm that sets up the sparsity pattern of T-1 using pedigree information. This algorithm provides positions of the elements of T-1 worth to be computed (i.e. different from zero). A recursive computation of is then achieved with or without information on the sparsity pattern and time required for each computation was recorded. For three numbers of selected animals (4000; 8000 and 12 000), A22 was computed using different pedigree extractions and the closeness of the resulting to the inverse computed using the fully extracted pedigree was measured by an appropriate norm. Results The use of prior information on the sparsity of T-1 decreased the computing time for inversion by a factor of 1.73 on average. Computational issues and practical uses of the different algorithms were discussed. Cases involving more than 12 000 selected animals were considered. Inclusion of 10 generations was determined to be sufficient when computing A22. Conclusions Depending on the size and structure of the selected sub-population, gains in time to compute are possible and these gains may increase as the number of selected animals increases. Given the sequential nature of most computational steps, the proposed algorithm can benefit from optimization and may be convenient for genomic evaluations.
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allfields	10.1186/1297-9686-45-45 doi (DE-627)SPR026807386 (SPR)1297-9686-45-45-e DE-627 ger DE-627 rakwb eng Faux, Pierre verfasserin aut Inversion of a part of the numerator relationship matrix using pedigree information 2013 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Faux and Gengler; licensee BioMed Central Ltd. 2013 Background In recent theoretical developments, the information available (e.g. genotypes) divides the original population into two groups: animals with this information (selected animals) and animals without this information (excluded animals). These developments require inversion of the part of the pedigree-based numerator relationship matrix that describes the genetic covariance between selected animals (A22). Our main objective was to propose and evaluate methodology that takes advantage of any potential sparsity in the inverse of A22 in order to reduce the computing time required for its inversion. This potential sparsity is brought out by searching the pedigree for dependencies between the selected animals. Jointly, we expected distant ancestors to provide relationship ties that increase the density of matrix A22 but that their effect on might be minor. This hypothesis was also tested. Methods The inverse of A22 can be computed from the inverse of the triangular factor (T-1) obtained by Cholesky root-free decomposition of A22. We propose an algorithm that sets up the sparsity pattern of T-1 using pedigree information. This algorithm provides positions of the elements of T-1 worth to be computed (i.e. different from zero). A recursive computation of is then achieved with or without information on the sparsity pattern and time required for each computation was recorded. For three numbers of selected animals (4000; 8000 and 12 000), A22 was computed using different pedigree extractions and the closeness of the resulting to the inverse computed using the fully extracted pedigree was measured by an appropriate norm. Results The use of prior information on the sparsity of T-1 decreased the computing time for inversion by a factor of 1.73 on average. Computational issues and practical uses of the different algorithms were discussed. Cases involving more than 12 000 selected animals were considered. Inclusion of 10 generations was determined to be sufficient when computing A22. Conclusions Depending on the size and structure of the selected sub-population, gains in time to compute are possible and these gains may increase as the number of selected animals increases. Given the sequential nature of most computational steps, the proposed algorithm can benefit from optimization and may be convenient for genomic evaluations. Genomic Prediction (dpeaa)DE-He213 Inversion Algorithm (dpeaa)DE-He213 Sparsity Pattern (dpeaa)DE-He213 Sparse Vector (dpeaa)DE-He213 Genomic Relationship Matrix (dpeaa)DE-He213 Gengler, Nicolas aut Enthalten in Genetics, selection, evolution London : BioMed Central, 1989 45(2013), 1 vom: 06. Dez. (DE-627)312849052 (DE-600)2012369-3 1297-9686 nnns volume:45 year:2013 number:1 day:06 month:12 https://dx.doi.org/10.1186/1297-9686-45-45 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA 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_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2003 GBV_ILN_2014 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_4338 GBV_ILN_4367 GBV_ILN_4700 AR 45 2013 1 06 12
spelling	10.1186/1297-9686-45-45 doi (DE-627)SPR026807386 (SPR)1297-9686-45-45-e DE-627 ger DE-627 rakwb eng Faux, Pierre verfasserin aut Inversion of a part of the numerator relationship matrix using pedigree information 2013 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Faux and Gengler; licensee BioMed Central Ltd. 2013 Background In recent theoretical developments, the information available (e.g. genotypes) divides the original population into two groups: animals with this information (selected animals) and animals without this information (excluded animals). These developments require inversion of the part of the pedigree-based numerator relationship matrix that describes the genetic covariance between selected animals (A22). Our main objective was to propose and evaluate methodology that takes advantage of any potential sparsity in the inverse of A22 in order to reduce the computing time required for its inversion. This potential sparsity is brought out by searching the pedigree for dependencies between the selected animals. Jointly, we expected distant ancestors to provide relationship ties that increase the density of matrix A22 but that their effect on might be minor. This hypothesis was also tested. Methods The inverse of A22 can be computed from the inverse of the triangular factor (T-1) obtained by Cholesky root-free decomposition of A22. We propose an algorithm that sets up the sparsity pattern of T-1 using pedigree information. This algorithm provides positions of the elements of T-1 worth to be computed (i.e. different from zero). A recursive computation of is then achieved with or without information on the sparsity pattern and time required for each computation was recorded. For three numbers of selected animals (4000; 8000 and 12 000), A22 was computed using different pedigree extractions and the closeness of the resulting to the inverse computed using the fully extracted pedigree was measured by an appropriate norm. Results The use of prior information on the sparsity of T-1 decreased the computing time for inversion by a factor of 1.73 on average. Computational issues and practical uses of the different algorithms were discussed. Cases involving more than 12 000 selected animals were considered. Inclusion of 10 generations was determined to be sufficient when computing A22. Conclusions Depending on the size and structure of the selected sub-population, gains in time to compute are possible and these gains may increase as the number of selected animals increases. Given the sequential nature of most computational steps, the proposed algorithm can benefit from optimization and may be convenient for genomic evaluations. Genomic Prediction (dpeaa)DE-He213 Inversion Algorithm (dpeaa)DE-He213 Sparsity Pattern (dpeaa)DE-He213 Sparse Vector (dpeaa)DE-He213 Genomic Relationship Matrix (dpeaa)DE-He213 Gengler, Nicolas aut Enthalten in Genetics, selection, evolution London : BioMed Central, 1989 45(2013), 1 vom: 06. Dez. (DE-627)312849052 (DE-600)2012369-3 1297-9686 nnns volume:45 year:2013 number:1 day:06 month:12 https://dx.doi.org/10.1186/1297-9686-45-45 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA 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_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2003 GBV_ILN_2014 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_4338 GBV_ILN_4367 GBV_ILN_4700 AR 45 2013 1 06 12
allfields_unstemmed	10.1186/1297-9686-45-45 doi (DE-627)SPR026807386 (SPR)1297-9686-45-45-e DE-627 ger DE-627 rakwb eng Faux, Pierre verfasserin aut Inversion of a part of the numerator relationship matrix using pedigree information 2013 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Faux and Gengler; licensee BioMed Central Ltd. 2013 Background In recent theoretical developments, the information available (e.g. genotypes) divides the original population into two groups: animals with this information (selected animals) and animals without this information (excluded animals). These developments require inversion of the part of the pedigree-based numerator relationship matrix that describes the genetic covariance between selected animals (A22). Our main objective was to propose and evaluate methodology that takes advantage of any potential sparsity in the inverse of A22 in order to reduce the computing time required for its inversion. This potential sparsity is brought out by searching the pedigree for dependencies between the selected animals. Jointly, we expected distant ancestors to provide relationship ties that increase the density of matrix A22 but that their effect on might be minor. This hypothesis was also tested. Methods The inverse of A22 can be computed from the inverse of the triangular factor (T-1) obtained by Cholesky root-free decomposition of A22. We propose an algorithm that sets up the sparsity pattern of T-1 using pedigree information. This algorithm provides positions of the elements of T-1 worth to be computed (i.e. different from zero). A recursive computation of is then achieved with or without information on the sparsity pattern and time required for each computation was recorded. For three numbers of selected animals (4000; 8000 and 12 000), A22 was computed using different pedigree extractions and the closeness of the resulting to the inverse computed using the fully extracted pedigree was measured by an appropriate norm. Results The use of prior information on the sparsity of T-1 decreased the computing time for inversion by a factor of 1.73 on average. Computational issues and practical uses of the different algorithms were discussed. Cases involving more than 12 000 selected animals were considered. Inclusion of 10 generations was determined to be sufficient when computing A22. Conclusions Depending on the size and structure of the selected sub-population, gains in time to compute are possible and these gains may increase as the number of selected animals increases. Given the sequential nature of most computational steps, the proposed algorithm can benefit from optimization and may be convenient for genomic evaluations. Genomic Prediction (dpeaa)DE-He213 Inversion Algorithm (dpeaa)DE-He213 Sparsity Pattern (dpeaa)DE-He213 Sparse Vector (dpeaa)DE-He213 Genomic Relationship Matrix (dpeaa)DE-He213 Gengler, Nicolas aut Enthalten in Genetics, selection, evolution London : BioMed Central, 1989 45(2013), 1 vom: 06. Dez. (DE-627)312849052 (DE-600)2012369-3 1297-9686 nnns volume:45 year:2013 number:1 day:06 month:12 https://dx.doi.org/10.1186/1297-9686-45-45 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA 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_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2003 GBV_ILN_2014 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_4338 GBV_ILN_4367 GBV_ILN_4700 AR 45 2013 1 06 12
allfieldsGer	10.1186/1297-9686-45-45 doi (DE-627)SPR026807386 (SPR)1297-9686-45-45-e DE-627 ger DE-627 rakwb eng Faux, Pierre verfasserin aut Inversion of a part of the numerator relationship matrix using pedigree information 2013 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Faux and Gengler; licensee BioMed Central Ltd. 2013 Background In recent theoretical developments, the information available (e.g. genotypes) divides the original population into two groups: animals with this information (selected animals) and animals without this information (excluded animals). These developments require inversion of the part of the pedigree-based numerator relationship matrix that describes the genetic covariance between selected animals (A22). Our main objective was to propose and evaluate methodology that takes advantage of any potential sparsity in the inverse of A22 in order to reduce the computing time required for its inversion. This potential sparsity is brought out by searching the pedigree for dependencies between the selected animals. Jointly, we expected distant ancestors to provide relationship ties that increase the density of matrix A22 but that their effect on might be minor. This hypothesis was also tested. Methods The inverse of A22 can be computed from the inverse of the triangular factor (T-1) obtained by Cholesky root-free decomposition of A22. We propose an algorithm that sets up the sparsity pattern of T-1 using pedigree information. This algorithm provides positions of the elements of T-1 worth to be computed (i.e. different from zero). A recursive computation of is then achieved with or without information on the sparsity pattern and time required for each computation was recorded. For three numbers of selected animals (4000; 8000 and 12 000), A22 was computed using different pedigree extractions and the closeness of the resulting to the inverse computed using the fully extracted pedigree was measured by an appropriate norm. Results The use of prior information on the sparsity of T-1 decreased the computing time for inversion by a factor of 1.73 on average. Computational issues and practical uses of the different algorithms were discussed. Cases involving more than 12 000 selected animals were considered. Inclusion of 10 generations was determined to be sufficient when computing A22. Conclusions Depending on the size and structure of the selected sub-population, gains in time to compute are possible and these gains may increase as the number of selected animals increases. Given the sequential nature of most computational steps, the proposed algorithm can benefit from optimization and may be convenient for genomic evaluations. Genomic Prediction (dpeaa)DE-He213 Inversion Algorithm (dpeaa)DE-He213 Sparsity Pattern (dpeaa)DE-He213 Sparse Vector (dpeaa)DE-He213 Genomic Relationship Matrix (dpeaa)DE-He213 Gengler, Nicolas aut Enthalten in Genetics, selection, evolution London : BioMed Central, 1989 45(2013), 1 vom: 06. Dez. (DE-627)312849052 (DE-600)2012369-3 1297-9686 nnns volume:45 year:2013 number:1 day:06 month:12 https://dx.doi.org/10.1186/1297-9686-45-45 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA 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_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2003 GBV_ILN_2014 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_4338 GBV_ILN_4367 GBV_ILN_4700 AR 45 2013 1 06 12
allfieldsSound	10.1186/1297-9686-45-45 doi (DE-627)SPR026807386 (SPR)1297-9686-45-45-e DE-627 ger DE-627 rakwb eng Faux, Pierre verfasserin aut Inversion of a part of the numerator relationship matrix using pedigree information 2013 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Faux and Gengler; licensee BioMed Central Ltd. 2013 Background In recent theoretical developments, the information available (e.g. genotypes) divides the original population into two groups: animals with this information (selected animals) and animals without this information (excluded animals). These developments require inversion of the part of the pedigree-based numerator relationship matrix that describes the genetic covariance between selected animals (A22). Our main objective was to propose and evaluate methodology that takes advantage of any potential sparsity in the inverse of A22 in order to reduce the computing time required for its inversion. This potential sparsity is brought out by searching the pedigree for dependencies between the selected animals. Jointly, we expected distant ancestors to provide relationship ties that increase the density of matrix A22 but that their effect on might be minor. This hypothesis was also tested. Methods The inverse of A22 can be computed from the inverse of the triangular factor (T-1) obtained by Cholesky root-free decomposition of A22. We propose an algorithm that sets up the sparsity pattern of T-1 using pedigree information. This algorithm provides positions of the elements of T-1 worth to be computed (i.e. different from zero). A recursive computation of is then achieved with or without information on the sparsity pattern and time required for each computation was recorded. For three numbers of selected animals (4000; 8000 and 12 000), A22 was computed using different pedigree extractions and the closeness of the resulting to the inverse computed using the fully extracted pedigree was measured by an appropriate norm. Results The use of prior information on the sparsity of T-1 decreased the computing time for inversion by a factor of 1.73 on average. Computational issues and practical uses of the different algorithms were discussed. Cases involving more than 12 000 selected animals were considered. Inclusion of 10 generations was determined to be sufficient when computing A22. Conclusions Depending on the size and structure of the selected sub-population, gains in time to compute are possible and these gains may increase as the number of selected animals increases. Given the sequential nature of most computational steps, the proposed algorithm can benefit from optimization and may be convenient for genomic evaluations. Genomic Prediction (dpeaa)DE-He213 Inversion Algorithm (dpeaa)DE-He213 Sparsity Pattern (dpeaa)DE-He213 Sparse Vector (dpeaa)DE-He213 Genomic Relationship Matrix (dpeaa)DE-He213 Gengler, Nicolas aut Enthalten in Genetics, selection, evolution London : BioMed Central, 1989 45(2013), 1 vom: 06. Dez. (DE-627)312849052 (DE-600)2012369-3 1297-9686 nnns volume:45 year:2013 number:1 day:06 month:12 https://dx.doi.org/10.1186/1297-9686-45-45 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA 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_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2003 GBV_ILN_2014 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_4338 GBV_ILN_4367 GBV_ILN_4700 AR 45 2013 1 06 12
language	English
source	Enthalten in Genetics, selection, evolution 45(2013), 1 vom: 06. Dez. volume:45 year:2013 number:1 day:06 month:12
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This algorithm provides positions of the elements of T-1 worth to be computed (i.e. different from zero). A recursive computation of is then achieved with or without information on the sparsity pattern and time required for each computation was recorded. For three numbers of selected animals (4000; 8000 and 12 000), A22 was computed using different pedigree extractions and the closeness of the resulting to the inverse computed using the fully extracted pedigree was measured by an appropriate norm. Results The use of prior information on the sparsity of T-1 decreased the computing time for inversion by a factor of 1.73 on average. Computational issues and practical uses of the different algorithms were discussed. Cases involving more than 12 000 selected animals were considered. Inclusion of 10 generations was determined to be sufficient when computing A22. 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author	Faux, Pierre
spellingShingle	Faux, Pierre misc Genomic Prediction misc Inversion Algorithm misc Sparsity Pattern misc Sparse Vector misc Genomic Relationship Matrix Inversion of a part of the numerator relationship matrix using pedigree information
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topic_title	Inversion of a part of the numerator relationship matrix using pedigree information Genomic Prediction (dpeaa)DE-He213 Inversion Algorithm (dpeaa)DE-He213 Sparsity Pattern (dpeaa)DE-He213 Sparse Vector (dpeaa)DE-He213 Genomic Relationship Matrix (dpeaa)DE-He213
topic	misc Genomic Prediction misc Inversion Algorithm misc Sparsity Pattern misc Sparse Vector misc Genomic Relationship Matrix
topic_unstemmed	misc Genomic Prediction misc Inversion Algorithm misc Sparsity Pattern misc Sparse Vector misc Genomic Relationship Matrix
topic_browse	misc Genomic Prediction misc Inversion Algorithm misc Sparsity Pattern misc Sparse Vector misc Genomic Relationship Matrix
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title	Inversion of a part of the numerator relationship matrix using pedigree information
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title_full	Inversion of a part of the numerator relationship matrix using pedigree information
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title_sort	inversion of a part of the numerator relationship matrix using pedigree information
title_auth	Inversion of a part of the numerator relationship matrix using pedigree information
abstract	Background In recent theoretical developments, the information available (e.g. genotypes) divides the original population into two groups: animals with this information (selected animals) and animals without this information (excluded animals). These developments require inversion of the part of the pedigree-based numerator relationship matrix that describes the genetic covariance between selected animals (A22). Our main objective was to propose and evaluate methodology that takes advantage of any potential sparsity in the inverse of A22 in order to reduce the computing time required for its inversion. This potential sparsity is brought out by searching the pedigree for dependencies between the selected animals. Jointly, we expected distant ancestors to provide relationship ties that increase the density of matrix A22 but that their effect on might be minor. This hypothesis was also tested. Methods The inverse of A22 can be computed from the inverse of the triangular factor (T-1) obtained by Cholesky root-free decomposition of A22. We propose an algorithm that sets up the sparsity pattern of T-1 using pedigree information. This algorithm provides positions of the elements of T-1 worth to be computed (i.e. different from zero). A recursive computation of is then achieved with or without information on the sparsity pattern and time required for each computation was recorded. For three numbers of selected animals (4000; 8000 and 12 000), A22 was computed using different pedigree extractions and the closeness of the resulting to the inverse computed using the fully extracted pedigree was measured by an appropriate norm. Results The use of prior information on the sparsity of T-1 decreased the computing time for inversion by a factor of 1.73 on average. Computational issues and practical uses of the different algorithms were discussed. Cases involving more than 12 000 selected animals were considered. Inclusion of 10 generations was determined to be sufficient when computing A22. Conclusions Depending on the size and structure of the selected sub-population, gains in time to compute are possible and these gains may increase as the number of selected animals increases. Given the sequential nature of most computational steps, the proposed algorithm can benefit from optimization and may be convenient for genomic evaluations. © Faux and Gengler; licensee BioMed Central Ltd. 2013
abstractGer	Background In recent theoretical developments, the information available (e.g. genotypes) divides the original population into two groups: animals with this information (selected animals) and animals without this information (excluded animals). These developments require inversion of the part of the pedigree-based numerator relationship matrix that describes the genetic covariance between selected animals (A22). Our main objective was to propose and evaluate methodology that takes advantage of any potential sparsity in the inverse of A22 in order to reduce the computing time required for its inversion. This potential sparsity is brought out by searching the pedigree for dependencies between the selected animals. Jointly, we expected distant ancestors to provide relationship ties that increase the density of matrix A22 but that their effect on might be minor. This hypothesis was also tested. Methods The inverse of A22 can be computed from the inverse of the triangular factor (T-1) obtained by Cholesky root-free decomposition of A22. We propose an algorithm that sets up the sparsity pattern of T-1 using pedigree information. This algorithm provides positions of the elements of T-1 worth to be computed (i.e. different from zero). A recursive computation of is then achieved with or without information on the sparsity pattern and time required for each computation was recorded. For three numbers of selected animals (4000; 8000 and 12 000), A22 was computed using different pedigree extractions and the closeness of the resulting to the inverse computed using the fully extracted pedigree was measured by an appropriate norm. Results The use of prior information on the sparsity of T-1 decreased the computing time for inversion by a factor of 1.73 on average. Computational issues and practical uses of the different algorithms were discussed. Cases involving more than 12 000 selected animals were considered. Inclusion of 10 generations was determined to be sufficient when computing A22. Conclusions Depending on the size and structure of the selected sub-population, gains in time to compute are possible and these gains may increase as the number of selected animals increases. Given the sequential nature of most computational steps, the proposed algorithm can benefit from optimization and may be convenient for genomic evaluations. © Faux and Gengler; licensee BioMed Central Ltd. 2013
abstract_unstemmed	Background In recent theoretical developments, the information available (e.g. genotypes) divides the original population into two groups: animals with this information (selected animals) and animals without this information (excluded animals). These developments require inversion of the part of the pedigree-based numerator relationship matrix that describes the genetic covariance between selected animals (A22). Our main objective was to propose and evaluate methodology that takes advantage of any potential sparsity in the inverse of A22 in order to reduce the computing time required for its inversion. This potential sparsity is brought out by searching the pedigree for dependencies between the selected animals. Jointly, we expected distant ancestors to provide relationship ties that increase the density of matrix A22 but that their effect on might be minor. This hypothesis was also tested. Methods The inverse of A22 can be computed from the inverse of the triangular factor (T-1) obtained by Cholesky root-free decomposition of A22. We propose an algorithm that sets up the sparsity pattern of T-1 using pedigree information. This algorithm provides positions of the elements of T-1 worth to be computed (i.e. different from zero). A recursive computation of is then achieved with or without information on the sparsity pattern and time required for each computation was recorded. For three numbers of selected animals (4000; 8000 and 12 000), A22 was computed using different pedigree extractions and the closeness of the resulting to the inverse computed using the fully extracted pedigree was measured by an appropriate norm. Results The use of prior information on the sparsity of T-1 decreased the computing time for inversion by a factor of 1.73 on average. Computational issues and practical uses of the different algorithms were discussed. Cases involving more than 12 000 selected animals were considered. Inclusion of 10 generations was determined to be sufficient when computing A22. Conclusions Depending on the size and structure of the selected sub-population, gains in time to compute are possible and these gains may increase as the number of selected animals increases. Given the sequential nature of most computational steps, the proposed algorithm can benefit from optimization and may be convenient for genomic evaluations. © Faux and Gengler; licensee BioMed Central Ltd. 2013
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title_short	Inversion of a part of the numerator relationship matrix using pedigree information
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These developments require inversion of the part of the pedigree-based numerator relationship matrix that describes the genetic covariance between selected animals (A22). Our main objective was to propose and evaluate methodology that takes advantage of any potential sparsity in the inverse of A22 in order to reduce the computing time required for its inversion. This potential sparsity is brought out by searching the pedigree for dependencies between the selected animals. Jointly, we expected distant ancestors to provide relationship ties that increase the density of matrix A22 but that their effect on might be minor. This hypothesis was also tested. Methods The inverse of A22 can be computed from the inverse of the triangular factor (T-1) obtained by Cholesky root-free decomposition of A22. We propose an algorithm that sets up the sparsity pattern of T-1 using pedigree information. This algorithm provides positions of the elements of T-1 worth to be computed (i.e. different from zero). A recursive computation of is then achieved with or without information on the sparsity pattern and time required for each computation was recorded. For three numbers of selected animals (4000; 8000 and 12 000), A22 was computed using different pedigree extractions and the closeness of the resulting to the inverse computed using the fully extracted pedigree was measured by an appropriate norm. Results The use of prior information on the sparsity of T-1 decreased the computing time for inversion by a factor of 1.73 on average. Computational issues and practical uses of the different algorithms were discussed. Cases involving more than 12 000 selected animals were considered. Inclusion of 10 generations was determined to be sufficient when computing A22. Conclusions Depending on the size and structure of the selected sub-population, gains in time to compute are possible and these gains may increase as the number of selected animals increases. 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Inversion of a part of the numerator relationship matrix using pedigree information

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