Monte Carlo approximation of the logarithm of the determinant of large matrices with applications for linear mixed models in quantitative genetics
Abstract Background Likelihood-based inferences such as variance components estimation and hypothesis testing need logarithms of the determinant (log-determinant) of high dimensional matrices. Calculating the log-determinant is memory and time-consuming, making it impossible to perform likelihood-ba...
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| Format: | Article |
| Language: | deu |
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BMC
2025-08-01
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| Series: | Genetics Selection Evolution |
| Online Access: | https://doi.org/10.1186/s12711-025-00991-1 |
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| author | Matias Bermann Alejandra Alvarez-Munera Andres Legarra Ignacio Aguilar Ignacy Misztal Daniela Lourenco |
| author_facet | Matias Bermann Alejandra Alvarez-Munera Andres Legarra Ignacio Aguilar Ignacy Misztal Daniela Lourenco |
| author_sort | Matias Bermann |
| collection | DOAJ |
| description | Abstract Background Likelihood-based inferences such as variance components estimation and hypothesis testing need logarithms of the determinant (log-determinant) of high dimensional matrices. Calculating the log-determinant is memory and time-consuming, making it impossible to perform likelihood-based inferences for large datasets. Results We presented a method for approximating the log-determinant of positive semi-definite matrices based on repeated matrix–vector products and complex calculus. We tested the approximation of the log-determinant in beef and dairy cattle, chicken, and pig datasets including single and multiple-trait models. Average absolute relative differences between the approximated and exact log-determinant were around 10–3. The approximation was between 2 and 500 times faster than the exact calculation for medium and large matrices. We compared the restricted likelihood with (approximated) and without (exact) the approximation of the log-determinant for different values of heritability for a single-trait model. We also compared estimated variance components using exact expectation–maximization (EM) and average information (AI) REML algorithms, against two derivative-free approaches using the restricted likelihood calculated with the log-determinant approximation. The approximated and exact restricted likelihood showed maxima at the same heritability value. Derivative-free estimation of variance components with the approximated log-determinant converged to the same values as EM and AI-REML. The proposed approach is feasible to apply to any data size. Conclusions The method presented in this study allows to approximate the log-determinant of positive semi-definite matrices and, therefore, the likelihood for datasets of any size. This opens the possibility of performing likelihood-based inferences for large datasets in animal and plant breeding. |
| format | Article |
| id | doaj-art-a93ec9ef0e1c4803a83a3a20bf8c0797 |
| institution | Kabale University |
| issn | 1297-9686 |
| language | deu |
| publishDate | 2025-08-01 |
| publisher | BMC |
| record_format | Article |
| series | Genetics Selection Evolution |
| spelling | doaj-art-a93ec9ef0e1c4803a83a3a20bf8c07972025-08-20T03:42:29ZdeuBMCGenetics Selection Evolution1297-96862025-08-0157111110.1186/s12711-025-00991-1Monte Carlo approximation of the logarithm of the determinant of large matrices with applications for linear mixed models in quantitative geneticsMatias Bermann0Alejandra Alvarez-Munera1Andres Legarra2Ignacio Aguilar3Ignacy Misztal4Daniela Lourenco5Department of Animal and Dairy Science, University of GeorgiaDepartment of Animal and Dairy Science, University of GeorgiaCDCBInstituto Nacional de Investigación Agropecuaria (INIA)Department of Animal and Dairy Science, University of GeorgiaDepartment of Animal and Dairy Science, University of GeorgiaAbstract Background Likelihood-based inferences such as variance components estimation and hypothesis testing need logarithms of the determinant (log-determinant) of high dimensional matrices. Calculating the log-determinant is memory and time-consuming, making it impossible to perform likelihood-based inferences for large datasets. Results We presented a method for approximating the log-determinant of positive semi-definite matrices based on repeated matrix–vector products and complex calculus. We tested the approximation of the log-determinant in beef and dairy cattle, chicken, and pig datasets including single and multiple-trait models. Average absolute relative differences between the approximated and exact log-determinant were around 10–3. The approximation was between 2 and 500 times faster than the exact calculation for medium and large matrices. We compared the restricted likelihood with (approximated) and without (exact) the approximation of the log-determinant for different values of heritability for a single-trait model. We also compared estimated variance components using exact expectation–maximization (EM) and average information (AI) REML algorithms, against two derivative-free approaches using the restricted likelihood calculated with the log-determinant approximation. The approximated and exact restricted likelihood showed maxima at the same heritability value. Derivative-free estimation of variance components with the approximated log-determinant converged to the same values as EM and AI-REML. The proposed approach is feasible to apply to any data size. Conclusions The method presented in this study allows to approximate the log-determinant of positive semi-definite matrices and, therefore, the likelihood for datasets of any size. This opens the possibility of performing likelihood-based inferences for large datasets in animal and plant breeding.https://doi.org/10.1186/s12711-025-00991-1 |
| spellingShingle | Matias Bermann Alejandra Alvarez-Munera Andres Legarra Ignacio Aguilar Ignacy Misztal Daniela Lourenco Monte Carlo approximation of the logarithm of the determinant of large matrices with applications for linear mixed models in quantitative genetics Genetics Selection Evolution |
| title | Monte Carlo approximation of the logarithm of the determinant of large matrices with applications for linear mixed models in quantitative genetics |
| title_full | Monte Carlo approximation of the logarithm of the determinant of large matrices with applications for linear mixed models in quantitative genetics |
| title_fullStr | Monte Carlo approximation of the logarithm of the determinant of large matrices with applications for linear mixed models in quantitative genetics |
| title_full_unstemmed | Monte Carlo approximation of the logarithm of the determinant of large matrices with applications for linear mixed models in quantitative genetics |
| title_short | Monte Carlo approximation of the logarithm of the determinant of large matrices with applications for linear mixed models in quantitative genetics |
| title_sort | monte carlo approximation of the logarithm of the determinant of large matrices with applications for linear mixed models in quantitative genetics |
| url | https://doi.org/10.1186/s12711-025-00991-1 |
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