Using recursion to compute the inverse of the genomic relationship matrix

Area of Science:

• Animal Breeding and Genetics
• Quantitative Genetics
• Bioinformatics

Background:

• Traditional methods for inverting the numerator relationship matrix rely on pedigree information.
• Genomic relationships require considering all genotyped animals, complicating direct inversion.
• Existing methods for genomic relationship matrix inversion can be computationally intensive for large datasets.

Purpose of the Study:

• To investigate recursive methods for computing the inverse of the genomic relationship matrix.
• To develop and evaluate algorithms for generating genomic recursions.
• To assess the computational efficiency and potential benefits of genomic recursions in genetic evaluations.

Main Methods:

• Development of algorithms for creating genomic recursions, including sequential and forward updates.
• Formulation of an algorithm for proven and young animals, distinguishing between their genomic recursions.
• Analysis of computational costs associated with different recursive algorithms compared to direct inversion.

Main Results:

• Genomic recursions allow solving mixed model equations without explicit inversion of the genomic relationship matrix.
• Sequential and forward update algorithms demonstrate lower or equal computing costs compared to regular inversion.
• The algorithm for proven and young animals provides exact genomic EBV in genomic BLUP with efficient cost scaling.

Conclusions:

• Genomic recursions offer a computationally efficient alternative for handling large genomic relationship matrices.
• These recursive approaches can provide novel insights into genomic evaluation processes.
• The developed methods have the potential to reduce the costs associated with genetic predictions in large-scale genomic studies.