Efficient Algorithms for Calculating Epistatic Genomic Relationship Matrices.

The genomic relationship matrix plays a key role in the analysis of genetic diversity, genomic prediction, and genome-wide association studies. The epistatic genomic relationship matrix is a natural generalization of the classic genomic relationship matrix in the sense that it implicitly models the epistatic effects among all markers. Calculating the exact form of the epistatic relationship matrix requires high computational load, and is hence not feasible when the number of markers is large, or when high-degree of epistasis is in consideration. Currently, many studies use the Hadamard product of the classic genomic relationship matrix as an approximation. However, the quality of the approximation is difficult to investigate in the strict mathematical sense. In this study, we derived iterative formulas for the precise form of the epistatic genomic relationship matrix for arbitrary degree of epistasis including both additive and dominance interactions. The key to our theoretical results is the observation of an interesting link between the elements in the genomic relationship matrix and symmetric polynomials, which motivated the application of the corresponding mathematical theory. Based on the iterative formulas, efficient recursive algorithms were implemented. Compared with the approximation by the Hadamard product, our algorithms provided a complete solution to the problem of calculating the exact epistatic genomic relationship matrix. As an application, we showed that our new algorithms easily relieved the computational burden in a previous study on the approximation behavior of two limit models.