Animal Science Unit, Gembloux Agro-Bio Tech, University of Liège, Passage des Déportés, 2, 5030 Gembloux, Belgium.
Genet Sel Evol. 2013 Dec 6;45(1):45. doi: 10.1186/1297-9686-45-45.
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 A22-1 might be minor. This hypothesis was also tested.
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 A22-1 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 A22-1 to the inverse computed using the fully extracted pedigree was measured by an appropriate norm.
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.
Depending on the size and structure of the selected sub-population, gains in time to compute A22-1 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.
在最近的理论发展中，可用信息（例如基因型）将原始群体分为两组：具有该信息的动物（选择的动物）和没有该信息的动物（排除的动物）。这些发展要求反转基于系谱的分子关系矩阵的一部分，该部分描述了选择动物之间的遗传协方差（A22）。我们的主要目标是提出和评估利用 A22 的逆的任何潜在稀疏性的方法，以减少反转所需的计算时间。这种潜在的稀疏性是通过在系谱中搜索选择动物之间的依赖关系而产生的。我们共同期望遥远的祖先提供关系联系，增加矩阵 A22 的密度，但它们对 A22-1 的影响可能很小。该假设也进行了测试。
可以通过对 A22 的 Cholesky 无根分解获得的三角因子（T-1）的逆来计算 A22 的逆。我们提出了一种使用系谱信息设置 T-1 的稀疏模式的算法。该算法提供了要计算的 T-1 元素的位置（即与零不同）。然后，可以在具有或不具有稀疏模式信息的情况下递归计算 A22-1，并记录每次计算所需的时间。对于选择的动物数量（4000；8000 和 12000），使用不同的系谱提取计算 A22，并通过适当的范数测量由此产生的 A22-1与使用完全提取的系谱计算的逆的接近程度。
使用 T-1 的稀疏性的先验信息将反转的计算时间平均减少了 1.73 倍。讨论了不同算法的计算问题和实际用途。考虑了涉及超过 12000 个选择动物的情况。当计算 A22 时，确定包含 10 代就足够了。
根据选择的亚群体的大小和结构，计算 A22-1 的时间可能会有所增加，并且随着选择动物数量的增加，这些收益可能会增加。考虑到大多数计算步骤的顺序性质，所提出的算法可以受益于优化，并且可能方便用于基因组评估。
