Using recursion to compute the inverse of the genomic relationship matrix
Using recursion to compute the inverse of the genomic relationship matrix
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DOI：
10.3168/jds.2013-7752
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发表时间：
2014-06-01
影响因子：
3.5
通讯作者：
Aguilar, I.
中科院分区：
文献类型：
--
作者：
Misztal, I.;Legarra, A.;Aguilar, I.
研究了使用递归计算基因组关系矩阵的逆。反转分子关系矩阵的传统算法基于以下观察：在给定所有其他动物的影响的情况下，1 只动物的相加效应的条件期望仅取决于其父系和母系的影响，每个动物的系数均为 0.5。对于基因组关系，这种期望取决于所有其他基因分型动物，并且系数没有任何设定值。对于每只动物，系数加上条件方差可以称为基因组递归。如果此类递归已知，则可以求解混合模型方程，而无需显式创建基因组关系矩阵的逆矩阵。开发了几种算法来创建基因组递归。在顺序更新的算法中，基因组递归是逐个动物创建的。该算法还可用于更新其他基因型的基因组关系矩阵的已知逆矩阵。在具有前向更新的算法中，新计算的递归立即应用于剩余动物的更新递归。两种算法的计算成本取决于基因组递归的稀疏模式，但低于或等于常规反演。针对经过验证的幼小动物的算法假设幼小动物的基因组递归仅包含经过验证的动物的系数。这种算法在基因组 BLUP 中生成精确的基因组 EBV，并且是单步基因组 BLUP 中的近似值。该算法对于经过验证的动物数量具有立方成本，对于幼年动物数量具有线性成本。基因组递归可以为基因组评估提供新的见解，并可能降低对大量基因型进行遗传预测的成本。
Computing the inverse of the genomic relationship matrix using recursion was investigated. A traditional algorithm to invert the numerator relationship matrix is based on the observation that the conditional expectation for an additive effect of 1 animal given the effects of all other animals depends on the effects of its sire and dam only, each with a coefficient of 0.5. With genomic relationships, such an expectation depends on all other genotyped animals, and the coefficients do not have any set value. For each animal, the coefficients plus the conditional variance can be called a genomic recursion. If such recursions are known, the mixed model equations can be solved without explicitly creating the inverse of the genomic relationship matrix. Several algorithms were developed to create genomic recursions. In an algorithm with sequential updates, genomic recursions are created animal by animal. That algorithm can also be used to update a known inverse of a genomic relationship matrix for additional genotypes. In an algorithm with forward updates, a newly computed recursion is immediately applied to update recursions for remaining animals. The computing costs for both algorithms depend on the sparsity pattern of the genomic recursions, but are lower or equal than for regular inversion. An algorithm for proven and young animals assumes that the genomic recursions for young animals contain coefficients only for proven animals. Such an algorithm generates exact genomic EBV in genomic BLUP and is an approximation in single-step genomic BLUP. That algorithm has a cubic cost for the number of proven animals and a linear cost for the number of young animals. The genomic recursions can provide new insight into genomic evaluation and possibly reduce costs of genetic predictions with extremely large numbers of genotypes.