International Maize and Wheat Improvement Center (CIMMYT), Km. 45, El Batán 56237 Texcoco, Mexico.
Universidad de Quintana Roo, Del Bosque, 77019 Chetumal, Q.R., Mexico.
Plant Genome. 2020 Nov;13(3):e20033. doi: 10.1002/tpg2.20033. Epub 2020 Jul 15.
When including genotype × environment interactions (G × E) in genomic prediction models, Hadamard or Kronecker products have been used to model the covariance structure of interactions. The relation between these two types of modeling has not been made clear in genomic prediction literature. Here, we demonstrate that a certain model based on a Hadamard formulation and another using the Kronecker product lead to exactly the same statistical model. Moreover, we illustrate how a multiplication of entries of covariance matrices is related to modeling locus × environmental-variable interactions explicitly. Finally, we use a wheat and a maize data set to illustrate that the environmental covariance E can be specified easily, also if no information on environmental variables - such as temperature or precipitation - is available. Given that lines have been tested in different environments, the corresponding environmental covariance can simply be estimated from the training set as phenotypic covariance between environments. To achieve a high level of increase in predictive ability, the environmental covariance has to be defined appropriately and records on the performance of the lines of the test set under different environmental conditions have to be included in the training set.
当在基因组预测模型中包含基因型与环境互作（G×E）时，Hadamard 或 Kronecker 积已被用于模拟互作的协方差结构。在基因组预测文献中，这两种建模类型之间的关系尚未明确。在这里，我们证明了基于 Hadamard 公式的特定模型和使用 Kronecker 积的另一个模型导致了完全相同的统计模型。此外，我们还说明了协方差矩阵的元素相乘如何与显式建模位点与环境变量的互作相关。最后，我们使用小麦和玉米数据集来说明，即使没有关于环境变量（如温度或降水）的信息，也可以轻松地指定环境协方差 E。如果已经在不同环境中测试了系谱，那么相应的环境协方差可以简单地从训练集中作为环境间的表型协方差进行估计。为了实现预测能力的显著提高，必须适当地定义环境协方差，并将测试集下线在不同环境条件下的性能记录包含在训练集中。
