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.

中文翻译：

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基因型×环境相互作用的协方差结构中的Hadamard和Kronecker乘积

当在基因组预测模型中包括基因型×环境相互作用（G×E）时，Hadamard或Kronecker乘积已用于建模相互作用的协方差结构。这两种类型的建模之间的关系尚未在基因组预测文献中阐明。在这里，我们证明了一个基于Hadamard公式的模型和另一个使用Kronecker产品的模型可以得出完全相同的统计模型。此外，我们说明了协方差矩阵项的乘法与显式建模基因座×环境变量相互作用之间的关系。最后，我们使用小麦和玉米数据集来说明环境协方差E如果没有关于环境变量（例如温度或降水）的信息，也可以轻松指定。假设已经在不同的环境中对线进行了测试，则可以简单地从训练集中将相应的环境协方差估算为环境之间的表型协方差。为了实现较高的预测能力增强，必须适当定义环境协方差，并且必须在训练集中包括测试集在不同环境条件下的性能记录。