Model Description

Genetic and environmental information were combined to construct trait models using a penalised linear-mixed model framework with a LASSO-type penalisation (Tibshirani, 1996) as implemented in the LMM-Lasso package (Rakitsch et al., 2013). Regularisation through LASSO-type penalization prevents potential overfitting caused by the large number of predictors. This linear-mixed model takes the form y = Σ + u + ε, where y is a vector of individual trait values, X is a matrix of daily minimum and maximum temperature with corresponding fixed effects β (fixed effect), u is the random effect of the genetic similarity between pairs of individuals, and ε is the vector of residuals.u is unobserved but assumed to be normally distributed with u ∼N(0 ,σg2K ), where K is the empirically computed GSM and σg2 is the variance explained by the genetic similarity. The residual vector ε is also normally distributedε ∼N(0, σe2I ), whereI is the identity matrix and σe2 is the residual variance.
The initial model considered genetic and environmental variation additively and independently (‘G+E model’), such that predicted reaction norms across environments were identical for all genotypes. In order to account for the non-linear influence of GxE on climate response, we computed ADMIXTURE proportions (Alexander & Lange, 2011) for each plant using k = 4 ancestral populations, which was found to be optimal (Appendix S4). ADMIXTURE proportions were used to generate additional predictors XADMIXTURE . For n genotypes andr environmental variables, XADMIXTURE is the column-wise Khatri-Rao product XADMIXTURE =(FTRT )T, where Fis the n x k matrix of ADMIXTURE proportions and R is then x r matrix of environmental predictors. This produces ann x kr matrix of additional predictors whose values are unique for each genotype-environment combination. These predictors were included alongside the minimum and maximum daily temperature (i.e.R ) in the design matrix X’ to create the ‘GxE model’ which takes the form y = ΣX’jβ j+ u + ε. We also created ‘G only’ and ‘E only’ models to determine the relative contribution of each component to prediction accuracy. These models are identical to the G + E model but used a column vector of ones as X and a square identity matrix asK respectively.

Assessing Model Performance

Internal Validation
Model performance was assessed through a random 10-fold cross validation (‘internal validation’) with 9 folds of the data used to train the model and the 10th fold used to test it. This was repeated 10 times, with each fold acting as the testing set once. Overall model performance was quantified using the root mean square error (RMSE) as a measure of error and the coefficient of determination between observed and predicted values (r2 ) as a measure of accuracy.
External Validation
External validation followed an ‘environmental blocking’ validation strategy (Roberts et al., 2017) designed to assess out-of-sample prediction accuracy. This involved training models on six plantings and testing on the seventh to mimic validation on independent data. Results of environmental blocking were also used to determine the effect of different training set compositions on model performance.
Finally, we performed an empirical external validation using data from an independent experiment. Korves and colleagues (2007) performed a planting of A. thaliana in Rhode Island, USA in Spring 2003 (RS) for which median DTB was reported. RS is geographically (North America vs. Europe) and temporally (2003 vs. 2006-2007) distant from the plantings in our data set, making it a novel environment. We predicted DTB in RS for 77 genotypes using a model trained on 100% of our data and 2 metre air temperature records sourced from DAYMET (Thornton et al., 2016).