GIS-FA: an approach to integrating thematic maps, factor-analytic, and envirotyping for cultivar targeting

Abstract

Key message

We propose an “enviromics” prediction model for recommending cultivars based on thematic maps aimed at decision-makers.

Abstract

Parsimonious methods that capture genotype-by-environment interaction (GEI) in multi-environment trials (MET) are important in breeding programs. Understanding the causes and factors of GEI allows the utilization of genotype adaptations in the target population of environments through environmental features and factor-analytic (FA) models. Here, we present a novel predictive breeding approach called GIS-FA, which integrates geographic information systems (GIS) techniques, FA models, partial least squares (PLS) regression, and enviromics to predict phenotypic performance in untested environments. The GIS-FA approach enables: (i) the prediction of the phenotypic performance of tested genotypes in untested environments, (ii) the selection of the best-ranking genotypes based on their overall performance and stability using the FA selection tools, and (iii) the creation of thematic maps showing overall or pairwise performance and stability for decision-making. We exemplify the usage of the GIS-FA approach using two datasets of rice [Oryza sativa (L.)] and soybean [Glycine max (L.) Merr.] in MET spread over tropical areas. In summary, our novel predictive method allows the identification of new breeding scenarios by pinpointing groups of environments where genotypes demonstrate superior predicted performance. It also facilitates and optimizes cultivar recommendations by utilizing thematic maps.

Appendix A: Partial least squares regression

Here, we employed the kernel PLS algorithm (Lindgren et al. 1993; Dayal and MacGregor 1997) to predict the factor loadings of untested environments. Details about this algorithm are presented below:

Take the following multiple regressions as a starting point:

$$\begin{aligned} \hat{\boldsymbol{\Lambda }}^\star = \textbf{W} \textbf{B} + \textbf{E} \end{aligned}$$

(A1)

where $\hat{\boldsymbol{\Lambda }}^\star$ is the $J \times K$ matrix of K rotated loadings for the J observed environments, $\textbf{W}$ is a $J \times P$ matrix of scaled values for P environmental features in the J observed environments, $\textbf{B}$ is a $P \times K$ vector of coefficients, and $\textbf{E}$ is a $J \times K$ matrix of lack of fit effects. Note that most of the environmental features are correlated (Supplementary Figure 4), so $\textbf{W}$ has multicollinearity problems, and $\textbf{B} = (\textbf{W}^\prime \textbf{W})^{-1} \textbf{W}^\prime \hat{\boldsymbol{\Lambda }}^\star$ does not yield a proper solution. To overcome this issue, we employed kernel PLS regression to transform $\textbf{B}$ into $\textbf{B}^*$, using the following equation:

$$\begin{aligned} \textbf{B}^\star = \boldsymbol{\Phi } (\boldsymbol{\Theta }^\prime \boldsymbol{\Phi })^{-1} \boldsymbol{\Xi }^\prime \end{aligned}$$

(A2)

where $\boldsymbol{\Phi }$ is a $P \times C$ matrix of weights for $\textbf{W}$ ($\boldsymbol{\Phi } = \{\boldsymbol{\phi }_1 \, \boldsymbol{\phi }_2 \, \ldots \boldsymbol{\phi }_C \}$), with C being the number of PLS components; $\boldsymbol{\Theta }$ is a matrix of loadings for $\textbf{W}$ ($\boldsymbol{\Theta } = \{\boldsymbol{\theta }_1 \, \boldsymbol{\theta }_2 \, \ldots \boldsymbol{\theta }_C \}$) and has the same dimension as $\boldsymbol{\Phi }$, and $\boldsymbol{\Xi }$ is a $K \times C$ matrix of weights for $\boldsymbol{\Lambda }$ ($\boldsymbol{\Xi } = \{\boldsymbol{\xi }_1 \boldsymbol{\xi }_2 \ldots \boldsymbol{\xi }_C\}$). We describe the CV procedure that defined the number of components ($c = 1, 2, \ldots , C$) in section Spatial predictions in the breeding zone. $\boldsymbol{\Phi }$, $\boldsymbol{\Theta }$, and $\boldsymbol{\Xi }$ were defined using an iterative process that leveraged the kernel functions of $\textbf{W}$ and $\boldsymbol{\Lambda }$. First, $\boldsymbol{\phi }_c$ is estimated as the eigenvector that is equivalent to the largest eigenvalue of the kernel $\textbf{W}^\prime \hat{\boldsymbol{\Lambda }}^\star \hat{\boldsymbol{\Lambda }}^{\star ^\prime } \textbf{W}$. We used this vector to initialize an iterative process whose number of repetitions is equivalent to C. Let $\textbf{R} = \boldsymbol{\Phi } (\boldsymbol{\Theta }^\prime \boldsymbol{\Phi })^{-1}$, with $\textbf{R} = \{\textbf{r}_1 \; \textbf{r}_2 \; \dots \; \textbf{r}_C\}$. In the first iteration, $\textbf{r}_1 = \boldsymbol{\phi }_1$. Subsequently, $\textbf{r}_c = \boldsymbol{\phi }_c - \boldsymbol{\theta }_{c-1}^\prime \boldsymbol{\phi }_c \boldsymbol{\xi }_{c-1}^\prime$. On each iteration, $\boldsymbol{\theta }_c$ and $\boldsymbol{\xi }_c$ are estimated as follows:

$$\begin{aligned} \boldsymbol{\theta }_c = \frac{ \textbf{r}_c^\prime (\textbf{W}^\prime \textbf{W}) }{ \textbf{r}_c^\prime (\textbf{W}^\prime \textbf{W}) \textbf{r}_c } \quad \boldsymbol{\xi }_c = \frac{ \textbf{r}_c^\prime (\textbf{W}^\prime \hat{\boldsymbol{\Lambda }}^\star ) }{ \textbf{r}_c^\prime (\textbf{W}^\prime \textbf{W}) \textbf{r}_c } \end{aligned}$$

(A3)

The solutions of these equations are stored in $\boldsymbol{\Theta }$ and $\boldsymbol{\Xi }$, respectively, and are used to update the covariance matrix for the next iteration as follows:

$$\begin{aligned} (\textbf{W}^\prime \hat{\boldsymbol{\Lambda }}^\star )_{c+1} = (\textbf{W}^\prime \hat{\boldsymbol{\Lambda }}^\star )_{c} - \boldsymbol{\theta }_c \boldsymbol{\xi }_c^\prime [(\textbf{W} \textbf{r}_c)^\prime \textbf{W} \textbf{r}_c] \end{aligned}$$

(A4)

When the iteration process is finished, $\textbf{B}^*$ provides a proper solution to Eq. (A1) and can be used for prediction purposes. We used $\textbf{B}^*$ in Eq. (17) to train the PLS model and in Eq. (18) to make predictions.