Prediction of maize single-cross hybrid performance: support vector machine regression versus best linear prediction

Abstract

Accurate prediction of the phenotypic performance of a hybrid plant based on the molecular fingerprints of its parents should lead to a more cost-effective breeding programme as it allows to reduce the number of expensive field evaluations. The construction of a reliable prediction model requires a representative sample of hybrids for which both molecular and phenotypic information are accessible. This phenotypic information is usually readily available as typical breeding programmes test numerous new hybrids in multi-location field trials on a yearly basis. Earlier studies indicated that a linear mixed model analysis of this typically unbalanced phenotypic data allows to construct ɛ-insensitive support vector machine regression and best linear prediction models for predicting the performance of single-cross maize hybrids. We compare these prediction methods using different subsets of the phenotypic and marker data of a commercial maize breeding programme and evaluate the resulting prediction accuracies by means of a specifically designed field experiment. This balanced field trial allows to assess the reliability of the cross-validation prediction accuracies reported here and in earlier studies. The limits of the predictive capabilities of both prediction methods are further examined by reducing the number of training hybrids and the size of the molecular fingerprints. The results indicate a considerable discrepancy between prediction accuracies obtained by cross-validation procedures and those obtained by correlating the predictions with the results of a validation field trial. The prediction accuracy of best linear prediction was less sensitive to a reduction of the number of training examples compared with that of support vector machine regression. The latter was, however, better at predicting hybrid performance when the size of the molecular fingerprints was reduced, especially if the initial set of markers had a low information content.

Appendix: Variance structure of the linear mixed models fitted to the phenotypic data of the validation field trial

The four random vectors $\user2{c}$, $\user2{a}_{\rm s},$ $\user2{a}_{\rm o}$ and $\user2{d}$ of Eq. 4 are assumed to be mutually independent. Furthermore, for each of these vectors $\user2{h} \in \{ \user2{c} , \user2{a}_{\rm s}, \user2{a}_{\rm o} , \user2{d} \}$ we assume that the variance has the separable form

$$ \hbox {Var}(\user2{h})=\user2{g}_e \otimes \user2{g}_{\rm v} , $$

(7)

where $\otimes$ denotes the Kronecker product. $\user2{g}_e$ represents a 3 × 3 symmetric matrix containing the covariance between environments while $\user2{g}_{\rm v}$ represents the covariance between the specified genetic components of the validation trial entries. We start by fitting a completely unstructured variance matrix for $\user2{g}_e$ while assuming an identity matrix for $\user2{g}_{\rm v}.$ In subsequent steps, the number of REML estimated variance components is reduced by fitting more parsimonious variance models for $\user2{g}_e$ using restricted maximum likelihood ratio tests in case of comparisons between nested models, or Akaike’s information criterion (AIC) otherwise. We attempt to fit a first-order factor analytic variance model such that $\user2{g}_e=\varvec{\lambda}\varvec{\lambda}^{\prime}+ \varvec{\Uppsi}$ where $\varvec{\lambda}$ is a vector of factor loadings and the matrix $\varvec{\Uppsi}$ is a diagonal matrix containing three location-specific variances (Smith et al. 2001). To obtain a more parsimonious model, the specific variances were sometimes made equal or zero (giving perfect correlation), and/or the loadings made equal (giving a common covariance (Cullis et al. 1998)). In a subsequent reduction, the variances on the diagonal are set equal which results in a compound symmetry model. The simplest model for $\user2{g}_e$ assumed zero covariance and equal variances.

Once the most parsimonious model for $\user2{g}_e$ is determined, we try different formulations for $\user2{g}_{\rm v}.$ We fit an identity matrix for the variance model of the six check varieties in vector $\user2{c}$ as no molecular marker or pedigree information is available for these varieties. For the vectors $\user2{a}_{\rm s}$ and $\user2{a}_{\rm o},$ containing the GCA effects of the inbred lines, we try to fit the different coefficient of coancestry derived matrices $\user2{a}$ described by Maenhout et al. (2009) or an identity matrix. In a similar way, we compare the different coefficient of fraternity-based matrices $\user2{d}$ for the variance matrix $\user2{g}_{\rm v}$ pertaining to the vector $\user2{d}.$ Sometimes, the most parsimonious model is obtained by not using the separable form of Eq. 7 but directly fitting a common GCA or SCA effect for all three locations.

The variance of each vector of residuals $\user2{e}_i$ that make up vector $\user2{e}$ in Eq. 3 is modeled as a separable process in the direction of rows and columns so we can write Var$(\user2{e}_i)=\varvec{\Upsigma}_{ic} \otimes \varvec{\Upsigma}_{ir}$ where ⊗ denotes the Kronecker product. The matrices $\varvec{\Upsigma}_{ic}$ and $\varvec{\Upsigma}_{ir}$ are either identity matrices or contain first order autoregressive correlations to account for spatial variation as described in Gilmour et al. (1997), Smith et al. (2001) and Oakey et al. (2007). Table 4 gives an overview of the final model for the variance structure of vectors $\user2{g}$ and $\user2{e}$ for each trait.

Table 4 Summary of the variance structures fitted on the measurements of the validation data set for the traits grain yield, grain moisture content and days until flowering