Genome-enabled prediction of genetic values using radial basis function neural networks

Abstract

The availability of high density panels of molecular markers has prompted the adoption of genomic selection (GS) methods in animal and plant breeding. In GS, parametric, semi-parametric and non-parametric regressions models are used for predicting quantitative traits. This article shows how to use neural networks with radial basis functions (RBFs) for prediction with dense molecular markers. We illustrate the use of the linear Bayesian LASSO regression model and of two non-linear regression models, reproducing kernel Hilbert spaces (RKHS) regression and radial basis function neural networks (RBFNN) on simulated data and real maize lines genotyped with 55,000 markers and evaluated for several trait-environment combinations. The empirical results of this study indicated that the three models showed similar overall prediction accuracy, with a slight and consistent superiority of RKHS and RBFNN over the additive Bayesian LASSO model. Results from the simulated data indicate that RKHS and RBFNN models captured epistatic effects; however, adding non-signal (redundant) predictors (interaction between markers) can adversely affect the predictive accuracy of the non-linear regression models.

Fig. 1

Graphical representation of a single hidden layer feed-forward neural network (NN). In the hidden layer, input variables ( markers) are combined using a linear function, (m = 1,…,M), and subsequently transformed using a non-linear activation function, , yielding a set of M (M = number of neurons) inferred scores, . These scores are used in the output layer as basis functions to regress the response using the linear activation function on the data-derived predictors ; could be either an identity or any other function

Fig. 2

Graphical representation of a single hidden layer (Gaussian) radial basis function neural network (RBFNN). In the hidden layer, information from input variables ( markers) is first summarized by means of the Euclidean distance between each of the input vectors {x _i} with respect to M (data-inferred) (M = number of neurons) centers {c _m}, that is . These distances are then transformed using the Gaussian function, , yielding M data-derived scores. These scores are used in the output layer as basis functions for the linear regression, ; is usually an identity function

Fig. 3

Plot of the correlation for each of 50 partitions in each of 21 trait–environment combinations for different combinations of models. In a when the best model is RKHS, this is represented by a white circle; when the best model is BL, this is represented by a black circle. In b when best model is RBFNN, this is represented by a white circle; when the best model is BL, this is represented by a black circle

Fig. 4

Histograms of the off-diagonal entries of each of the three kernels used (K ₁, K ₂, K ₃) in the RKHS model for the ASI-SS maize data set

Fig. 5

Plot of the PMSE for each of the 50 partitions in each of 21 trait–environment combinations for different combinations of models. In a when the best model is RKHS, this is represented by a white circle; when the best model is BL, this is represented by a black circle. In b when best model is RBFNN, this is represented by a white circle; when the best model is BL, this is represented by a black circle