Reproducing Kernel Hilbert Spaces Regression Research Articles

Haplotype blocks for genomic prediction: a comparative evaluation in multiple crop datasets.

In modern plant breeding, genomic selection is becoming the gold standard for selection of superior genotypes. The basis for genomic prediction models is a set of phenotyped lines along with their genotypic profile. With high marker density and linkage disequilibrium (LD) between markers, genotype data in breeding populations tends to exhibit considerable redundancy. Therefore, interest is growing in the use of haplotype blocks to overcome redundancy by summarizing co-inherited features. Moreover, haplotype blocks can help to capture local epistasis caused by interacting loci. Here, we compared genomic prediction methods that either used single SNPs or haplotype blocks with regards to their prediction accuracy for important traits in crop datasets. We used four published datasets from canola, maize, wheat and soybean. Different approaches to construct haplotype blocks were compared, including blocks based on LD, physical distance, number of adjacent markers and the algorithms implemented in the software "Haploview" and "HaploBlocker". The tested prediction methods included Genomic Best Linear Unbiased Prediction (GBLUP), Extended GBLUP to account for additive by additive epistasis (EGBLUP), Bayesian LASSO and Reproducing Kernel Hilbert Space (RKHS) regression. We found improved prediction accuracy in some traits when using haplotype blocks compared to SNP-based predictions, however the magnitude of improvement was very trait- and model-specific. Especially in settings with low marker density, haplotype blocks can improve genomic prediction accuracy. In most cases, physically large haplotype blocks yielded a strong decrease in prediction accuracy. Especially when prediction accuracy varies greatly across different prediction models, prediction based on haplotype blocks can improve prediction accuracy of underperforming models. However, there is no "best" method to build haplotype blocks, since prediction accuracy varied considerably across methods and traits. Hence, criteria used to define haplotype blocks should not be viewed as fixed biological parameters, but rather as hyperparameters that need to be adjusted for every dataset.

Genomic prediction in plants: opportunities for ensemble machine learning based approaches.

Background: Many studies have demonstrated the utility of machine learning (ML) methods for genomic prediction (GP) of various plant traits, but a clear rationale for choosing ML over conventionally used, often simpler parametric methods, is still lacking. Predictive performance of GP models might depend on a plethora of factors including sample size, number of markers, population structure and genetic architecture. Methods: Here, we investigate which problem and dataset characteristics are related to good performance of ML methods for genomic prediction. We compare the predictive performance of two frequently used ensemble ML methods (Random Forest and Extreme Gradient Boosting) with parametric methods including genomic best linear unbiased prediction (GBLUP), reproducing kernel Hilbert space regression (RKHS), BayesA and BayesB. To explore problem characteristics, we use simulated and real plant traits under different genetic complexity levels determined by the number of Quantitative Trait Loci (QTLs), heritability ( h 2 and h 2 e ), population structure and linkage disequilibrium between causal nucleotides and other SNPs. Results: Decision tree based ensemble ML methods are a better choice for nonlinear phenotypes and are comparable to Bayesian methods for linear phenotypes in the case of large effect Quantitative Trait Nucleotides (QTNs). Furthermore, we find that ML methods are susceptible to confounding due to population structure but less sensitive to low linkage disequilibrium than linear parametric methods. Conclusions: Overall, this provides insights into the role of ML in GP as well as guidelines for practitioners.

Genomic prediction in plants: opportunities for ensemble machine learning based approaches.

Background: Many studies have demonstrated the utility of machine learning (ML) methods for genomic prediction (GP) of various plant traits, but a clear rationale for choosing ML over conventionally used, often simpler parametric methods, is still lacking. Predictive performance of GP models might depend on a plethora of factors including sample size, number of markers, population structure and genetic architecture. Methods: Here, we investigate which problem and dataset characteristics are related to good performance of ML methods for genomic prediction. We compare the predictive performance of two frequently used ensemble ML methods (Random Forest and Extreme Gradient Boosting) with parametric methods including genomic best linear unbiased prediction (GBLUP), reproducing kernel Hilbert space regression (RKHS), BayesA and BayesB. To explore problem characteristics, we use simulated and real plant traits under different genetic complexity levels determined by the number of Quantitative Trait Loci (QTLs), heritability ( h 2 and h 2 e ), population structure and linkage disequilibrium between causal nucleotides and other SNPs. Results: Decision tree based ensemble ML methods are a better choice for nonlinear phenotypes and are comparable to Bayesian methods for linear phenotypes in the case of large effect Quantitative Trait Nucleotides (QTNs). Furthermore, we find that ML methods are susceptible to confounding due to population structure but less sensitive to low linkage disequilibrium than linear parametric methods. Conclusions: Overall, this provides insights into the role of ML in GP as well as guidelines for practitioners.

The Goldenshluger–Lepski method for constrained least-squares estimators over RKHSs

We study an adaptive estimation procedure called the Goldenshluger–Lepski method in the context of reproducing kernel Hilbert space (RKHS) regression. Adaptive estimation provides a way of selecting tuning parameters for statistical estimators using only the available data. This allows us to perform estimation without making strong assumptions about the estimand. In contrast to procedures such as training and validation, the Goldenshluger–Lepski method uses all of the data to produce non-adaptive estimators for a range of values of the tuning parameters. An adaptive estimator is selected by performing pairwise comparisons between these non-adaptive estimators. Applying the Goldenshluger–Lepski method is non-trivial as it requires a simultaneous high-probability bound on all of the pairwise comparisons. In the RKHS regression context, we choose our non-adaptive estimators to be clipped least-squares estimators constrained to lie in a ball in an RKHS. Applying the Goldenshluger–Lepski method in this context is made more complicated by the fact that we cannot use the L2 norm for performing the pairwise comparisons as it is unknown. We use the method to address two regression problems. In the first problem the RKHS is fixed, while in the second problem we adapt over a collection of RKHSs.

KTBoost: Combined Kernel and Tree Boosting

We introduce a novel boosting algorithm called ‘KTBoost’ which combines kernel boosting and tree boosting. In each boosting iteration, the algorithm adds either a regression tree or reproducing kernel Hilbert space (RKHS) regression function to the ensemble of base learners. Intuitively, the idea is that discontinuous trees and continuous RKHS regression functions complement each other, and that this combination allows for better learning of functions that have parts with varying degrees of regularity such as discontinuities and smooth parts. We empirically show that KTBoost significantly outperforms both tree and kernel boosting in terms of predictive accuracy in a comparison on a wide array of data sets.

Multi-omics-based prediction of hybrid performance in canola

Key messageComplementing or replacing genetic markers with transcriptomic data and use of reproducing kernel Hilbert space regression based on Gaussian kernels increases hybrid predictionaccuracies for complex agronomic traits in canola.In plant breeding, hybrids gained particular importance due to heterosis, the superior performance of offspring compared to their inbred parents. Since the development of new top performing hybrids requires labour-intensive and costly breeding programmes, including testing of large numbers of experimental hybrids, the prediction of hybrid performance is of utmost interest to plant breeders. In this study, we tested the effectiveness of hybrid prediction models in spring-type oilseed rape (Brassica napus L./canola) employing different omics profiles, individually and in combination. To this end, a population of 950 F1 hybrids was evaluated for seed yield and six other agronomically relevant traits in commercial field trials at several locations throughout Europe. A subset of these hybrids was also evaluated in a climatized glasshouse regarding early biomass production. For each of the 477 parental rapeseed lines, 13,201 single nucleotide polymorphisms (SNPs), 154 primary metabolites, and 19,479 transcripts were determined and used as predictive variables. Both, SNP markers and transcripts, effectively predict hybrid performance using (genomic) best linear unbiased prediction models (gBLUP). Compared to models using pure genetic markers, models incorporating transcriptome data resulted in significantly higher prediction accuracies for five out of seven agronomic traits, indicating that transcripts carry important information beyond genomic data. Notably, reproducing kernel Hilbert space regression based on Gaussian kernels significantly exceeded the predictive abilities of gBLUP models for six of the seven agronomic traits, demonstrating its potential for implementation in future canola breeding programmes.

Genomic Selection for Ascochyta Blight Resistance in Pea.

Genomic selection (GS) is a breeding tool, which is rapidly gaining popularity for plant breeding, particularly for traits that are difficult to measure. One such trait is ascochyta blight resistance in pea (Pisum sativum L.), which is difficult to assay because it is strongly influenced by the environment and depends on the natural occurrence of multiple pathogens. Here we report a study of the efficacy of GS for predicting ascochyta blight resistance in pea, as represented by ascochyta blight disease score (ASC), and using nucleotide polymorphism data acquired through genotyping-by-sequencing. The effects on prediction accuracy of different GS models and different thresholds for missing genotypic data (which modified the number of single nucleotide polymorphisms used in the analysis) were compared using cross-validation. Additionally, the inclusion of marker × environment interactions in a genomic best linear unbiased prediction (GBLUP) model was evaluated. Finally, different ways of combining trait data from two field trials using bivariate, spatial, and single-stage analyses were compared to results obtained using a mean value. The best prediction accuracy achieved for ASC was 0.56, obtained using GBLUP analysis with a mean value for ASC and data quality threshold of 70% (i.e., missing SNP data in <30% of lines). GBLUP and Bayesian Reproducing kernel Hilbert spaces regression (RKHS) performed slightly better than the other models trialed, whereas different missing data thresholds made minimal differences to prediction accuracy. The prediction accuracies of individual, randomly selected, testing/training partitions were highly variable, highlighting the effect that the choice of training population has on prediction accuracy. The inclusion of marker × environment interactions did not increase the prediction accuracy for lines which had not been phenotyped, but did improve the results of prediction across environments. GS is potentially useful for pea breeding programs pursuing ascochyta blight resistance, both for predicting breeding values for lines that have not been phenotyped, and for providing enhanced estimated breeding values for lines for which trait data is available.

Predictive ability of genome-assisted statistical models under various forms of gene action

Recent work has suggested that the performance of prediction models for complex traits may depend on the architecture of the target traits. Here we compared several prediction models with respect to their ability of predicting phenotypes under various statistical architectures of gene action: (1) purely additive, (2) additive and dominance, (3) additive, dominance, and two-locus epistasis, and (4) purely epistatic settings. Simulation and a real chicken dataset were used. Fourteen prediction models were compared: BayesA, BayesB, BayesC, Bayesian LASSO, Bayesian ridge regression, elastic net, genomic best linear unbiased prediction, a Gaussian process, LASSO, random forests, reproducing kernel Hilbert spaces regression, ridge regression (best linear unbiased prediction), relevance vector machines, and support vector machines. When the trait was under additive gene action, the parametric prediction models outperformed non-parametric ones. Conversely, when the trait was under epistatic gene action, the non-parametric prediction models provided more accurate predictions. Thus, prediction models must be selected according to the most probably underlying architecture of traits. In the chicken dataset examined, most models had similar prediction performance. Our results corroborate the view that there is no universally best prediction models, and that the development of robust prediction models is an important research objective.

Predicting Growth Traits with Genomic Selection Methods in Zhikong Scallop (Chlamys farreri).

Selective breeding is a common and effective approach for genetic improvement of aquaculture stocks with parental selection as the key factor. Genomic selection (GS) has been proposed as a promising tool to facilitate selective breeding. Here, we evaluated the predictability of four GS methods in Zhikong scallop (Chlamys farreri) through real dataset analyses of four economical traits (e.g., shell length, shell height, shell width, and whole weight). Our analysis revealed that different GS models exhibited variable performance in prediction accuracy depending on genetic and statistical factors, but non-parametric method, including reproducing kernel Hilbert spaces regression (RKHS) and sparse neural networks (SNN), generally outperformed parametric linear method, such as genomic best linear unbiased prediction (GBLUP) and BayesB. Furthermore, we demonstrated that the predictability relied mainly on the heritability regardless of GS methods. The size of training population and marker density also had considerable effects on the predictive performance. In practice, increasing the training population size could better improve the genomic prediction than raising the marker density. This study is the first to apply non-linear model and neural networks for GS in scallop and should be valuable to help develop strategies for aquaculture breeding programs.

Historical Datasets Support Genomic Selection Models for the Prediction of Cotton Fiber Quality Phenotypes Across Multiple Environments.

Genomic selection (GS) has successfully been used in plant breeding to improve selection efficiency and reduce breeding time and cost. However, there has not been a study to evaluate GS prediction models that may be used for predicting cotton breeding lines across multiple environments. In this study, we evaluated the performance of Bayes Ridge Regression, BayesA, BayesB, BayesC and Reproducing Kernel Hilbert Spaces regression models. We then extended the single-site GS model to accommodate genotype × environment interaction (G×E) in order to assess the merits of multi- over single-environment models in a practical breeding and selection context in cotton, a crop for which this has not previously been evaluated. Our study was based on a population of 215 upland cotton (Gossypium hirsutum) breeding lines which were evaluated for fiber length and strength at multiple locations in Australia and genotyped with 13,330 single nucleotide polymorphic (SNP) markers. BayesB, which assumes unique variance for each marker and a proportion of markers to have large effects, while most other markers have zero effect, was the preferred model. GS accuracy for fiber length based on a single-site model varied across sites, ranging from 0.27 to 0.77 (mean = 0.38), while that of fiber strength ranged from 0.19 to 0.58 (mean = 0.35) using randomly selected sub-populations as the training population. Prediction accuracies from the M×E model were higher than those for single-site and across-site models, with an average accuracy of 0.71 and 0.59 for fiber length and strength, respectively. The use of the M×E model could therefore identify which breeding lines have effects that are stable across environments and which ones are responsible for G×E and so reduce the amount of phenotypic screening required in cotton breeding programs to identify adaptable genotypes.

Genome-wide association mapping and genomic prediction for CBSD resistance in Manihot esculenta

Cassava (Manihot esculenta Crantz) is an important security crop that faces severe yield loses due to cassava brown streak disease (CBSD). Motivated by the slow progress of conventional breeding, genetic improvement of cassava is undergoing rapid change due to the implementation of quantitative trait loci mapping, Genome-wide association mapping (GWAS), and genomic selection (GS). In this study, two breeding panels were genotyped for SNP markers using genotyping by sequencing and phenotyped for foliar and CBSD root symptoms at five locations in Uganda. Our GWAS study found two regions associated to CBSD, one on chromosome 4 which co-localizes with a Manihot glaziovii introgression segment and one on chromosome 11, which contains a cluster of nucleotide-binding site-leucine-rich repeat (NBS-LRR) genes. We evaluated the potential of GS to improve CBSD resistance by assessing the accuracy of seven prediction models. Predictive accuracy values varied between CBSD foliar severity traits at 3 months after planting (MAP) (0.27–0.32), 6 MAP (0.40–0.42) and root severity (0.31–0.42). For all traits, Random Forest and reproducing kernel Hilbert spaces regression showed the highest predictive accuracies. Our results provide an insight into the genetics of CBSD resistance to guide CBSD marker-assisted breeding and highlight the potential of GS to improve cassava breeding.

&lt;b&gt;Comparison of whole genome prediction accuracy across generations using parametric and semi parametric methods

Accuracy of genomic prediction was compared using three parametric and semi parametric methods, including BayesA, Bayesian LASSO and Reproducing kernel Hilbert spaces regression under various levels of heritability (0.15, 0.3 and 0.45), different number of markers (500, 750 and 1000) and generation intervals of validating set. A historical population of 1000 individuals with equal sex ratio was simulated for 100 generations at constant size. It followed by 100 extra generations of gradually reducing size down to 500 individuals in generation 200. Individuals of generation 200 were mated randomly for 10 more generations applying litter size of 5 to expand the historical generation. Finally, 50 males and 500 females chosen from generation 210 were randomly mated to generate 10 more generations of recent population. Individuals born in generation 211 considered as the training set while the validation set was composed of individuals either from generations 213, 215 or 217. The genome comprised one chromosome of 100 cM length carrying 50 QTLs. There was no significant difference between accuracy of investigated methods (p > 0.05) but among three methods, the highest mean accuracy (0.659) was observed for BayesA. By increasing the heritability, the average genomic accuracy increased from 0.53 to 0.75 (p < 0.05). The number of SNPs affected the accuracy and accuracies increased as number of SNPs increased; therefore, the highest accuracy was for the case number of SNPs=1000. With getting away from validating set, the accuracies decreased and the most severe decay observed in the case of low heritability. Decreasing the accuracy across generations affected by marker density but was independent from investigated methods.

A Unified and Comprehensible View of Parametric and Kernel Methods for Genomic Prediction with Application to Rice.

One objective of this study was to provide readers with a clear and unified understanding of parametric statistical and kernel methods, used for genomic prediction, and to compare some of these in the context of rice breeding for quantitative traits. Furthermore, another objective was to provide a simple and user-friendly R package, named KRMM, which allows users to perform RKHS regression with several kernels. After introducing the concept of regularized empirical risk minimization, the connections between well-known parametric and kernel methods such as Ridge regression [i.e., genomic best linear unbiased predictor (GBLUP)] and reproducing kernel Hilbert space (RKHS) regression were reviewed. Ridge regression was then reformulated so as to show and emphasize the advantage of the kernel “trick” concept, exploited by kernel methods in the context of epistatic genetic architectures, over parametric frameworks used by conventional methods. Some parametric and kernel methods; least absolute shrinkage and selection operator (LASSO), GBLUP, support vector machine regression (SVR) and RKHS regression were thereupon compared for their genomic predictive ability in the context of rice breeding using three real data sets. Among the compared methods, RKHS regression and SVR were often the most accurate methods for prediction followed by GBLUP and LASSO. An R function which allows users to perform RR-BLUP of marker effects, GBLUP and RKHS regression, with a Gaussian, Laplacian, polynomial or ANOVA kernel, in a reasonable computation time has been developed. Moreover, a modified version of this function, which allows users to tune kernels for RKHS regression, has also been developed and parallelized for HPC Linux clusters. The corresponding KRMM package and all scripts have been made publicly available.

Assessing Predictive Properties of Genome-Wide Selection in Soybeans.

Many economically important traits in plant breeding have low heritability or are difficult to measure. For these traits, genomic selection has attractive features and may boost genetic gains. Our goal was to evaluate alternative scenarios to implement genomic selection for yield components in soybean (Glycine max L. merr). We used a nested association panel with cross validation to evaluate the impacts of training population size, genotyping density, and prediction model on the accuracy of genomic prediction. Our results indicate that training population size was the factor most relevant to improvement in genome-wide prediction, with greatest improvement observed in training sets up to 2000 individuals. We discuss assumptions that influence the choice of the prediction model. Although alternative models had minor impacts on prediction accuracy, the most robust prediction model was the combination of reproducing kernel Hilbert space regression and BayesB. Higher genotyping density marginally improved accuracy. Our study finds that breeding programs seeking efficient genomic selection in soybeans would best allocate resources by investing in a representative training set.

Modeling Epistasis in Genomic Selection.

Modeling epistasis in genomic selection is impeded by a high computational load. The extended genomic best linear unbiased prediction (EG-BLUP) with an epistatic relationship matrix and the reproducing kernel Hilbert space regression (RKHS) are two attractive approaches that reduce the computational load. In this study, we proved the equivalence of EG-BLUP and genomic selection approaches, explicitly modeling epistatic effects. Moreover, we have shown why the RKHS model based on a Gaussian kernel captures epistatic effects among markers. Using experimental data sets in wheat and maize, we compared different genomic selection approaches and concluded that prediction accuracy can be improved by modeling epistasis for selfing species but may not for outcrossing species.

Genome-wide prediction of maize single-cross performance, considering non-additive genetic effects.

The prediction of single-cross hybrids in maize is a promising technique for optimizing the use of financial resources in a breeding program. This study aimed to evaluate Genomic Best Linear Unbiased Predictors models for hybrid prediction and compare them with the Bayesian Ridge Regression, Bayes A, Bayesian LASSO, Bayes C, Bayes B, and Reproducing Kernel Hilbert Spaces Regression models, with inclusion or absence of non-additive effects under three heritability scenarios. Data from a maize germplasm bank belonging to USDA were used to determine the effects of molecular markers, which were considered to be parametric, to build 400 single-cross hybrids between two line groups via simulation. The following parameters were used to compare the models: predictive ability, estimation of variance components, heritability of genetic effects present in all situations, and the sum of squares of the predicted errors. The models responded positively when dominance effects were included in non-additive models, with all models tending to show an increase in the values of heritability parameters under all scenarios. Differences occur between models depending on the heritability range considered. Estimates of additive and dominant effects were better than estimates of epistatic effects. Estimates increased in accuracy for all models when non-additive effects for maize cob weight were considered.

Exploring the areas of applicability of whole-genome prediction methods for Asian rice (Oryza sativa L.)

Our simulation results clarify the areas of applicability of nine prediction methods and suggest the factors that affect their accuracy at predicting empirical traits. Whole-genome prediction is used to predict genetic value from genome-wide markers. The choice of method is important for successful prediction. We compared nine methods using empirical data for eight phenological and morphological traits of Asian rice cultivars (Oryza sativa L.) and data simulated from real marker genotype data. The methods were genomic BLUP (GBLUP), reproducing kernel Hilbert spaces regression (RKHS), Lasso, elastic net, random forest (RForest), Bayesian lasso (Blasso), extended Bayesian lasso (EBlasso), weighted Bayesian shrinkage regression (wBSR), and the average of all methods (Ave). The objectives were to evaluate the predictive ability of these methods in a cultivar population, to characterize them by exploring the area of applicability of each method using simulation, and to investigate the causes of their different accuracies for empirical traits. GBLUP was the most accurate for one trait, RKHS and Ave for two, and RForest for three traits. In the simulation, Blasso, EBlasso, and Ave showed stable performance across the simulated scenarios, whereas the other methods, except wBSR, had specific areas of applicability; wBSR performed poorly in most scenarios. For each method, the accuracy ranking for the empirical traits was largely consistent with that in one of the simulated scenarios, suggesting that the simulation conditions reflected the factors that affected the method accuracy for the empirical results. This study will be useful for genomic prediction not only in Asian rice, but also in populations from other crops with relatively small training sets and strong linkage disequilibrium structures.

Machine learning methods and predictive ability metrics for genome-wide prediction of complex traits

Genome-wide prediction of complex traits has become increasingly important in animal and plant breeding, and is receiving increasing attention in human genetics. Most common approaches are whole-genome regression models where phenotypes are regressed on thousands of markers concurrently, applying different prior distributions to marker effects. While use of shrinkage or regularization in SNP regression models has delivered improvements in predictive ability in genome-based evaluations, serious over-fitting problems may be encountered as the ratio between markers and available phenotypes continues increasing. Machine learning is an alternative approach for prediction and classification, capable of dealing with the dimensionality problem in a computationally flexible manner. In this article we provide an overview of non-parametric and machine learning methods used in genome wide prediction, discuss their similarities as well as their relationship to some well-known parametric approaches. Although the most suitable method is usually case dependent, we suggest the use of support vector machines and random forests for classification problems, whereas Reproducing Kernel Hilbert Spaces regression and boosting may suit better regression problems, with the former having the more consistently higher predictive ability. Neural Networks may suffer from over-fitting and may be too computationally demanded when the number of neurons is large.We further discuss on the metrics used to evaluate predictive ability in model comparison under cross-validation from a genomic selection point of view. We suggest use of predictive mean squared error as a main but not only metric for model comparison. Visual tools may greatly assist on the choice of the most accurate model.

Combining Genomic and Genealogical Information in a Reproducing Kernel Hilbert Spaces Regression Model for Genome-Enabled Predictions in Dairy Cattle

Genome-enhanced genotypic evaluations are becoming popular in several livestock species. For this purpose, the combination of the pedigree-based relationship matrix with a genomic similarities matrix between individuals is a common approach. However, the weight placed on each matrix has been so far established with ad hoc procedures, without formal estimation thereof. In addition, when using marker- and pedigree-based relationship matrices together, the resulting combined relationship matrix needs to be adjusted to the same scale in reference to the base population. This study proposes a semi-parametric Bayesian method for combining marker- and pedigree-based information on genome-enabled predictions. A kernel matrix from a reproducing kernel Hilbert spaces regression model was used to combine genomic and genealogical information in a semi-parametric scenario, avoiding inversion and adjustment complications. In addition, the weights on marker- versus pedigree-based information were inferred from a Bayesian model with Markov chain Monte Carlo. The proposed method was assessed involving a large number of SNPs and a large reference population. Five phenotypes, including production and type traits of dairy cattle were evaluated. The reliability of the genome-based predictions was assessed using the correlation, regression coefficient and mean squared error between the predicted and observed values. The results indicated that when a larger weight was given to the pedigree-based relationship matrix the correlation coefficient was lower than in situations where more weight was given to genomic information. Importantly, the posterior means of the inferred weight were near the maximum of 1. The behavior of the regression coefficient and the mean squared error was similar to the performance of the correlation, that is, more weight to the genomic information provided a regression coefficient closer to one and a smaller mean squared error. Our results also indicated a greater accuracy of genomic predictions when using a large reference population.

Model averaging for genome-enabled prediction with reproducing kernel Hilbert spaces: a case study with pig litter size and wheat yield

Predictive ability of yet-to-be observed litter size (pig) grain yield (wheat) records of several reproducing kernel Hilbert spaces (RKHS) regression models combining different number of Gaussian or t kernels was evaluated. Predictive performance was assessed as the average (over 50 replicates) predictive correlation in the testing set. Predictions from these models were combined using three different types of model averaging: (i) mean of predicted phenotypes obtained in each model, (ii) weighted average using mean squared error as weight or (iii) using the marginal likelihood as weight. (ii) and (iii) were obtained in a validation set with 5% of the data. Phenotypes consisted of 2598, 1604 and 1879 average litter size records from three commercial pig lines and wheat grain yield of 599 lines evaluated in four macro-environments. SNPs from the PorcineSNP60 BeadChip and 1447 DArT markers were used as predictors for the pig and wheat data analyses, respectively. Gaussian and univariate t kernels led to same predictive performance. Multikernel RKHS regression models overcame shortcomings of single kernel models (increasing the predictive correlation of RKHS models by 0.05 where 3 Gaussian or t kernels were fitted in the RKHS models simultaneously). None of the proposed averaging strategies improved the predictive correlations attained with single models using multiple kernel fitting.