# A Multi-Trait Gaussian Kernel Genomic Prediction Model under Three Tunning Strategies

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## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Dataset 1. Japonica

#### 2.2. Dataset 2. Indica

#### 2.3. Dataset 3. Groundnut

#### 2.4. Dataset 4. Cotton

#### 2.5. Dataset 5. Disease

#### 2.6. Multi-Trait Kernel Model

#### 2.7. Evaluation of Prediction Performance

## 3. Results

#### 3.1. Dataset 1 Japonica

#### 3.2. Dataset 2 Indica

#### 3.3. Dataset 3 Groundnut

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**The prediction performance for every environment and across environments (Global) of the dataset 1 (Japonica) in terms of normalized root mean square error (NRMSE) under three methods of tuning (BO, GrS and NT) for the Japonica dataset. RE denotes relative efficiency. The RE in the rows corresponding to BO were computed dividing the NRMSE under NT by the NRMSE under BO. The RE in rows corresponding to GrS were computed dividing the NRMSE under NT by the NRMSE under GrS. RE in the rows corresponding to the NT strategy were computed dividing the NRMSE under GrS by the NRMSE under BO.

Tuning Type | Year | NRMSE_GC | NRMSE_GY | NRMSE_PH | NRMSE_PHR | NRMSE | RE |
---|---|---|---|---|---|---|---|

Bayesian Optimization | 2009 | 1.1256 | 0.9846 | 0.7289 | 0.8383 | 0.919350 | 1.0603959 |

Bayesian Optimization | 2010 | 0.9377 | 0.8181 | 0.6398 | 0.8298 | 0.806350 | 1.0849197 |

Bayesian Optimization | 2011 | 0.7741 | 0.8752 | 0.7574 | 0.8119 | 0.804650 | 1.0283353 |

Bayesian Optimization | 2012 | 0.8877 | 0.8585 | 0.6541 | 0.9253 | 0.831400 | 1.0304306 |

Bayesian Optimization | 2013 | 0.7327 | 0.8658 | 0.6643 | 0.8374 | 0.775050 | 1.0427714 |

Bayesian Optimization | Global | 0.4048 | 0.4919 | 0.4255 | 0.5328 | 0.463750 | 1.0310512 |

Grid Search | 2009 | 1.1246 | 1.0185 | 0.7283 | 0.8445 | 0.928975 | 1.0494093 |

Grid Search | 2010 | 0.9130 | 0.8299 | 0.6436 | 0.8270 | 0.803375 | 1.0889373 |

Grid Search | 2011 | 0.7787 | 0.8943 | 0.7585 | 0.8156 | 0.811775 | 1.0193095 |

Grid Search | 2012 | 0.8877 | 0.8689 | 0.6530 | 0.9298 | 0.834850 | 1.0261724 |

Grid Search | 2013 | 0.7305 | 0.9151 | 0.6578 | 0.8401 | 0.785875 | 1.0284078 |

Grid Search | Global | 0.4053 | 0.5049 | 0.4278 | 0.5348 | 0.468200 | 1.0212516 |

No Tuning | 2009 | 1.3539 | 0.9841 | 0.6954 | 0.8661 | 0.974875 | 1.0104694 |

No Tuning | 2010 | 1.1494 | 0.8527 | 0.6445 | 0.8527 | 0.874825 | 0.9963105 |

No Tuning | 2011 | 0.8013 | 0.9069 | 0.7661 | 0.8355 | 0.827450 | 1.0088548 |

No Tuning | 2012 | 0.9123 | 0.8961 | 0.6673 | 0.9511 | 0.856700 | 1.0041496 |

No Tuning | 2013 | 0.7988 | 0.9049 | 0.6741 | 0.8550 | 0.808200 | 1.0139668 |

No Tuning | Global | 0.4246 | 0.5088 | 0.4306 | 0.5486 | 0.478150 | 1.0095957 |

**Table A2.**The prediction performance for every environment and across environments (Global) of the dataset 2 (Indica) in terms of normalized root mean square error (NRMSE) under three methods (BO, GrS and NT) for the Indica dataset. RE denotes relative efficiency. The RE in rows corresponding to BO were computed dividing the NRMSE under NT by the NRMSE under BO. The RE in rows corresponding to GrS were computed dividing the NRMSE under NT by the NRMSE under GrS. Those RE in the rows corresponding to NT strategy were computed dividing the NRMSE under GrS by the NRMSE under BO.

Tuning Type | Year | NRMSE_GC | NRMSE_GY | NRMSE_PH | NRMSE_PHR | NRMSE | RE |
---|---|---|---|---|---|---|---|

Bayesian Optimization | 2010 | 0.9201 | 0.9234 | 0.4439 | 0.8195 | 0.776725 | 1.1207956 |

Bayesian Optimization | 2011 | 0.9305 | 0.8324 | 0.8707 | 0.8687 | 0.875575 | 1.0563344 |

Bayesian Optimization | 2012 | 0.9492 | 0.7529 | 0.6981 | 0.8116 | 0.802950 | 1.0391058 |

Bayesian Optimization | Global | 0.9287 | 0.7190 | 0.6117 | 0.8006 | 0.765000 | 1.0645425 |

Grid Search | 2010 | 0.9181 | 0.9154 | 0.4229 | 0.8253 | 0.770425 | 1.1299607 |

Grid Search | 2011 | 0.9206 | 0.8250 | 0.8728 | 0.8668 | 0.871300 | 1.0615173 |

Grid Search | 2012 | 0.9433 | 0.7542 | 0.6926 | 0.8004 | 0.797625 | 1.0460429 |

Grid Search | Global | 0.9248 | 0.7165 | 0.6070 | 0.8002 | 0.762125 | 1.0685583 |

No Tuning | 2010 | 0.9470 | 1.0435 | 0.5777 | 0.9140 | 0.870550 | 0.9918890 |

No Tuning | 2011 | 0.9390 | 0.8394 | 0.9642 | 0.9570 | 0.924900 | 0.9951175 |

No Tuning | 2012 | 0.9461 | 0.8031 | 0.7495 | 0.8387 | 0.834350 | 0.9933682 |

No Tuning | Global | 0.9409 | 0.7669 | 0.6816 | 0.8681 | 0.814375 | 0.9962418 |

**Table A3.**The prediction performance for every environment and across environments (Global) of the dataset 3 (Groundnut) in terms of normalized root mean square error (NRMSE) under three tuning methods (BO, GrS and NT) for Groundnut dataset. RE denotes relative efficiency. The RE in rows corresponding to BO were computed dividing the NRMSE under NT by the NRMSE under BO. While the RE in rows corresponding to GrS were computed dividing the NRMSE under NT by the NRMSE under GrS. While those RE in the rows corresponding to NT strategy were computed dividing the NRMSE under GrS by the NRMSE under BO.

Tuning Type | Environment | NRMSE_NPP | NRMSE_PYPP | NRMSE_SYPP | NRMSE_YPH | NRMSE | RE |
---|---|---|---|---|---|---|---|

Bayesian Optimization | ALIYARNAGAR_R15 | 0.8992 | 0.9442 | 0.9452 | 0.8242 | 0.9032 | 1.0139781 |

Bayesian Optimization | ICRISAT_R15 | 0.7872 | 0.7729 | 0.786 | 0.6701 | 0.75405 | 1.0458524 |

No Tuning | ALIYARNAGAR_R15 | 0.9152 | 0.9554 | 0.9597 | 0.833 | 0.915825 | 0.9879318 |

Bayesian Optimization | ICRISAT_PR15-16 | 0.9025 | 0.9547 | 0.9469 | 0.9191 | 0.9308 | 1.068194 |

Bayesian Optimization | JALGOAN_R15 | 0.8081 | 0.8361 | 0.8383 | 0.7674 | 0.812475 | 1.0423705 |

No Tuning | ICRISAT_PR15-16 | 0.9331 | 1.0378 | 1.0184 | 0.9878 | 0.994275 | 0.9975827 |

Grid Search | ALIYARNAGAR_R15 | 0.8902 | 0.9342 | 0.9337 | 0.8111 | 0.89230 | 1.0263645 |

Grid Search | ICRISAT_R15 | 0.7862 | 0.7755 | 0.7873 | 0.6737 | 0.755675 | 1.0436034 |

No Tuning | ICRISAT_R15 | 0.7866 | 0.823 | 0.8324 | 0.7125 | 0.788625 | 1.002155 |

Grid Search | ICRISAT_PR15-16 | 0.9026 | 0.9517 | 0.9441 | 0.9158 | 0.92855 | 1.0707824 |

Grid Search | JALGOAN_R15 | 0.8091 | 0.8377 | 0.8404 | 0.7671 | 0.813575 | 1.0409612 |

No Tuning | JALGOAN_R15 | 0.827 | 0.8753 | 0.8758 | 0.8095 | 0.8469 | 1.0013539 |

Bayesian Optimization | Global | 0.7726 | 0.7841 | 0.7952 | 0.7871 | 0.78475 | 1.0440268 |

Grid Search | Global | 0.7707 | 0.7825 | 0.7925 | 0.7845 | 0.78255 | 1.0469619 |

No Tuning | Global | 0.7912 | 0.8229 | 0.8324 | 0.8307 | 0.8193 | 0.9971966 |

## Appendix B

**Table A4.**The standard error of prediction performance for each year and across years (Global) of the dataset 1 (Japonica) in terms of normalized root mean square error (NRMSE) under three methods of tuning (BO, GrS and NT). RE denotes relative efficiency.

Tuning Type | Year | NRMSE_SE_GC | NRMSE_SE_GY | NRMSE_SE_PH | NRMSE_SE_PHR | NRMSE_SE |
---|---|---|---|---|---|---|

Bayesian Optimization | 2009 | 0.1457 | 0.0531 | 0.1201 | 0.0303 | 0.0436500 |

Bayesian Optimization | 2010 | 0.0475 | 0.0239 | 0.0580 | 0.0244 | 0.0192250 |

Bayesian Optimization | 2011 | 0.0344 | 0.0405 | 0.0606 | 0.0173 | 0.0191000 |

Bayesian Optimization | 2012 | 0.0176 | 0.0408 | 0.0630 | 0.0317 | 0.0191375 |

Bayesian Optimization | 2013 | 0.0594 | 0.0850 | 0.1139 | 0.0211 | 0.0349250 |

Bayesian Optimization | Global | 0.0251 | 0.0129 | 0.0184 | 0.0240 | 0.0100500 |

Grid Search | 2009 | 0.1326 | 0.0701 | 0.1150 | 0.0328 | 0.0438125 |

Grid Search | 2010 | 0.0378 | 0.0296 | 0.0580 | 0.0227 | 0.0185125 |

Grid Search | 2011 | 0.0353 | 0.0399 | 0.0619 | 0.0152 | 0.0190375 |

Grid Search | 2012 | 0.0182 | 0.0410 | 0.0623 | 0.0310 | 0.0190625 |

Grid Search | 2013 | 0.0599 | 0.1058 | 0.1102 | 0.0246 | 0.0375625 |

Grid Search | Global | 0.0266 | 0.0187 | 0.0186 | 0.0253 | 0.0111500 |

No Tuning | 2009 | 0.2366 | 0.0583 | 0.0931 | 0.0406 | 0.0535750 |

No Tuning | 2010 | 0.0825 | 0.0189 | 0.0522 | 0.0315 | 0.0231375 |

No Tuning | 2011 | 0.0325 | 0.0328 | 0.0587 | 0.0164 | 0.0175500 |

No Tuning | 2012 | 0.0122 | 0.0419 | 0.0702 | 0.0332 | 0.0196875 |

No Tuning | 2013 | 0.0614 | 0.0880 | 0.1179 | 0.0201 | 0.0359250 |

No Tuning | Global | 0.0246 | 0.0141 | 0.0162 | 0.0307 | 0.0107000 |

**Table A5.**The standard error of prediction performance for each year and across years (Global) of the dataset 2 (Indica) in terms of normalized root mean square error (NRMSE) under three methods of tuning (BO, GrS and NT). RE denotes relative efficiency.

Tuning Type | Year | NRMSE_SE_GC | NRMSE_SE_GY | NRMSE_SE_PH | NRMSE_SE_PHR | NRMSE_SE |
---|---|---|---|---|---|---|

Bayesian Optimization | 2010 | 0.0272 | 0.0275 | 0.0314 | 0.0428 | 0.0161125 |

Bayesian Optimization | 2011 | 0.0300 | 0.0300 | 0.0333 | 0.0574 | 0.0188375 |

Bayesian Optimization | 2012 | 0.0261 | 0.0125 | 0.0297 | 0.0457 | 0.0142500 |

Bayesian Optimization | Global | 0.0225 | 0.0201 | 0.0172 | 0.0292 | 0.0111250 |

Grid Search | 2010 | 0.0280 | 0.0264 | 0.0299 | 0.0481 | 0.0165500 |

Grid Search | 2011 | 0.0245 | 0.0270 | 0.0356 | 0.0557 | 0.0178500 |

Grid Search | 2012 | 0.0253 | 0.0133 | 0.0317 | 0.0398 | 0.0137625 |

Grid Search | Global | 0.0197 | 0.0184 | 0.0177 | 0.0295 | 0.0106625 |

No Tuning | 2010 | 0.0328 | 0.0168 | 0.0382 | 0.0528 | 0.0175750 |

No Tuning | 2011 | 0.0235 | 0.0271 | 0.0366 | 0.0521 | 0.0174125 |

No Tuning | 2012 | 0.0295 | 0.0077 | 0.0228 | 0.0387 | 0.0123375 |

No Tuning | Global | 0.0207 | 0.0157 | 0.0176 | 0.0273 | 0.0101625 |

**Table A6.**The standard error of prediction performance for each environment and across environments (Global) of the dataset 3 (Groundnut) in terms of normalized root mean square error (NRMSE) under three methods of tuning (BO, GrS and NT). RE denotes relative efficiency.

Tuning Type | Environment | NRMSE_SE_NPP | NRMSE_SE_PYPP | NRMSE_SE_SYPP | NRMSE_SE_YPH | NRMSE_SE |
---|---|---|---|---|---|---|

Bayesian Optimization | ALIYARNAGAR_R15 | 0.0269 | 0.0164 | 0.0185 | 0.0287 | 0.0113125 |

Bayesian Optimization | ICRISAT_PR15-16 | 0.0299 | 0.0342 | 0.0386 | 0.0283 | 0.0163750 |

Bayesian Optimization | ICRISAT_R15 | 0.0228 | 0.0282 | 0.0255 | 0.0255 | 0.0127500 |

Bayesian Optimization | JALGOAN_R15 | 0.0249 | 0.0089 | 0.0110 | 0.0169 | 0.0077125 |

Bayesian Optimization | Global | 0.0094 | 0.0105 | 0.0114 | 0.0222 | 0.0066875 |

Grid Search | ALIYARNAGAR_R15 | 0.0312 | 0.0228 | 0.0253 | 0.0310 | 0.0137875 |

Grid Search | ICRISAT_PR15-16 | 0.0322 | 0.0366 | 0.0403 | 0.0288 | 0.0172375 |

Grid Search | ICRISAT_R15 | 0.0223 | 0.0288 | 0.0268 | 0.0241 | 0.0127500 |

Grid Search | JALGOAN_R15 | 0.0246 | 0.0077 | 0.0086 | 0.0166 | 0.0071875 |

Grid Search | Global | 0.0106 | 0.0119 | 0.0126 | 0.0241 | 0.0074000 |

No Tuning | ALIYARNAGAR_R15 | 0.0294 | 0.0205 | 0.0202 | 0.0262 | 0.0120375 |

No Tuning | ICRISAT_PR15-16 | 0.0429 | 0.0555 | 0.0581 | 0.0452 | 0.0252125 |

No Tuning | ICRISAT_R15 | 0.0239 | 0.0198 | 0.0216 | 0.0176 | 0.0103625 |

No Tuning | JALGOAN_R15 | 0.0224 | 0.0086 | 0.0078 | 0.0195 | 0.0072875 |

No Tuning | Global | 0.0104 | 0.0113 | 0.0113 | 0.0205 | 0.0066875 |

## References

- Caamal-Pat, D.; Pérez-Rodríguez, P.; Crossa, J.; Velasco-Cruz, C.; Pérez-Elizalde, S.; Vázquez-Peña, M. lme4GS: An R-Package for Genomic Selection. Front. Genet.
**2021**, 12, 680569. [Google Scholar] [CrossRef] [PubMed] - Montesinos-López, O.A.; Montesinos-López, A.; Cano-Paez, B.; Hernández-Suárez, C.M.; Santana-Mancilla, P.C.; Crossa, J. A Comparison of Three Machine Learning Methods for Multivariate Genomic Prediction Using the Sparse Kernels Method (SKM) Library. Genes
**2022**, 13, 1494. [Google Scholar] [CrossRef] [PubMed] - Cordell, H.J. Epistasis: What it means, what it doesn’t mean, and statistical methods to detect it in humans. Hum. Mol. Genet.
**2002**, 11, 2463–2468. [Google Scholar] [CrossRef] [PubMed][Green Version] - Golan, D.; Rosset, S. Effective genetic-risk prediction using mixed models. Am. J. Hum. Genet.
**2014**, 95, 383–393. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gianola, D.; Fernando, R.L.; Stella, A. Genomic-assisted prediction of genetic value with semi parametric procedures. Genetics
**2006**, 173, 1761–1776. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gianola, D.; van Kaam, J.B.C.H.M. Reproducing kernel Hilbert spaces regression methods for genomic assisted prediction of quantitative traits. Genetics
**2008**, 178, 2289–2303. [Google Scholar] [CrossRef] [PubMed][Green Version] - Vapnik, V. The Nature of Statistical Learning Theory; Springer: New York, NY, USA, 1995. [Google Scholar]
- Long, N.; Gianola, D.; Rosa, G.J.; Weigel, K.A.; Kranis, A.; González-Recio, O. Radial basis function regression methods for predicting quantitative traits using SNP markers. Genet. Res.
**2010**, 92, 209–225. [Google Scholar] [CrossRef] [PubMed][Green Version] - Crossa, J.; de los Campos, G.; Pérez, P.; Gianola, D.; Burgueño, J.; Araus, J.L. Prediction of genetic values of quantitative traits in plant breeding using pedigree and molecular markers. Genetics
**2010**, 186, 713–724. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cuevas, J.; Montesinos-López, O.A.; Juliana, P.; Guzmán, C.; Pérez-Rodríguez, P.; González-Bucio, J.; Burgueño, J.; Montesinos-López, A.; Crossa, J. Deep Kernel for Genomic and Near Infrared Predictions in Multi-environment Breeding Trials. G3-Genes Genomes Genet.
**2019**, 9, 2913–2924. [Google Scholar] [CrossRef] [PubMed][Green Version] - Tusell, L.; Pérez-Rodríguez, P.; Wu, S.F.X.-L.; Gianola, D. Genome-enabled methods for predicting litter size in pigs: A comparison. Animal
**2013**, 7, 1739–1749. [Google Scholar] [CrossRef] [PubMed] - Morota, G.; Koyama, M.; Rosa, G.J.M.; Weigel, K.A.; Gianola, D. Predicting complex traits using a diffusion kernel on genetic markers with an application to dairy cattle and wheat data. Genet. Sel. Evol.
**2013**, 45, 17. [Google Scholar] [CrossRef] [PubMed][Green Version] - Arojju, S.K.; Cao, M.; Trolove, M.; Barrett, B.A.; Inch, C.; Eady, C.; Stewart, A.; Faville, M.J. Multi-Trait Genomic Prediction Improves Predictive Ability for Dry Matter Yield and Water-Soluble Carbohydrates in Perennial Ryegrass. Front. Plant Sci.
**2020**, 11, 1197. [Google Scholar] [CrossRef] [PubMed] - Montesinos-López, O.A.; Montesinos-López, A.; Crossa, J.; Cuevas, J.; Montesinos-López, J.C.; Salas-Gutiérrez, Z.; Lillemo, M.; Philomin, J.; Singh, R. A Bayesian Genomic Multi-output Regressor Stacking Model for Predicting Multi-trait Multi-environment Plant Breeding Data. G3-Genes Genomes Genet.
**2019**, 9, 3381–3393. [Google Scholar] [CrossRef] [PubMed][Green Version] - Monteverde, E.; Gutierrez, L.; Blanco, P.; Pérez de Vida, F.; Rosas, J.E.; Bonnecarrère, V.; Quero, G.; McCouch, S. Integrating Molecular Markers and Environmental Covariates To Interpret Genotype by Environment Interaction in Rice (Oryza sativa L.) Grown in Subtropical Areas. G3 Genes Genomes Genet.
**2019**, 9, 1519–1531. [Google Scholar] [CrossRef][Green Version] - Pandey, M.K.; Chaudhari, S.; Jarquin, D.; Janila, P.; Crossa, J.; Patil, S.C.; Sundravadana, S.; Khare, D.; Bhat, R.S.; Radhakrishnan, T.; et al. Genome-based trait prediction in multi- environment breeding trials in groundnut. Theor. Appl. Genet.
**2020**, 133, 3101–3117. [Google Scholar] [CrossRef] [PubMed] - Gapare, W.; Liu, S.; Conaty, W.; Zhu, Q.H.; Gillespie, V.; Llewellyn, D.; Stiller, W.; Wilson, I. Historical Datasets Support Genomic Selection Models for the Prediction of Cotton Fiber Quality Phenotypes Across Multiple Environments. G3 Genes Genomes Genet.
**2018**, 8, 1721–1732. [Google Scholar] [CrossRef] [PubMed][Green Version] - VanRaden, P.M. Efficient Methods to Compute Genomic Predictions. J. Dairy Sci.
**2008**, 91, 4414–4423. [Google Scholar] [CrossRef] [PubMed][Green Version] - Montesinos-López, O.A.; Montesinos-López, A.; Crossa, J. (Eds.) Multivariate Statistical Machine Learning Methods for Genomic Prediction; Springer International Publishing: Cham, Switzerland, 2022; ISBN 978-3-030-89010-0. [Google Scholar]
- Montesinos-López, O.A.; Carter, A.H.; Bernal-Sandoval, D.A.; Cano-Paez, B.; Montesinos-López, A.; Crossa, J. A Comparison Between Three Tuning Strategies for Gaussian kernels in the Context of Univariate Genomic Prediction. Genes, 2022; submitted for publication. [Google Scholar]
- R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2022. [Google Scholar]
- Pérez, P.; de los Campos, G. Genome-Wide Regression and Prediction with the BGLR Statistical Package. Genetics
**2014**, 198, 483–495. [Google Scholar] [CrossRef] [PubMed] - Meuwissen, T.H.E.; Hayes, B.J.; Goddard, M.E. Prediction of total genetic value using genome-wide dense marker maps. Genetics
**2001**, 157, 1819–1829. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**A**) The prediction performance for dataset 1, Japonica dataset in terms of normalized root mean squared error (NRMSE) for each year (2009–2013), across years (Global), and across traits with three strategies of tuning (BO, GrS and NT) under 7 Fold Cross-Validation (7FCV). (

**B**) The relative efficiency for each environment (2009–2013) and across environments (Global) and across traits (CG, GY, PH and PHR) with three strategies of tuning (BO, GrS and NT) under 7 Fold Cross-Validation (7FCV). (

**C**) The prediction performance in terms of normalized root mean squared error (NRMSE) for each trait (CG, GY, PH and PHR) across years with three strategies of tuning (BO, GrS and NT) under 7 Fold Cross-Validation (7FCV). (

**D**) The relative efficiency for each trait (CG, GY, PH and PHR) across years with three strategies of tuning (BO, GrS and NT) under 7 Fold Cross-Validation (7FCV). When RE > 1 the denominator method outperforms the numerator in terms of prediction performance.

**Figure 2.**(

**A**) Prediction performance for dataset 2, Indica dataset in terms of normalized root mean squared error (NRMSE) for each year (2010–2012) across traits (CG, GY, PH and PHR), across years and across traits (Global) with three strategies of tuning (BO, GrS and NT) under 7 Fold Cross-Validation (7FCV). (

**B**) The relative efficiency for each environment (2010–2012) across traits and across environments and across traits (Global) with three strategies of tuning (BO, GrS and NT) under 7 Fold Cross-Validation (7FCV). (

**C**) Prediction performance in terms of normalized root mean squared error (NRMSE) for each trait (CG, GY, PH and PHR) across years with three strategies of tuning (BO, GrS and NT) under 7 Fold Cross-Validation (7FCV). (

**D**) The relative efficiency for each trait (CG, GY, PH and PHR) across environments with three strategies of tuning (BO, GrS and NT) under 7 Fold Cross-Validation (7FCV). When RE > 1 the denominator method outperforms the numerator in terms of prediction performance.

**Figure 3.**(

**A**) The prediction performance for dataset 3, Groundnut dataset in terms of normalized root mean squared error (NRMSE) for each environment across traits (ALIYARNAGAR_R15, ICRISAT_PR15-16 ICRISAT_R15 and JALGOAN_R15), across environments and traits (Global) with three tuning strategies (BO, GrS and NT) under 7 Fold Cross-Validation (7FCV). (

**B**) The relative efficiency for each environment across traits (ALIYARNAGAR_R15, ICRISAT_PR15-16 ICRISAT_R15 and JALGOAN_R15), across environments and traits (Global) with three tuning strategies (BO, GrS and NT) under 7 Fold Cross-Validation (7FCV). (

**C**) The prediction performance in terms of normalized root mean squared error (NRMSE) for each trait (NPP, PYPP, SYPP, YPH)) across environments with three tuning strategies (BO, GrS and NT) under 7FCV. (

**D**) The relative efficiency for each trait (NPP, PYPP, SYPP, YPH) across environments with three tuning strategies (BO, GrS and NT) under 7 Fold Cross-Validation (7FCV). When RE > 1, the denominator method outperforms the numerator in terms of prediction performance.

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## Share and Cite

**MDPI and ACS Style**

Kismiantini; Montesinos-López, A.; Cano-Páez, B.; Montesinos-López, J.C.; Chavira-Flores, M.; Montesinos-López, O.A.; Crossa, J. A Multi-Trait Gaussian Kernel Genomic Prediction Model under Three Tunning Strategies. *Genes* **2022**, *13*, 2279.
https://doi.org/10.3390/genes13122279

**AMA Style**

Kismiantini, Montesinos-López A, Cano-Páez B, Montesinos-López JC, Chavira-Flores M, Montesinos-López OA, Crossa J. A Multi-Trait Gaussian Kernel Genomic Prediction Model under Three Tunning Strategies. *Genes*. 2022; 13(12):2279.
https://doi.org/10.3390/genes13122279

**Chicago/Turabian Style**

Kismiantini, Abelardo Montesinos-López, Bernabe Cano-Páez, J. Cricelio Montesinos-López, Moisés Chavira-Flores, Osval A. Montesinos-López, and José Crossa. 2022. "A Multi-Trait Gaussian Kernel Genomic Prediction Model under Three Tunning Strategies" *Genes* 13, no. 12: 2279.
https://doi.org/10.3390/genes13122279