# 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 |

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**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