# Automated Fitting Process Using Robust Reliable Weighted Average on Near Infrared Spectral Data Analysis

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Partial Least Squares Regression

#### 2.2. Partial Robust M-Regression

#### 2.3. Weighted Average PLS

## 3. Robust Reliable Weighted Average

## 4. Monte Carlo Simulation Study

^{2}), and Standard Error (SE). The RMSE measures the absolute error of the predicted model; R

^{2}is the proportion of variation in the data summarized by the model and indicates the reliability of the goodness of fit for model; and SE measures the uncertainty in the prediction. Here, the RPD parameter has no more used because it is not different than R

^{2}to classify the model is poor or not [44]. Using the classical PLSR, the RMSECV which is the RMSE obtained through cross-validation, is calculated, along with the increasing number of PLS components. The RMSEP value is the RMSE obtained using the fitted model. In the simulation study, the maximum number of PLS components used was limited up to 20. Some different scenarios were applied to see the stability of classical PLSR model based on sample size, number of predictors, number of important variables, and the contamination of outlier and high leverage points in the dataset. In Figure 1, with no contamination in the data it can be seen that using small sample size ($n$ = 60), small number of predictors ($m$ = 41), and 10% relevant variable ($IV$ = 10%) the discrepancy between RMSECV and RMSEP is about two to five times. While using higher number of predictors ($m$ = 101) the discrepancy then become larger. Another scenario using bigger sample size ($n$ = 200), small number of predictors ($m$ = 41), and 30% relevant variable ($IV$ = 30%) the discrepancy between RMSECV and RMSEP relatively smaller. While using higher number of predictors ($m$ = 101) the discrepancy increases about two times. This shows that the classical PLS become instable and loss it accuracy when the number of sample size is small and number of predictor higher than sample size. In addition, with less number of relevant variable in the predictor variable also impacts to decrease the model accuracy. Using bigger sample size (for example $n$ = 200) as the number of PLS components increases the discrepancy between RMSECV and RMSEP become smaller hence improve the model accuracy and reliability. The rule is the gap between RMSEC and RMSEP values should very small and close to 0. This condition guarantees the reliability of the calibrated model and prevent the model becomes over-under fitting.

^{2}values. The classical PLS fails to find the optimal number of PLS components due to the influence of 5–10% contamination of outliers and HLP during the fitting process. The WA-PLS also fails to fit the predicted model due to the impact of the contamination. The proposed RRWA-PLS consistently has the lowest RMSE, SE, and better R

^{2}compared to the other methods, irrespective of the sample sizes, number of important variables, and percentages of contamination of outliers and HLP in the dataset.

## 5. NIR Spectral Dataset

#### 5.1. Oil to Dry Mesocarp

#### 5.2. Oil to Wet Mesocarp

^{2}, and SE values in Table 3, it can be concluded that the proposed RRWA-PLS produces better accuracy than the other methods. The modified weight in MWA-PLS has improved the accuracy of the predicted model; however, it cannot outperform the RRWA-PLS. The robust weighted-average strategy prevents the PLSR model from depending on the specific number of PLS components used in the fitting process.

#### 5.3. Fat Fatty Acids

## 6. Reliability Values

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Declaration

## References

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**Figure 1.**The RMSECV and RMSEP of the classical PLSR on the simulated data with no contamination of outlier and high leverage points.

**Figure 2.**The RMSECV and RMSEP of the classical PLSR on the simulated data with contamination of outlier and high leverage points.

**Figure 3.**SEP values in the RRWA-PLS using different approach on the simulated data with contamination of outlier and HLP.

**Figure 4.**The mean weights of the WA-PLS and RRWA-PLS on the simulated data with and without contamination of outlier and HLP.

**Figure 5.**The RMSEP values of the classical PLS, WA-PLS, and RRWA-PLS on the simulated data with and without contamination of outlier and HLP.

**Figure 6.**Predicted against actual values on the simulated data using PLS with opt., WA-PLS, MWA-PLS, and RRWA-PLS.

**Figure 7.**The RMSE of the fitted PLSR through cross validation and the prediction ability using %ODM dataset.

**Figure 10.**The RMSE of the fitted PLSR through cross validation and the prediction ability using %OWM dataset.

**Figure 13.**The RMSE of the fitted PLSR through cross validation and the prediction ability using %FFA dataset.

**Figure 16.**Reliability values using RRWA-PLS method on different datasets: (

**a**) artificial data; NIR spectral dataset: (

**b**) %ODM; (

**c**) %OWM; (

**d**) %FFA.

**Table 1.**The RMSE, R

^{2}, and SE in the weighted methods using the Monte Carlo Simulation with different dataset scenarios.

Outlier and HLP | n | m | IV | Methods | nPLS | RMSE | R^{2} | SE |
---|---|---|---|---|---|---|---|---|

No outlier and HLP | 60 | 41 | 10% | PLS with opt. | 9 | 2.752 | 0.980 | 2.776 |

WA-PLS | 15 | 2.496 | 0.984 | 2.517 | ||||

MWA-PLS | 15 | 3.318 | 0.972 | 3.305 | ||||

RRWA-PLS | 15 | 2.495 | 0.983 | 2.497 | ||||

60 | 101 | 10% | PLS with opt. | 3 | 9.348 | 0.903 | 9.427 | |

WA-PLS | 15 | 2.759 | 0.993 | 2.782 | ||||

MWA-PLS | 15 | 8.181 | 0.931 | 8.250 | ||||

RRWA-PLS | 15 | 2.702 | 0.960 | 2.708 | ||||

60 | 201 | 10% | PLS with opt. | 1 | 18.717 | 0.859 | 18.875 | |

WA-PLS | 15 | 2.333 | 0.998 | 2.352 | ||||

MWA-PLS | 15 | 5.542 | 0.908 | 5.543 | ||||

RRWA-PLS | 15 | 2.460 | 0.984 | 2.480 | ||||

200 | 41 | 30% | PLS with opt. | 6 | 6.707 | 0.969 | 6.723 | |

WA-PLS | 15 | 6.532 | 0.970 | 6.548 | ||||

MWA-PLS | 15 | 6.799 | 0.968 | 6.816 | ||||

RRWA-PLS | 15 | 6.594 | 0.970 | 6.610 | ||||

200 | 101 | 30% | PLS with opt. | 10 | 7.926 | 0.980 | 7.946 | |

WA-PLS | 15 | 7.915 | 0.981 | 7.935 | ||||

MWA-PLS | 15 | 12.621 | 0.951 | 12.653 | ||||

RRWA-PLS | 15 | 7.860 | 0.988 | 7.862 | ||||

200 | 201 | 30% | PLS with opt. | 9 | 12.995 | 0.973 | 13.028 | |

WA-PLS | 15 | 9.237 | 0.988 | 9.260 | ||||

MWA-PLS | 15 | 15.163 | 0.965 | 15.201 | ||||

RRWA-PLS | 15 | 9.582 | 0.985 | 9.601 | ||||

400 | 41 | 50% | PLS with opt. | 4 | 9.213 | 0.967 | 9.224 | |

WA-PLS | 15 | 9.062 | 0.968 | 9.073 | ||||

MWA-PLS | 15 | 9.522 | 0.965 | 9.534 | ||||

RRWA-PLS | 15 | 9.108 | 0.968 | 9.109 | ||||

400 | 101 | 50% | PLS with opt. | 7 | 12.727 | 0.972 | 12.733 | |

WA-PLS | 15 | 12.611 | 0.973 | 12.627 | ||||

MWA-PLS | 15 | 18.812 | 0.939 | 18.836 | ||||

RRWA-PLS | 15 | 12.787 | 0.972 | 12.803 | ||||

400 | 201 | 50% | PLS with opt. | 10 | 14.244 | 0.981 | 14.262 | |

WA-PLS | 15 | 14.343 | 0.981 | 14.361 | ||||

MWA-PLS | 15 | 31.060 | 0.910 | 31.099 | ||||

RRWA-PLS | 15 | 14.153 | 0.983 | 14.172 | ||||

With outlier and HLP (5%) | 60 | 41 | 10% | PLS with opt. | 0 | N/A | N/A | N/A |

WA-PLS | 15 | 24.139 | 0.869 | 24.343 | ||||

MWA-PLS | 15 | 3.160 | 0.975 | 3.188 | ||||

RRWA-PLS | 15 | 3.042 | 0.976 | 3.069 | ||||

60 | 101 | 10% | PLS with opt. | 0 | N/A | N/A | N/A | |

WA-PLS | 15 | 16.559 | 0.892 | 16.699 | ||||

MWA-PLS | 15 | 9.156 | 0.931 | 9.241 | ||||

RRWA-PLS | 15 | 5.068 | 0.984 | 5.116 | ||||

60 | 201 | 10% | PLS with opt. | 0 | N/A | N/A | N/A | |

WA-PLS | 15 | 15.156 | 0.998 | 15.284 | ||||

MWA-PLS | 15 | 9.500 | 0.936 | 9.591 | ||||

RRWA-PLS | 15 | 8.580 | 0.973 | 8.662 | ||||

200 | 41 | 30% | PLS with opt. | 1 | 151.317 | 0.494 | 151.697 | |

WA-PLS | 15 | 175.959 | 0.603 | 176.400 | ||||

MWA-PLS | 15 | 6.441 | 0.970 | 6.458 | ||||

RRWA-PLS | 15 | 6.267 | 0.971 | 6.284 | ||||

200 | 101 | 30% | PLS with opt. | 2 | 331.650 | 0.734 | 332.482 | |

WA-PLS | 15 | 258.614 | 0.835 | 259.263 | ||||

MWA-PLS | 15 | 10.679 | 0.960 | 10.707 | ||||

RRWA-PLS | 15 | 8.195 | 0.976 | 8.217 | ||||

200 | 201 | 30% | PLS with opt. | 1 | 462.150 | 0.855 | 462.307 | |

WA-PLS | 15 | 226.599 | 0.969 | 227.167 | ||||

MWA-PLS | 15 | 17.791 | 0.952 | 17.839 | ||||

RRWA-PLS | 15 | 11.602 | 0.979 | 11.634 | ||||

400 | 41 | 50% | PLS with opt. | 2 | 304.843 | 0.516 | 305.225 | |

WA-PLS | 15 | 336.519 | 0.533 | 336.941 | ||||

MWA-PLS | 15 | 8.841 | 0.964 | 8.853 | ||||

RRWA-PLS | 15 | 8.383 | 0.968 | 8.394 | ||||

400 | 101 | 50% | PLS with opt. | 2 | 569.727 | 0.718 | 570.441 | |

WA-PLS | 15 | 537.184 | 0.776 | 537.857 | ||||

MWA-PLS | 15 | 17.678 | 0.941 | 17.702 | ||||

RRWA-PLS | 15 | 12.664 | 0.970 | 12.681 | ||||

400 | 201 | 50% | PLS with opt. | 2 | 808.964 | 0.836 | 809.977 | |

WA-PLS | 15 | 620.385 | 0.899 | 621.161 | ||||

MWA-PLS | 15 | 29.338 | 0.920 | 29.377 | ||||

RRWA-PLS | 15 | 17.163 | 0.973 | 17.186 | ||||

With outlier and HLP (20%) | 60 | 41 | 10% | PLS with opt. | 2 | 94.896 | 0.718 | 95.697 |

WA-PLS | 15 | 72.625 | 0.903 | 73.238 | ||||

MWA-PLS | 15 | 9.731 | 0.878 | 9.825 | ||||

RRWA-PLS | 15 | 8.689 | 0.897 | 8.774 | ||||

60 | 101 | 10% | PLS with opt. | 2 | 121.598 | 0.872 | 122.624 | |

WA-PLS | 15 | 29.488 | 0.905 | 29.737 | ||||

MWA-PLS | 15 | 12.795 | 0.932 | 12.924 | ||||

RRWA-PLS | 15 | 10.488 | 0.934 | 10.596 | ||||

60 | 201 | 10% | PLS with opt. | 2 | 209.076 | 0.721 | 210.841 | |

WA-PLS | 15 | 26.243 | 0.899 | 26.464 | ||||

MWA-PLS | 15 | 27.206 | 0.791 | 27.496 | ||||

RRWA-PLS | 15 | 25.145 | 0.878 | 25.204 | ||||

200 | 41 | 30% | PLS with opt. | 1 | 254.290 | 0.719 | 254.928 | |

WA-PLS | 15 | 237.919 | 0.783 | 238.516 | ||||

MWA-PLS | 15 | 7.848 | 0.956 | 7.872 | ||||

RRWA-PLS | 15 | 7.383 | 0.961 | 7.406 | ||||

200 | 101 | 30% | PLS with opt. | 1 | 438.504 | 0.855 | 439.604 | |

WA-PLS | 15 | 353.163 | 0.928 | 354.049 | ||||

MWA-PLS | 15 | 16.810 | 0.911 | 16.863 | ||||

RRWA-PLS | 15 | 16.105 | 0.924 | 16.155 | ||||

200 | 201 | 30% | PLS with opt. | 2 | 692.302 | 0.792 | 693.037 | |

WA-PLS | 15 | 294.979 | 0.799 | 295.719 | ||||

MWA-PLS | 15 | 121.881 | 0.740 | 122.262 | ||||

RRWA-PLS | 15 | 34.982 | 0.891 | 35.091 | ||||

400 | 41 | 50% | PLS with opt. | 1 | 443.979 | 0.740 | 444.535 | |

WA-PLS | 15 | 396.425 | 0.767 | 396.921 | ||||

MWA-PLS | 15 | 10.339 | 0.957 | 10.356 | ||||

RRWA-PLS | 15 | 10.059 | 0.958 | 10.074 | ||||

400 | 101 | 50% | PLS with opt. | 1 | 773.558 | 0.865 | 774.527 | |

WA-PLS | 15 | 655.858 | 0.903 | 656.679 | ||||

MWA-PLS | 15 | 23.244 | 0.912 | 23.281 | ||||

RRWA-PLS | 15 | 23.066 | 0.913 | 23.102 | ||||

400 | 201 | 50% | PLS with opt. | 1 | 944.986 | 0.792 | 945.425 | |

WA-PLS | 15 | 803.520 | 0.796 | 804.526 | ||||

MWA-PLS | 15 | 40.656 | 0.859 | 40.720 | ||||

RRWA-PLS | 15 | 35.121 | 0.894 | 35.176 |

Dataset | Methods | nPLS | RMSEP | R^{2} | SE |
---|---|---|---|---|---|

%ODM | PLS with opt. | 27 | 3.139 | 0.648 | 3.141 |

WA-PLS | 30 | 3.316 | 0.603 | 3.317 | |

MWA-PLS | 30 | 3.315 | 0.644 | 3.317 | |

RRWA-PLS | 30 | 3.071 | 0.661 | 3.072 |

Dataset | Methods | nPLS | RMSEP | R^{2} | SE |
---|---|---|---|---|---|

%OWM | PLS with opt. | 22 | 4.442 | 0.668 | 4.444 |

WA-PLS | 30 | 4.520 | 0.672 | 4.522 | |

MWA-PLS | 30 | 4.239 | 0.708 | 4.241 | |

RRWA-PLS | 30 | 4.185 | 0.718 | 4.187 |

Dataset | Methods | nPLS | RMSEP | R^{2} | SE |
---|---|---|---|---|---|

%FFA | PLS with opt. | 27 | 0.287 | 0.729 | 0.288 |

WA-PLS | 30 | 0.324 | 0.658 | 0.324 | |

MWA-PLS | 30 | 0.311 | 0.683 | 0.312 | |

RRWA-PLS | 30 | 0.275 | 0.747 | 0.276 |

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Silalahi, D.D.; Midi, H.; Arasan, J.; Mustafa, M.S.; Caliman, J.-P.
Automated Fitting Process Using Robust Reliable Weighted Average on Near Infrared Spectral Data Analysis. *Symmetry* **2020**, *12*, 2099.
https://doi.org/10.3390/sym12122099

**AMA Style**

Silalahi DD, Midi H, Arasan J, Mustafa MS, Caliman J-P.
Automated Fitting Process Using Robust Reliable Weighted Average on Near Infrared Spectral Data Analysis. *Symmetry*. 2020; 12(12):2099.
https://doi.org/10.3390/sym12122099

**Chicago/Turabian Style**

Silalahi, Divo Dharma, Habshah Midi, Jayanthi Arasan, Mohd Shafie Mustafa, and Jean-Pierre Caliman.
2020. "Automated Fitting Process Using Robust Reliable Weighted Average on Near Infrared Spectral Data Analysis" *Symmetry* 12, no. 12: 2099.
https://doi.org/10.3390/sym12122099