# Introductory Engineering Mathematics Students’ Weighted Score Predictions Utilising a Novel Multivariate Adaptive Regression Spline Model

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## Abstract

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## 1. Introduction

## 2. Theoretical Overview and Methodology

#### 2.1. Objective Model: Multivariate Adaptive Regression Splines (MARS)

#### 2.2. The Benchmark Model 1: Kernel Ridge Regression (KRR)

#### 2.3. The Benchmark Model 2: k-Nearest Neighbour (KNN)

#### 2.4. The Benchmark Model 3: Decision Tree (DT)

## 3. Research Context, Project Design, and Model Performance Criteria

#### 3.1. Engineering Mathematics Student Performance Data

#### 3.2. Model Development Stages

#### 3.3. Performance Evaluation Criteria

## 4. Results and Discussion

## 5. Further Discussion, Limitations of This Work, and Future Research Direction

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MARS | multivariate adaptive regression splines |

KNN | k-nearest neighbour |

KRR | kernal ridge regression |

DTR | decision tree regression |

MOOCs | massive open online courses |

SVM | support vector machine |

$GCV$ | generalized cross-validation |

$BF$ | basis function |

$MSE$ | mean square error |

ADNG | Associate Degree of Engineering |

B.CON | Bachelor of Construction Management |

$A1$ | Assignment 1 |

$A2$ | Assignment 2 |

$Q1$ | Quiz 1 |

$Q2$ | Quiz 2 |

$EX$ | examination score |

$WS$ | weighted score |

$RMSE$ | root mean square error |

$MAE$ | mean absolute error |

$WI$ | Willmott’s index |

$NSE$ | Nash–Sutcliffe coefficient |

$LM$ | Legates and McCabe’s index |

$RRMSE$ | relative RMSE |

$RMAE$ | relative MAE |

$W{S}_{obs}$ | observed (real) weighted score |

$W{S}_{pred}$ | predicted weighted score |

${U}_{95}$ | expanded uncertainty |

r | correlation coefficient |

${r}^{2}$ | coefficient of determination |

DR | discrepancy ratio |

ECDF | empirical cumulative distribution function |

|$PE$| | predicted error |

## References

- Curran, C. Strategies for E-Learning in Universities; University of California: Sacramento, CA, USA, 2004. [Google Scholar]
- Nguyen-Huy, T.; Deo, R.C.; Khan, S.; Devi, A.; Adeyinka, A.A.; Apan, A.A.; Yaseen, Z.M. Student Performance Predictions for Advanced Engineering Mathematics Course With New Multivariate Copula Models. IEEE Access
**2022**, 10, 45112–45136. [Google Scholar] [CrossRef] - Deo, R.C.; Yaseen, Z.M.; Al-Ansari, N.; Nguyen-Huy, T.; Langlands, T.A.M.; Galligan, L. Modern artificial intelligence model development for undergraduate student performance prediction: An investigation on engineering mathematics courses. IEEE Access
**2020**, 8, 136697–136724. [Google Scholar] [CrossRef] - Alhothali, A.; Albsisi, M.; Assalahi, H.; Aldosemani, T. Predicting Student Outcomes in Online Courses Using Machine Learning Techniques: A Review. Sustainability
**2022**, 14, 6199. [Google Scholar] [CrossRef] - Cendon, E. Lifelong learning at universities: Future perspectives for teaching and learning. J. New Approaches Educ. Res.
**2018**, 7, 81–87. [Google Scholar] [CrossRef] - Alrashdi, S.M.; Elshaiekh, N.E.M. Designing an IoT framework to improve student assessment performance in the Oman educational portal. Int. J. Innov. Digit. Econ. (IJIDE)
**2022**, 13, 1–12. [Google Scholar] [CrossRef] - Eudaley, S.T.; Farland, M.Z.; Melton, T.; Brooks, S.P.; Heidel, R.E.; Franks, A.S. Student Performance With Graded vs. Ungraded Readiness Assurance Tests in a Team-Based Learning Elective. Am. J. Pharm. Educ.
**2022**, 86. [Google Scholar] [CrossRef] [PubMed] - Oscarson, M.; Apelgren, B.M. Mapping language teachers’ conceptions of student assessment procedures in relation to grading: A two-stage empirical inquiry. System
**2011**, 39, 2–16. [Google Scholar] [CrossRef] - Leighton, J.; Gierl, M. Cognitive Diagnostic Assessment for Education: Theory and Applications; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Petscher, Y.; Schatschneider, C. A simulation study on the performance of the simple difference and covariance-adjusted scores in randomized experimental designs. J. Educ. Meas.
**2011**, 48, 31–43. [Google Scholar] [CrossRef] [PubMed] - Burgis-Kasthala, S.; Elmitt, N.; Smyth, L.; Moore, M. Predicting future performance in medical students. A longitudinal study examining the effects of resilience on low and higher performing students. Med Teach.
**2019**, 41, 1184–1191. [Google Scholar] [CrossRef] [PubMed] - Patil, P.; Hiremath, R. Big Data Mining—Analysis and Prediction of Data, Based on Student Performance. In Pervasive Computing and Social Networking; Springer: Berlin/Heidelberg, Germany, 2022; pp. 201–215. [Google Scholar]
- Stecker, P.M.; Fuchs, L.S.; Fuchs, D. Using curriculum-based measurement to improve student achievement: Review of research. Psychol. Sch.
**2005**, 42, 795–819. [Google Scholar] [CrossRef] - Mitrovic, A. Investigating students’ self-assessment skills. In Proceedings of the International Conference on User Modeling; Springer: Berlin/Heidelberg, Germany, 2001; pp. 247–250. [Google Scholar]
- Guzmán, E.; Conejo, R.; Pérez-de-la Cruz, J.L. Improving student performance using self-assessment tests. IEEE Intell. Syst.
**2007**, 22, 46–52. [Google Scholar] [CrossRef] - Do, Q.H.; Chen, J.F. A comparative study of hierarchical ANFIS and ANN in predicting student academic performance. WSEAS Trans. Inf. Sci. Appl.
**2013**, 10, 396–405. [Google Scholar] - Yusof, N.; Zin, N.A.M.; Yassin, N.M.; Samsuri, P. Evaluation of Student’s Performance and Learning Efficiency based on ANFIS. In Proceedings of the 2009 International Conference of Soft Computing and Pattern Recognition, Malacca, Malaysia, 4–7 December 2009; pp. 460–465. [Google Scholar]
- Alkhasawneh, R.; Hobson, R. Modeling student retention in science and engineering disciplines using neural networks. In Proceedings of the 2011 IEEE Global Engineering Education Conference (EDUCON), Amman, Jordan, 4–6 April 2011; pp. 660–663. [Google Scholar]
- Guarín, C.E.L.; Guzmán, E.L.; González, F.A. A model to predict low academic performance at a specific enrollment using data mining. IEEE Rev. Iberoam. De Tecnol. Del Aprendiz.
**2015**, 10, 119–125. [Google Scholar] - Al-Shehri, H.; Al-Qarni, A.; Al-Saati, L.; Batoaq, A.; Badukhen, H.; Alrashed, S.; Alhiyafi, J.; Olatunji, S.O. Student performance prediction using support vector machine and k-nearest neighbor. In Proceedings of the 2017 IEEE 30th Canadian Conference on Electrical and Computer Engineering (CCECE), Windsor, ON, Canada, 30 April–3 May 2017; pp. 1–4. [Google Scholar]
- Alshabandar, R.; Hussain, A.; Keight, R.; Khan, W. Students performance prediction in online courses using machine learning algorithms. In Proceedings of the 2020 International Joint Conference on Neural Networks (IJCNN), Glasgow, UK, 19–24 July 2020; pp. 1–7. [Google Scholar]
- Czibula, G.; Mihai, A.; Crivei, L.M. S PRAR: A novel relational association rule mining classification model applied for academic performance prediction. Procedia Comput. Sci.
**2019**, 159, 20–29. [Google Scholar] [CrossRef] - Goga, M.; Kuyoro, S.; Goga, N. A recommender for improving the student academic performance. Procedia-Soc. Behav. Sci.
**2015**, 180, 1481–1488. [Google Scholar] [CrossRef] - Fariba, T.B. Academic performance of virtual students based on their personality traits, learning styles and psychological well being: A prediction. Procedia-Soc. Behav. Sci.
**2013**, 84, 112–116. [Google Scholar] [CrossRef] - Taylan, O.; Karagözoğlu, B. An adaptive neuro-fuzzy model for prediction of student’s academic performance. Comput. Ind. Eng.
**2009**, 57, 732–741. [Google Scholar] [CrossRef] - Ashraf, M.; Zaman, M.; Ahmed, M. An intelligent prediction system for educational data mining based on ensemble and filtering approaches. Procedia Comput. Sci.
**2020**, 167, 1471–1483. [Google Scholar] [CrossRef] - Pallathadka, H.; Wenda, A.; Ramirez-Asís, E.; Asís-López, M.; Flores-Albornoz, J.; Phasinam, K. Classification and prediction of student performance data using various machine learning algorithms. Mater. Today Proc. 2021; in press. [Google Scholar] [CrossRef]
- Mubarak, A.A.; Cao, H.; Hezam, I.M. Deep analytic model for student dropout prediction in massive open online courses. Comput. Electr. Eng.
**2021**, 93, 107271. [Google Scholar] [CrossRef] - Bravo-Agapito, J.; Romero, S.J.; Pamplona, S. Early prediction of undergraduate Student’s academic performance in completely online learning: A five-year study. Comput. Hum. Behav.
**2021**, 115, 106595. [Google Scholar] [CrossRef] - Zeineddine, H.; Braendle, U.; Farah, A. Enhancing prediction of student success: Automated machine learning approach. Comput. Electr. Eng.
**2021**, 89, 106903. [Google Scholar] [CrossRef] - Pandey, M.; Taruna, S. Towards the integration of multiple classifier pertaining to the Student’s performance prediction. Perspect. Sci.
**2016**, 8, 364–366. [Google Scholar] [CrossRef] - Yang, F.; Li, F.W. Study on student performance estimation, student progress analysis, and student potential prediction based on data mining. Comput. Educ.
**2018**, 123, 97–108. [Google Scholar] [CrossRef] - Xing, W.; Guo, R.; Petakovic, E.; Goggins, S. Participation-based student final performance prediction model through interpretable Genetic Programming: Integrating learning analytics, educational data mining and theory. Comput. Hum. Behav.
**2015**, 47, 168–181. [Google Scholar] [CrossRef] - Hamsa, H.; Indiradevi, S.; Kizhakkethottam, J.J. Student academic performance prediction model using decision tree and fuzzy genetic algorithm. Procedia Technol.
**2016**, 25, 326–332. [Google Scholar] [CrossRef] - Gonzalez-Nucamendi, A.; Noguez, J.; Neri, L.; Robledo-Rella, V.; García-Castelán, R.M.; Escobar-Castillejos, D. The prediction of academic performance using engineering student’s profiles. Comput. Electr. Eng.
**2021**, 93, 107288. [Google Scholar] [CrossRef] - Huang, S.; Fang, N. Predicting student academic performance in an engineering dynamics course: A comparison of four types of predictive mathematical models. Comput. Educ.
**2013**, 61, 133–145. [Google Scholar] [CrossRef] - Pallathadka, H.; Sonia, B.; Sanchez, D.T.; De Vera, J.V.; Godinez, J.A.T.; Pepito, M.T. Investigating the impact of artificial intelligence in education sector by predicting student performance. Mater. Today Proc.
**2022**, 51, 2264–2267. [Google Scholar] [CrossRef] - Santhosh, R.; Mohanapriya, M. Generalized fuzzy logic based performance prediction in data mining. Mater. Today Proc.
**2021**, 45, 1770–1774. [Google Scholar] [CrossRef] - Bhatt, R.; Bhatt, D. Fuzzy logic based student performance evaluation model for practical components of engineering institutions subjects. Int. J. Technol. Eng. Educ.
**2011**, 8, 1–7. [Google Scholar] - Wang, X.; Mei, X.; Huang, Q.; Han, Z.; Huang, C. Fine-grained learning performance prediction via adaptive sparse self-attention networks. Inf. Sci.
**2021**, 545, 223–240. [Google Scholar] [CrossRef] - Khan, A.; Ghosh, S.K.; Ghosh, D.; Chattopadhyay, S. Random wheel: An algorithm for early classification of student performance with confidence. Eng. Appl. Artif. Intell.
**2021**, 102, 104270. [Google Scholar] [CrossRef] - Cheng, M.Y.; Cao, M.T. Accurately predicting building energy performance using evolutionary multivariate adaptive regression splines. Appl. Soft Comput.
**2014**, 22, 178–188. [Google Scholar] [CrossRef] - Krzyścin, J. Nonlinear (MARS) modeling of long-term variations of surface UV-B radiation as revealed from the analysis of Belsk, Poland data for the period 1976–2000. Ann. Geophys.
**2003**, 21, 1887–1896. [Google Scholar] [CrossRef] - Friedman, J.H. Multivariate adaptive regression splines. Ann. Stat.
**1991**, 19, 1–67. [Google Scholar] [CrossRef] - Zakeri, I.F.; Adolph, A.L.; Puyau, M.R.; Vohra, F.A.; Butte, N.F. Cross-sectional time series and multivariate adaptive regression splines models using accelerometry and heart rate predict energy expenditure of preschoolers. J. Nutr.
**2013**, 143, 114–122. [Google Scholar] [CrossRef] - Zhang, W.; Goh, A.T.C. Multivariate adaptive regression splines for analysis of geotechnical engineering systems. Comput. Geotech.
**2013**, 48, 82–95. [Google Scholar] [CrossRef] - Zhang, Y.; Duchi, J.; Wainwright, M. Divide and conquer kernel ridge regression. In Proceedings of the 26th Annual Conference on Learning Theory, PMLR, Princeton, NJ, USA, 12–14 June 2013; pp. 592–617. [Google Scholar]
- Exterkate, P. Model selection in kernel ridge regression. Comput. Stat. Data Anal.
**2013**, 68, 1–16. [Google Scholar] [CrossRef] - Ahmed, A.M.; Sharma, E.; Jui, S.J.J.; Deo, R.C.; Nguyen-Huy, T.; Ali, M. Kernel ridge regression hybrid method for wheat yield prediction with satellite-derived predictors. Remote Sens.
**2022**, 14, 1136. [Google Scholar] [CrossRef] - You, Y.; Demmel, J.; Hsieh, C.J.; Vuduc, R. Accurate, fast and scalable kernel ridge regression on parallel and distributed systems. In Proceedings of the 2018 International Conference on Supercomputing, Beijing, China, 12–15 June 2018; pp. 307–317. [Google Scholar]
- Saunders, C.; Gammerman, A.; Vovk, V. Ridge regression learning algorithm in dual variables. In Proceedings of the 15th International Conference on Machine Learning, Madison, WI, USA, 24–27 July 1998. [Google Scholar]
- Cover, T.; Hart, P. Nearest neighbor pattern classification. IEEE Trans. Inf. Theory
**1967**, 13, 21–27. [Google Scholar] [CrossRef] - Yakowitz, S. Nearest-neighbour methods for time series analysis. J. Time Ser. Anal.
**1987**, 8, 235–247. [Google Scholar] [CrossRef] - Farmer, J.D.; Sidorowich, J.J. Predicting chaotic time series. Phys. Rev. Lett.
**1987**, 59, 845. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shih, Y.S. Families of splitting criteria for classification trees. Stat. Comput.
**1999**, 9, 309–315. [Google Scholar] [CrossRef] - Loh, W.Y. Classification and Regression Tree Methods. In Encyclopedia of Statistics in Quality and Reliability; Wiley: Hoboken, NJ, USA, 2008. [Google Scholar]
- Willmott, C.J.; Ackleson, S.G.; Davis, R.E.; Feddema, J.J.; Klink, K.M.; Legates, D.R.; O’donnell, J.; Rowe, C.M. Statistics for the evaluation and comparison of models. J. Geophys. Res. Ocean.
**1985**, 90, 8995–9005. [Google Scholar] [CrossRef] - Legates, D.R.; McCabe, G.J., Jr. Evaluating the use of “goodness-of-fit” measures in hydrologic and hydroclimatic model validation. Water Resour. Res.
**1999**, 35, 233–241. [Google Scholar] [CrossRef] - Albreiki, B.; Zaki, N.; Alashwal, H. A systematic literature review of student’performance prediction using machine learning techniques. Educ. Sci.
**2021**, 11, 552. [Google Scholar] [CrossRef] - Kotsiantis, S.; Pintelas, P. Local voting of weak classifiers. Int. J. Knowl.-Based Intell. Eng. Syst.
**2005**, 9, 239–248. [Google Scholar] [CrossRef] - Yuan, X.; Chen, G.; Jiao, P.; Li, L.; Han, J.; Zhang, H. A neural network-based multivariate seismic classifier for simultaneous post-earthquake fragility estimation and damage classification. Eng. Struct.
**2022**, 255, 113918. [Google Scholar] [CrossRef]

**Figure 1.**The architecture of the newly proposed multivariate adaptive regression splines (MARS) model used to predict undergraduate Introductory Engineering Mathematics student performance at the University of Southern Queensland, Australia.

**Figure 2.**Exploring the relationships between each predictor variable and the respective target variable. $Q1$ = Quiz 1; $Q2$ = Quiz 2; $A1$ = Assignment 1; $A2$ = Assignment 2; $EX$ = exam score; $WS$ = weighted score. A least-square regression line with a best fit equation and the coefficient of determination (${r}^{2}$) is shown.

**Figure 3.**Comparative analysis of machine learning methods (i.e., MARS, vs. KNN, KRR, and DTR) employing Nash and Sutcliffe’s coefficient ($NSE$) computed between the predicted $WS$ and the observed $WS$ in the testing phase.

**Figure 4.**Change in the predicted value of the root mean square error ($RMSE$) deduced by comparing the $RSME$ for the proposed MARS model relative to the $RSME$ generated by the benchmark (i.e., DTR, KNN, and KRR) model. Note:% Change = |($RMS{E}_{MARS}$ – $RMS{E}_{DTR,KNN,KRR}$)/$RMS{E}_{MARS}$| × 100.

**Figure 5.**Evaluation of the predictive skill of all machine learning models with various input combinations developed to predict the weighted score, shown in terms of expanded uncertainty (${U}_{95}$) and the Legates and McCabe index ($LM$) in the testing phase. Note that the proposed MARS model attains the highest value of $LM$ and the lowest value of ${U}_{95}$.

**Figure 6.**Taylor diagram showing the correlation coefficient between the predicted and the observed weighted scores, including the standard deviation and root mean square centred difference for the machine learning models (i.e., MARS, KNN, KRR, and DTR) and including different feature (or input) combinations M1–M5, and M9–M12.

**Figure 7.**Scatter plot of the predicted weighted score ($WS$) versus the observed $WS$ in the testing phase in terms of the nine different sets of feature (input) combinations used to predict $WS$. Least-square regression line y = $mx$ + C and the coefficient of determination (${r}^{2}$) are shown in each sub-panel. (

**a**) MARS, (

**b**) KNN.

**Figure 9.**Discrepancy ratio, $DR$ (i.e., the predicted $WS$ divided by the observed $WS$), for the proposed MARS model within the ±10% and ±20% error bands for all tested data points.

**Figure 10.**Empirical cumulative distribution function ($CDF$) showing the predicted error |$PE$| for the MARS, versus DTR, KNN, and KRR models for the model denoted as M12. Note that the MARS model converges more rapidly for |$[PE$| > 2.5, compared to the benchmark models.

**Table 1.**Descriptive statistics of ENM1500 Introductory Engineering Mathematics student performance (2015–2019) used to construct the proposed MARS model with the predictors (inputs) as: $A1$: Assignment 1, $A2$: Assignment 2, $A3$: Assignment 3, $Q1$: Quiz 1, and $Q2$: Quiz 2 with the target. The weighted score ($WS$) represents the overall score used to allocate a course grade. Note that a raw mark for each assessment had a different total with a certain percentage contribution towards the final grade.

Statistical Property | Predictors | Target | ||||
---|---|---|---|---|---|---|

$\mathit{Q}1$/50 | $\mathit{A}1$/150 | $\mathit{Q}2$/50 | $\mathit{A}2$/150 | $\mathbf{EX}$/600 | $\mathbf{WS}$/100 | |

5% | 15% | 5% | 15% | 60% | 100% | |

Mean | 46.6 | 120.5 | 46.3 | 119.9 | 359.1 | 69.3 |

Median | 50.0 | 127.0 | 50.0 | 126.0 | 360.0 | 70.0 |

Standard Deviation | 5.5 | 26.0 | 6.7 | 26.4 | 141.1 | 17.3 |

Minimum | 8.0 | 15.0 | 0.0 | 0.0 | 0.0 | 20.0 |

Maximum | 50.0 | 150.0 | 50.0 | 150.0 | 600.0 | 100.0 |

Skewness | −2.7 | −1.2 | −3.4 | −1.3 | −0.2 | −0.2 |

Flatness | 10.1 | 1.4 | 15.7 | 1.9 | −0.9 | −0.8 |

**Table 2.**Cross-correlation coefficients (r) of predictor and target variables and the rank of model inputs based on strength of associations between inputs and the target.

Predictor versus Target | Assessment in Teaching Week | r-Value | Input Rank |
---|---|---|---|

$Q1$ versus $WS$ | 2 | 0.407 | 2 |

$Q2$ versus $WS$ | 10 | 0.606 | 3 |

$A1$ versus $WS$ | 5 | 0.262 | 1 |

$A2$ versus $WS$ | 12 | 0.640 | 4 |

$EX$ versus $WS$ | 13 | 0.967 | 5 |

**Table 3.**Input combinations based on first-year undergraduate engineering mathematics student performance data used to construct the proposed MARS model. Note that Models M1 to M5 are based on single predictor variables, and M6 to M12 are based on multiple predictors used to model the weighted score ($WS$).

Designated Model | Input Combinations | Data Points/Period | ||||
---|---|---|---|---|---|---|

(Using Predictors in Table 1) | Data Period S1, S2, S3 | Total Data | Training (60%) | Validation | Testing (40%) | |

M1 | $WS$ = f{$A1$} | |||||

M2 | $WS$ = f{$Q1$} | |||||

M3 | $WS$ = f{$Q2$} | |||||

M4 | $WS$ = f{$A2$} | |||||

M5 | $WS$ = f{$EX$} | |||||

M6 | $WS$ = f{$A1$, $Q1$} | 2015–2019 | 739 records | 444 | 145 (∼33%) of training set | 295 |

M7 | $WS$ = f{$A1$, $Q1$, $Q2$} | |||||

M8 | $WS$ = f{$A1$, $Q1$, $Q2$, $A2$} | |||||

M9 | $WS$ = f{$EX$, $A2$} | |||||

M10 | $WS$ = f{$EX$, $A2$, $Q2$} | |||||

M11 | $WS$ = f{$EX$, $A2$, $Q2$, $Q1$} | |||||

M12 | $WS$ = f{$EX$, $A2$, $Q2$, $Q1$, $A1$} |

**Table 4.**The optimal hyperparameter of the proposed (i.e., MARS) and benchmark machine learning models (i.e., DTR, KNN, and KRR).

Model Name | Hyper-Parameters | Acronym | Optimum |
---|---|---|---|

MARS | Maximum degree of terms | max_degree | 1 |

Smoothing parameter used to calculate $GCV$ | penalty | 3.0 | |

KRR | Regularization strength | alpha | 1.5 |

Kernel mapping | kernel | linear | |

Gamma parameter | gamma | None | |

Degree of the polynomial kernel | degree | 3 | |

Zero coefficient for polynomial and sigmoid kernels | coef0 | 1.2 | |

DTR | Maximum depth of the tree | max_depth | None |

Minimum number of samples for an internal node | min_sample_split | 2 | |

Number of features for the best split | max_features | Auto | |

KNN | Number of neighbours | n_neighbors | 5 |

Weights | Weights | uniform | |

The algorithm used to compute the nearest neighbours | algorithm | auto | |

Leaf-size passed | leaf_size | 25 | |

Power parameter for the Minkowski metric | p | 2 | |

The distance metric to use for the tree | metric | minkowski | |

Additional keyword arguments for the metric | metric_params | none | |

The number of parallel jobs | n_jobs | int |

**Table 5.**Architecture of the proposed MARS model with the basis functions ($BF$), ${C}_{o}$ = y-intercept, y = ${C}_{o}\pm {BF}_{x}$, in terms of the coefficient of determination (${r}^{2}$), the mean square error ($MSE$), and the generalized cross-validation statistic ($GCV$) in the model’s training phase.

Model | MARS Model Equation: y = ${\mathit{C}}_{\mathit{o}}\pm {\mathbf{BF}}_{\mathit{x}}$ | BF | $\mathit{MSE}$ | ${\mathit{R}}^{2}$ | GCV |
---|---|---|---|---|---|

M1 | y = 61.98 + 0.5219 $B{F}_{1}$ − 0.364 $B{F}_{2}$ $B{F}_{1}$ = max(0, x1 − 109); $B{F}_{2}$ = max(0, 109 − x1) | 3 | 178.9 | 0.38 | 183.8 |

M2 | y = 50.8 + 2.29 $B{F}_{1}$ + 0.936 $B{F}_{2}$ $B{F}_{1}$ = max(0, x1 − 46); $B{F}_{2}$ = max(0, 40 − x1) | 3 | 243.3 | 0.182 | 248.87 |

M3 | y = 51.7 + 0.943 ${B}_{1}$ $B{F}_{1}$ = max(0, x1 − 28) | 2 | 276.39 | 0.10 | 283.00 |

M4 | y = 25.49 + 0.642 $B{F}_{1}$ − 0.516 $B{F}_{2}$ + 0.333 $B{F}_{3}$ $B{F}_{1}$ = max(0, x1 − 57.5); $B{F}_{2}$ = max(0, 61 − x1); $B{F}_{3}$ = max(0, 120 − x1) | 4 | 167.76 | 0.429 | 173.63 |

M5 | y = 48.33 − 0.161 $B{F}_{1}$ − 0.220 $B{F}_{2}$ + 0.339 $B{F}_{3}$ $B{F}_{1}$ = max(0, 155 − x1); $B{F}_{2}$ = max(0, x1 − 115); $B{F}_{3}$ = max(0, x1 − 138) | 4 | 17.43 | 0.939 | 18.50 |

M6 | y = 72.45 + 1.878 $B{F}_{1}$ − 0.531 $B{F}_{2}$ − 0.822$B{F}_{3}$ − 0.346 $B{F}_{4}$ $B{F}_{1}$ = max(0, x2 − 47); $B{F}_{2}$ = max(0, 47 − x2); $B{F}_{3}$ = max(0, x1 − 139); $B{F}_{4}$ = max(0, 139 − x1) | 5 | 162.2 | 0.442 | 169.78 |

M7 | y = 71.48 + 2.777 $B{F}_{1}$ + 307.38 $B{F}_{2}$ − 0.5777 $B{F}_{3}$ − 3.348 $B{F}_{4}$ − 3.313 $B{F}_{5}$ + 2.196 $B{F}_{6}$ − 0.054 $B{F}_{7}$ − 2.122 $B{F}_{8}$ 0.0757 $B{F}_{9}$ $B{F}_{1}$ = max(0, x1 − 144); $B{F}_{2}$ = max(0, x2 − 47); $B{F}_{3}$ = max(0, 47 − x2); $B{F}_{4}$ = $B{F}_{2}$ max(0, 149 − x1); $B{F}_{5}$ = $B{F}_{2}$ max(0, x1 − 57); $B{F}_{6}$ = max(0, 36 − x3); $B{F}_{7}$ = max(0, x3 − 36) max(0, 122 − x1); $B{F}_{8}$ = max(0, 43 − x3); $B{F}_{9}$ = max(0, x3 − 43) max(0, 101 − x1); | 10 | 154.144 | 0.442 | 169.90 |

M8 | y = 72.62 + 0.645 $B{F}_{1}$ 0.267 $B{F}_{2}$ +2.209 $B{F}_{3}$ − 3.928 $B{F}_{4}$ − 0.345 $B{F}_{5}$ + 0.002 $B{F}_{6}$ − 0.313 $B{F}_{7}$ + 1.187 $B{F}_{8}$ $B{F}_{1}$ = max(0, x4 − 33); $B{F}_{2}$ = max(0, 33 − x4); $B{F}_{3}$ = max(0, x1 − 47); $B{F}_{4}$ = $B{F}_{3}$ max(0, x2 − 149); $B{F}_{5}$ = max(0, 137 − x2); $B{F}_{6}$ = $B{F}_{5}$ max(0, 146 − x4); $B{F}_{7}$ = max(0, x2 − 137) max(0, x3 − 47); $B{F}_{8}$ = max(0, x2 − 145); | 9 | 124.145 | 0.547 | 137.82 |

M9 | y = 46.838 + 0.105 $B{F}_{1}$ − 0.133 $B{F}_{2}$ + 0.151 $B{F}_{3}$ − 0.152 $B{F}_{4}$ + 0.002 $B{F}_{5}$ + 0.001 $B{F}_{6}$ $B{F}_{1}$ = max(0, x2 − 205); $B{F}_{2}$ = max(0, 205 − x2); $B{F}_{3}$ = max(0, x1 − 77); $B{F}_{4}$ = max(0, 77 − x1); $B{F}_{5}$ = $B{F}_{2}$ max(0, x1 − 109); $B{F}_{6}$ = $B{F}_{2}$ max(0, 109 − x1); | 7 | 5.081 | 0.982 | 5.60 |

M10 | y = 39.665 + 0.103 $B{F}_{1}$ + 2.375 $B{F}_{2}$ + 0.001 $B{F}_{3}$ − 0.013 $B{F}_{4}$ − 0.015 $B{F}_{5}$ +0.004 $B{F}_{6}$ +0.016 $B{F}_{7}$ − 1.307 $B{F}_{8}$ − 0.009 $B{F}_{9}$ − 0.018 $B{F}_{10}$ + 1.427 $B{F}_{11}$ − 2.465 $B{F}_{12}$ +0.010 $B{F}_{13}$ $B{F}_{1}$ = max(0, x3 − 205); $B{F}_{2}$ = max(0, 77 − x2); $B{F}_{3}$ = max(0, 205 − x3) max(0, x2 − 115); $B{F}_{4}$ = max(0, 44 − x1) max(0, x2 − 57.5); $B{F}_{5}$ = max(0, 44 − x1) max(0, 57.5 − x2); $B{F}_{6}$ = max(0, x2 − 77) max(0, x1 − 44); $B{F}_{7}$ = max(0, x2 − 77) max(0, 44 − x1); $B{F}_{8}$ = max(0, x2 − 80); $B{F}_{9}$ = max(0, 205 − x3) max(0, x1 − 30); $B{F}_{10}$ = max(0, 205− x3) max(0, 30 − x1); $B{F}_{11}$ = max(0, x2 − 74); $B{F}_{12}$ = max(0, 74 − x2); $B{F}_{13}$ = max(0, x1 − 44) max(0, 210 − x1) | 14 | 4.45 | 0.985 | 3.565 |

M11 | y = 45.628 + 0.102 $B{F}_{1}$ − 0.115 $B{F}_{2}$ + 0.494 $B{F}_{3}$ −0.259 $B{F}_{4}$ + 0.106 $B{F}_{5}$ − 0.042 $B{F}_{6}$ + 0.003 ${B}_{7}$ + 0.006 ${B}_{8}$ + 0.007 $B{F}_{9}$ − 0.005 $B{F}_{10}$ − 0.356 $B{F}_{11}$ + 0.063 $B{F}_{12}$ − 0.015 $B{F}_{13}$ $B{F}_{1}$ = max(0, x4 − 200); $B{F}_{2}$ = max(0, 200 − x4); $B{F}_{3}$ = max(0, x3 − 77); $B{F}_{4}$ = max(0, 44 − x2); $B{F}_{5}$ = $B{F}_{2}$ max(0, x1 − 44) max(0, x1 − 47.5); $B{F}_{6}$ = max(0, 77 − x3) max(0, x1 − 43); $B{F}_{7}$ = $B{F}_{4}$ max(0, x3 − 106); $B{F}_{8}$ = $B{F}_{4}$ max(0, 106 − x3); $B{F}_{9}$ = $B{F}_{2}$ max(0, 42 − x2); $B{F}_{10}$ = $B{F}_{2}$ max(0, 37 − x1); $B{F}_{11}$ = max(0, x3 − 81); $B{F}_{12}$ = max(0, 81 − x3) max(0, x1 − 46.67); $B{F}_{13}$ = max(0, 81 − x3) max(0, 46.67 − x1); | 14 | 3.187 | 0.986 | 4.11 |

M12 | y = 41.719 + 0.0999 $B{F}_{1}$ − 0.1000 $B{F}_{2}$ + 0.101 $B{F}_{3}$ − 0.0999 $B{F}_{4}$ + 0.100 $B{F}_{5}$ − 0.102 ${B}_{6}$ + 0.0987 $B{F}_{7}$ − 0.0978 $B{F}_{8}$ + 0.0989 $B{F}_{9}$ − 0.0938 $B{F}_{10}$ $B{F}_{1}$ = max(0, x5 − 200); $B{F}_{2}$ = max(0, 200 − x5); $B{F}_{3}$ = max(0, x4 − 77); $B{F}_{4}$ = max(0, 77 − x4); $B{F}_{5}$ = $B{F}_{2}$ max(0, x2 − 80); $B{F}_{6}$ = max(0, 80 − x2); $B{F}_{7}$ = $B{F}_{4}$ max(0, x3 − 26); $B{F}_{8}$ = $B{F}_{4}$ max(0, 26 − x3); $B{F}_{9}$ = max(0, x1 − 33.33); $B{F}_{10}$ = max(0, 33.33 − x1); $B{F}_{11}$ = max(0, x3 − 81); $B{F}_{12}$ = max(0, 81 − x3) max(0, x1 − 46.67); $B{F}_{13}$ = max(0, 81 − x3) max(0, 46.67 − x1); | 11 | 0.079 | 0.997 | 0.0902 |

**Table 6.**Root mean square error ($RMSE$) and correlation coefficient (r) between observed $WS$ and predicted $WS$ generated by the proposed MARS model compared with three different benchmark (i.e., DTR, KNN, KRR) models.

Designated Model | Predicted Error: $\mathit{RMSE}$ | Correlation Coefficient (r) | ||||||
---|---|---|---|---|---|---|---|---|

MARS | DTR | KNN | KRR | MARS | DTR | KNN | KRR | |

M01 | 14.26 | 16.06 | 15.74 | 14.30 | 0.574 | 0.472 | 0.452 | 0.568 |

M02 | 16.07 | 16.37 | 15.88 | 16.01 | 0.401 | 0.373 | 0.438 | 0.408 |

M03 | 16.93 | 17.21 | 17.66 | 16.81 | 0.269 | 0.222 | 0.184 | 0.285 |

M04 | 13.81 | 14.96 | 14.52 | 13.75 | 0.622 | 0.524 | 0.556 | 0.628 |

M05 | 5.76 | 6.54 | 5.95 | 5.89 | 0.963 | 0.950 | 0.961 | 0.960 |

M06 | 13.69 | 16.79 | 14.55 | 13.80 | 0.620 | 0.478 | 0.580 | 0.607 |

M07 | 13.69 | 16.73 | 14.32 | 13.77 | 0.620 | 0.496 | 0.597 | 0.608 |

M08 | 12.64 | 15.95 | 13.28 | 12.66 | 0.688 | 0.536 | 0.655 | 0.686 |

M09 | 4.58 | 5.14 | 4.75 | 4.67 | 0.986 | 0.978 | 0.985 | 0.985 |

M10 | 4.30 | 5.24 | 4.79 | 4.66 | 0.990 | 0.978 | 0.986 | 0.988 |

M11 | 4.21 | 5.05 | 5.21 | 4.64 | 0.991 | 0.978 | 0.984 | 0.990 |

M12 | 3.29 | 4.39 | 4.60 | 3.89 | 0.998 | 0.987 | 0.990 | 0.994 |

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**MDPI and ACS Style**

Ahmed, A.A.M.; Deo, R.C.; Ghimire, S.; Downs, N.J.; Devi, A.; Barua, P.D.; Yaseen, Z.M.
Introductory Engineering Mathematics Students’ Weighted Score Predictions Utilising a Novel Multivariate Adaptive Regression Spline Model. *Sustainability* **2022**, *14*, 11070.
https://doi.org/10.3390/su141711070

**AMA Style**

Ahmed AAM, Deo RC, Ghimire S, Downs NJ, Devi A, Barua PD, Yaseen ZM.
Introductory Engineering Mathematics Students’ Weighted Score Predictions Utilising a Novel Multivariate Adaptive Regression Spline Model. *Sustainability*. 2022; 14(17):11070.
https://doi.org/10.3390/su141711070

**Chicago/Turabian Style**

Ahmed, Abul Abrar Masrur, Ravinesh C. Deo, Sujan Ghimire, Nathan J. Downs, Aruna Devi, Prabal D. Barua, and Zaher M. Yaseen.
2022. "Introductory Engineering Mathematics Students’ Weighted Score Predictions Utilising a Novel Multivariate Adaptive Regression Spline Model" *Sustainability* 14, no. 17: 11070.
https://doi.org/10.3390/su141711070