# 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 |

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**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