Statistical modelling of a tractor tractive performance during ploughing operation on a tropical Alfisol

2.1 Research conceptual framework

The need to improve operational efficiency and tractive performance of a tractor-implement combination during the initial soil loosen-up process is the focus of this study. Four input variables were investigated to obtain the measured parameters which were statistically analysed using the R software package. Figure 1 presents the experimental flowchart.

Figure 1

Research procedural approach.

2.2 Soil properties and description of the study area

The research farm of National Centre for Agricultural Mechanization, Ilorin was used for the study under real working conditions. The location is 370 m above sea level, longitude 4°30′E and latitude 8°26′N along Ilorin-Lokoja highway, Idofian, Kwara State, Nigeria. It is found under the southern Guinea Savannah agro-ecological zone characterized by a tropical climate with distinct wet and dry seasons, mean annual rainfall of about 1,000 mm, and daily temperature range of 20–35°C [40]

The dominant soil type is a tropical sandy loam classified as Alfisol developed on parent material from the basement complex made up of gneisses and granite. The texture of the ploughed layer (0–21 cm) was established to contain 67% sand, 16% silt, and 17% clay with 0.21% organic matter (OM). Experimental site topography is relatively flat and confirmed to be fallow land as it has not been used for any intensive farming for two farming seasons.

2.3 Machine specifications and experimental equipment

Process parameters for the investigation were measured and recorded following standard procedures [41,42]. The tractors used for the research were TS 604 (60 hp), John Deere 5503 (65 hp), and Shuhe SH750 (70 hp) while JS-800 (80 hp) was the auxiliary tractor. The choice was premised on availability and wide acceptability of these power ranges among the farmers and stakeholders in tractor hiring services. The corresponding implements attachment were tractor mounted 3-bottom and Mex Rav, 2-bottom disc ploughs. The specifications of the tractors and the matching implements are presented in Tables 1 and 2. Other equipment used were digital dynamometer, digital pressure gauge, digital weighing balance, cone penetrometer, shear vane apparatus, Memmert electrical oven, toolbox, measuring tape, metre rule, ranging pole, chain, tray, airtight polythene material, cutlass, rope, and pegs.

2.4 Field layout, experimental structure, and procedures

A 5 ha fallow parcel of land was used for the fieldwork under real-time conditions. Preliminary field preparations and marking out of 270 sample plots were done. Each experimental plot measured 35 m × 4 m considering the plough width. The 35 m length was partitioned into 15 and 20 m. The measured variables were taken in the 20 m portion while the 15 m length was for the tractor speed pick up and stability maintenance. Each tractor was prepared before the fieldwork and the required pressures, 83, 124, and 165 kPa were put into the tractor rear wheel being the traction wheel using a digital pressure gauge, while the front wheels were maintained at the manufacturer’s recommended pressures. The tractor-implement combination was investigated at five levels of speed, 6.3, 7.9, 9.5, 11.1, and 12.6 km/h at gear 2 H using the hand throttle.

2.5 Statistical analysis and algorithmic approach

Analysis of Variance (ANOVA) at 95% confidence level was used to examine which of the measured variables contributed significantly to the response variable. Multiple linear regression analysis provided regression coefficient estimation, t-test, and F-test values. These analyses helped to establish the linear relationship existing between the response variable (TF) and the predictors using the R software package. The experimental design used for data collection has both nested and factorial factors of a linear model structure equation (4).

for i = 1 , 2 , 3 , 4 , 5 j = 1 , 2 , 3 k = 1 , 2 l = 1 , 2 , 3 , where y i j k l = tractive force measurement arising from ith speed, jth tractor type with kth implement attached at lth inflation pressure (replicate), μ = overall mean tractive force, α i = ith speed effect, β j = jth tractor hp effect, γ k ( j ) = effect of the kth implement within the jth tractor level, ( α β ) i j = speed and tractor hp interaction, ( α γ ) i k ( l ) = speed and implement within inflation pressure level interaction, and ϵ ( i j k ) l = random error term.

Regression analysis help to infer causal relationships between the predictors and response variable. The relationship existing between the TF and the other variables identified to influence tractor tractive performance during ploughing operation was established using the multiple linear regression technique. This is a supervised statistical learning process where both input and output variables were observed and the best mathematical equation that describes the existing relationship was established. Using the least regression approach [50], the mathematical form of the multiple linear regression models in terms of the variables is as follows:

where Y = TF (kN) of ith tractor, X ₁ = D (kN) of ith tractor, X ₂ = WS (%) of ith tractor, X ₃ = DC (cm) of ith tractor, X ₄ = WC (cm) of ith tractor, X ₅ = MC (%) of ith tractor, X ₆ = BD (g/cm³) of ith tractor, X ₇ = CI (kPa) of ith tractor, X ₈ = SS (kPa) of ith tractor, ϵ = random error term; and β 0 , β 1 , β 2 , β 3 , β 4 , β 5 , β 6 , β 7 , and β 8 are the regression coefficients.

2.6 Model validation

This is to affirm model usefulness and real-life applicability [56]. It was done using a 10-fold cross-validation structure as provided in the DAAG library and the cv.glm function of ISLR library of the R software [50]. This validation structure randomly divides dataset into approximately ten equal subsets or size. At each run, nine subsets were used to construct the model while the remaining subset was used for validation. The average accuracy for ten iterations was recorded as the final measurement [57]. The first fold was treated as a validation set, and the least squares regression was fitted on the remaining folds (ten less one). The Mean square error (MSE) was then computed on the observations in the held-out folds. This procedure was repeated ten times, and each time, a different group of observations was treated as a validation (test) set. The output is in ten estimates of the test errors, MSE 1 , MSE 2 , … , MSE 10 . The 10-fold cross-validation (CV) estimate was computed based on the average of these values equation (12).

This validation method is an extremely flexible, powerful, and widely used technique in validation work for estimating prediction error. The measure of error for cross-validation is the MSE for a quantitative response [57].

2.7 Multi-collinearity effect estimation

To avoid multi-collinearity effect in the predictors, only models whose variance inflation factor (VIF) is smaller than 10 were retained [51,58,59]. VIF defined the ratio of the variance of β̂ _j when fitting the full model divided by the variance of β̂ _j if fit on its own equation (13). This has been reported to be a better technique to evaluate multi-collinearity in regression models [50],

where R X j | X − j 2 – R 2 from a regression of X j onto all of the other predictors. Estimation having R X j | X − j 2 value close to 1 indicates the presence of multi-collinearity, and hence, a large VIF.

3.1 Experimental site soil textural properties

This established the soil conditions for the tractor-implement working environment, a principal factor that influences machine field performance. Soil data collected included soil grain, BD, MC, cone penetrometer (CI), and SS. The values obtained are indicators to affirm the suitability of the experimental site for the study because agricultural field machines perform well within certain soil properties or conditions [60]. Textural characteristic of the experimental site was found to be sandy loam based on United States Department of Agriculture (USDA) classification (Table 3).

These soil properties affect traction and tyre performance, tractor fuel efficiency, e, wheel slip (WS), tillage energy, and soil morphology [37,61,62].

3.2 Collected experimental field data characteristics

Descriptive statistics of the field data revealed the minimum, maximum, mean value, and standard deviation of the 270 treatments obtained from the fieldwork (Table 4). These variables have a direct impact on tractor performance in the field.

Quantitative output from these indices have impact on the soil, machine, and operational characteristics. Soil BD (ρ _b) has a direct effect on soil properties such as porosity, soil moisture availability, and hydraulic conductivity, with an indirect effect on root growth and crop yield [61,62]. A BD greater than 1.2 mg/m³ (clayey soil), 1.6 mg/m³ (loam soil), and 1.8 mg/m³ (sandy loam soil) may adversely affect paddy root growth [60]. Most tillage exercise for crop sowing are usually done at depths of 10–30 cm because average depths of roots penetration for food crops are within this range. The mean DC for the investigations was 20.51 cm which implies a favourable condition for the exercise as the deeper DC has effect on tractive performance indices such as the D, WS, and speed of operation. A CI of 246 kPa was considered a soft soil condition and a CI of 492 kPa was considered a firm soil [63]. Overall, these are pertinent to good traction force and tractor efficiency during field operations.

3.3 Data pre-processing of experimental results for predictive analysis and inference

Based on standard established procedures followed in parametric model assumptions checking and data inspection for the anomaly that may impact the reliability of the predictive inference on the study data, the dataset was checked for missing observations and correlation between the response variable and among the predictors, using version 3.5.1 of the R software for statistical computing and graphics. The dataset run for all the observations produced no missing reading or value. A check for bivariate (Pearson) linear correlation between the variables is as shown in Table 5. Clearly, D, WS, and WC have a significantly strong relationship with tractor TF. The other predictors show no enough data evidence to reject null hypothesis of no significant correlation with tractor TF in ploughing operation. Significant correlations among predictors may suggest multi-collinearity which can be checked only when classical test for multi-collinearity on fitted models is performed.

3.4 Multiple linear regression modelling of TF in ploughing operation

To predict TF in the 270 runs included in the study datasets with the eight traction inputs, model was fitted into the data using the lm() function in the R software [64]. Data indications in Table 6 reveal that only least squares estimate of model parameters D, WC, MC, BD, and CI, respectively, are non-zero since their p-values are much smaller than the α = 0.05 rejection region. Hence, a clear statistical relationship exists between them and the response variable. The variables with value greater than p-values as observed for the intercept, WS, DC, and SS indicate that the least squares estimate for them should be zero. This confirmed the suitability of regression through the origin for the data and that WS, DC, and SS do not have significant predictive relationship with the TF. The fitted least squares regression model is

Based on equation (14), average TF (kN) is −0.0837 when zero unit of D ( X 1 ), WS ( X 2 ), DC ( X 3 ), WC ( X 4 ), MC ( X 5 ), BD ( X 6 ), CI ( X 7 ), and SS ( X 8 ) are applied.

The positive least squares estimate for D, WS, WC, BD, CI, and SS, respectively, implies that they have a direct relationship with TF of tractors in ploughing operations. Similarly, the DC and MC are negatively correlated depicting an inverse relationship with the response variable in ploughing operations.

3.5 ANOVA for the fitted regression model

To investigate marginal effects of the individual traction input variables, the ANOVA was obtained and the summary is presented in Table 7. This is to test the null hypothesis of non-dependence of TF (y) on individual traction inputs at the 5% critical rejection region. Data evidence in Table 7 reveals that traction input variables WS, DC, and SS may not have a significant marginal effect on tractor TF in ploughing operation.

3.6 Variable selection to identify the best model for TF prediction

To determine which subset of these predictors and their interactions are significantly associated with the response, variable selection using a forward stepwise selection algorithm was performed to ascertain the best model for predicting TF, based on the Residual sum of squares (RSS), and AIC criterion [64,65],. In Table 8, all asterisked inputs and interaction terms are said to have marginal significant effects on tractor TF in ploughing operation. They are, therefore, said to be important in the final model.

Based on obtained minimum average AIC, the final model includes D, WC, MC, WS, D:MC interaction, and WC:BD interaction, respectively. The final model produces significant improvement by deleting the intercept term to include CI (Table 9). The no-intercept term model also known as regression-through-the-origin is pronounced in engineering and science phenomena depicting input defines the output. Having intercept in the model signify that when all inputs are zero, the output will assume the intercept value.

The final estimated multiple linear regression model including these input variables and their important combinations is as presented in equation (15).

x 1 = D, x 2 = WC, x 3 = MC, x 4 = WS, x 5 = CI, x 6 = D:MC, x 7 = WC:BD.

It can be observed from equation (15) that holding all other variables constant, a unit change in the D will result in 1.51% change in the response variable. Similarly, a unit change in WC will result in a 0.0068% reduction in the TF. This expression depicts the dynamic contribution of the predictors on TF either in the positive or negative trends during ploughing operation on a tropical Alfisol.

Summarily, the adoption of these contributing indices will help to integrate only variables that significantly influence the response variable so that energy and resources can be maximally applied. The use of only significant variables permit the development of simplified models that not only reduce the cost of collecting irrelevant data but also lessen the risk of overfitting the model, which reduces the prediction accuracy for unseen (new) data [66,67]. Mahore et al. [4] reported a perfect R ² score of 1.000 on the training set based on Decision Tree algorithm, indicating an excellent fit to the data. However, a higher RMSE score on the testing set was affirmed, potentially suggesting overfitting.

The best model quality of fit as investigated using the coefficient of multiple determination ( R 2 ), achieved an adjusted ( R 2 ), of 87.54%. This established the fact that using the seven terms in the final model, the response variable can be predicted by the model up to 87.54%, which depicts the level of how the variation in the dependent variable can be explained by the regression model. Correlating with the ASABE draft prediction model that integrate soil texture, implement width, tillage depth, and speed of operation with R ² = 0.62, the model performs more fairly.

Inferences from the R ² may depict the need to establish the presence of multicollinearity as the independent variables may be related [58,59,68,69]. The R ² is not a sufficient method to diagnose if the model suffers from multicollinearity, hence the model was further diagnosed using the VIF. Those with a VIF greater than 10 show that the model suffers from multicollinearity. One of the consequences of this problem is that ordinary least square (OLS) estimates are no longer reliable because the variance of the regression coefficients becomes very large and sometimes possesses a wrong sign [70,71]. One of the estimators suggested as an alternative to the OLS is the ridge regression [72]. The VIF result in Table 10 shows that there is multicollinearity. Therefore, the model was analysed using the ridge regression estimator, and the results are depicted in Table 9. Though the estimates produced by both estimators are not too far from each other, the result of the ridge regression estimate is more reliable, especially when there is severe multicollinearity. The MSE is the criteria used to compare the two estimators. The estimator with the minimum MSE is generally preferred. The model developed for tractive performance investigations during ploughing operations can be a viable tool for use in the planning of operational characteristics of a tractor-implement matching within the investigated tractor horsepower ranges and input parameters on a tropical Alfisol. The ANOVA for the final fitted regression model is as shown in Table 10. All the terms in equation (12) have a significant effect on the tractor TF during ploughing operations.

3.7 Model diagnostics to investigate the validity of regression assumptions

Before checking the prediction performances of the final model on test/unseen data, there is a need to check whether required regression assumptions hold for the model by testing the null hypothesis of normally distributed residuals using Kolmogorov–Smirnov (K–S) and Shapiro–Wilk (S–W) test of normality [72,73] as presented in Table 11. The data evidence reveals that residuals of the final model are normally distributed since the p-value 0.4756 of S–W test is larger than the critical value ( α = 0.05 ). The same also applies to the value obtained with the K–S test with a p-value of 0.200.

Regression diagnostic plots for ploughing operation for the developed model is as shown in Figure 2. It is evident from residuals against fitted values plot that there is no pattern in the residual plot. This suggests that a linear relationship between the response variables and the predictors can be assumed. Because the horizontal line in bottom left plot has approximately evenly distributed points, the residuals are equally spread along the ranges of the independent variables. Hence, assumptions of homogeneity of error variances hold on the fitted model. Again, the inference from test of normality assumption for fitted model residuals is further supported by the top right normal quantile plot, the normal probability plot of residuals approximately follows a straight line. Supportively, the bottom right plot highlights the top three most extreme points (No. 2, No. 53, and No. 223), with a standardized residuals below −2 or above 2. However, there is no outliers that exceed 3 standard deviations, which is good and acceptable.

Figure 2

Regression diagnostic plots for ploughing operation showing residuals vs fitted values plot (top left), Spread-location plot for homogeneity of variance (bottom left), the normal quantile plot (top right), and the residuals vs leverage plot (bottom right).

Furthermore, no high leverage point can be found in the data. This implies that all data points have a leverage statistic below 2(p + 1)/n = 16/270 = 0.0593 (since there are p = 7 predictors in the final model).

As evident in Figure 3, observations number 53, 223, and 243 cannot be removed from the model because they have high Cook’s distance scores and they are to the upper right of the leverage plot. They have leverage, signifying they are important to the regression results. The regression results will be altered if these cases are excluded.

Figure 3

Cook’s distance plot (left) and leverage plot (right).

3.8 Evaluating model predictive performance

The quality of the fitted model in terms of predicting tractor tractive performance of future ploughing operation runs with the identified significant input variables as investigated in terms of MSEs and RMSEs over 10-fold cross-validation iterations as presented in Table 12. The results obtained are within the range of values obtained in a previous study [74] that investigated the development of empirical regression equations for predicting the performances of disc plough and harrow in clay-loam soil.

Based on the cross-validated standard error of prediction (CV. RMSE), the final model has a good prediction performance over all the ten folds. This inference is further supported by the graph of predicted vs observed ploughing operational runs presented in Figure 4. The predicted TF by the model traced out the observed/measured TF as closely as possible. Hence, an estimation of the predicting model obtained in the study reveals a good agreement with the measured or experimental data obtained from the field performance. Multiple regression analysis was employed to predict a describing model for TF as influenced by the predictors with a reliable coefficient of determination (R ² = 0.8858). This implies that the comparison of the predicted model with the experimental data for TF achieved 88.58% accuracy of prediction during ploughing operations. The result is similar to what was established by Mahore et al. [4] in a study that used machine learning to predict the draft of a mouldboard ploughing in a sandy clay loam soil. The study affirmed that the algorithms linear regression, ridge, and support vector machine exhibit R ² scores ranging from 0.801 to 0.808, indicating moderate to good predictive performance.

Figure 4

The accuracy of the predicted model and measured values for the TF during ploughing operation.

3.9 Confidence interval for the model coefficients

The fact that none of the confidence intervals for the parameters contains zero further confirms that all estimated parameters are significantly different from zero. Hence, all input variables remaining in the model are significant predictors of TF at 95% confidence level (Table 13).