Linear and Non-Linear Regression Analysis on the Prediction of Compressive Strength of Sodium Hydroxide Pre-Treated Crumb Rubber Concrete ^†

Abstract

This research focuses on dataset development using NaOH treatment period (NaTP), NaOH concentration (NaCon), coarse aggregates (gravel), fine aggregates (sand), water, water–cement ratio (w/c), crumb rubber percentage (CR%), and equations to predict the CS of concrete. The criteria for the model accuracy included the coefficient of regression (R²), mean absolute error (MAE), and root mean square deviation (RMSE). In this study, Multiple Non-Linear Regression (MNLR) performed better compared to Multiple Linear Regression (MLR). The MNLR values obtained for R², MAE, and RMSE were 0.88, 4.64, and 6.15; and the MLR values were 0.82, 5.86, and 7.43 for R², MAE, and RMSE, respectively.

1. Introduction

Concrete structures play an essential role in providing shelter, housing, transportation, and various aspects of construction [1]. Concrete is made up of fine aggregate, coarse aggregate, and cement when mixed with water [2]. Aggregates in the construction and mining processes are depleting natural resources [2]. The current scenario faces a major problem with industrial waste, posing a threat to the environment. Researchers use waste products to create sustainable cementitious composites to address these major problems. [3]. Waste in excessive amounts contributes to pollution, which in turn is harmful to living habitats [4,5,6].

A recent study used 152 datasets to forecast 28 days of compressive strength of high-performance concrete with metakaolin. The models used were Linear Regression (LR), Multi-Logistic Regression (MLR), Response Surface Methodology (RSM), and Non-Linear Regression (NLR). The RSM model performed best, providing results close to those of laboratory testing. The sequence of accuracy was RSM > NLR >LR >MLR [7]. A study on the compressive strength of geopolymer mortar using statistical predictive models was performed using 247 datasets. The models used were LR, MLR, and NLR. The NLR outperformed the other two models in forecasting and real-time analysis, indicating a greater reliance on NLR [8].

This research focuses on predicting the compressive strength of NaOH pre-treated crumb rubber concrete using statistical models with basic evaluation criteria for the accurate forecasting of CS. This approach saves time, reduces laboratory testing, and is convenient for designers.

2. Research Procedure

2.1. Dataset Development

The dataset for the linear and non-linear regressions was obtained from international journals [9,10,11,12,13,14,15,16,17]. In this dataset, different parameters were used to create a linear and non-linear model in order to avoid the hectic task of casting samples. The dataset included the NaOH-treated crumb rubber data with 115 entries. The input variables used in the dataset were the NaOH treatment period (NaTP), NaOH concentration (NaCon), crumb rubber percentage (CR%), water-to-cement ratio (w/c), cement, water, fine aggregates (sand), and coarse aggregates (gravel). NaTP is the period during which the crumb rubber is placed in the prepared solution of sodium hydroxide and water to make its surface rough for better bonding, NaCon is the amount of sodium hydroxide in the water for the preparation of the solution: 10% of NaCon indicates that 10% sodium hydroxide was added to water to prepare the solution as the treatment solution, and CR% is the percentage of the tire rubber, which is shredded and to be used as a replacement for sand in the mix. The output parameter used in the dataset was the compressive strength of NaOH-treated crumb rubber after 28 days of curing (Table 1).

Table 1. Input and output parameters.

Parameters	NaTP	NaCon	%CR	w/c	Cement	Water	Sand	Gravel	CS
Mean	7.40	8.07	10.00	0.45	377.55	173.92	699.46	607.81	36.41
Standard Error	1.01	0.81	0.75	0.01	8.59	4.45	20.94	36.28	1.21
Median	0.50	10.00	10.00	0.48	396.00	175.00	685.00	416.00	36.72
Mode	0.00	10.00	0.00	0.50	400.00	175.00	990.00	311.00	35.00
Standard Deviation	10.81	8.64	8.08	0.06	92.11	47.74	224.58	389.09	12.96
Sample Variance	116.85	74.59	65.25	0.00	8483.41	2279.06	50,437.86	151,393.70	167.99
Kurtosis	−1.20	4.85	−0.84	−0.25	8.03	5.01	2.42	−1.52	−0.18
Skewness	0.90	1.77	0.27	−0.36	−2.35	−1.23	−0.24	0.15	−0.29
Range	24.00	50.00	30.00	0.25	481.20	284.60	1333.20	1257.00	61.30
Minimum	0.00	0.00	0.00	0.35	18.80	9.20	41.80	0.00	1.70
Maximum	24.00	50.00	30.00	0.60	500.00	293.80	1375.00	1257.00	63.00
Sum	851.34	928.00	1150.00	52.29	43,418.00	20,001.09	80,438.20	69,898.40	4187.19
Count	115.00	115.00	115.00	115.00	115.00	115.00	115.00	115.00	115.00

2.2. Error Evaluation

After the model development, the accuracy of the model depends on the error, which can be calculated with the help of statistics. Each error has different criteria to check the accuracy of the model. In this model, the first error evaluation method that was used was the R² coefficient of determination; its value normally ranges between 0 and 1, and a value close to 1 indicates that the error is less. ‘A’ represents the observed or actual value while ‘F’ represents the forecasted value.

$${\mathrm{R}}^2=\frac{{\sum }_{i=1}^n\left(A_i-{\overline{A}}_i\right)\left(F_i-{\overline{F}}_i\right)}{\sqrt{{\sum }_{i=1}^n{\left(A_i-{\overline{A}}_i\right)}^2{\sum }_{j=1}^p{\left(F_i-{\overline{F}}_i\right)}^2}}$$

(1)

The second error valuation criterion used in this research was the root mean square deviation (RMSE). In the case of RMSE, a value closer to 0 is good as compare to a value farther from 0, so 0 is considered the benchmark. In the equation of RMSE, the actual value is represented by ‘A’ and the forecasted value by ‘F’.

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{\frac{{\sum }_{i=1}^n{(A_i-F_i)}^2}{n}}$$

(2)

The mean absolute error was also used in this research; the closer the value to zero, the better the results predicted.

$$\mathrm{M}\mathrm{A}\mathrm{E}=\frac{1}{n}\sum _{i=1}^n|F_i-A_i|$$

(3)

3. Results

3.1. MLR Model

The general representation of linear and non-linear regression is represented in Figure 1. With the help of the input parameters, i.e., NaTP, NaCon, CR%, w/c, water, sand, and gravel, the equation was developed for the MLR model in an Excel data sheet for forecasting and results extraction for the dependent variable. The equation developed for the prediction of NaOH-pre-treated crumb rubber is shown in Equation (4).

$$\begin{gathered} \mathrm{C}\mathrm{S}=-28.966-0.571\mathrm{N}\mathrm{a}\mathrm{T}\mathrm{P}+0.136\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{o}\mathrm{n}-0.564\mathrm{\%}\mathrm{C}\mathrm{R}+120.039\raisebox{1ex}{$\mathrm{w}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{c}$}\right.+0.303\mathrm{C}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}-0.496\mathrm{W}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}+0.006\mathrm{S}\mathrm{a}\mathrm{n}\mathrm{d} \\ -0.021\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l} \end{gathered}$$

(4)

Figure 1. General figure for (a) linear and (b) non-linear regression.

3.2. MNLR Model

Independent variables, i.e., NaTP, NaCon, CR%, w/c, water, sand, and gravel, were used for the development of an equation for forecasting the compressive strength of NaOH-pre-treated crumb rubber concrete. The equation developed using the Excel dataset for the prediction is shown in Equation (5).

$$\begin{gathered} \mathrm{C}\mathrm{S}=77.540-2.795\mathrm{N}\mathrm{a}\mathrm{T}\mathrm{P}+0.093{\mathrm{N}\mathrm{a}\mathrm{T}\mathrm{P}}^2+1.239\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{o}\mathrm{n}-0.025{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{o}\mathrm{n}}^2-1.395\mathrm{\%}\mathrm{C}\mathrm{R}+0.019{\mathrm{\%}\mathrm{C}\mathrm{R}}^2-243.610\raisebox{1ex}{$\mathrm{w}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{c}$}\right. \\ +253.984{\raisebox{1ex}{$\mathrm{w}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{c}$}\right.}^2+0.105\mathrm{C}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}+0.259\mathrm{W}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}-0.001{\mathrm{W}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}^2-0.008\mathrm{S}\mathrm{a}\mathrm{n}\mathrm{d}-0.025\mathrm{C}\mathrm{o}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{e} \end{gathered}$$

(5)

The error results from the calculations using the modeled equations are shown in Table 2. As the value of R2 of Multiple Linear Regression (MLR) is 0.8177 and the Multiple Non-Linear Regression value is 0.8791, which is close to 1, the MLNR is more accurate in terms of model development and the reliance on MLNR is preferred. In the case of MAE, the MLNR value is 4.642 and the MLR error value is 5.855; according to the criteria, lower values are preferred, which is why the MLNR is the leading model. The RMSE calculations show that the value of MLNR is 6.15, whereas the MLR error value is 7.43; in this error evaluation, the MLNR is more reliable because the RMSE also prefer values near 0.

Table 2. Error evaluation.

Model	R2	MAE	RMSE
MLR	0.8177	5.855	7.43
MNLR	0.8791	4.642	6.15

4. Conclusions

The dataset is created using 115 results from the literature to train the model. The equations are developed separately for MLR and MNLR for the prediction of the compressive strength of NaOH-pre-treated crumb rubber concrete using different independent variables. The accuracy is analyzed using different errors like R², MAE, and RMSE. This study is conducted for the convenience of researchers to avoid laboratory procedures and extract direct results in a smart way.

The following conclusions can be drawn from this study:

• Several evaluation criteria are used for checking the accuracy of the models, including R², MAE, and RMSE. The MNLR performs the best compared to MLR;
• MLNR obtains the value of 0.8791, 4.642, and 6.15 for R², MAE, and RMSE, respectively;
• The determined sequence with respect to accuracy in this research is MNLR > MLR;
• Using a comprehensive set of variables for concrete mixture design, including alternative waste materials, was found to be feasible for predicting the strength of sustainable concrete.

The research needs some more studies, like parametric and sensitivity analyses, for more precise and practical implications in the real engineering world. After performing a detailed analysis using the parametric effect and sensitivity analysis; design engineers can use it with no objection.