Implementing The Multi Expression Programming, Nonlinear Regression, Articial Neural Network, and M5P-Tree Models to Predict The Long-Term of Compressive Strength of Cement- Mortar Modied With Fly Ash

In the recent decade, supplementary cementing ingredients have become an essential part of various strength ranges of concrete and cement-mortar mix design. Examples are natural materials, by-products, industrial wastes, and materials that require less energy and time to generate. Fly ash is one of the most widely utilized additional cementing ingredients. Fly ash is a by-product substance produced by coal combustion. It's being used in cement mortar and concrete as a pozzolanic substance. It has demonstrated signicant inuence in improving liquid and solid properties of cement mortar, such as compressive strength. Multi Expression Programming (MEP) is employed in this study to estimate the compressive strength (CS) of cement mortar modied with y ash. The outcomes of this model were compared and evaluated with several other models such as the Nonlinear Regression model (NLR), Articial Neural Network (ANN), and M5P-tree models that have been used in the construction elds. The input parameters included water/cement ratio (w/c), curing time (t days), and y ash content (FA %), while the target property was compressive strength up to 360 days of curing. Four hundred fty (450) data are collected from previous literature on modifying cement mortar with y ash for that purpose. The water/cement ratio ranged from 0.24 to 1.2, and the y ash was used to replace cement up to 55% (%wt. of dry cement). Based on the Coecient of Determination (R 2 ), Root Mean Squared Error (RMSE), Scatter Index (SI), Objective (OBJ), Mean Absolute Error (MAE), t-test value, the uncertainty of 95%, Performance Index (ρ), and boxplot for actual and predicted compressive strength. The MEP model performed better than other developed models according to evaluation tools. The compressive strength was also correlated with exural and splitting tensile strengths using different nonlinear models.


Introduction
Management of industrial waste materials is a global problem; y ash (FA) is a waste -product of power plants resulting from coal combustion. Supplementary cementitious materials (SCM) are those materials that are used in concrete plants to replace Portland cement in cement-based mortar and cement-based concrete. The hydration of cement with water forming calcium silicate hydroxide gel (C-S-H) and calcium silicate (C-H), SCM like y ash reacted with C-H and resulted in the formation of further C-S-H and solving the durability problems related to C-H which is vulnerable to chemical attack (Abed 2018). Fly ash modi ed cementitious material generate less heat during the hydration process; therefore, it is suitable for mass concrete (Fraay et al. 1989;Gartner 2004; Rahhal and Talero 2004;Peter 2005;Sakai et al. 2005), and their strength is greatly in uenced by the physical characteristics and chemical composition of the y ash; those properties depend on the coal type and the equipment used in the power plant and the reactivity of the y ash (Sakai et al. 2005; Otsuka et al. 2009). The pozzolanic reactivity of y ash has been investigated in various research. Pozzolanic reactivity of y ash can be measured through chemical analysis to determine the quantity of silica or measuring the heat developed at the hydration time. However, since the silicate forms a gel at a pH greater than 10, the amount of silica used in the gel formation must be considered (Hassett and Eylands 1997;Bumrongjaroen et al. 2007). After 28 days of curing, the consumption of C-H through a pozzolanic reaction of y ash can be measured by X-Ray diffraction (XRD) or thermal analysis (Giergiczny 2004; Baert et al. 2008). Cho et al. (2019) evaluated the effect of y ash chemical composition on the compressive strength of y ash modi ed cement mortar using sixteen different types of y ashes for replacing cement in cement mortar. They concluded that the pozzolanic reactively of y ash is mainly affected by the percentage of SiO 2 , Al 2 O 3 , and Fe 2 O 3 , and the y ash effect on compressive strength at 90 days of curing is greater than compressive strength at 28 days of curing. Chindaprasirt et al. (2004) studied the effect of y ash neness on the mechanical properties, sulfate resistance, and drying shrinkage of cement mortar. The study results showed that y ash with higher neness improves strength, drying shrinkage, and sulfate attack. Chindaprasirt et al. (2008) evaluated workability and chloride ion resistance of cement mortar modi ed with y ash. Replacement of cement with y ash improved resistance to Chloride ion penetration and better workability for the cement mortar.
Modeling the properties of materials can be performed in various ways, including computational modeling, statistical techniques, and newly created tools like Regression analysis, M5P-tree, and arti cial neural networks (ANN) ( Mohammed et al. (2020a) used ANN, M5P tree, and nonlinear regression to predict the compressive strength of cement-based mortar modi ed with y ash. They have concluded that the ANN model can be used e ciently with a high correlation coe cient (R) and minimum RMSE. ANN model was also used by Apostolopoulou et al. (2019) to predict the compressive strength of natural hydraulic lime; the results revealed that ANN could accurately forecast the CS of natural hydraulic lime mortars, implying that they can be used as a decision-making tool when developing natural hydraulic lime mortars. Also, Armaghani and Asteris (2021) investigated the application of ANN and adaptive neuro-fuzzy inference system (ANFIS) models to predict the compressive strength of cement mortar with or without metakaolin concluded that ANFIS performed better than ANN. At the same time, over tting was observed for some of the data.
Moreover, Suba (2009) employed ANN for mechanical properties prediction and compared the result with linear regression, the forecast of the ANN model was pretty close to actual work. MEP was used to estimate the mechanical properties of concrete and provide acceptable results as implemented by Shah et al. (2021). The parametric study revealed the accuracy of the MEP model, with a high correlation coe cient (R). As a result, several techniques were used in the literature to forecast the mechanical properties of cement-based mortar, but MEP has not been used for that purpose.
In this study, the MEP model was used to predict the compressive strength of the y ash modi ed cementmortar using 450 data collected from previous research related to modi ed cement-based mortar, and outcomes were compared with different approaches, including ANN, nonlinear regression, M5P-tree, and nonlinear model. The various statistical evaluations were applied to assess the accuracy of the models. The correlation between the compressive with exural and splitting strengths of y ash-modi ed cement-based mortar using different nonlinear models.

Objectives
This study is aimed to investigate the application of the MEP model to forecast the compressive strength of cement-based mortar with or without y ash up to 360 days curing; the followings are the main objectives: (i). Statistically analyze the collected data to evaluate the effect of the mix proportion of cement-based mortar modi ed with y ash on the compressive strength.
(ii). Developing a reliable model to predict the compressive strength of cement mortar modi ed with y ash and obtaining the sensitivity of the models using different statistical approaches.
(iii). Correlating compressive strength of the cement mortar with exural and splitting tensile strengths of the cement mortar modi ed with y ash. Figure 1 presents the steps that have been followed during this study. The following are steps of the current study methodology:

Methodology
Collecting a considerable number of the datasets (450 datasets) from different published studies in reputable journals.
Considering w/c, curing time, and y ash content as independent variables for predictors and compressive strength of the cement-based mortar as a target.
Dividing the collected data into three datasets, 70% for training, 30% for testing and validation.
Statistical analysis, visualizing data, and determining the correlation between independent and dependent variables.
Modeling the compressive strength using MEP, NLR, ANN, and M5P-tree models.
Evaluating developed models based on R 2 , RMSE, SI, MAE, OBJ, t-test, 95% uncertainty, and performance index for actual and predicted compressive strength.
Performing sensitivity analysis to detect the most dominant parameter on the compressive strength of cement-based mortar modi ed with y ash. divided into three groups (training, testing, and validating) randomly using the Rand function in Microsoft Excel. The largest group included 70% of the dataset (300 data), and each of the other two groups had 15% of the dataset (75 data). The training data is used to develop the model while validating and testing data is provided to test the developed model against unseen data. The over tting of the developed model can be minimized (Rai et al. 2014).The summary of statistical analysis on the input and output parameters with detail of the collected data is shown in Table 1.   Table 2.

Data collection
CS, w/c, t, and FA are compressive strength, water to cement ratio, curing time, y ash content, and β 1 to β 7 are model parameters.

ANN model
The Arti cial Neural Network (ANN) is a computing system that resembles the human brain and its information analyzes. In addition, this model is a performance. In more technical terms, the topology of the network and connection weights change repeatedly such that the error at each output layer node is minimized (Armaghani and Asteris 2021). In this study, a multi-layer feed-forward network was designed with mortar composition (w/c, t, FA) as input and CS as output, and a sigmoid activation function is used in the output layer.
( ) ( ) Output = f ∑ n j = 1 w j x j + bias (2) Where j is the number of input variables, x j is the input number j, and bias is the threshold for sigmoid activation function. The typical process of the result of ANN is illustrated in Fig. 5. 3.3.4. M5P-tree model Quinlan (1992) rst devised the M5 algorithm, which was developed into the M5P-tree algorithm (Wang and Witten 1996). One of the most signi cant advantages of model trees is their ability to e ciently solve problems, dealing with many data sets with a substantial number of attributes and dimensions. They are also noted for being powerful while dealing with missing data. The M5P-tree approach establishes a linear regression at the terminal node by classifying or partitioning diverse data areas into numerous separate spaces. It ts on each sub-location in a multivariate linear regression model. The error is estimated based on the default variance value inserted into the node. The general formula for the M5P-tree model is shown in Eq. 3.
CS, w/c, t, and FA are compressive strength, water to cement ratio, curing time, y ash content, and a, b, c, and d are model parameters (Table 3).

Performance Criteria For Model Evaluation
The developed models are evaluated based on different assessment tools to choose the best model to predict the CS of the mortar; the following are e ciency measurements for the models: ( )  Fig. 6. The model had a good performance with R 2 of 0.87, 0.87, and 0.897 for training, testing, and validating, respectively. Figure 6 (a) contained -20 and +25% error lines in the training phase and -10 and 15% for testing and validating (Fig. 6b &c). for the training data set and -15 to 20% for testing, and -15 to 25% for validating datasets. Figure 11 shows the pruned M5P-tree, which classi ed the training dataset into four parts based on the criteria shown in the gure; each part of the divided dataset resulted in a single regression model as mentioned in Eq. 3, the model parameters for the M5P-tree model are summarized in Table 3.

Relationship between compressive, exural, and tensile strengths
Based on the collected data, three different models were developed to predict exural and splitting tensile strengths from measured compressive strength using the Vipulanandan correlation model, Exponential association-2, DR-Hill-Zero background, and Power model, as illustrated in Eqs. 20 to 25. Figure 13 (a) shows the variation of FS with CS for data collected from literature and predicted FS using developed models. The residual error for predicted FS from CS ranged between 1 MPa to -1 MPa is shown in Fig. 13 (b). Variation of splitting tensile strength with CS is shown in Fig. 13 (c), and the residual errors for predicted STS from CS using ranged between 0.15 MPa to -0.35 MPa (Fig. 13 (d)). Based on the R 2 and RMSE, the DR-Hill-Zero background model is better than other models for predicting exural strength from compressive strength; on the other hand, the best model for correlation of splitting tensile strength with compressive strength is DR-Hill-Zero background and Power Models.

Model Evaluations
The proposed models are compared according to the relationship between predicted and measured CS for testing data set; the MEP model had less variation; the plotted data are near the Y=X line, which indicates a minor error in predicted values, as shown in Fig. 14 (a). Furthermore, the maximum and minimum residual errors for the MEP model were -19 and 18 MPa. Residual error of NLR, ANN, and M5P-tree model was -12 to 14 MPa, -14 to 14 MPa, and -21 to 19 MPa, respectively. The residual error indicates better performance of the NLR model than other developed models, as shown in Fig. 14 (b).
The SI value of the MEP model, NLR, ANN, and M5P-tree model for the training dataset was 0.148, 0.16, 0.155, and 0.173. When comparing SI value for validating datasets, the SI value for the MEP model is less than NLR, ANN, and M5P-tree model by 8, 6, and 16.5%, respectively. For the testing dataset, the SI value of the MEP model is equal to 0.159 and less than ANN, and M5P-tree model by 10 and 5%, and more signi cant than the NLR model by 5%, as shown in Fig. 15 (a).
The comparison of developed models based on MAE is presented in Fig. 15 (b). The MAE for MEP models is less than the MAE of other developed models for training and validating datasets; however, the MAE of MEP model value for testing is less than ANN, and M5P-tree model by 8 and 4%, and greater than the NLR model by 6%.
The OBJ values for the proposed models are also evaluated; the OBJ for the MEP model is less than NLR, ANN, and M5P-tree models by 7, 6, and 14, as displayed in Fig. 16 (a).
The t-test and U 95 values comparison for the developed models is illustrated in Fig. 16 (b). as can be seen from the gure, the uncertainty of the predicted compressive strength for 95% con dence level of MEP model is less than ANN and M5P-tree models by 2 and 6%, and greater than NLR model by 4%. However, the t-test value of the MEP model is less than other developed models. The t-test value results in a probability of accepting or rejecting the null hypothesis. The larger t-test value indicates a signi cant difference in the measured and predicted CS of the cement mortar.
Also, the performance index for the MEP model was less than other developed models for training and validating data. At the same time, it is greater than the NLR model in testing the data set by 4%, as presented in Fig. 17 (a).
The box plot for actual and predicted CS is drawn as shown in Fig. 18 (a, b & c). The boxplot for the MEP model had the same pattern for the minimum and maximum CS values, Mean and median. According to the box plot MEP model is better than other developed models.
Summary of model evaluation for R 2 , RMSE, and MAE of the developed models is presented in Table 4.

Sensitivity evaluation
The most in uential parameter on the compressive strength of cement-based mortar modi ed with y ash is determined using the MEP model. Every time a single input parameter is removed from the training dataset, regression is run again in the process. MAE for the model is recorded, the trial with maximum MAE (MPa) and RMSE (MPa) is chosen, and the trials ranked according to the recorded MAE the more sensitive variable in predicting the compressive strength of cement mortar modi ed with y ash is the removed parameter from the trial with the highest MAE. Based on the sensitivity analysis, the most in uential parameter is the curing time of the tested samples, as summarized in Table 5.

Declarations Author Declarations
We wish to draw the attention of the Editor to the following facts, which may be considered as potential con icts of interest and to signi cant nancial contributions to this work.
[OR] We wish to con rm that there are no known con icts of interest associated with this publication, and there has been no signi cant nancial support for this work that could have in uenced its outcome. We con rm that the manuscript has been read and approved by all named authors and that there are no other persons who satis ed the criteria for authorship but are not listed. We further con rm that all have approved the order of authors listed in the manuscript of us. We con rm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing, we con rm that we have followed the regulations of our institutions concerning intellectual property. We further con rm that any aspect of the work covered in this manuscript that has involved either experimental animals or human patients has been conducted with the ethical approval of all relevant bodies and that such approvals are acknowledged within the manuscript. We understand that the Corresponding Author is the sole contact for the Editorial process Ash ). I have read the nal version and consent for the article to be published in this Journal.

Availability of data and materials
The data supporting the conclusions of this article are included with the article.

Competing interests
The authors declare that they have no competing interests.

Funding
This work had no nding.

Authors' contributions
Aso and Ahmed S. Mohammed are collecting data, planning, and writing. Aso, results, and analysis. Aso and Ahmed did the conclusions and editing. Figure 1 Page 24/28

Figures
Methodology owchart of the current study  Correlation matrix for independent variables and dependent variable Figure 5 Typical procedure for output of ANN network in a single node  Optimal ANN network selection based on RMSE and MAE

Figure 10
Page 27/28 Variation of CS Predicted with CS Measured using ANN model (a) training data, (b) testing data, and (c) validating data Figure 11 Pruned M5P-tree model  Comparing developed models using performance index Figure 18 Comparing developed models using boxplot for actual and predicted compressive strength values (a) training data, (b) testing data, and (c) validating data