Multiple linear regression, artificial neural network, and fuzzy logic prediction of 28 days compressive strength of concrete

Abstract

Evaluating the in situ concrete compressive strength by means of cores cut from hardened concrete is acknowledged as the most ordinary method, however, it is very difficult to predict the compressive strength of concrete since it is affected by many factors such as different mix designs, methods of mixing, curing conditions, compaction, etc. In this paper, considering the experimental results, three different models of multiple linear regression model (MLR), artificial neural network (ANN), and adaptive neuro-fuzzy inference system (ANFIS) are established, trained, and tested within the Matlab programming environment for predicting the 28 days compressive strength of concrete with 173 different mix designs. Finally, these three models are compared with each other and resulted in the fact that ANN and ANFIS models enables us to reliably evaluate the compressive strength of concrete with different mix designs, however, multiple linear regression model is not feasible enough in this area because of nonlinear relationship between the concrete mix parameters. Finally, the sensitivity analysis (SA) for two different sets of parameters on the concrete compressive strength prediction are carried out.

Graphical abstract

Keywords

concrete / 28 days compressive strength / multiple linear regression / artificial neural network / ANFIS / sensitivity analysis (SA)

1 Introduction

The compressive strength as the most fundamental mechanical property of concrete and performance indicator for structural design is generally examined after the standard concrete is kept in water having specific laboratory conditions for 28 days. The concrete compressive strength can be achieved by increasing the uniaxial compressive load gradually until the specimen fails, and this maximum load signifies the 28 days compressive strength of concrete. Concrete compressive strength is affected by several parameters like age, water to cement ratio, ingredients, curing conditions, compaction, etc. Prediction of compressive strength of concrete using the traditional methods is usually based on the linear and nonlinear regression methods. In recent years, there has been increasing interest in evaluating the concrete compressive strength using data-driven models. Currently, many researchers have productively considered different elaborate methods in this area such as artificial intelligence based techniques like artificial neural network [1–3] and ANFIS which is the short-term of adaptive network-based fuzzy inference systems [3–5].

Prediction of various concrete properties have been investigated by many scientists. Bal & Buyle-Bodin estimated the drying shrinkage of concrete using artificial neural network [6]. Atici utilized multivariable regression analysis and artificial neural network to evaluate the strength of mineral admixture concrete [7]. Sadowski determined the pull-off adhesion between concrete layers with a self-organization feature map [8]. Nikoo et al. used evolutionary artificial neural networks to determine the displacement in concrete reinforcement building [9]. Amani & Moeini approximated the shear strength of reinforced concrete beams by adaptive neuro-fuzzy inference system and artificial neural network [10]. Ramezanianpour predicted the 28 days compressive strength of high strength concrete using ANFIS [5]. Sadowski used imperialist competitive algorithm to predict the corrosion current density in reinforced concrete [11]. Khademi and Behfarnia estimated the 28 days compressive strength of concrete using artificial neural network and multiple linear regression models [12]. Nikoo et al. used self-organization feature map (SOFM) to determine the compressive strength of concrete [13].

In addition to the importance of predictor models in affecting the accuracy of estimating the compressive strength of concrete, sensitivity analysis is another important aspect that should be considered. Vu-Bac et al. have determined the basic effects on the mechanical models to recognize the significant parameters in the context of averaged Local SA. They have concluded that the key factor affecting the ISS is the SWCNT radius followed by the temperature and pulling velocity, respectively [14]. Vu-Bac et al. also have carried out the sensitivity analysis based on the molecular dynamics to determine the effect of the uncertain input parameters on the predicted elastic modulus and yield stress [15]. Furthermore, Vu-Bac et al. have investigated different sensitivity analysis to predict the influence of some of the uncertain dependent elements on both the Young’s modulus and Poisson’s ratio [16].

This research focus is on the prediction of the 28 days compressive strength of concrete with three methods of Multiple Linear Regression (MLR), Artificial Neural Network (ANN), and Adaptive Network-based Fuzzy Inference Systems (ANFIS). Briefly, Multiple Linear Regression model challenges modeling two or more input variables by fitting a linear equation to observed data. This model includes two subsets of training and test. In addition, Artificial Neural Networks are a family of models containing three subsets of training, validation, and test. ANN utilized biologic neural networks to evaluate functions that are dependent on numerous input variables. Moreover, ANFIS is a specific type of artificial neural network which is based on Takagi-Sugeno fuzzy inference system [3]. Furthermore, the sensitivity analysis (SA) for two different sets of parameters are performed on the concrete compressive strength prediction. The overview of this study is shown in Fig. 1.

2 Materials and mix design

Usually, an acceptable data set is the one which contains comprehensive information regarding the material characteristics and behavior. To apply the three different models of MLR, ANN, and ANFIS on the prediction of compressive strength of concrete, corresponding data should be provided. Availability of large variety of experimental data are essential to estimate the compressive strength based on concrete parameters. Therefore, 173 concrete specimens having different mix designs are constructed in the laboratory. The specimens were cylindrical with a diameter of 15 cm and height of 30 cm. The basic parameters considered in this study were namely 3/4 sand, 3/8 sand, cement, gravel in kilograms, maximums and size of aggregate in millimeter, fineness modulus of sand, and water-cement ratio. Each specimen is pounded with 25 blows using a hydraulic jack, and intended data set are collected [2,12]. The characteristics of concrete cylindrical samples are shown in Table 1 and the mean and standard deviation of each parameter is shown in Table 2 [2,12,13].

The data are collected from [2,12,13,17] and have not been previously used in the presented numerical modeling.

3 Methodology

It is worth mentioning that prediction of the concrete compressive strength is an important fact in the quality assurance of the produced concrete. Although there are numerous methods of predicting the mechanical properties of concrete, not all of them are valuable since they are in accordance with many trial and errors. As mentioned earlier, this study compares the feasibility of three different models of multiple linear regressing (MLR), artificial neural network (ANN), and ANFIS to predict the compressive strength of concrete. These models are defined comprehensively in the following sections.

3.1 Multiple linear regression model (MLR)

Many engineering problems involve determining the correlation between two or more variables. Regression analysis is one of the statistical tools which has always been the interest of scientists in this area. Generally, regression models can be considered as the process of fitting models to data. In a linear regression model which is a specific form of regression models, the linear predictor functions would be utilized to model the data, and output parameters are estimated from the data. It is worth mentioning that in many applications of regression analysis, more than one input variables is involved which would lead to having the “multiple linear regression” function. In this case, MLR evaluates the correlation between two or more input variables by fitting a linear equation to observed data [12,18]. Multiple linear regression involves summarizing data as well as investigating the relationship between variables. The general form of a multiple linear regression model is given in Eq. (1) below [1,3]:

(1)\widehat{\mathrm{Y}}=a_{\mathrm{0}}+\sum _{j=\mathrm{1}}^m{\displaystyle a_j{X}_j},

where Ŷ is the model`s output, X_j’s are the independent input variables to the model, and a₀, a₁, a₁,…, a_m are partial regression coefficients. The parameters are trained in such a way that leads to the most similarity between the model’s output and output on the training data set. Therefore, just one optimization model would be employed which minimizes the sum of the squares of the vertical deviations from each data point to the regression equation. It is worth mentioning that if a data point lays on the fitted line completely, then the vertical deviation would be equal to zero. The multiple linear regression model constructed in this study demonstrates the correlation between the concrete characteristics and 28 days compressive strength.

3.2 Artificial neural networks model (ANN)

Artificial neural network which can be identified as the data processing systems, are algorithms simulating the functioning of the biologic neurons. ANN is able to learn from experiences and information in order to develop its performance in the environment [17,19]. The ANN is a collection of simple neurons that operate locally. In other words, the artificial neural network consists of numerous neurons working in union to solve particular problems [1,20]. Neural networks represents simplified methods of a human brain and can be replaced with the customary computations which finds the problems difficult to solve.

The artificial neural network obtains knowledge through learning. The same way as human brain, ANN utilizes examples to learn. Artificial neural networks have been used broadly in the various engineering applications because of their ability to offer a worldwide practical method for real-valued, discrete-valued, and vector valued functions from examples [1,12,17].

Figure 2 shows the general structure of the ANN model used in this research. The multi-layer feed-forward network is comprised of numbers of layers of neurons which is also called nodes. The layers consists of an input layer and output layer with one or more hidden layers in between [1,12,17].

It is highly suggested to use a single hidden layer since scientists have demonstrated that for the purpose of estimating any measurable functional relationship between input data and output variable for any preferred accuracy, a single layer network which is comprised of a sufficiently large number of neurons can be utilized [1,21].

The weights have capabilities to determine strength of connections between interconnected nodes. Therefore the weighted connections are used to find the connections between each particular node in a layer to many other nodes. Wherever the number of weights exceeds the sample size, the early stopping method can be used [12,22].

Normally, the data in ANN are divided into three different subsets of

1) Training; in this step the subset is trained and learned from examples, just like what happens to human brain. The number of epochs is repeated until the accepted accuracy of the model is obtained.

2)Validation; this subset would be able to present how well your model is trained and estimate model properties such as classification errors, mean error for numerical predictors, etc.

3) Test; this subset would be able to verify the performance of the constructed training subset in the ANN model.

3.3 Adaptive neuro-fuzzy inference system model (ANFIS)

ANFIS which is the short-term of adaptive neuro-fuzzy inference system, first was introduced by Jang. This model is a widespread approximation methodology and a hybrid predictive model and, as such, is talented to estimate any real continuous function on a complex set to any degrees of accuracy. In other words, in order to generate mapping relationships between input variables and output variables based on human knowledge, ANFIS which is a combination of fuzzy logic and neural network can be utilized.

ANFIS has the ability to seek interpretable IF_THEN rules which leads this model to have a significant advantage in comparison to other models [3,23,24]. Figure 3 shows the architecture of an ANFIS model with two input variables.

According to Jang, ANFIS model and Takagi-Sugeno-Kang (TSK) are similar with each other in terms of functionality. Having two fuzzy IF_THEN rules of TSK in the ANFIS model, we would have the following statements [3,24]:

Rule 1 → if x is A₁ and y is B₁, then f₁ = p₁x + q₁y + r₁.

Rule 2 → if x is A₂ and y is B₂, then f₂ = p₂x + q₂y + r₂.

As it is shown in Fig. 3, the architecture of the ANFIS model consists of 5 different layers which each of them are identified in the following [3]:

1) Layer 1: This layer is the fuzzification layer. Using the membership function at any node i in this layer, is transformed to membership values as it is illustrated in Eq. (2) [3,24]:

(2)O_i^{\mathrm{1}}={\mu }_{A_{\mathrm{i}}}(x),

in which x is the input to node i and A_i is the linguistic label associated with this node function.

2) Layer 2: Any node in this layer multiplies the incoming signals and sends the results out. In other words, each specific node existing in this layer is able to determine the firing power of each rule. The example for this layer is shown in Eq. (3) [3,24].

(3)w_i={\mu }_{A_{\mathrm{i}}}(y)\times {\mu }_{B_{\mathrm{i}}}(y),\hspace{1em}i=\mathrm{1},\mathrm{2.}

3) Layer 3: This layer is able to normalize the membership values. The ith node in this layer determines the ratio of the ith rule`s firing strength to the sum of all rule’s firing strength. Equation (4) shows the calculations of normalized firing strength for node ith in this layer [3,24].

(4){\bar{w}}_i={\displaystyle \frac{w_i}{(w_{\mathrm{1}}+w_{\mathrm{2}})}},\hspace{1em}i=\mathrm{1},\mathrm{2.}

4) Layer 4: This layer which is also called the adaptive layer, would be able to determine the relationship between the input and output values as shown in Eq. (5) [3,24].

(5)O_i^{\mathrm{4}}={\bar{w}}_i(p_{i}x+q_{i}y+r_i),

where {\bar{w}}_iis the output resulted from layer 3, and {p_i, q_i, r_i}is the parameter set.

5) Layer 5: This layer is also called the de-fuzzification layer. The signal node in this later is a circle node labeled ∑ that computed the overall output as the sum of all input signals shown in Eq. (6) [3,24].

(6)O_i^{\mathrm{5}}={\displaystyle {\sum }_i{\bar{w}}_i{f}_i={\displaystyle \frac{{\displaystyle {\sum }_i{w}_i{f}_i}}{{\displaystyle {\sum }_i{w}_i}}}}.

The hybrid learning algorithm combines the least-squares and gradient descent estimation to find a practical set of antecedent and consequent parameters. Lately, ANFIS has adapted hybrid-learning method which is a rapid learning method for its predicting purposes. The hybrid algorithm has been identified as an accurate algorithm by many scientists [3,25].

3.4 Performance criteria

For comparing the results between the three different models in Matlab environment, statistical goodness-of-fit criteria have been offered in this study. To check the correlation performance of the model the coefficient of determination (R²) is the best indicator, shown in Eq. (7) [1].

(7)R^{\mathrm{2}}={\displaystyle \frac{{\left[\underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{\displaystyle (y_i-\overline{y})({\widehat{y}}_i-\overline{\widehat{y}})}\right]}^{\mathrm{2}}}{{\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{(y_i-\overline{y})}^{\mathrm{2}}{\displaystyle {\sum }_{i=\mathrm{1}}^n{({\widehat{y}}_i-\overline{\widehat{y}})}^{\mathrm{2}}}}}},

where y_iis the experimental strength of ith specimen, \bar{y} is the averaged experimental strength, {\widehat{y}}_i is the calculated compressive strength of ith specimen, and \bar{\widehat{y}}is the averaged calculated compressive strength.

To evaluate how well observed outcomes are replicated by the model, the R² coefficient which varies from 0 to 1 is provided. It is worth mentioning that the higher the quantity of R² is, the better the model would be.

4 The models application

It is worth mentioning that in order to establish reliable predictive models, the choice of input variables is one of the most important concepts in modeling the system. The input variables should be chosen in such a way that the model performs the most accurate and powerful functions between the input and output variables. To predict the 28-day compressive strength of concrete, seven different concrete characteristics have been identified as the input variables. Therefore, in Eq. (1) which represents the prediction of the 28-days compressive strength of concrete (\widehat{y}), by the multiple linear regression model, X₁ is the W/C, X₂ is the maximum size of aggregate, X₃ is the amount of gravel, X₄ is the amount of cement, X₅ is the amount of sand 3/4, X₆ is the amount of sand 3/8, and X₇ is the fineness modulus of sand. The same parameters are used as input variables in ANN and ANFIS Models. All selected parameters were found to be statistically significant and comprehensive in the case of predicting the 28-days compressive strength of concrete.

In the multiple linear regression model, the data were divided into two subsets of: 1) training which contains 85% of total data (i.e., 147 samples), and 2) test data which contains 15% of total data (i.e., 26 samples).

In ANN model the data set was divided into three subsets of: 1) training which contains 70% of total data (i.e., 121 samples), 2) validation which contains 15% of total data (i.e., 26 samples), and 3) test data which contains 15% of total data (i.e., 26 samples). This division is due to both increase the generalization capacity of the ANN model and overcome over-fitting.

In ANN model the sigmoidal tangent function was considered for the hidden nodes and a linear activation function was employed for the output layer. ANN uses backpropagation algorithm (BP) to simulate the results. Backpropagation algorithm determines the error in the model, and based on the result, equilibrate the weights in the output layer. Different algorithms were tried in this study and among all of them, the Levenberg-Marquardt (LM) algorithm was chosen as the best one. Also, the network was formed by only one hidden layer and one output layer. In addition, the number of hidden neurons in the hidden layer was chosen as 15. This number can be estimated as 2(n + 1) in which nis the number of input variables. The final structure of the neural network used in this study is shown in Fig. 4 which is established in Matlab environment [25].

In ANFIS, the same for ANN, the data set was divided into three subsets of: 1) training which contains 70% of total data (i.e., 121 samples), 2) validation which is also called check data contains 15% of total data (i.e., 26 samples), and 3) Test data which contains 15% of total data (i.e., 26 samples).

In the architecture of the adaptive neuro-fuzzy inference system the number of membership functions, type of membership function, and the optimization method play important rules. The number of membership functions is chosen as 3 for each input. The sub-clustering is chosen for the type of membership function. Also, To train the model the hybrid training algorithm was used. The structure of the ANFIS model is shown in Fig. 5 which is established in Matlab environment [25].

5 Results and discussion

In recent years, scientists have performed various investigations on different types of civil engineering materials, particularly concrete and cement [26–29]. The 28 days compressive strength of concrete is assumed as the standard compressive strength of concrete, and as a result, in both the engineering decisions and concrete constructions, approximating the compressive strength of concrete is a major fact. Following, the 28 days compressive strength of concrete is estimated using three different models of MLR, ANN, and ANFIS.

5.1 Regression

In this study, the multiple linear regression model was performed in Matlab software. Figure 6 shows the training data set results for the MLR in Matlab software.

5.2 Artificial neural network

In this study, the ANN model was performed in Matlab software. Figure 7 shows the training state for the ANN model which is established in Matlab environment [25]. As it is shown in the Figure, after epoch number 5, the errors are happened 6 times and the test is stopped in epoch number 11. The repetition in number of errors implies over-fitting of the data. Therefore, the epoch number 5 is considered as the basis and the weights of epoch 5 is considered as the final weights. In addition, since the errors are happened 6 times before stopping the test, the validation checks would be equal to 6.

In addition, Fig. 8 shows the validation performance which is established in Matlab environment [25]. According to this Figure, it is illustrated that the best validation performance is happened at epoch 5, and the test is stopped at epoch 11 after 6 error repetitions. These numbers imply the same numbers presented by Fig. 7.

Figures 9 and 10 show the relationships between target and output variables for training and validation, respectively. These two Figures are established in Matlab software [25]. The “target” values imply the “measured compressive strength” and the “output” values imply the “predicted compressive strength by Matlab software.” The term “R²” is obtained by Matlab software which implies the accuracy of the model. In both of the Figures, the term “R²” illustrates acceptable accuracies of training and validation.

5.3 ANFIS

In this study, the ANFIS model was performed in Matlab software. Figure 11 shows the target and output data for compressive strength of concrete for training data. Blue and red nodes in the Figure indicate target and output data, respectively. The Figure represents good coincidence of target and output data which indicates the qualified ANFIS model. Furthermore, Figs. 12, and 13 shows the comparison between the “target” and “output”parameters for validation and test data in ANFIS modeling, respectively. These two figures as well as Fig. 11 demonstrate qualified coincidence of target and output data in all the three steps of “training,” “validation,” and “testing.” Figs. 11–13 are established in Matlab environment [25].

5.4 Comparison of results for the 3 models

Figures 14–16 show the performance of predicting the concrete compressive strength for the test data set for three different models of MLR, ANN, and ANFIS, respectively. As it is shown in the Figures, the ANN and ANFIS perform better than MLR in terms of R². However, the ANN is more accurate in comparison with ANFIS. It can be concluded that the ANN and ANFIS show a closer match between the target and output values for the test data. The reason might be due to the fact that in both the models of ANN and ANFIS the nonlinear relation between the input variables is involved, however in the MLR model, the relation between the variables is linear.

5.5 Sensitivity analysis (SA)

Generally, sensitivity analysis (SA) can be explained as how much the model response is influenced by changes in the model input parameters [30]. In this study, two different sets of parameters are chosen to perform the sensitivity analysis. Set 1 includes 7 different parameters i.e., 3/4 sand, 3/8 sand, cement, gravel in kilograms, maximums and size of aggregate in millimeter, fineness modulus of sand, and water-cement ratio. Set 2 includes cement, water-cement ratio, gravel, and sands. The coefficient of determination of prediction models based on these two different sets are obtained and compared with each other, shown in Table 3.

As it is shown in the Table 3, coefficient of determinations for set 1 are higher than the ones for set 2. Therefore, based on the performed sensitivity analysis, it can be concluded that the models are sensitive to the number of their input variables, and therefore, the more the number of these input elements are, the more accurate the results would be obtained.

6 Conclusion

In this research, multiple linear regression, artificial neural network, and ANFIS models were developed to predict the 28 days compressive strength of 173 different concrete specimens. The compressive strength was influenced by different concrete characteristics namely 3/4 sand, 3/8 sand, cement, gravel in kilograms, maximums and size in millimeter, fineness modulus of sand, and water-cement ratio. It is concluded that the ANN and ANFIS models could predict the 28 days compressive strength of concrete more accurately. However, it is observed that the ANN model could perform better in comparison with ANFIS model in terms of R². On the other hand, it was resulted that the regression model which is a familiar method in modeling of engineering systems for its closed-form representation, failed to be reliable and therefore, advanced models like ANN and ANFIS models are preferred. This advantage may be because of the nonlinear correlation between the input variables which can be presented better by both ANN and ANFIS models. Generally, the regression model is a suitable model for proposing the preliminary mix design, however for higher accuracy requirements, the ANN and ANFIS models are suggested.

To overall, both the methods of ANN and ANFIS can be used to predict the 28 days compressive strength of concrete with high reliability, instead of using costly experimental investigations. The outcomes of this research will present helpful information which can be used at the time scientists are dealing with choosing appropriate concrete mix designs. The ANN and ANFIS can be used as capable tools in order in the selection of the most optimized concrete mix design.

Finally, the sensitivity analysis on two different sets of input variables was performed for all the three models. Results indicate that the accuracy of the concrete compressive strength prediction is highly dependent on the number of input variables. In other words, the more the number of input parameters are, the more accurate the results of the predictor models would be.

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