Prediction of the resilient modulus of flexible pavement subgrade soils using adaptive neuro-fuzzy inference systems

Construction and Building Materials 123 (2016) 235–247

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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

Prediction of the resilient modulus of flexible pavement subgrade soils using adaptive neuro-fuzzy inference systems Ehsan Sadrossadat a,⇑, Ali Heidaripanah b, Saeedeh Osouli c a

Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran Department of Civil Engineering, Kerman Graduate University of Technology, Kerman, Iran c Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran b

h i g h l i g h t s

g r a p h i c a l a b s t r a c t


This study explores the application an

a r t i c l e

i n f o

Article history: Received 18 January 2016 Received in revised form 3 July 2016 Accepted 5 July 2016

Keywords: Resilient modulus Flexible pavements Prediction Artificial intelligence Adaptive neuro-fuzzy inference system

Direct estimation

Traffic Live Load

Surface Base Subbase

Model Processing Input Variables

Artificial Neural Networks

Output Variable

Fuzzy Inference system

a b s t r a c t The resilient modulus (MR) of pavement subgrade soils essentially describes the structural response of pavements for a reliable design. Due to the elaborate, expensive and complex experimental estimation of MR factor, several models are proposed for indirect estimation of it which are mostly established based on statistical analyses e.g. regression analyses. The deficiencies of existing models in addition to the complexity of resilient behavior of soils indicate the necessity to develop better models. This study investigates the potential of a powerful hybrid artificial intelligence paradigm, i.e. adaptive neuro-fuzzy inference system (ANFIS), for prediction of MR of flexible pavements subgrade soils. A comprehensive database which comprises a total of 891 experimental datasets conducted on different Ohio cohesive soils is taken from the literature for evolving models. In ANFIS modeling, P#200, LL, PI, wopt, wc, Sr, qu, r3, rd are considered as input variables and correspondingly the output is MR. Several statistical criteria, validation and verification studies are used for evaluating the performance capability of the obtained model. A sensitivity analysis is utilized to demonstrate the effectiveness of the considered input variables for characterizing MR. Besides, the response of ANFIS based MR model to variations of each input variable is examined using a parametric study and results are compared to those experimentally provided in the literature. Eventually, the obtained results approve the robustness of ANFIS approach for indirect estimation of MR of subgrade soils. Ó 2016 Published by Elsevier Ltd.

⇑ Corresponding author. E-mail address: [email protected] (E. Sadrossadat). http://dx.doi.org/10.1016/j.conbuildmat.2016.07.008 0950-0618/Ó 2016 Published by Elsevier Ltd.

Experimental studies

Subgrade

Indirect estimation

artificial intelligence method, namely adaptive neuro fuzzy system (ANFIS) for prediction of resilient modulus of flexible pavement subgrade soils.
The structure of ANFIS is described for utilizing in different complex prediction problems.
Various validation and verification study phases are represented for evaluating the performance and accuracy of a model.
The robustness of ANFIS, as a predictive tool, is confirmed for indirect estimation of resilient modulus of pavement subgrade soils.

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1. Introduction The main purpose of a pavement design is to provide structural and economical combinations of materials so that it can serve the applied traffic loads for a specific duration of time. Flexible pavement structures commonly consist of four principal layers including surface, base and two unbound layers, i.e. sub-base and subgrade. As vehicles pass over the pavement surface, different dynamic and cyclic loads are applied to pavement layers. All pavements gain their ultimate support from the underlying subgrade which usually includes different combination of soils [1,2]. Subgrade soils play the role of being a flexible foundation via deformation, compaction and distortion. At the initial stage of vehicular traffic loadings, i.e. the deviator stress (rd), a considerable deformation happens in subgrade soil structure. As the number of stress repetitions increases, the plastic strain decreases until it becomes practically all recoverable. The elastic modulus based on the recoverable strain under repeated stresses is referred to as the resilient modulus (MR). As illustrated in Fig. 1, MR is considered as the ratio of applied deviator stress (rd) to the recoverable strain (er) [3]. MR can be defined as the structural response of pavements. It is a measure of the elastic modulus of recognizing nonlinear stressstrain characteristics of subgrade materials [4]. MR takes into account the significant effects of traffic loading, deviator, cyclic and confining stress states. MR can be determined by conducting various in-situ and cyclic tri-axial load, torsional shear and resonant column laboratory testing methods [5–11]. But, experimental estimation of the MR is definitely expensive, elaborate and complex [12]. On the other hand, various codes such as MechanisticEmpirical Pavement Design Guide (M-E PDG), American Association of State Highways and Transportation Officials (AASHTO) and National Cooperative Highway Research Program (NCHRP) have recommended considering MR for structural analysis and design of multi-layered pavement systems [9,11,13–15]. Regarding to M-E PDG procedure for determining MR value through a numerical model, numerous properties of unbound layers materials must be taken into account for describing the resilient behavior of pavement. Nevertheless, it is crucial to understand the factors affecting MR for a precise estimation due to the changes in traffic patterns and environmental properties [16–18]. Accordingly, numerous studies have been conducted and several constitutive models have been suggested for estimation of MR in terms of applied loads, stress states and soil physical and hydrological properties [5,17,19–24]. These models are often obtained via regression analyses. General forms of some constitutive MR equations of subgrade

1-

3

d

= Deviator Stress

1=

M R=

Major Stress

3=

MR = Resilient Modulus

soils which are made by regression analysis are represented in Table 1. Regarding to the equations represented in Table 1, main variables are related to stress states such as rd and r3 which vary in laboratory tests for different soil samples to obtain a new MR value. k1, k2 and k3 are obtained by using either linear or nonlinear regression analyses to fit the prediction model to laboratorygenerated MR test data. Coefficient k1 cannot take a negative value since MR cannot be negative. Besides, k2 should be positive since increasing the bulk stress or confining stress should produce stiffening effect on the material, which leads to higher MR. In contrast, k3 must take a negative value because increasing the shear or deviator stress should produce softening effect on the material [14,23,24]. Afterwards, a second set of regression analyses are carried out to relate these k-coefficients with the soil physical properties such as moisture or water content (wc), specimen dry density (cd), maximum dry density (cd max), liquid limit (LL), plasticity index (PI), uniformity coefficient (CU), coefficient of curvature (CC), percent passing #200 sieve (P#200) and etc. A summary of obtaining these models can be found in the previously published literatures, including [7,17–20,23,24,27]. It can be realized that aforementioned analysis procedure is definitely complicated. Moreover, there are other disadvantages. Although the model developed by regression analysis performs reasonably well on selected data sets, the capability of them is thoroughly restricted to the range of utilized datasets used for the analysis; besides, they are not validated and not tested on new data. The deficiencies of existing models in addition to the complexity of MR behavior indicate the necessity to develop better prediction models for indirect estimation of MR of pavement subgrade soils. During past two decades, many computational intelligence techniques such as artificial neural networks (ANNs), fuzzy inference system (FIS), genetic programming (GP), support vector machines (SVM) and such methods are proposed based on biological theories and activities for tackling real world problems. These computing techniques have a lot of features that have made them attractive choices for using in different problems. The main feature is that they are data-driven self-adaptive methods. They can automatically learn from data to determine the structure of a prediction model. Besides, these techniques have been successfully employed to solve various problems in civil engineering domains [28–46]. Despite extending promotions in such artificial intelligence (AI) methods, there are few attempts in the literature for using them in predicting the MR of pavement subgrade soils.

Confining Stress

3

r

Elastic Strain

Total Strain

(Recoverable Strain)

Plastic Strain

Accumulated Plastic Strain

Plastic Strain Fig. 1. Resilient modulus of subgrade soils under repeated traffic loading (Huang, 2004).

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cohesive soils via the same database collected by [25] and compared it with the results obtained by SVM approach. Kim and his colleagues [49] studied on the development of an ANN model to estimate subgrade resilient modulus relating it into the associated stress states and soil physical properties. Complex real-world problems may require intelligent systems that possess humanlike expertise within a specific domain, adapt themselves for modifying environments and be able to explain how they make decisions or take actions. The objective of this paper is to utilize adaptive neuro-fuzzy inference system (ANFIS), as a powerful branch of AI methods, for indirect estimation of MR of flexible pavement subgrade soils. A comprehensive database which comprises a total of 891 experimental data sets conducted on Ohio cohesive A-4, A-6, and A-7-6 soil types is taken from the literature for evolving models. It is worth mentioning that the database is the same as that used for modeling in the previous works of [25,35]. Herein, the MR factor is characterized in terms of soil physical and hydrological properties as well as unconfined compressive strength, confining stress and deviator stress. Moreover, in order to evaluate the performance of the obtained ANFIS model, several validation and verification study phases are considered. At last, results are discussed.

Table 1 General forms of regression based equations for resilient modulus of soils. Reference

Soil

Equation

Kim [19]

A-4 and A-6 soils A-7-6 soil

Universal model [19,24,25] Pezo and Hudson [26]

MR Pa

¼ k1

h

Pa :roct

MR ¼ k1 :P a

All types of soils

ik2

s2oct

MR ¼ k1 :P a





9Pa 2

1 3rd

 k2  h Pa

þ rr32 d

k2 k3

soct þ 1 P a

MR ¼ k1 rk32 rkd3

Fine grain materials

MR: resilient modulus; Pa: atmosphere pressure (=101.4 kPa); h: bulk stress (= [r1 + r2 + r3]); roct: octahedral normal stress (=[r1 + r2 + r3]/3); soct: octahedral shear stress (=[(r1  r3)+ (r1  r2)+ (r2  r3)]0.5/3); r1: major principal stress; r2: intermediate principal stress (=r3 for MR tests on cylindrical specimens); r3: minor principal stress or confining stress; rd: deviator stress (=[r1  r3] for MR test on cylindrical specimens) and k1, k2 and k3: regression coefficients.

Recently, Hanittinan [25] utilized ANNs for prediction of the resilient modulus of pavement subgrade soils through compiling a comprehensive datasets of Ohio cohesive A-4, A-6 and A-7-6 class of soils. Furthermore, Zaman et al. [47] represented the development of different ANN models to correlate the MR factor with routine properties of subgrade soils and states of stress for pavement design application. Nazzal and Tatari [48] evaluated the use of ANNs and genetic algorithms to improve the accuracy of the prediction of subgrade resilient modulus based on soil index properties. More recently, Pal and Deswal [35] investigated the potential of two variants of extreme learning machine based regression approaches in predicting the resilient modulus of

2. Review of ANFIS Adaptive neuro-fuzzy inference system (ANFIS) is a fuzzy inference system (FIS) that trains their learning parameters in the artificial neural network (ANN) architecture [50,51].

Premise Part

Consequent Part

f = p x+q y+r 1 1 1 1

f =

w1 f 1 + w2 f 2 = w f1 + w f w1 + w2

f = p x+q y+r 2 2 2 2

Layer 3 Normaliation

Layer 1 Fuzzification

Layer 4 Defuzzification

Layer 2 Implication A1 x

Fixed Node

N

Adaptive Node Layer 5 Combination

A2 Output

B1 y

N B2 Fig. 2. A typical first-order TS model reasoning and basic ANFIS architecture.

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ANNs are the artificial intelligence systems which are motivated from the neural networks of the human brain for modeling different real-world problems. An ANN structure is comprised of interconnected artificial neurons that mimic some properties of biological neurons. The interest in ANNs is largely due to their ability in learning from experiences [29,33,52,53]. On the other hand, FIS is another technique which utilizes the human expertise and linguistic knowledge for modeling problems [54]. FIS includes premise and consequent parameters connected by fuzzy rules. In FIS the crisp input variables are fuzzified by the membership functions and are submitted to the fuzzy inference block, which is a decision-making unit and generates fuzzy variables through fuzzy reasoning models [55]. Lastly, the resulted fuzzy output is mapped to a crisp output using the membership functions, in the defuzzification step [52,56,57]. Takagi–Sugeno (TS) model is a type of fuzzy approaches that utilizes optimization and adaptive techniques to design the membership functions and IF–THEN rules [58,59]. The output membership function in TS is designed as either linear or constant which are referred to as first and zero order TS type, respectively. Considering two input variables (x, y) and one output (f), the two if–then rules in zero and first-order TS type can be represented as the following expressions: Zero-order: if x = A1 and y = B1, then f1 = c1 First-order: if x = A2 and y = B2, then f2 = p2x + q2y + r2

1 XX n n 2 ðt  hk Þ 2 n k k

O1;i ¼ lAi ðxÞ for

i ¼ 1; 2

ð2Þ

O1;i ¼ lBi ðxÞ for

i ¼ 3; 4

ð3Þ

In equations above, x is the input to node i, and Ai is the linguistic label associated with this node function. Parameters in this layer are called premise parameters. So, the O1,i (x) is essentially the membership grade for x and y. Layer two: Every node in this layer is fixed and labeled by ‘AND’. These nodes multiply incoming signals from layer one, and the product represents the firing strength of a rule (wi). This is similar to the fulfillment degree of a fuzzy rule. This is where the normalization is performed to ‘AND’ the membership grades for example the product:

O2;i ¼ wi ¼ lAi ðxÞ:lBi ðyÞ for i ¼ 1; 2

where ci, pi, qi and ri are the consequent parameters obtained from the training, A and B labels of fuzzy set defined suitable membership function. Determination of the number and form of fuzzy rules for a system is a crucial phase and requires expert knowledge about the system data. Hence, several algorithm-based methods have been proposed to automate this process. Those methods are developed by dividing data into different groups, namely clustering. The process of clustering in FIS is mostly done by utilizing methods such as grid partitioning, subtractive clustering, fuzzy c-means clustering and k-means clustering [60]. Finally, the output of FIS is fuzzy. Hence, defuzzification should be implemented to obtain a real-valued output which is mostly done by using different algorithms e.g. center of gravity, weighted average, constraint decision defuzzification, center of area, fuzzy clustering defuzzification, mean of maxima [61]. Although a fuzzy system cannot be trained, neural networks are able to do self-training by utilizing datasets. ANFIS is a hybrid modeling technique to overcome deficiencies of fuzzy inference system (FIS) and artificial neural networks (ANNs). ANFIS combines aspects of human linguistic knowledge and reasoning representation of fuzzy systems with the learning power of multi-layer perceptron (MLP) as a well-known branch of ANN structures [50–52,62]. ANFIS uses a hybrid learning rule combining the back propagation gradient descent error and a least-squares method (LSM) for updating weights and membership functions for obtaining the fittest input-output model similar to MLP architecture [31,63,64]. Accordingly, in ANFIS, the input variables are propagated forward in a network layer by layer and optimal consequent parameters are identified by the LSM, while the premise parameters are assumed to be fixed for the current cycle through the training set. Next, the error values propagate backward to modify or update the premise parameters, using back propagation gradient descent method. In this algorithm, the weight values are changed to minimize the following error function (E):

E¼

m

where tnk and hk are, respectively, the calculated output and the actual output value. n is the number of samples and k is the number of output neurons. As illustrated in Fig. 2, there are five essential layers in which the mathematical computations in ANFIS are performed. The process in each layer may be described as follows: Layer one: Each node i in this layer produces a membership grade of a linguistic label (fuzzification). The output of the ith node of the first layer may be selected by a MF such as linear, triangular, trapezoidal, Gaussian, generalized bell or several other functions.

ð1Þ

ð4Þ

Layer three: Each node in this layer is a fixed node labeled N. The ith node calculates the ratio of the ith ratio of the firing strengths of the rules as follows:

O3;i ¼ wi ¼

wi w1 þ w2

for i ¼ 1; 2

ð5Þ

Layer four: The fourth layer is the second adaptive layer of ANFIS architecture, i.e., defuzzification layer.

O4;i ¼ wi f i ¼ wi ðpi x þ qi y þ ri Þ

ð6Þ

The parameters in this layer (pi, qi, ri) are to be determined and are called consequent parameters. Layer five: The single node in this layer is a fixed node labeled which calculates the overall output as the summation of all incoming signals.

Overall output ¼ O5;i ¼

X X wi f i wi f i ¼ Xi w i i i

ð7Þ

3. Database and model variables A comprehensive database is utilized to generate the ANFIS model for indirect estimation of MR of subgrade soils. The database consists of 891 datasets from three different types of soils which are classified as A-4 or A-6 and A-7-6 based on AASHTO soil classification code. The database is acquired from Hanittinan [25] which is a data collection of previous studies conducted on three types of cohesive Ohio soils at the soil mechanics research laboratory of the Ohio State University, Purdue University and the University of Mississippi [25,35]. A total of 418 experimental datasets were collected from nine different A-4 soil locations, about 283 data for the A-6 soils from seven different locations and 190 datasets from A-7-6 soil type samples came from four different locations [25]. There are several protocols for obtaining MR value from an experimental test. AASHTO has adopted several test protocols over the last two decades, e.g., T292-91, T294-92, TP46-94 and T307-03 [10]. In the considered database, all MR tests were performed in

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E. Sadrossadat et al. / Construction and Building Materials 123 (2016) 235–247 Table 2 Descriptive statistics of variables used to develop ANFIS Model for all data. Statistical index

P#200 (%)

LL (%)

PI (%)

Wopt (%)

Wc (%)

Sr (%)

qu (kPa)

r3 (kPa)

rd (kPa)

MR (MPa)

Mean Standard Deviation Sample Variance Minimum Maximum

75.22 17.64 311.34 42 100

32.60 9.92 98.42 21 59

12.69 8.69 75.49 2 36

15.18 3.03 9.18 9.4 24.2

15.33 3.53 12.46 7.53 27.2

81.43 11.17 124.78 42.92 100

311.11 162.80 26502.58 54.3 715.74

20.97 16.36 267.56 0 41.4

40.31 18.22 332.09 10 71.69

54.82 29.68 880.91 6.4 179.44

accordance with the AASHTO designation T294-94, standard methods of experimental testing for resilient modulus of subgrade soils [25,35]. The testing system contains equipment of applying loads, a tri-axial pressure chamber and a computer with data acquisition system [6]. When the purpose is to estimate MR, the tested functional parameters are deviator pressure or major principal stress (rd) and confining or minor principal stress (r3). Besides, soil shear strength is proportional to MR, and so the unconfined compressive strength (qu) which is the undrained shear strength of cohesive soils must be taken into account [2]. In fact, MR highly depends on variations of soil physical and hydrological properties and environmental changes which cannot be described by stress-based parameters [5]. Hence, many other associated parameters should be involved. Percent of soil particles passing through a #200 sieve (P#200), liquid limit (LL), and plastic index (PI) are the soil parameters used for classifying a soil type. On the other hand, changes in soil water conditions in pavement layers result in dynamic changes in environments. Therefore, moisture or water content (wc) and the degree of soil saturation (Sr) which is defined as a function of gravimetric density of purified water of soil have to be involved as input parameters [25,42]. In addition to aforementioned parameters, soil compacting condition effects MR which is desirable for unbound layers in pavement design. Herein, optimum moisture content (wopt) which is an important index of a soil at its maximum dry density is taken into account [9,25,35,49]. According to the literature and the existing database, nine parameters obtained from experimental results are chosen as input variables for the modeling development. Furthermore, these variables are also selected as the effective parameters in previously published literature [7,9,21,25,35,49,65,66]. Consequently, the proposed model for prediction of the MR factor is considered to be a function of the parameters as follows:

M R ¼ f ðP#200 ; LL; PI; wopt ; wc ; Sr ; qu ; r3 ; rd Þ

presented to the model, the error becomes very large. This phenomenon is referred to as overfitting. Overfitting decreases the predictive performance of models, as they can exaggerate minor fluctuations in the datasets. A traditional approach to avoid overfitting is to test the model on a validation dataset to find a better generalization. To cope with these issues, the available database is classified into three sets: (1) training, (2) validation, and (3) test subsets. The training set is utilized to fit the models, the validation

Fig. 3. The schematic representation of obtained ANFIS structure.

ð8Þ

The descriptive statistics of the variables is represented in Table 2. For more details and analyses on the datasets, respected readers are invited to refer directly to [7,9,19,25,35,49]. The validity of the models obtained by artificial intelligence methods depends upon the number of data used for the training process. The more data used in training process, the safer model can be reached. In this context, researchers describe that the minimum ratio of the number of objects over the number of selected variables for model acceptability is 3 [32]. Also, they suggest that considering a higher ratio equal to 5 is safer. In the present study, this ratio is much higher and is equal to 891/9 = 99. 4. Data classification Machine learning and statistical based models are typically evolved by optimizing their performance on some set of training data. Hence, attempting to make the model to conform closely to a limited range of data can affect the model and reduce its predictive power. A major problem in generalization of those techniques is a case in which the error on the datasets obtained by the model is driven to a very small value, but when new datasets are

Fig. 4. The generated Sugeno-type FIS used in ANFIS structure: 9 inputs, 1 output and 8 rules.

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E. Sadrossadat et al. / Construction and Building Materials 123 (2016) 235–247

set is used to estimate prediction error for model selection and the test set is used for evaluation of the generalization error of the selected model. The training, validation and testing data are usually taken as 50–70%, 15–25% and 15–25% of all data, respectively, for the development of models in artificial intelligence (AI) techniques. These ratios are selected based on previously suggested values [34,53,63,67]. In the present study, among 891 data, 60%, i.e. 535 data vectors for the training process and 20%, i.e., 178 data as the validation data. The 20% remaining datasets, i.e. 178 data, are used for the testing of the obtained model.

5. Development of the ANFIS model In order to establish the ANFIS-based model for prediction of the MR factor, nine variables are considered as inputs. A code was written in MATLAB 2011b environment using Genifis3 command. The obtained ANFIS structure is demonstrated in Fig. 3. In inference method which is used in the present study, AND is prod, OR is probor, implication is prod and aggregation is sum. The detailed definitions of these expressions are described in Matlab functions.

Fig. 5. The initial and obtained Gaussian MFs generated by the optimal ANFIS model before and after the training process.

E. Sadrossadat et al. / Construction and Building Materials 123 (2016) 235–247

Fig. 5 (continued)

241

242

E. Sadrossadat et al. / Construction and Building Materials 123 (2016) 235–247 Table 3 Statistical parameters of the ANFIS model for the external validation using all data. Item

Formula

Condition

ANFIS model

1 2 3

R RMSE  P MAE ¼ 1n ni¼1 jhi  t i j   varianceðhi t i Þ VAF ¼ 1  varianceðhi Þ Pn ðhi t i Þ k ¼ i¼1 2 h Pn i ðhi t i Þ 0 k ¼ i¼1t2

0:8 < R

0.982 5.682 3.95

4 5 6

96.33 0:85 < k < 1:15 0

0.9979

0:85 < k < 1:15

0.9938

m < 0:1

-0.038

n < 0:1

-0.0379

i

7

m¼R

8

n¼R

2

9

2

2

Ro R2 Ro02 R2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rm ¼ R ð1  jR2  Ro2 jÞ Pn 2 ðt i hoi Þ o Ro2 ¼ 1  Pi¼1 n 2 hi ¼ k  t i ðt i t i Þ Pi¼1 n o 2 ðhi t i Þ 0 Ro02 ¼ 1  Pi¼1 and t oi ¼ k  hi n  2 2

where

i¼1

ðhi hi Þ

0:5 < Rm

0.779

Should be close to 1 Should be close to 1

1.00 0.9998

hi: The actual outputs for the ith output. ti: The calculated outputs for the ith output. n: The number of samples. hi : The average of the actual outputs. ti : The average of the calculated outputs.

Fig. 6. Predicted versus experimental MR values using selected ANFIS model for (a) training data, (b) validation data and (c) test data.

By default, Genfis3 considers a Gaussian and a Linear function for input and output MFs, respectively. General form of the Gaussian membership function is represented as below [68]: ðxci Þ2 2r2 i

lAi ðxÞ ¼ e

where ci and ri are the center and width of the ith fuzzy set Ai, respectively. Genfis3 automatically generates a Sugeno-type FIS structure using fuzzy c-means (FCM) clustering and extracting a set of rules that models the data behavior. In FCM technique each data point belongs to a cluster with some degree of MF. The algorithm begins with an initial guess for the cluster centers, which are almost incorrect. Then every data point is assigned a membership grade for each cluster. By iteratively updating the cluster centers and the membership grades for each data point, the correct mean location of each cluster is identified. This iteration is based on minimizing a differential function that calculates the distance from any given data point to a cluster center weighted by that data point‘s membership grade [69]. The generated Sugeno-type FIS is represented in Fig. 4. In addition, Fig. 5 represents generated initial and obtained MFs by ANFIS before and after the training process. The number of clusters, which are determined in genfis3, is equal to the number of rules and membership functions in the generated FIS. A fuzzy rule which is extracted from ANFIS model can be represented as follows: If (P#200 is in1cluster1) and (LL is in2cluster1) and (PI is in3cluster1) and (Wopt is in4cluster1) and (Wc is in5cluster1) and (Sr is in6cluster1) and (qu is in7cluster1) and (r3 is in8cluster1) and (rd is in9cluster1) then (MR is out1cluster1). In expression above, P#200 is in1cluster1 indicates that P#200 is considered as the first input variable which is selected from the first cluster. In this study, the weighted average method (wave) is utilized as the defuzzification method. It is one of the most frequently used methods in fuzzy applications [70]. Wave is typically applied to symmetrical output MFs such as those provided in this study, i.e. Gaussian MF. It is formed by weighting each membership function in the output, using its respective maximum membership value which is the center of symmetrical MF. The algebraic expression of wave is given as follows:

Xn 

ð9Þ

lAi ðci Þ:ci l ðci Þ i¼1 Ai

c ¼ Xi¼1 n

ð10Þ

243

In equation above, c⁄ is the defuzzified real-valued output where lAi(x) is the ith MF and ci is the center of the ith fuzzy set Ai, respectively. In the written code, the correlation coefficient (R) value has been used as the termination criteria via software modeling and after that the model is selected based on considering the lowest root mean square error (RMSE) value. According to suggested criteria by various researchers, if a model gives R more than 0.8, there is a strong correlation between the predicted and measured values. These parameters are calculated using the following equations:

MR (MPa)

E. Sadrossadat et al. / Construction and Building Materials 123 (2016) 235–247

180 160 140 120 100 80 60 40 20 0

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sX n 2 ðhi  t i Þ i¼1 n

0

100

200

300

400

500

600

700

800

900

600

700

800

900

40

ð11Þ

ð12Þ

30

Residual Error

 Þðt  t Þ ðhi  h i i i R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn 2 Xn 2  ðhi  hi Þ ðt  ti Þ i¼1 i¼1 i

ANFIS (This Work)

Test No.

Xn

i¼1

Experimental

Max RE =31.25 Min RE =-31.76

20 10 0 -10 -20 -30 -40

th

where hi and ti are the actual and predicted output values for the i

0

100

200

300

400

output, respectively, hi and t i are the average of the actual and predicted outputs, and n is the number of samples. In addition, another termination criterion, i.e. the maximum number of epochs, is considered equal to 100. Therefore, several runs should be implemented to obtain a robust and generalized model by changing different parameters via several trial and error processes. Herein, the optimization process has been stopped after 51 iterations and the optimal model is selected with R and RMSE equal to 0.98 and 5.68, respectively. It is noteworthy that the model which is introduced and considered as the optimal model in this paper is selected through making several preliminary runs as well as conducting comparative studies based on validation criteria as will be discussed in following sections.

500

Test No.

6. Results and discussion In order to validate and generalize the model after it is selected based on different criteria, supplementary studies must be conducted for more precise assessment of the model performance [28,30,33,34,63]. 6.1. Model validation and generalization A plot of the predicted versus measured MR values obtained by ANFIS is illustrated in Fig. 4 for different classifications of data. Regarding to the high R and low RMSE values, it may be realized that the chosen ANFIS model is able to predict the MR with an acceptable degree of accuracy. Moreover, close R and RMSE values of the training, validation and accordingly the testing data sets confirm that overfitting is avoided and the predictive abilities and generalization performance of the model are sufficient (Fig. 6). Traditionally, several statistical indices are proposed in the literature such as correlation coefficient (R), the root-mean-square error (RMSE), variance account for (VAF) and mean absolute error (MAE) to evaluate the model performance. Recently, another index is introduced as a confirm indicator of the external predictability of models (Rm). For Rm > 0.5, the condition is satisfied. Either the squared correlation coefficient (through the origin) between predicted and experimental values (R2o), or the coefficient between experimental and predicted values (R0o 2) should be close to R2, and close to 1. Besides, new criteria recommended by researchers are checked for validation of the model on data sets. It is suggested that at least one slope of regression lines (k or k0 ) through the origin should be close to 1. Also, the performance indexes of m and n should be lower than 0.1. These criteria are adopted by

Fig. 7. Detecting different values (a) experimental and predicted values of MR in ascending form, (b) residual error value for each datum and (c) histogram of errors.

1

Relative Importance (rij) 0.85

0.84

0.81

200P # (%)

LL

PI

0.8

0.88

0.84

0.83

0.86

0.87

0.85

0.6 0.4 0.2 0

Wopt (%)

(%) Wc (%) Sr (qu (kPa ((kPa3 σ (σd (kPa

Input Variable Fig. 8. Contributions of the each variable in obtained ANFIS model.

various researchers [32,33,71]. The noted criteria and the relevant results obtained by the optimal ANFIS model are summarized in Table 3.

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Although considering statistical measure parameters can be used to realize the overall error value in different ways, they are not comprehensible or easily interpretable [72,73]. Those values may highly be affected by different variables, e.g. the number, biased or unbiased distribution, standard deviation of datasets, existence and number of outliers and the frequency of error values. A precise way of observing the systematic mismatches between the predictions and real experimental values is to consider the errors, defined as the residual error (RE) between observed and

predicted values [33,34,38]. Regarding to the great number of datasets, the experimental and predicted values of MR are sorted from the smallest to the largest values and depicted in Fig. 7(a). For a specified value, this figure can show the viewer the difference between the predicted and experimental value. Furthermore, the ascending arrangement of values can illustrate the trend of experimental results. Considering the same arrangement of MR values in Fig. 7(a), accurate residual error value of each datum is represented in Fig. 7(b).

Fig. 9. Parametric analysis of MR in ANFIS model.

E. Sadrossadat et al. / Construction and Building Materials 123 (2016) 235–247

245

Fig. 9 (continued)

Besides, an error histogram of errors in predicting MR by ANFIS is utilized to demonstrate the frequency, range and distribution of errors in different intervals. As illustrated in Fig. 7(c), the frequency of errors distributed in low intervals is high which indicates the precise measurement capability of the ANFIS model. It is suggested that these three figures be used and represent together so that the user can obtain more precise information about the obtained error and prediction performance of the model. As can be seen, representing residual error values in charts or diagrams has to be identified to be precisely useful for detecting error diversities. The indicated validation procedure can give the user an idea about the amount of reliability, accuracy or the factor of safety when using the model or selecting the best model among different models. In the present study, the results in aforementioned studies represent that the obtained model satisfies the required calibration and validation conditions. Also, it is strongly capable in computational modeling. 6.2. Sensitivity analysis

Cosine amplitude method (CAM) is one of the sensitivity analysis techniques proposed to determine the significance of each input variables [74,75]. Assume that n data samples are collected from a common data array, X. The data pairs used to construct a data array X defined as:

X ¼ fX 1 ;X 2 ; X 3 ; . . . X m g

ð13Þ

Each of the elements, xi, in the data array X is a vector of lengths of m, that is:

X i ¼ fxi1 ;xi2 ; xi3 ; . . . xim gi

ð14Þ

Thus, each of the data samples can be considered as an element in m-dimensional space, where each element requires m coordinates for a full description. Each element in space, rij, has relation with results in a pair wise comparison. In CAM method, the strength of the relation between the data pairs, xi and xj, are calculated and illustrated by the following equation:

rij ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X Xm Xm xik :xjk = x2 : k¼1 x2jk k¼1 ik

ð15Þ

k¼1

Sensitivity analysis is utilized to determine the contribution and relative importance of variables. It is another important concern in selecting an input variable for the aim of future experiments, model developments or field investigations. Sensitivity analysis aims to describe how model output values are affected by changes in input values.

The value of strength ratio (rij) close to one represents the impact of the input parameter on the model output and how much it increases indicates that the input parameter is a more effective than other parameters. Here, rij values between the MR and related input parameters using the CAM method are shown in Fig. 8. As can be seen, the obtained rij values of variables are not far different

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and approximately equal. This means that all variables are almost equally contributed in the obtained ANFIS-based model; in addition, considering all these input variables are important in characterizing MR and none of them can be omitted. The obtained results are conformed to those results provided by of [7,9,19,25,35,49]. 6.3. Parametric analysis Regarding to the literature, several experimental tests are conducted to study the behavior and response of MR to variations of different affecting parameters. In general, the MR of subgrade soils is very sensitive to wc. MR decreases as soil water content increases [19,49]. Correspondingly, MR decreases with increasing Sr but in pavement construction, Sr can be controlled by adjusting the amount of water added to soil and the percent of soil compaction. On the other hand, the wopt represents soil water content at its maximum dry density. The MR factor increases with increasing the degree of compaction [19,49]. Therefore, it can be concluded that MR tends to increase with increasing wopt. In previous researches, it is noted that the effect of the dry density decreases with increase of fine content [19,49]. In general, when the P#200 increases, MR would decrease. However, it is considered that a complicated relationship between the percent of soil particles passing through a #200 sieve and MR, aside from the influences of other factors [7,25]. On the other hand, when LL and PI increases, MR tends to decline. Nevertheless, the relationship between soil PI and MR varies for different soils [25]. On the other hand, soils with high shearing resistance often have higher MR, i.e., MR increases with increasing qu factor. Moreover, it is precisely shown that the MR increases with increasing r3 and decreasing rd [49]. In order to ensure the capability of prediction models, the behavior of variables should be assessed and compared to the results experimentally or theoretically provided in the literature. To do so, a parametric analysis is performed in this study. The parametric analysis represents the response of the ANFIS model to variations of variables. The methodology is based on changing only one predictor variable at a time while the other variables are kept constant at a specified value. A set of synthetic data is generated for each variable regarding to its range in the database. These variables are presented to the prediction model and MR is calculated. This procedure is repeated using another variable until the model response is obtained for all of the predictor variables [32,33,38]. Regarding to the procedure of calculating MR factor via regression based models, it is worth mentioning that the model trends is just checked for k-coefficients to represent that MR increases with increasing r3 and decreasing rd but usually the effects of other variables on the model are not examined [49]. Herein, the parametric analysis is performed for all variables and the effects of all variables on ANFIS model are considered. Fig. 9 presents the tendency of the MR predictions to variations of the affecting parameters, i.e. P#200, LL, PI, wopt, wc, Sr, qu, r3 and rd. Fig. 9 depicts that the MR value continuously increases with increasing qu, r3 and wopt and decreases with increasing P#200, LL, PI, wc, Sr and rd. As can be observed, the results of parametric study on ANFIS model conforms to those experimentally provided in the literature. It approves that the behavior of variables in predicting MR is acceptable and the ANFIS model is capable of capturing the important soil physical and hydrological properties in addition to the applied stress states. 7. Conclusion In this paper, the architecture and potential of ANFIS approach was evaluated for indirect estimation of MR of pavements subgrade soils. ANFIS solves prediction problems with a high degree of accuracy by utilizing human’s knowledge or expe-

rience in modeling and extracting rules from the data without any prior assumption. Regarding to the employed database and also the considered variables in the literature for predicting MR as the output, nine input variables (P#200, LL, PI, Wopt, Wc, Sr, qu, r3, rd) were considered in modeling process. Various studies were conducted for validation and generalization of the model. The optimal ANFIS model, which was considered in this paper, was gained after several preliminary runs and supplementary studies as discussed. A sensitivity analysis method, i.e. CAM, was utilized to check the contribution of each variable in the modeling process. Results show the high accuracy of the ANFIS model as verified by different criteria. Furthermore, a parametric analysis is performed to examine the effects of variables on the obtained MR model. Parametric study demonstrated that the MR value continuously increases with increasing qu, r3 and wopt and decreases with increasing P#200, LL, PI, wc, Sr and rd. The results of aforementioned studies approve the robustness and potential of ANIFS in predicting MR of subgrade soils as well as the capability of capturing the important soil physical and hydrological properties in addition to the applied stress states. Despite the good performance of ANFIS in most cases, it is considered as a black-box predictive tool. That is, such tools are not capable of generating practical prediction equations or if they are, the obtained model is definitely complex and does not have the worth to be used. Moreover, it is worth mentioning that the predictive capability of such computational methods is mostly limited to the range of the data used for its modeling process. To cope with this limitation, the model can be easily retrained and improved to make more precise predictions for a wider range. Consequently, this study represents that ANFIS approach can be employed as a reliable and powerful alternative to traditional methods for solving high nonlinear engineering problems such as indirect estimating the MR of pavement subgrade soils.

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