Estimation of uniaxial compressive strength of pyroclastic rocks (Cappadocia, Turkey) by gene expression programming

ORIGINAL PAPER

Estimation of uniaxial compressive strength of pyroclastic rocks

(Cappadocia, Turkey) by gene expression programming

İsmail İnce1,2_{&Ali Bozdağ}1,2_{&Mustafa Fener}3_{&Sair Kahraman}4

Received: 10 January 2019 / Accepted: 31 October 2019 / Published online: 5 December 2019 # Saudi Society for Geosciences 2019

Abstract

Compressive strength of rocks is an important factor in structural design in rock engineering. Compressive strength can be determined in the laboratory by means of the uniaxial compressive strength (UCS) test, or it can be estimated indirectly by simple experiments such as point load strength (PLT) test and Schmidt hammer rebound test. Although the UCS test method is time-consuming and expensive, it is simple when compared to other methods. Therefore, many studies have been performed to estimate UCS values of rocks. Studies indicated that correlation coefficient of rock groups is low unless they are classified as metamorphic, sedimentary, or volcanic. Pyroclastic rocks are widely used as construction materials because of the fact that they crop out over extensive areas in the world. To estimate the UCS values of pyroclastic rocks in Central and Western Anatolia region, Turkey, multiple linear regression (MLR) analysis and gene expression programming (GEP) were employed and during the analysis, and PLT, ρd,ρs, and n were used as the independent variables. Based on the analysis results, it was detected that the

GEP methods gave better results than MLR method. Additionally, the correlation coefficient (R2) values of training and sets of validation of the GEP-I model are 0.8859 and 0.9325, respectively, and this model, thereby, is detected the best of generation individuals for prediction of the UCS.

Keywords Uniaxial compressive strength (UCS) . Pyroclastic rocks . Gene expression programming (GEP) . Multiple linear regression (MLR) . Construction materials

Introduction

Intact rock strength which is a principal rock property is described as the strength of the rock material separated by discontinuities. It is determined by the uniaxial compres-sive strength (UCS) test in the laboratory. The UCS is used in the different rock engineering projects such as tunneling,

excavation, drilling, slope stability, etc. It is also used as main input parameter for the rock mass classification sys-tems and of failure criteria of intact rock. There are six major intact rock failure criteria in the literature: Mohr-Coulomb, Hoek and Brown (1980a), modified Lade (1977), modified Wiebols and Cook (1968), Mogi (1971), and Drucker and Prager (1952). Although some of the criteria have not incorporated the intermediate principal stress, the others incorporate the intermediate principal stress as well as the minimum and maximum principal stresses. Unlike all other major failure criteria, the Mohr-Coulomb and Hoek and Brown (1980a) criteria do not incorporate the intermediate principal stress. Mehranpour and Kulatilake (2016) investigated the suitability of six major intact rock failure criteria for representing the intact rock strength under the true-triaxial stress condition. They recommended modified Lade, Mogi, and modified Wiebols and Cook criteria for predicting the intact rock strength under true-triaxial stress condition. They also stat-ed that the Mogi criterion is the most stable among the three criteria with respect to the prediction accuracy.

Responsible Editor: Murat Karakus * İsmail İnce

iince@selcuk.edu.tr

1 _{Department of Geological Engineering, Konya Technical University,}

42250 Konya, Turkey

2

Department of Geological Engineering, Selçuk University, 42250 Konya, Turkey

3

Department of Geological Engineering, Ankara University, 06100 Ankara, Turkey

4 _{Department of Mining Engineering, Hacettepe University,}

06800 Ankara, Turkey

Rock mass strength depends on the mechanical behavior of intact rock and the characteristics of discontinuities. Jointed rock mass indicates lower strength and stiffness compared to intact rock. The determination of the mechanical behavior of a jointed rock mass is comparatively more complicated com-pared to intact rock (Kulatilake1985; Yu 2001). As stated by Mehranpour et al. (2018), there are analytical, empirical, and numerical approaches for estimating the mechanical be-havior of rock masses. Analytical approaches are seldom ap-plicable to field rock masses which are usually more compli-cated than the assumed simplified models. Although the em-pirical approaches based on a rock mass classification system can be used for complicated rock masses, they have some limitations since they do not incorporate anisotropic behavior of rock masses and the effect of the intermediate principal stress on the rock mass strength. In order to overcome the inadequacy of the analytical and empirical approaches, nu-merical methods have been used by different researchers (Zheng et al.2014; Kulatilake and Shu2015; He et al.2016,

2017; Shreedharan and Kulatilake2016; Mehranpour and Kulatilake2017; Mehranpour et al.2018; etc.).

The UCS is widely used for designing the civil and mining engineering projects performed in rock masses. However, pre-paring rock samples and running experiments are both costly and time-consuming in some situations. To estimate the UCS values of rocks, thereby, some alternative tests (such as point load strength test, Schmidt hammer rebound test) and/or ana-lytical or empirical relations of various mechanical and phys-ical properties have been developed. These analytphys-ical and em-pirical relations can be detected by using simple regression analysis, multiple regression analysis, fuzzy inference system, a neural network, genetic programming and genetic expres-sion programming, etc.

Initially, many researchers studied on the method of esti-mating the UCS via simple regression analysis by using one of the abovementioned rock properties (Deere and Miller1966; Broch and Franklin1972; Bieniawski1975; Sachpazis1990; Ghosh and Srivastava1991; Ulusay et al. 1994; Kahraman

2001, 2014; Dinçer et al. 2004; Basu and Kamran 2010; Török and Vásárhelyi2010). Later, many other researchers predicted UCS by multiple regression analysis using more rock properties (Aggistalis et al.1980; Dinçer et al.2008; Diamantis et al.2009). In the 2000s, new methods such as fuzzy inference system, neural network, and genetic program-ming, which replaced of traditional methods, were used to estimate the UCS values, and better results were obtained (Gokceoglu2002; Karakus and Tutmez 2006; Zorlu et al.

2008; Dehghan et al. 2010; Karakus 2011; Manouchehrian et al. 2012; Ceryan et al. 2013; Majdi and Rezaei 2013; Mishra and Basu2013; Yesiloglu-Gultekin et al. 2013; Ceryan2014; Armaghani et al.2015).

Gene expression programming (GEP) proposed by Ferreira (2001) is another method mostly used for estimation

problems. It has been increasingly used in many engineering disciplines such as geological, environmental, and civil engi-neering because of the fact that it has produced very simple and effective mathematical formulas. A lot of researchers have estimated the UCS values of different rocks by using GEP (Baykasoğlu et al.2008; Çanakcı et al.2009; Ozbek et al.

2013; Dindarloo and Siami-Irdemoosa 2015; Behnia et al.

2017; Armaghani et al.2018).

Baykasoğlu et al. (2008) estimated strength value of lime-stone, and Çanakcı et al. (2009) estimated strength value of basalt by means of GEP using different rock properties like water absorption by weight, dry density, saturated density, bulk density, and P-wave velocity. Ozbek et al. (2013) esti-mated the UCS values of rocks by using index features (i.e., porosity, water absorption by weight and unit weight) of four ignimbrite and basalt samples as inputs. Additionally, Dindarloo and Siami-Irdemoosa (2015) predicted the UCS values of rocks by means of GEP, using total porosity and P-wave velocity of carbonate rocks. Furthermore, Armaghani et al. (2018) and Behnia et al. (2017) estimated the UCS values of rocks by using GEP based on their some engineering properties.

Cappadocia located in the Central Anatolia (Turkey) is one of the oldest sites of the world and has become prominent with its natural, historical, and cultural heritage. In the region, the unique landforms by eroding of pyroclastic rocks were formed, and the rocks were also carved out as Underground City, semi-Underground City since ancient times (Fig. 1). In the previous studies, the UCS values of various rock groups were tried to be estimated by using of different method types. However, the studies related to the prediction of the UCS values of pyroclas-tic rocks are very limited. This study aims to predict the UCS of pyroclastic rocks, the strength values of which change in a wide range, by using a software program called Gene X pro Tools 4.0 and by means of point load strength (PLT), porosity (n), dry density (ρd), and

satu-rated density (ρs).

Sampling area and geology

Forty-four of the fifty pyroclastic samples were collected from the Cappadocia Volcanic Province (CVP) in Central Anatolia (Fig.2). The Neogene-Quaternary aged volcanic field is rang-ing from 6.000 to 15.000 km2(Toprak1998). The covered surface area of the province is 40.000 square kilometers, and it has about 1.000 cubic-kilometer pyroclastic materials (Temel et al.1998).

In the study area, the pre-Miocene rocks form the base-ment. The basement rocks are unconformably overlain by the Lower Miocene Yeşilhisar formation which is composed of red color sandstone, conglomerate, and mudstone

alternation (Temel1992). The unit is unconformably overlain by the Upper Miocene-Pliocene aged Urgüp formation, which

consists of ignimbrite and lava flows interbedded with lacus-trine and fluvial sediments (Temel 1992). These units are

a

b

d

c

Fig. 1 Cultural heritages and modern building in the Cappadocia region a Fairy chimneys, b Kaymaklı Underground City (Nevşehir), c Gümüşler Monastery (Niğde), and d modern building (Nevşehir)

Fig. 2 Simplified geological map of Cappadocian Volcanic Province (CVP) (modified after Toprak1998)

transitive with the Quaternary Kumtepe formation composed of air-fall deposits. All these units are covered unconformably by the recent alluvial deposits.

For this study, the rock samples were collected from the ignimbrite member belonging to the Ürgüp formation. The name of the sample locations and their some features are shown in Table1.

Laboratory studies

Density test

According to the ISRM (2007) standards, the core samples were used to detect dry and saturated densities of rocks. The volume of the core samples was calculated from the average

Table 1 Index and mechanical properties of samples

Sample no Location Color

PLT-MPa ρd-g/ cm3 ρs-g/ cm3 n-% UCS-MPa 1 Derbent-1/Nigde Beige 1.58 1.43 1.78 34.10 12.56

2 Dere Mah.-1/Nigde Gray 1.34 1.53 1.79 26.60 8.09

3 Dere Mah.-2/Nigde Yellowish 2.12 1.43 1.72 29.40 17.61

4 Kiziloren/Nigde Beige 2.11 1.74 1.99 24.60 26.08

5 Ozbelde/Nigde Dark yellow 2.19 1.65 1.9 25.10 15.70

6 Aktas-1/Nigde Gray-white 0.97 1.17 1.59 41.80 4.95

7 Aktas-2/Nigde Black 0.53 1.45 1.55 35.10 3.71

8 Derinkuyu/Nevsehir White 1.39 1.51 1.79 27.60 8.00

9 Karayazi/Nevsehir Old rose 1.47 1.56 1.83 27.50 12.09

10 Akkaya/Nigde White 2.19 1.47 1.75 20.30 20.37 11 Bahceli/Nigde Beige-white 2.22 1.89 1.95 14.70 15.04 12 Gumusler/Nigde Beige 1.13 1.44 1.73 28.90 9.42 13 Tomarza/Kayseri Yellow 2.37 1.55 1.81 26.00 26.46 14 Tomarza/Kayseri Black 1.83 1.23 1.69 35.80 13.42 15 Avanos/Nevsehir Pink 2.02 1.46 1.71 25.30 14.94 16 Derbent-2/Nigde Gray 2.18 1.60 1.88 27.60 17.42 17 Gulluce/Nigde Yellow 3.25 1.68 1.83 15.00 39.02 18 Arapli/Kayseri Beige 1.42 1.62 1.85 23.80 11.80 19 Mustafapasa/Nevsehir Pink-white 0.99 1.21 1.50 39.10 7.04 20 Incesu/Kayseri Black 1.06 1.52 1.84 31.90 11.80 21 Avanos/Nevsehir White 0.26 0.81 1.31 40.00 2.09

22 Karayazi/Nevsehir Dark pink 2.00 1.58 1.84 25.40 9.80

23 Karayazi/Nevsehir Light yellow 1.23 1.41 1.70 28.90 8.66

24 Karayazi/Nevsehir White 1 1.57 1.42 1.68 26.10 8.09 25 Karayazi/Nevsehir Mixed 1.26 1.39 1.66 27.60 7.52 26 Karayazi/Nevsehir Yellow 1.13 1.40 1.70 29.30 8.38 27 Selime/Aksaray Beige 2.19 1.65 1.91 26.20 24.46 28 Taspinar/Aksaray Beige 2.16 1.75 1.91 16.00 17.89 29 Altunhisar/Nigde Gray 2.19 1.85 2.04 19.00 22.75 30 Persek/Kayseri Brownish 3.25 1.76 1.94 18.50 42.16

31 Karayazı/Nevsehir Cherry color 1.24 1.66 1.97 30.76 15.68

32 Demirciler/Aksaray Dark lilac 3.27 1.75 1.99 23.89 48.63

33 Selime/Aksaray Grayish 1.23 1.54 1.81 24.81 10.55

34 Gumuslu/Nigde Light pink 1.03 1.30 1.67 36.83 7.57

35 Emmiler/Kayseri Brownish 2.68 1.82 2.08 26.21 36.64

36 Tomarza/Kayseri Black 1.88 1.42 1.75 33.05 28.27

37 Karayazı/Nevsehir White 2 1.02 1.75 1.91 21.09 16.86

38 Karayazı/Nevsehir Rose red 1.18 1.58 1.90 32.20 12.36

39 Karayazi/Nevsehir Purple 1.13 1.44 1.75 31.31 6.87

40 Karayazi/Nevsehir Light pink 1.04 1.56 1.86 29.25 8.76

41 Karayazi/Nevsehir Yellow 1.18 1.64 1.93 28.68 13.57 42 Kayseri Gray 3.26 2.01 2.14 12.67 42.13 43 Karayazi/Nevsehir Cream-orange 1.79 1.72 1.93 20.55 12.50 44 Basakpinar/Kayseri Black 2.60 1.91 2.08 16.97 26.41 45 *Koccagiz/Kayseri Gray 0.82 1.38 1.68 30.27 8.89 46 *Karayazi/Nevsehir Pink 1.09 1.61 1.88 26.60 11.05

47 **Urgup/Nevsehir Dirty white to pink 0.48 1.39 1.81 38.29 6.53

48 ***Yazilikaya/Eskisehir White 0.80 1.25 1.63 38.82 10.00

49 ***Yazilikaya/Eskisehir Pink 1.32 1.47 1.81 33.48 16.95

50 ****Alacatı/Izmir Cream 1.47 1.52 1.79 26.26 14.90

of several caliper tests. The dry and saturated masses of the core samples were detected by means of a balance hav-ing an accuracy of 0.01 g. Additionally, the density values were also detected by calculating the ratio of the weight of the samples to volume of the samples. In this study, the average value of the least three samples tested for each rock type was used

Dry density of samples varies from 0.81 to 2.01 g/cm3. The minimum and maximum saturated densities were determined

as 1.31 g/cm3and 2.14 g/cm3(Table1). Histogram graphics prepared for dry and saturated density values are shown in Fig.

3a and b.

Porosity test

Effective porosity values of the core samples were determined by the saturation and caliper tests (ISRM2007). Volumes of the sample were calculated by the caliper method, and

Fig. 3 Histograms of index and mechanical properties a dry density, b saturated density, c porosity, d UCS, and e PLT

volumes of the pore were also determined by terms of calcu-lating of the dry and saturated weights. The porosity was ob-tained from the ratio of the pore volumes to the volume of the core samples. The tests were performed at least three times for each rock sample, and the average of the test results was used. Accordingly, the porosity values were found ranging from 12.67% to 41.80% (Table1, Fig.3c).

Uniaxial compressive strength test

The UCS tests were performed on the core samples having 38 mm in diameter and a length-to-diameter ratio of 2.0–2.5 (ASTM1995). The stress within the limits of 0.5–1.0 MPa/s

was applied. In the study, the average value of the tests repeat-ed at least five times for each rock sample was taken into consideration. The UCS value was adjusted in ac-cordance with the sample having an equivalent diameter to 50 mm (Hoek and Brown1980b) (Table1). Accordingly, it obtained the UCS values ranging from 2.09 MPa to 48.63 MPa (Fig.3d).

Point load test

Diametric point load test was performed on the core samples having 38 mm in diameter and a length-to-diameter

ratio of 1.2 (ASTM 2005). The results were corrected to core samples having an equivalent diameter of 50 mm. The average value of the tests performed at least seven times for each rock sample was taken into con-sideration and recorded as the point load strength value (Table 1). The PLT values of samples are determined ranging from 0.26 to 3.27 MPa (Fig.3e).

Determination of empirical equations

MLR analysis and genetic programming approaches were employed to predict the UCS values of pyroclastic rocks. During the analysis,PLT, ρd,ρs, and n were selected as

inde-pendent variables. Statistical results of index (ρd,ρs, n) and

mechanical (PLT) values are given in Table2.

Multiple linear regression analysis

The MLR analysis was performed using the Statistical Package for the Social Sciences version 21 (SPSS Inc.), and the confidence interval for the analysis was selected as 95%.

Table 2 Descriptive statistics of data used in the analysis Data used in the models

Variables Minimum Maximum Mean Std. deviation COV

PLT - MPa 0.26 3.27 1.64 0.74 0.45

ρd - g/cm3 0.81 2.01 1.54 0.21 0.14

ρs - g/cm3 1.31 2.14 1.81 0.15 0.08

n - % 12.67 41.80 27.58 6.85 0.25

UCS - MPa 2.09 48.63 16.05 10.64 0.66

COV Coefficient of variation

Table 3 Results of subsets MLR

Independent variables Equation R2

PLT, ρd UCS = − 9.426 + 12.142 ∗ PLT + 3.600 ∗ ρd 0.795 PLT, ρs UCS = − 19.128 + 11.655 ∗ PLT + 8.856 ∗ ρs 0.802 PLT, n UCS = − 12.716 + 14.190 ∗ PLT + 19.801 ∗ n 0.800 ρd,ρs UCS = − 52.754 + 14.344 ∗ ρd+ 25.816∗ ρs 0.426 ρd, n UCS = − 13.159 + 24.211 ∗ ρd− 29.074 ∗ n 0.421 ρs, n UCS = − 28.683 + 31.176 ∗ ρs− 42.496 ∗ n 0.455 PLT, ρd,ρs UCS = − 28.050 + 12.119 ∗ PLT − 13.210 ∗ ρd+ 24.579∗ ρs 0.810 PLT, ρd, n UCS = − 42.863 + 13.635 ∗ PLT + 14.306 ∗ ρd+ 52.654∗ n 0.824 PLT, ρs, n UCS = − 44.252 + 13.318 ∗ PLT + 15.498 ∗ ρs+ 37.578∗ n 0.824 ρd,ρs, n UCS = − 27.312 − 15.910 ∗ ρd+ 46.519∗ ρs− 59.498 ∗ n 0.460 PLT, ρd,ρs, n UCS = − 45.006 + 13.437 ∗ PLT + 7.150 ∗ ρd+ 8.463∗ ρs+ 45.930∗ n 0.825

Table 4 Parameters of GEP approach models

Parameter definition GEP

Function set +, -, *, /, Sqrt, ln 10x

Chromosomes 30

Head size 8

Number of genes 3

Linking function Multiplication

Mutation rate 0.044

Inversion rate 0.1

One-point recombination rate 0.3

Two-point recombination rate 0.3

Gene recombination rate 0.1

Additionally, an enter analysis was performed to determine the best subset in SPSS program. In the analysis, 11 subsets were formed and different models were also produced (Table3). To determine the best subset, then, the model with the highest R2value was selected. The highest R2value in the analysis was detected as 0.825 (Table3). Independent vari-ables in the model having the highest R2value are PLT, ρd,ρs,

and n (Eq.1):

UCS ¼ −45:006 þ 13:437*PLT þ 7:150*ρdþ 8:463*ρs

þ 45:930*n ð1Þ

Genetic expression programming approach

The GEP was developed by Ferreira by using fundamental principles of the genetic algorithm (GA) and genetic program-ming (GP) in the year 2001. To evaluate any knowledge, the GEP methodology is used like that of the biological evalua-tion. In this application, individuals are encoded as linear strings of fixed-size, genome, described later as nonlinear as-sets having different shapes and sizes (Ferreira2001). These assets are known as expression trees (ETs), which are the phrase of a chromosome. GEP chromosomes usually consist of more than one gene of equal length, and each gene is di-vided into two parts consisting of head and tail. ETs undergo

the selection procedure and guided by their fitness value to generate new individuals (Ferreira 2002). The assets are encoded in linear chromosomes of fixed length. A mathemat-ical function, in another saying, is defined as a chromosome with multigene and developed by using the data submitted.

The GEP methodology is composed of five main compo-nents: (i) function set, (ii) terminal set, (iii) control parameters, (iv) fitness function, and (v) stop condition (Ferreira2002). Function set is a set of field-specific basic mathematical func-tions used in conjunction with terminal set to install potential solutions for a given problem. The set of terminals forms the input variables and set of constants. Fitness function is a nu-meric value defined for each member of a population to ensure a measure of compatibility of a solution to the problem in any question. The control parameters include the population den-sity, crossover, and mutation probabilities. And, the stop con-dition is commonly specified as a predefined number of generations or an error tolerance on the fitness (Willis et al. 1997).

In this study, expression tree of GEP models was developed as seen in Figs.4and5. In the GEP, the number of gene and length of the head was detected as 3 (Sub-ETs) and 8,

respectively. During this process, multiplication was used for linking function.

During the training and validation of the GEP approach models and linking function, PLT,ρd,ρs, and n values were

entered as input variables, while the UCS value was used as output variable. Fifty pyroclastic rock samples were used in this study. Six of these samples were taken from previous studies (Bozdag 2013; Topal and Doyuran1998; Topal and Sozmen

2003; Yavuz 2012). Thirty-eight samples, which were chosen randomly among these fifty samples used in the study, were used as education set for GEP approaches. The rest twelve samples were used for validation set in the GEP approaches.

In the application, firstly, the fitness (fi) of an individual

program (i) was detected using Eq. (2):

fi¼ ∑ j¼1 Ct M − C

ð Þij−Tj

  ð2Þ

where M is the selection range; C(ij)is the value, which is

returned by the individual chromosome i for fitness case j (except Ct fitness cases); and Tjis the target value for the

fit-ness case. If the precision obtained for |C(ij)− Tj| is less than or

Fig. 5 Expression tree of GEP-II approach

equal to 0.01, the precision is considered as equal to zero, and, thereby, fi= fmax= CtM. Thus, M and fmaxvalues are used as

100 and 1000, respectively. That the system can find the op-timal solution for itself is the most important advantage of this kind of fitness functions (Ferreira2001,2002).

To create the chromosomes, T = { PLT,ρd,ρs, n}, later, the

set of functions (F) and the set of terminals (T) were selected, and four basic arithmetic operators (+, _,*, /) and some basic mathematical functions such as Sqrt, ln, and 10xwere used.

Finally, a combination of all genetic operators including mutation, crossover, and transposition was used as set of ge-netic operators. In Table4, the training parameters for the GEP approaches were given. According to the GEP approaches for the UCS, explicit formulations were obtained by

UCS ¼ f PLT ; ρð d; ρs; nÞ ð3Þ

The best of generation individuals for prediction of the UCS is the GEP-I approach, and fitness value of the model was detected as 766. In the GEP-I approach, Fig.4shows the expression trees of formulation given in Eq. (4).

UCS ¼½½ð2:43*nÞ−ρs* n−PLT½ð Þ*PLT þ ρs * 3:01− PLT= 10½ ½ ½ð n−2:51Þ*10ρs * ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρs− ρð d−nÞ* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:66 þ PLT p q   _ð4Þ

Another generation individual for prediction of the UCS is the GEP-II approach. In the GEP-II approach, the expression trees of formulation given in Eq. (5) are shown in Fig.5.

UCS ¼½½ð2:43*nÞ−ρ_s* n−PLT½ð Þ*PLT þ ρ_s * 3:03½ 

* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρs− ρð d−nÞ*ln 4:95 þ PLTð Þ p

h i ð5Þ

Results and discussion

In this study, in the MLR and GEP approaches developed to predict the UCS values of the pyroclastic samples, dry density, saturated density, porosity, and PLT values of fifty pyroclastic samples, the strength value of which change approximately between 2 and 50 MPa, were used as input value.

The most convenient subset determined to estimate the UCS by means of MLR analysis is made up of independent variables of PLT, ρd,ρs, and n (Table5). To check the

valida-tion of the regression equavalida-tion formed by PLT, ρd,ρs, and n,

variance analysis was performed, and the results of analysis are given in Table6. Additionally, the confidence interval for the analysis was selected as 95%. The developed statistical model was able to prove for estimation of the UCS (F = 53.072, p = 0.000 < α) on the basis of 5% significance (α = 0.05). The R2, which showed the relationship between

Table 5 The statistical results of

MLR analysis Model Unstandardized coefficients Standardized coefficients t p

β Std. error Beta (Constant) − 45.006 14.035 − 3.207 0.002 n 45.930 22.933 0.296 2.003 0.051 ρs 8.463 15.240 0.123 0.555 0.581 ρd 7.150 14.128 0.143 0.506 0.615 PLT 13.437 1.386 0.931 9.692 0.000

Table 6 The analysis of variance of MLR model for UCS

Model Sum of squares df Mean square F p

Regression 4576.389 4 1144.097 53.072 0.000 Residual 970.081 45 21.557 Total 5546.470 49 y = 0.825x + 2.805 R² = 0.825 0 10 20 30 40 50 0 10 20 30 40 50 a P M , S C U Predicted

Observed UCS, MPa

Fig. 6 Comparison of UCS experimental results with results of MLR model

estimated values obtained from MLR equation and experi-mental values, was detected as 0.825 (Fig.6).

The PLT and n independent variables used in the subsets are directly related to the UCS (Kahraman et al. 2005; Kahraman2014), whileρdandρsindependent variables are

related to n. The highest R2value of the subsets created to estimate the UCS value by the MLR method was obtained from Eq. (1). While the low correlation coefficient (R2) was determined in the subsets formed by n,ρd, andρs, the high

correlation coefficient was determined in the subsets formed by PLT and other independent variables (ρd,ρs, n) separately

(Table3). According to the equations given in Table3, it was detected that index parameters such as n,ρd, andρsare not

sufficient for estimating of the UCS due to having low corre-lation coefficients of 0.421 to 0.460. It was also determined that the mechanical properties of the rocks such as PLT gave high significance correlation coefficients of 0.795 to 0.824 for UCS estimation. Consequently, the highest correlation coeffi-cient for estimating of the UCS was determined by means of

using parameters such as n,ρd, andρsin addition to PLT in the

equations (Table3).

Samples used in the GEP approaches were classified as training and validation. Estimation values obtained from the models for these sets of samples are given in Table7. Comparison between the UCS values estimated by the GEP approaches and by experimental tests is also given in Fig. 7, which also show the most correct equation and R2values for each of the sets used in the model. For these comparisons, three norms, which are mean absolute per-centage error (MAPE), root mean squared error (RMSE), and R2, were used. Equations of these norms are given in Eqs.6–8. MAPE ¼100 n ∑n i¼1joi−eij ∑n i¼1oi   ð6Þ RMSE ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n∑ n i¼1ðoi−eiÞ2 r ð7Þ

Table 7 GEP model results compared with experimental results are used as test (training and validation) sets

Training

Sample no Experimental GEP-I GEP-II Sample no Experimental GEP-I GEP-II

1 12.56 12.57 12.57 20 11.80 8.37 8.69 2 8.09 9.98 10.24 21 2.09 1.32 4.80 3 17.61 18.46 18.18 22 9.80 18.49 18.49 4 26.08 22.28 22.28 23 8.66 8.65 8.89 5 15.70 22.31 22.20 24 8.09 11.34 11.47 6 4.95 6.17 7.02 25 7.52 8.52 8.74 7 3.71 4.05 4.29 26 8.38 8.00 8.26 8 8.00 10.58 10.82 27 24.46 22.50 22.38 9 12.09 11.58 11.81 28 17.89 20.37 20.37 10 20.37 21.47 21.32 29 22.75 23.45 23.46 11 15.04 17.98 17.97 30 42.16 46.05 44.60 12 9.42 8.12 8.39 31 15.68 10.55 10.84 13 26.46 24.12 23.75 32 48.63 50.28 48.60 14 13.42 15.31 14.79 33 10.55 9.32 9.62 15 14.94 16.93 16.83 34 7.57 7.60 7.66 16 17.42 22.02 21.86 35 36.64 36.30 35.70 17 39.02 41.35 39.97 36 28.27 15.59 15.40 18 11.80 10.97 11.25 37 16.86 7.22 7.56 19 7.04 7.11 6.10 38 12.36 9.61 9.92 Validation

Sample no Experimental GEP-I GEP-II Sample no Experimental GEP-I GEP-II

39 6.87 8.31 8.58 45 8.89 6.17 6.48 40 8.76 8.26 8.60 46 11.05 8.54 8.89 41 13.57 9.66 10.00 47 6.53 6.21 6.54 42 42.13 47.79 45.55 48 10.00 6.76 6.52 43 12.50 15.94 16.14 49 16.95 10.29 10.48 44 26.41 32.04 31.70 50 14.90 11.34 11.57

R2¼ ðn∑oiei−∑oi∑eiÞ 2 n∑o2 i− ∑oð iÞ2  n∑e2 i− ∑eð iÞ2  ð8Þ

where e is a predicted value, o is an observed value, and n is a total data number.

Statistical parameters belonging to the training and valida-tion sets used in the GEP approaches and MLR are given in Table8. MAPE and RMSE values for the GEP-I approach model training set and validation set are 0.41 and 3.7693 and 1.85 and 3.8136, respectively. The R2value is determined as 0.8859 in training set and 0.9325 in validation set of GEP-I approach (Table8, Fig.7a).

The MAPE, RMSE, and R2 values for training set of the GEP-II approach are 0.40, 3.7164, and 0.8826, re-spectively. The MAPE, RMSE, and R2 values for the validation set are 1.66, 3.5779, and 0.9300, respectively (Table 8, Fig. 7b).

Comparison between estimation values obtained from all these sets and experimental results verifies the high generalization capacity of the proposed model. The statis-tical values given in Table 8 indicate that the proposed models are very convenient, and predicted UCS results are very close to the experimental results. All these results indicate highly successful performance of the GEP-I ap-proach used in estimation of the UCS during training and validation phases.

Additionally, it obtained the high positive relationship (R2 = 0.825) between estimated values obtained from MLR methods and experimental results. The detected high R2value is close to the R2value obtained from GEP-I approach model, but it is lower. Therefore, in this study, GEP-I approach model having the highest R2value gave the best results for using in estimation of the UCS.

In current study, the GEP-I approach used to estimate the UCS value was performed mostly on the pyroclastic rocks collected from the CVP. However, obtained results should be checked with further researches whether be va-lidity of the results for the pyroclastic rocks in the differ-ent regions.

Conclusion

Pyroclastic rocks are commonly used as construction materials for many years because they are malleable and crop out over extensive areas in the world. These rocks have been preferred as building stone because of their variable strength values.

In this paper, multiple linear regression analysis and genetic programming approaches were employed to predict the UCS values of pyroclastic rocks collected from fifty different locations in Central and Western Anatolia region in Turkey. In the

GEP-I training y = 0.974x + 0.504 R² = 0.8859 GEP-I validation y = 1.2103x - 3.7334 R² = 0.9325 0 10 20 30 40 50 60 0 10 20 30 40 50 60 a P M, S C U de tc i de r P

Observed UCS, MPa

GEP-I tra ining results GEP-I va lida tion results

a

Observed UCS, MPa

GEP-II tra ining results GEP-II va lida tion results

b

Fig. 7 Comparison of UCS experimental results with results of GEP approaches a GEP-I and b GEP-II

Table 8 The UCS statistical values of GEP approach

Model Statistical parameters Training set Validation set GEP-I MAPE 0.41 1.85 RMSE 3.7693 3.8136 R2 _{0.8859 0.9325} GEP-II MAPE 0.40 1.66 RMSE 3.7164 3.5779 R2 _{0.8826 0.9300} MLR Sample set* MAPE 0.43 RMSE 4.4047 R2 0.8250 *50 sample

analysis,PLT, ρd, ρs, and n were used as the independent

variables.

According to the analysis, obtained main conclusions are the following:

a. The best R2

value obtained for the estimation of UCS value by MLR analysis used in the study was detected as 0.825.

b. The number of gene in the GEP-I and GEP-II ap-proaches used in the study is chosen as 3, and mul-tiplication was used as linking function. The results obtained from the GEP approaches and experiments are very compatible with each other. However, the R2 values of training and sets of validation of the GEP-I approach are higher than those of GEP-II approach. The R2 values of training and sets of validation of the proposed model are 0.8859 and 0.9325, respectively. Additionally, the MAPE, RMSE, and R2 values of each sets of GEP-I ap-proach support the consistency of the model. Therefore, the models produced by GEP method give better results than models developed from MLR method.

Accordingly, in this study, it is concluded that the best of generation individuals for prediction in the UCS is the GEP-I approach. Furthermore, it is shown that hard and time-consuming UCS can be estimated without experiment by the virtue of the proposed mathematical model.

Equations developed for pyroclastic rocks can be used to determine UCS both in construction materials produced by pyroclastic rocks in CVP in Anatolia and pyroclastic rocks having similar index and mechanical features in other regions of the world.

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