Generic compressional strength prediction model applicable to multiple lithologies based on a broad global database

In this study, we present a global database of ten parameters, which include measurements of rock index properties, strength stiffness and dynamic properties. Hoek–Brown constant m i , is included, and was estimated, using Hoek and Brown proposed guidelines for determining m i values for different rock types that can be used for preliminary design when triaxial tests are not available. This broad database is compiled from 96 studies and is labelled as ‘‘ROCK/10/4025’’, to describe the type of geomaterial, the number of the parameters, and the number of the data samples included. It consists of 35.4 % igneous, 54.8 % sedimentary, and 9.2% metamorphic rocks. The purpose of this paper is to propose a generic soft computing model applicable to multiple lithologies, that can become more reliable and perhaps more suitable for a specific site study when used in order to densify often limited similar site-specific data. To this end four broad samples of data were selected, and served as training data sets for developed machine learning models, to develop a generic compression strength prediction model applicable to multiple lithologies. The suggested algorithms in this study are Back-Propagation Artificial Neural Networks, Artificial Neuro-Fuzzy Inference Systems, Support Vector Machines, Nearest Neighbour classifiers and Ensemble Bagged Trees. According to the findings of this study, Artificial Neuro-Fuzzy Inference Systems model performance was found to be marginally superior, while Back Propagation Artificial Neural Networks, Support Vector Machines and Ensemble Bagged Trees models were found to have good performance. Constant m i seems to be an important training parameter when training predictive models centred on data from multiple lithologies. As a result, we can suggest that these models are powerful tools that allow for a reliable estimation of compressive strength, based on the performance indicators. The performance was found to be 70%–82% when the problem of compressive strength prediction was approached as a classification problem (that is successful prediction of class from very weak to very strong), and 80%–96% when solved as a function approximation problem.


Introduction
The mechanical properties of intact rock, compressive strength ( c ) and Young's modulus (E), are essential parameters required for most geotechnical projects.However, it is not always possible for direct determination of these parameters.Although the uniaxial compressive strength  c , of intact rock is a significant parameter in rock engineering projects, it is challenging to obtain a representative value due to the need for high-quality core samples, and/or limited budget, especially during the preliminary design stage of a project.There are in addition, many uncertainties associated with the sampling of data when predicting  c .Consequently, established transformation models need to be constantly updated when applied to multiple lithologies, or sites.Hence, the numerous empirical equations that are proposed in the literature seem to be inadequate in estimating the  c reliably, in a large application range, since the material properties are affected by many The proposed models usually suffer from an inherent weakness, that they can be suitable only for the range of the conditions that exist in the calibration dataset and therefore could be dataset specific or site specific [96].These models may not perform in a satisfactory way, if applied in another dataset originated from a different site.Most of these models are univariate or multivariate regression equations, but there are also several models based on soft computing methods.These models could not be characterized as generic since their performance beyond the calibration range is somehow questionable.
Dataset specific or site-specific models have their advantages when applied during the preliminary design stage, for a project, and adopted to estimate  c or E. They can serve as tools that allow, for initial estimations at preliminary design stage, but suffer from inherent limited and questionable application, outside the calibration range.Therefore, a generic transformation model, calibrated by a global rock database maybe necessary.
Ching et al. [96], compiled a global rock database, labelled as ROCK/9/4069 which has a wider coverage than existing previously proposed transformation models.14 out of 96 case studies, are common in both databases.These are original contributions that are presented for the first time in the current paper.The proposed database was compared to other transformation models and was adopted to calibrate the bias and variability of proposed transformation models.The investigation of the named database suggested that most probably the proposed in literature transformation models were not dependent on the rock type, and that the models predicting E as opposed to the models that predict  c suffer from smaller transformation uncertainties.It was also found that the I s( 50) is the most effective parameter into predicting the  c while S h (shore scleroscope hardness) and V p (shear wave velocity) are the most effective at estimating E.  available, the selection of the characteristic value may be mainly based on results of regional sampling or other relevant experience''.This challenge is called ''site challenge'' [98] or site recognition challenge [99].In the current study we want to develop a model that can reliably predict  c applicable in multiple lithologies.We believe that this is a first step towards exploring further the idea of densifying a limited local or site-specific population, with a larger more generic but similar population, to augment the available sample population and estimate reliably the compressional strength or the Young's modulus of intact rocks on a specific site.To this end we compiled a new generic database and applied intelligent bioinspired computational algorithms: Back-Propagation Artificial Neural Networks (BP-ANN), Artificial Neuro-Fuzzy Inference Systems (ANFIS), Support Vector Machines (SVM), and in future can be adopted to develop quasi-specific models [101].

Table 2
Data map for ROCK/10/4025 and available in the literature.

Authors Nr of data groups
x Afolagboye et al. [3] x x x Aggistalis et al. [4] x x x x Akram and Bakar [5] x x Aliyu et al. [6] x x x x x x Armaghani et al. [8] x x x x Jahed Armaghani et al. [7] x x x Jahed Armaghani et al. [94] x x x x x Aydin and Basu [9] x x x x x x Azmian (2017) x x x Basu and Kamran [11] x x Basu et al. [12] x x x x x Bieniawski [14] x x Bilgin et al. [15] x x x x Bell and Lindsay [13] x x x x x x Briševac et al. [16] x x x x x x Bruno et al. [17] x x Cargill and Shakoor [18] x x x x x Çelik [19] x x x x x Ceryan et al. (2012) x x x x Cheshomi et al. [21] x x x Çobanoğlu and Çelik [22] x x x x Dehghan et al. [23] x x x x x Demirdag et al. [24] x x x Diamantis et al. [25] x x x x x Dinçer et al. [26] x x x x Dinçer et al. [27] x x x x x x x Ersoy and Acar [28] x x x Fakir et al. [29] x x x x x x x Fener et al. [30] x x x Ferentinou and Fakir [31] x x x x Ghasemi et al. [32] x x x x x x Gomez-Heras et al. [33] x x x x Gonz'lez et al. [34] x x x x Guney and Altindag [35] x x x x Hebib et al. [36] x x x x Heidari et al. [37] x x x Heidari et al. [95] x x Heidari et al. [38] x x x x x Ince and Fener [39] x x x x x x İnce et al. [40] x x x x Jamshidi et al. [41] x x x Jalali et al. (2017) x x x x x Kahraman and Gunaydin [42] x x Kahraman [43] x x x x Kahraman [45] x x x x Kahraman et al. [44] x x x x x x x Kahraman et al. [46] x x x x Kahraman et al. [47] x x x x Kainthola et al. [48] x x x Kamani and Ajalloeian [49] x x Karakus [51] x x x x x Karakus et al. [50] x x x x x x Karaman and Kesimal [53] x x Karaman et al. [52] x x x x Kasim and Shakoor [54] x x Khandelwal and Singh [55] x x x x x Kılıç and Teymen [56] x x x x x x Korkanç and Solak [57] x x x x x Madhubabu et al. [59] x x x x x x Mahdiabadi and Khanlari [58] x x x x Mahmoodzadeh et al. [60] x x x x x Ludovico-Marques et al. [61] x x Martins et al. [62] x x x x x Mehrabi Mazidi et al. [63] x x Mishra and Basu [64] x x x x x x x x Mohamad et al. [65] x x x x x Momeni et al. [66] x x x x x Naijib et al. (2015) x x x x Nefeslioglu [67] x x x (continued on next page)

Database ROCK/10/4025
A generic global database ROCK/10/4025 is compiled from published case studies, in the literature, and includes 9 measured intact rock parameters, and a 10 th parameter Hoek-Brown constant m i , which was estimated for all the data, to indirectly include the lithology in the database.The current database is labelled, (material type)/(number of parameters of interest)/(number of data points) following generic databases compiled for clays [102][103][104][105], sands [106], rocks [96], and rock mass [101].
There are 4025 records in the database.Each data set is stored in Excel worksheet that consists of a set of values measured for the same intact rock sample.The resulting database is not a genuine multivariate database in the way that all the 10 parameters are populated for all the rows in the worksheet, (i.e each row is not in full, completed with values).The percentage of completeness, of the data base defined as ''(number of filled values)/[(number of parameters) × (number of rows)].The percentage of completeness is 50% for ROCK/10/4025, while the percentage of completeness for ROCK/9/4069 is 34.2% [101] Ten parameters commonly measured and identified in the literature to indirectly estimate  c and E, are included in the data base.The parameters of interest are , n, R L , BPI, I S50 ,  bt ,  c , E, V p , m i .They can be categorized in the following groups: 1. Index properties: porosity (n), unit weight (), L-type Schmidt hammer hardness (R L ), and Block Bunch Index (BPI).The Schmidt rebound hammer is a practical non-destructive test method for the evaluation initially of rock strength and, subsequently, rock quality [108].The surface hardness of rock is measured using a portable device with a non-destructive application called the Schmidt rebound hammer [109].The ISRM has standardized the recommended test procedure for the Schmidt rebound hammer test.Researchers have indicated that the Schmidt rebound hammer could be a good indicator for determining the  c of rock [108,109].There are two types of Schmidt hammer: L-type and N-type.In ROCK/10/4025, R L data dominate (96% are R L ) because the L-type hammer is recommended for rocks as per [110].There are studies where both R L and R N were measured Aydin and Basu [9], Basu et al. [12], Bilgin et al. [15], but in ROCK/10/4025, only R L was included.2. Strengths: Brazilian tensile strength ( bt ), point load strength index (I s50 ), and uniaxial compressive strength ( c ) strengths.Index tests like the I s50 ,  bt , have helped researchers develop models to predict the  c of rocks because index tests require relatively small data set samples.Grasso et al. [111] state that despite the limitations associated with index tests, when coupled with experienced judgement, they provide initial estimates of rock properties required at the feasibility and preliminary design stage.The point load index has long been used as the most suitable intermediary for the  c .It is evident from published research that the equations exhibit a wide range varying from linear to quadratic and power laws, but the issue is that there is no agreement between authors on a specific conversion factor [31].The strength parameters of rock depend on specimen size, sample geometry, loading rate.I s50 is size dependent, it is customary to correct I s50 to a standard diameter of 50 mm [112], because point load testing can be conducted over a wide range of diameters.Prakoso [113] did not observe an effect of sample diameter on the coefficient of variation (COV) of rock strengths ( c ,  bt , and I s( 50) ), therefore sample size correction is not adopted for  c parameter or  bt in the compiled data base ROCK/10/4025.3. Stiffness: Young modulus (E), the database contains average values of E as reported in the various case studies.ISRM suggested methods could be determined using tangent, secant, and average modulus.Ching et al. [96], report that the difference between (tangent modulus at 50% (E t50 ) and average modulus for the linear portion of the stress strain curve (E av ) is not significant compared to the transformation uncertainty.Thus, average values of Young's modulus are recorded in the database.4. Dynamic property: P-wave velocity (V p ). Determination of dynamic properties of rock using ultrasonic pulse or sound tests, also known as P-wave velocity.It is regarded as a non-destructive test as it emits low-amplitude waves whose stress is below that of the yield stress of rock [64].Ferentinou and Fakir [31] mention that P-wave velocity is an important parameter that should be measured for reliable  c prediction.The P-wave velocity of rock is affected by lithology, formation porosity, pore pressure, matrix, and temperature, rock mass weathering and alteration zones, bedding planes, and joint properties [114].

Hoek-Brown constant m i estimation
The constant m i is a fundamental parameter essential for the Hoek-Brown (HB) failure criterion developed for estimating the strength of rock mass properties.Hoek-Brown constant parameter m i is estimated from a series of triaxial compression tests [115].The constant m i depends on the frictional characteristics of the component minerals in the intact rock, and it has a significant influence on rock strength, depends on, grain size, and cementation of rocks, [116].Based on the general pattern of the correlations between m i and rock types, Hoek and Brown [117] proposed guidelines for determining m i values for different rock types that can be used for preliminary design when triaxial tests are not available.An updated version of these guidelines was proposed by Hoek [118] based on a more complete and detailed lithologic classification of rocks, and the range of m i values depended on the accuracy of the geologic description of rock types.These guidelines were adopted in the current study, for the estimation of m i .There are alternative methods suggested in the literature for the estimation of m i in the absence of laboratory triaxial tests, which can be useful at the early stages of various design applications.An alternative way to estimate m i values in the absence of triaxial tests is the R index, [117,[119][120][121]. R index, is the ratio of  c to tensile strength  bt .Research by Read and Richards [122] investigated the relation between m i and R values that were calculated from direct and Brazilian tensile tests which indicated that the use of direct tensile tests does not improve the ability to predict m i values compared with Brazilian tests.R index was not adopted in the current study due to the non-completeness of the data base in terms of tensile and uniaxial and compressive strength.Another method proposed by Cai [120], Peng et al. [123], for the prediction of m i is directly from the  c of the intact rock in which the m i values depend on the ratio of crack-initiation stress obtained using acoustic emission techniques to the peak strength.Crack-initiation stress of rock samples data were not included in the studies that were used for the compilation of the ROCK/10/4025 database.As a result, the model was not used.
Shen and Karakus [124], proposed a simplified method for m i values estimation directly from  c values for specific rock types in the absence of triaxial test data.They considered, correlations for five common rock types (sandstone, limestone, marble, granite, coal), and proposed a simplified method Eq. (1) together with the rock-specific relations that can estimate m i (or normalized   ) values using only  c and rock types.They used 112 groups of data for five common rock types in an existing database together with laboratory tests.
The proposed regression equation is where m in is a normalized m i ∕ c where a and b, are constants, and their values depend on rock types, Shen and Karakus (2014).The results revealed that there is a close agreement between estimated and experimental rock strength values.The reliability of the proposed method was evaluated and compared with existing methods ( [125] guidelines and R index) that are commonly used for estimating m i values when triaxial test data are not accessible.According to their results, the simplified method could be used reliably in the HB criterion to estimate intact rock strength with small discrepancies between estimated and experimental strengths.This simplified method was used in this study to develop machine learning (ML) models that can predict  c and was included in the database only for the common type of rocks (sandstone, limestone, marble, granite and coal).
The basic statistics for the ten parameters in the ROCK/9/4069 database are listed in Table 3.The statistics are the mean value, coefficient of variation (COV), minimum value (min), and maximum value (max).It should be noticed that the mean and COV are not for a specific site, but for the entire ROCK/10/4025 database that covers numerous sites.They should not be used for design, which requires statistics at the site level.The number of data points is further subdivided into the number of igneous, sedimentary, and metamorphic data points.

Basic statistics
The number of cases is shown in the third column in the format of ''number of tests, (number of igneous + sedimentary and metamorphic).The COV and range are large in the global database and are comparable to the ROCK/9/4069 [96].In ROCK/10/4025 the range of n,  bt and  c is higher, which is well illustrated in the series of correlations in Fig. 1.The COV value is broadly in agreement in the two databases.Tables 4,5,6 show the site level statistics for the igneous, sedimentary, and metamorphic rock intact rock properties.The site level statistics are broadly consistent with those in ROCK/9/4069.COV values and range for the specific sedimentary igneous and metamorphic groups of rocks is narrower than at global level, yet the site-specific range is larger in ROCK/10/4025 than in ROCK/9/4069.It was found that unit weight) generally exhibit low variability while porosity exhibit high variability across the three types of rocks considered for both data bases.There is high variability in strength properties across the three rock types, with upper COV values greater than 30%.Young's modulus is found to be highly variable, with high COV values across the three types of rock, with the lowest value at metamorphic rocks.In general, deformation and strength properties are more variable rock properties than index properties.Also, properties of sedimentary rocks tend to have higher COV values in most of the properties considered.Indicating that possibly sedimentary rocks exhibit greater variability than other types of rocks.
Fig. 1 presents the correlations between the index properties of the behaviour of intact rock properties as measured for both ROCK/9/2069 and ROCK/10/4025.The general trends are in agreement, and ROCK/10/4025 seems to be broader in terms of range.We could also suggest agreeing with [96] that there is not strong evidence indicating that the transformation relationship exhibited by the data points depend on the rock classes (i.e sedimentary, igneous or metamorphic).

Data preparation
Several, machine learning techniques, including artificial neural networks (ANN), adaptive neuro-fuzzy inference system (ANFIS), and classification learner algorithms (SVM, KNN, EBGM) were utilized, to predict the value of  c in this study.The use of these soft computational techniques enabled the comparison of different machine learning models in  c estimation from rock index properties, included in the ROCK/10/4025.The data samples were normalized with respect to maximum and minimum values to adapt to the interval [0, 1].Normalization is conducted to ensure that all variables receive equal attention, soften the training procedure, and improve the accuracy of the results according to the linear mapping function [20,31].The data were separated into training and testing data sets before the model was built.The training data set is used to create the model, and the testing dataset is used to validate the model.

Combination summary of generic data sets
Four combinations of parameters included in the ROCK/10/4025 were selected to develop and train predictive models.The selection of the four combinations of input parameters was based on four criteria.Predominantly, on the completeness of the database, incorporation of a physical property, strength, preferably through indexes resulting from non-destructive test methods, inclusion of lithology through m i in the training.The authors selected one physical property as essential to be represented in the ML models, the intention was to investigate whether ML trained with a physical property as an input parameter, would have a better performance as opposed to the ML models that would not include a physical property as an input parameter.The second criterion was to comprise a non-destructive, easy, and cost-effective method like V p and R L which are present in all combinations.Point load index has long been regarded as the best intermediary for the  c , and therefore was comprised as a training parameter in all combinations.The constant m i is incorporated as input parameter in all models as the purpose of the study is to propose a method that would allow to include qualitative properties such as lithology in the training of an ML model towards  c prediction for multiple lithologies.
1.The completeness of the data base: The intention of the authors was to compile a large representative database with typical ranges of rock properties that could be used as an approximation of rock property variability, when rock property data are not available or are very limited.The aim is to use ML algorithms and train with many representative educational examples which serve to build prior knowledge in the ML models.The authors performed a thorough check in the database to examine which combinations would allow for the larger data sets and concluded into the selected presented in Table 7.This strategy, of sample and parameter selection, allowed us to investigate the importance of sample data in comparison to the selected parameters in the generalization capacity of the proposed ML models.2. Inclusion of the physical properties: Porosity n was selected representing the physical property of the intact rock sample.Unit weight  has the lowest COV value of 0.12% and low variability and therefore was not selected.Porosity n, on the other hand is found on the upper bound of COV values greater than 60% which are considered with high variability, and therefore would allow for adequate representative educational examples included in ML models.3. Inclusion of non-destructive test methods: The P-wave velocity has been successful as a non-destructive test for the prediction of mechanical properties of rocks easy and cost effective, the COV value is 0.30 which shows average variability.The relationship between  c and V p has been investigated by many of researchers and a high correlation is identified.The Schmidt hammer is a hand-held portable device which is commonly used to assess the strength of rocks and concrete.It is a non-destructive costeffective method used to assess the mechanical properties and therefore R L index is included as an input parameter although the COV value is close to 30% which is considered low to medium.4. Inclusion of strength index: Point load index I s50 has long been regarded as the best intermediary for the  c .It is relatively easy to conduct and economical, and thus widely applied both in the field and laboratory.It is evident from literature that the equations published exhibit a wide range, varying from linear to quadratic, and power laws.COV value is 0.80 which shows high variability and therefore adequate representation of a variability in the model.Ching et al. [96] report that Is 50 is the most effective parameter into predicting the  c .The Brazilian tensile strength has been widely used as an indirect test to measure tensile strength ( bt ).It has also been employed to produce estimates of  c strength as these two parameters are commonly required and determined in most geotechnical projects.As  bt can be easily determined from the Brazilian tensile strength, it is useful to find strong conversion factors between the two parameters.Brazilian tensile strength test is also associated to the constant m i .
The selected combinations are the following four R L -Is (50) - bt -mi, V p -R L -Is (50) -mi, n-V p -R L -Is (50) -m i , V p -Is (50) -mi.Combinations of physical and index strength parameters of rock are developed from the data available in the database.Table 7 presents the different combinations and number of samples used for testing and validating.
The models were trained with and without the inclusion of m i constant as an input parameter to verify the importance of m i constant in the convergence and prediction capability of ML models.To further compare the effect of m i estimation method on the generalization performance of ML models, four additional models were developed, m i estimation was based on Shen and Karakus (2014), for the common type of rocks (sandstone, limestone, marble, granite and coal), in the data base, summarized in Table 7.The training data sets in this case were smaller.
R L -I s( 50) - bt -m i : This combination includes a total of 227 rock samples.Many of the samples are from Turkey, and some are from South Africa.The combination contains 118 igneous, 92 sedimentary, and 17 metamorphic rock type samples.
V p -R L -I s( 50) -m i : This combination includes 1192 intact rock samples.The set contains 511 igneous rocks, 102 metamorphic, 574 sedimentary and 5 unclassified.The samples are from Malaysia, Croatia, Iran, Turkey, India, and China.n-V p -R L -I s(50) -m i : This combination includes 575 intact rock samples.The set contains 282 igneous rocks, 41 metamorphic and 251 sedimentary intact rock samples.The samples are from Malaysia, Croatia, Iran, Turkey, India, and China.
V p -I s(50) -m i : This combination includes a total of 1454 intact rock samples.The set contains 564 igneous, 752 sedimentary, 123 metamorphic, and 15 non classified.This combination contains samples from France, Iran, Greece, Malaysia, Turkey, the United Kingdom, Denmark, India, Croatia, and China.

Machine learning methods
During the last two decades ML have been successfully applied to solve various problems in geotechnical engineering applications.In a survey of 444 papers by ISSMGE TC304/309 in 2021, both supervised and unsupervised algorithms are used with ANN, Support vector machines, nearest neighbour classifiers and Bayesian networks, being the most popular amongst them employed to solve problems such as site characterization, geomaterial behaviour modelling, foundations, retaining structures, slope stability, landslides, tunnels and underground openings, liquefaction assessment, etc, [126].
ML methods have also proven to be very effective in solving nonlinear and complicated problems in geotechnical engineering, such as the settlement of shallow foundations on cohesionless soils [127], thermohydromechanical behaviour of hydrate reservoirs [128], non-stationary and non-Gaussian geotechnical properties [129], soil constitutive modelling [130] suction distribution in shallow soil layers [131] and soils' air entry value [132].
Some of the advantages of ML methods that guided the authors to develop a generic compressional strength prediction model for multiple lithologies is that they conserve the complexity of the systems they model because they have complex structures themselves, they recognize different sets of data within a whole data set, they do not require pre-existing knowledge or experience, they do not require a statistical pre-existing model in order to train data and they give reasonable results even when data are inaccurate and incomplete which is typical for geotechnical data.Phoon et al. [99] presented a successful mnemonic, MUSIC-X (Multivariate, Uncertain and Unique, Sparse, Incomplete, and potentially Corrupted with ''X'' denoting the spatial/ temporal dimension) to highlight seven common ugly attributes in real site geotechnical data.
In this study compressional strength  c prediction using a generic database was approached as a function approximation problem and as a classification problem.For the function approximation problem, Back-Propagation Artificial Neural Networks (BP-ANN) and Artificial Neuro-Fuzzy Inference Systems (ANFIS), were employed whereas to solve the classification problem binary and multiclass classification algorithms, such as decision trees, nearest neighbour classifiers (KNN), support vector machines, (SVM), ensemble classifiers (EBGM) and neural networks classifiers algorithms were selected based on their accuracy.
Artificial Neural Networks (ANNs) are supervised machine learning techniques that evaluate the relationships between a known set of observations (thus the training data) to predict unknown data sets.ANN consists of multiple interconnected processors, called neurons which are inspired by biological neurons.The neurons are logically arranged in two or more layers and interact with each other via multiple weighted connections.The links that connect neurons carry a numerical weight that is associated with the neuron.These weights express the strength, thus the importance of input neurons.A neural network learns through the constant/regular adjustment of these weights.Training of a network is modifying the network input/output behaviour to align with the external stimulus.An ANN learns through the process of reducing and minimizing the difference between the actual output and the desired output through adjusting the weights/synapses of the network.A widely used type of activation function is the continuous one designed to respond to the magnitude of the received excitation.The sigmoid function is one of the continuous transfer functions typically used in modelling the activity of neurons.(Eq.( 2)) where f is the amount of activation, x is the net excitation and ''a'' is the slope function.These functions are also known as squashing functions since their output is limited in a finite range of values.
A simple BP-ANN consists of three layers: the input layer, the hidden layer, and the output layer, as illustrated in Fig. 2. BP-ANN is usually layered, with each layer fully connected to the layers below and above.The first layer is the input layer, the only layer in the network that can receive external input.The second layer is the hidden layer in which the processing units are interconnected to layers below and above.The third layer is the output layer.Each unit of the second hidden layer is interconnected to the units in the output layer.Units are not connected to other units in the same layer.Each interconnection has associative connection strength, depicted as weight in Fig. 2. When an output array is presented the error vector, which represents the difference between the desired value and the actual output, is calculated.Using a ''wide range'' and ''representative data sets'', allows for a better representation of the input space and therefore, better generalization capability.A network is said to generalize when it appropriately classifies vectors not included in the training set.The generalization ability is measured by the accuracy of these classifications.It is important for the number of training input vectors to be greater than the number of degrees of freedom (the number of variable weights) of the network.The basic mathematical concepts of the backpropagation algorithm are found in literature [133].
The main hyperparameters of a multi-layer perceptron (BP-ANN) are the learning rule, the number of neurons in the hidden layer, and the transfer function.An optimum ANN architecture selection is an essential step in building a model that is ideal for prediction purposes.
For this study, a trial and error, approach is adopted to select the number of neurons in the hidden layer starting from a small number and increasing gradually according to the received accuracy index for the 12 developed ANNs.The size of the BP-ANN model was verified following Baum and Haussler [134], who suggest the bounds on appropriate sample vs. network size, for a feedforward network of linear threshold, with  nodes and W weights.The tangent sigmoid nonlinear transfer function is used between the input and hidden layer and the purelin transfer function between the hidden and output layer.

Adaptive Neuro-Fuzzy Inference System (ANFIS)
The artificial neural networks learning ability is combined with the fuzzy logic's decision-making mechanism in ANFIS.It is a hybrid soft computing technique that combines the best of ANN and fuzzy inference system (FIS).Fuzzy inference is the process of mapping a given set of inputs to an output using fuzzy logic [135].ANFIS eliminates the primary problem in FIS defining membership function (MF) parameters and obtaining the fuzzy if-then rules through the effective use of ANN learning capability for automated fuzzy rule generation and parameter optimization, Maiti and Tiwari [136].This method's innovation is derived from the fact that it does not require expert knowledge for the assignment of parameters of the FIS but utilizes neural network algorithms to adjust parameters.This method uses representative sets of input and output to generate a FIS whose parameters are adjusted using backpropagation or a combination with the least-squares method.A FIS develops ''if. . .then. . .'' fuzzy rules and determines MFs to map the input and output data of the system.ANFIS automatically transforms a training set into a set of fuzzy rules, thus reducing the dependence on expert knowledge for building intelligent models.The linguistic variables are subdivided into a specified number of sets during the initialization step of ANFIS.
The structure of an ANFIS model is similar to that of a multilayer perceptron neural network, Fig. 3.In general, a neuro-fuzzy system has five layers, thus with one input, one output, and hidden layers representing the membership functions and fuzzy rules.The Takagi and Sugeno [137] type FIS is employed in this research as it uses a systematic proposition to generate fuzzy rules from the given input/output sets.
Details on the mathematical formulation of ANFIS can be found in [138].A trial and error approach is taken to select whether grid partition or subtractive clustering is used to generate the FIS structure.In the current study Subtractive clustering first introduced by Chiu [139], was applied.To build an ANFIS model for this combination of parameters, a trial and error approach was applied to select the best model features.
To obtain an optimum Adaptive Neuro-Fuzzy Inference System, 34 models are developed.All the input membership functions are of Gaussian type, and the subtractive clustering method was applied for the proposition of fuzzy rules and to generate the desired training architecture.In the FIS, five descriptive linguistic labels (i.e., very low, low, moderate, high, and very high) were assigned to the input membership functions.The hybrid rule was employed in the learning procedure, and training was conducted until the error measure of the output was tolerant.Testing and checking data pairs were used to characterize the model.The range of influence (ROI) and squash factor (SF) is varied until the optimum Artificial Neuro-Fuzzy Inference System is obtained to develop the models.The ROI and SF influence the number of fuzzy rules and membership functions, thus affect the ANFIS network structure.The structure details of the models is presented in Table 8.
A total number of five rules show the best performance in predicting  c .An example of one of the ''IF-THEN'' rules is illustrated in linguistic terms is as follows: 1. ''IF ( c is low) and (I s(50) is low) and ( bt is low) and (m i is low) THEN (is cluster4) (1)''.2. ''IF (R L is very_high) and (I s(50) is very_high) and ( bt is very_high) and ( i is very_high) THEN ( c is cluster5) (1)''.

Classification models
Classification is a supervised machine learning method that allows for the prediction of discrete responses or cluster generation according to common features.The models are trained to classify data into categories.For the specific problem, the data outputs are first turned into categorical data.The  c value is translated into seven classes from 0 to 6 as per Hoek and Brown (2007)).The values of  c in the Rock-96/10/4025 range from 0.27 MPa<UCS<560.31MPa, from extremely weak to extremely strong rock, Table 9, according to field estimates of uniaxial compressive strength [117] As can per frequency distribution, 25% of the samples have  c ranging from 50 MPa-250 MPa, corresponding to strong and very strong.A 20% of samples have  c in the range 5 MPa-50 MPa, which are weak and medium-strong rocks.whereas 10% samples have  c of 0 MPa-5 MPa and 8%  c of greater than 250 MPa.
The ML models simulate multiclass classification problems since there are seven desired output categories for all datasets to be classified.Determining which classification algorithm to use is often very  Naïve Bayes Classifiers (NB) are used for classification where the observations are differentiated using specified features.This model is a probabilistic classifier based on strong independence assumptions between features.In this study kernel naïve bayes variant is applied.
Support Vector Machine (SVM), Cortes and Vapnik [140] is a type of supervised learning ML technique.The supervised learning algorithm selects the hyper-plane or the decision boundary defined by the solution vector w to determine the maximum margins between the training data samples and the test data.The variants of SVM used in the current research are medium Gaussian SVM, Linear SVM, Cubic SVM, Quadratic SVM.
Nearest Neighbour classifiers (KNN), the K-Nearest neighbour model is successfully used in previous studies in solving non-linear problems.
KNN is used to assign a class label using the smallest Euclidean distance between the target point and the training point in the feature space.The variants of KNN used in the current training are Weighted KNN, Medium KNN, Cubic KNN, and Fine KNN.
Ensemble classifiers is a ML approach in which numerous models (called ''weak learners'') are trained to tackle the same problem and then combined to achieve superior results [141].Usually, these base models do not perform well individually, either because they contain too much bias or too much variation to be robust.Ensemble techniques are used to try to reduce then bias and variance of weak learners by stacking many of them to generate a strong learner.Base models and a meta-learner (or a second-stage model) that uses base-model predictions are used to design a stacking ensemble model.Variants of ensemble models used in the classification process are Bagged Trees and subspace KNN.

Statistic performance evaluation -Model performance indicators
In order to check the overall performance of the developed predictive ML models, some statistical performance indices were calculated for each model according to the Eqs.( 3)-(4) given below.Root mean square error (RMSE) which evaluates the residual between desired and output data, and R 2 which evaluates the linear relation between desired and output data.where n is the number of training or testing samples, d t and y t are the measured and predicted values, respectively.
The performance of all the 12 ANN models is presented through Fig. 4a and 4b and in Table 8.
In classification algorithms accuracy is the statistical indicator used to quantify the performance of the four compression strength class classifiers.It is the total number of classes correctly predicted.The mathematical expression of accuracy is given in Eq. ( 5): where TP is the true positive and TN is the true negative.

𝑅 L -I s(50) -𝜎 bt -m i
The network topology adopted in this model is 4:5:1, corresponding to four predictors input neurons, five neurons in the hidden layer, and one output layer node.The training parameter used was 1000 epochs.The MSE for training is 0.0037 at epoch 52.The regression plots Fig. 4a show that the model is a good fit as for all data the R values are above 0.87, meaning the ANN model accurately represents more than 87% of the data.

𝑉 p -𝑅 L -𝐼 s(50) -m i
The network topology adopted in this model is 4:4:1, corresponding to four predictors input neurons, four neurons in the hidden layer, and one output layer node.The training process stopped when the minimum gradient was reached (8.22e−08) at epoch 74.The regression plot shows that the model is a good fit as for all data the R values are 0.81, meaning the ANN model accurately represents more than 81% of the data, as shown in Fig. 4a.

n-𝑉 p -𝑅 L -I s(50) -m i
The network topology adopted in this model is 5:4:1, corresponding to four predictors input neurons, four neurons in the hidden layer, and one output layer node.The training process stopped when the maximum epoch number was reached at 10,000.The regression plot shows that the model is a good fit as all the R values are above 0.88, meaning the ANN model accurately represents more than 88% of the data, Fig. 4a.

𝑉 p -I s(50) -m i
For this combination of parameters, the network topology adopted in this model is 3:5:1.The training process stopped when the minimum gradient was reached (9.80e−08) at epoch 338.The MSE for training is 0.0027 at epoch 29.The regression plots Fig. 4a shows that the model is a good fit as for all data the R values are 0.90, meaning the ANN model accurately represents more than 90% of the data.
In order to investigate the importance of m i constant as a training parameter, and therefore verify if models that included m i constant as input parameter outperform the models that do not include m i as an input parameter 8 additional models were developed.
The models that were trained with the inclusion of m i constant in the input parameters seem to have systematically higher performance than the models that did not include m i parameter as an input parameter as summarized in Fig. 4b (R 2 ), and Table 10, for all the four selected combinations.This applies to training validation and test data, with the exception of V p -I s(50) -m i combination, where no significant improvement is recorded.To further compare the effect of m i estimation method on the generalization, m inab was estimated according to simplified method proposed by Shen and Karakus (2014), for the common type of rocks (sandstone, limestone, marble, granite and coal), in the data sets, and therefore four more models were developed.These models presented a clearly higher performance four all the selected combinations in above 85% for broadly all the models.R L -I s(50)−  bt -m i predictive model and n-V p -R L -I s(50)− m i based on the performance indicators seem to have higher performance, than V p R L -I s(50) -m i .

Ranking of the training parameters
A powerful feature of neural networks is their ability to perform parametric analysis through manipulation of the connection weights.The developed ANN model can also provide the parameter relative importance by partitioning the hidden output neuron connection weights into components connected with each input neuron [142][143][144].This parametric analysis evaluates which parameters are more important in the prediction of  c .Fig. 5 shows the parametric analysis and relative importance of the input variables for the neural network models.The results indicate that the most important parameters that affect the compression strength prediction are the P-wave velocity (V p ), and Schmidt hammer rebound index, (R L ).According to the same analysis the point load index I s (50) , and constant mi are equally important.
The MRMR algorithm [145] which finds an optimal set of features that is mutually and maximally dissimilar and can represent the response variable effectively was also applied to assess the importance of the parameters in the classification problem.The algorithm minimizes the redundancy of a feature set and maximizes the relevance of a feature set to the response variable.MRMR algorithm was found to broadly produce similar ranking with partitioning of connection weights method as illustrated in Fig. 5. Training is a process of adjusting the pre-set membership functions and fuzzy rules to model the training dataset.The epoch number is set to 1000, and the error tolerance is set to 0.01 to allow the network to train until the error tolerance of the training is to acceptable limits and there is no overfitting.The graph in Fig. 6 presents the plot of a trained data set, for n-V p -R L -I s(50) -m i combination the blue circles represent the training data while the red asterisks the output data after training.The plot shows that the training pattern is comparable to the desired output pattern.

ANFIS model validation
The testing data set is used to validate FIS structure.After the model is trained, a set of data points which were not used for training are used for data validation.The trained model is used for simulation of the testing datasets.Table 11 presents the testing error which is an indication of the ability of the model to predict  c .According to the obtained results R L -I s(50) - bt -m i predictive model seems to yield the higher performance of 0.089502, followed by n-V p -R L -I s(50) -m i with 0.076215.

Discussion
In this paper, a global intact rock database is compiled (ROCK/10/4025) and used to train generic predictive models for  c prediction for multiple lithologies.The database includes samples from igneous, sedimentary and metamorphic intact rocks with a wide range of characteristics from a wide range of geographical regions.
According to the findings of this paper the lithology seems to be an important factor contributing to the estimation of  c .Based on the study of this data base and the associated transformation models presented in the cases that included these results, there seems to be agreement with the findings of [96] that suggest that there is no strong evidence that the transformation relationships among intact rock properties exhibited by ROCK/10/4025 data points depend on rock classes (igneous, sedimentary, and metamorphic) or the degree of weathering and metamorphism.
Analysis of partitioning of connection weights and analysis minimum redundancy maximum relevance algorithm, indicate that R L , and V p parameters seem to be the most significant in terms of estimating  c , while m i parameter is equally important as I s(50) .
Computational intelligence algorithms were employed to predict  c of intact rock, and classify rock samples to a reliable grade of strength from extremely weak to extremely strong.Regression and classification machine learning techniques including artificial neural networks (ANN), artificial neuro-fuzzy inference systems (ANFIS), support vector machines (SVM) and ensemble bagged tress models (EBTM) were tested.Multiple statistical evaluation criteria were used to assess the performance of the machine learning algorithms on prediction results.
For the performance evaluation of the prediction models,  c values were estimated from different constructed combinations of physical and mechanical rock properties by the machine learning algorithms.A comparison of the performance of the results for all suggested models are summarized in Table 13.The R 2 value of 0.85-0.96shows a good prediction performance of the ANFIS model for all combinations.ANN shows an equally satisfactory performance with RMSE values of 0.80-0.96,while the classifiers, seem to be less adequate with 0.70-0.82.
A qualitative analysis of the results suggests that certain combinations of parameters resulted in predictive models of higher performance.Next high performer model/combination includes n-V p -R L -I s(50) -m i (index property n porosity, and R L Schmidt rebound index, dynamic property V p , and constant m i ).This model is systematically ranked high in terms of performance capability when ANN, ANFIS or Cubic SVM algorithms are used.This combination includes 575 number of sets.It does not seem however that inclusion of physical properties in training affect the performance of the ML models.
The other two combinations V p -R L -I s(50) -mi and V p -I s(50) m i in descending order, show a lower prediction capability according to the statistic metrics, for ANN, ANFIS, KNN, ensemble classifiers and Bagged Trees.It is interesting though to underline that these classification models performed in a more balanced level, similar prediction accuracy for all the classes showing less false misclassifications.It is also of note that the number of sample data was higher almost double or triple than the latter models.This perhaps suggests that the performance capability is not necessarily associated to the sample of data, but to the quality of data, and the inclusion of specific parameters in the predictive models.This links to what Phoon and Zhang [126] suggest that a deep appreciation of the geotechnical context is critical to the development of novel ML methods that can lead to 'data-centric geotechnics' as a distinctive field that can transform practice.

Conclusions
In this paper, a global intact rock database (ROCK/10/4025) is developed and compared with ROCK/9/4069 database.This global database contains igneous, sedimentary, and metamorphic intact rocks which cover a wide range of lithologies and selected from a wide range of geographical locales.The database includes rock index properties, strength stiffness and dynamic properties.For the compiled data sets, Hoek-Brown constant mi, was estimated, using Hoek and Brown proposed guidelines for determining mi values for different rock types that can be used for preliminary design when triaxial tests are not available.
The main objective of this paper is to produce generic model that can estimate a reliable value for  c , for multiple lithologies.The proposed models used a function approximation approach and a classification approach.The results of the analysis indicate that the function approximation approach yields more reliable results with ANFIS algorithm to appear marginally superior.The classification approach suggests that support vector machines, nearest neighbour and ensemble classifiers performed better.The details of performance index indicators suggest the highest 90% performance for ANN model and 96% performance for ANFIS.The classifiers' accuracy was 82% for the best performer Cubic SVM.
The selected parameters and sample data suggest that for the selected combinations, and training data sets, the model that includes data on a physical parameter n, P-wave velocity, R L , I s(50) and m i ,

Fig. 2 .
Fig. 2. Typical BP-ANN architecture of a generalized multilayer perceptron.Simulation is applied to the inputs of the first layer, and signals propagate through the middle (hidden) layer to the output layer.Each link between the neurons has a unique weighting value.

Fig. 3 .
Fig. 3.A typical structure of first order Takagi and Sugeno model.

Fig. 4 .
Fig. 4. Left: Performance of the ANN algorithm, parity plots showing the ANN predictions against the corresponding measured values for training, validation, target and all data.Right: Performance of all models measured through R 2 index performance indicators in relation to m i impact (without m i ), with m i estimated through[118] guidelines, mi ab estimated through(Shen and Karakus, 2014).

Fig. 5
Fig. 5 presents a complete constructed fuzzy model for R L -I s(50) - btm i combination.The yellow plots show the antecedent part, which are the input membership functions representing the IF-part of the rule.The consequent part is shown by the blue plots and defines the membership functions are referenced as the THEN part of the rule.All the plots are filled since these correspond to the characterization of 'none' for the variable for that rule.The aggregate weighted decision for the FIS is plotted on the final plot of the last column.The bold red line on this plot represents the defuzzified output.Training is a process of adjusting the pre-set membership functions and fuzzy rules to model the training dataset.The epoch number is set to 1000, and the error tolerance is set to 0.01 to allow the network to train until the error tolerance of the training is to acceptable limits and there is no overfitting.The graph in Fig.6presents the plot of a trained data set, for n-V p -R L -I s(50) -m i combination the blue circles represent the training data while the red asterisks the output data after training.The plot shows that the training pattern is comparable to the desired output pattern.The testing data set is used to validate FIS structure.After the model is trained, a set of data points which were not used for training are used for data validation.The trained model is used for simulation of the testing datasets.Table11presents the testing error which is an indication of the ability of the model to predict  c .According to the obtained results R L -I s(50) - bt -m i predictive model seems to yield the higher performance of 0.089502, followed by n-V p -R L -I s(50) -m i with 0.076215.

Fig. 6 .
Fig. 6.ANFIS results for training, checking and testing data for n-V p -R L -I s(50) -m i combination.

Fig. 7 .
Fig. 7. Comparing the training and testing performance of for the applied classifiers, for the four selected combinations of parameters.

Table 1
Transformation models for c in the case studies from the literature used to compile ROCK/10/4025.
n, , V p , I s(50) ,  t , Abrasion R, c = 39.91 + 42.25Vp −12.50I s(50) + 95.29 d + 2.76Id 4 Turkey 40 İnce et al. [40] n, I s(50) ,  dry ,  sat , (continued on next page) Engineers were traditionally screening data from previous studies on similar or adjacent sites, to support geotechnical analysis on a new site.This practice was more common, in the case of limited ground investigation.It is not rare for geotechnical practitioners to inform their analysis based on experience and knowledge of the regional and site geology and engineering geological ground conditions, as wellinformed assumptions.It is interesting to refer to [97] suggestion on characteristic value selection.''Parameter values are gathered in statistical
c = (6222/ (88.15−RL )) −70.38, simulated annealing-gene expression programming (SA-GEP) China 91 Wen et al. [89] , Is(50), V p ,  40 Cylindrical dolomitic limestone Sedimentary  c = 0.034Vp −86.36,E = 0.013Vp −30.71,  c = 20.91Is −4.79 c Turkey Nearest Neighbour classifiers (KNN) and Ensemble Bagged Trees Model (EBTM) to develop models that can learn from existing paradigms and subsequently make reliable predictions of rock compressive strength for a site of concern.Based on the developed model's performance indicators on testing data, we could present some confidence, that the applied machine learning (ML) algorithms, generalize successfully and can be used to predict successfully in future the  c for multiple rock types.As far as we are concerned there are no studies that propose a common regression equation or a soft computing algorithm that can predict the compressional strength of intact rock  c based on physical and index properties for multiple rock types.Rahman and Sarkar [100] concluded that a common regression equation cannot be used to predict the  c from for multiple rock types, based on their study for  c prediction from P-wave velocity values of 12 different rocks.This paper further investigates the performance of machine learning algorithms (BP-ANN, ANFIS, SVM, KNN, EBGM), that can predict  c

Table 3
Summary statistics of the 10 parameters in the ROCK/10/4025 database at global level.

Table 5
Summary statistics of ROCK/10/4025 for sedimentary intact rocks.

Table 6
Summary statistics of ROCK/10/4025 for metamorphic intact rocks.

Table 8
Structure details for the ANFIS models developed.R L -I s(50) - bt -m i V p -R L -I s(50) m i n-V p -R L -Is (50) -m i V p -I s(50) -m i

Table 11
ANFIS training results.-I s(50) - bt −  i combination has the highest performance, and includes index properties and strength properties (non-destructive R L index-Point load index I s50 , Brazilian indirect tensional strength  bt , and constant m i ).The model systematically outperformed, both in function approximation, with ANN, ANFIS algorithms, and in classification problem with Linear SVM.The specific combination has the smallest data set 227 no of sets.The number of data satisfies the lower bound of number of samples vs.net size needed such that valid generalization can be expected.