Enhancing of uniaxial compressive strength of travertine rock prediction through machine learning and multivariate analysis

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Results in Engineering 20 (2023) 101593

Available online 17 November 2023

Enhancing of uniaxial compressive strength of travertine rock prediction through machine learning and multivariate analysis

Dima A. Malkawi

^a

, Samer R. Rabab ’ ah

^b^,^*

, Abdulla A. Sharo

^b^,^c

, Hussein Aldeeky

^d

, Ghada K. Al-Souliman

^b

, Haitham O. Saleh

^b

aDepartment of Civil and Environmental Engineering, School of Natural Resources Engineering & Management, German Jordanian University, Amman Madaba Street, P.

O. Box 35247, Amman, 11180, Jordan

bDepartment of Civil Engineering, Faculty of Engineering, Jordan University of Science &Technology, P.O. Box 3030, Irbid, 22110, Jordan

cCivil Engineering Program, Al Ain University, P.O. Box 112612, Abu Dhabi, United Arab Emirates

dDepartment of Civil Engineering, Faculty of Engineering, The Hashemite University P.O. Box 150459, Zarqa, 13115, Jordan

A R T I C L E I N F O Keywords:

Travertine rock

Uniaxial compressive strength (UCS) Tree model

K-nearest neighbors (KNN) Artificial neural networks (ANN)

A B S T R A C T

Indirect methods for predicting material properties in rock engineering are vital for assessing elastic mechanical properties. Accurately predicting material properties holds significant importance in rock and geotechnical engineering, as it strongly influences decisions about the design and construction of infrastructure projects. Uni- axial compressive strength (UCS) is one of the most important elastic mechanical properties for understanding how rocks and geological formations respond to stress and deformation. However, the standard UCS test faces several challenges, including its destructive nature, high costs, time-consuming procedures, and the requirement for high-quality samples. Therefore, there is a growing demand for indirect methods to estimate UCS, which are invaluable tools for evaluating the elastic mechanical properties of materials. The study aimed to comprehen- sively analyze the relationships between UCS of travertine rock samples collected from the Dead Sea and Jordan Valley formations and seven different rock indices by utilizing parametric and non-parametric methods. The laboratory results indicate that the study area’s travertine rock possesses high-quality and desirable properties.

The results reveal that certain rock indices, such as Schmidt hammer, Leeb rebound hardness, and Point Load, strongly correlate with Uniaxial Compressive Strength (UCS). Conversely, other indices, specifically dry density, absorption, pulse velocity, and porosity, exhibit a considerably weaker or very weak relationship with UCS. The paper employs three machine learning techniques, namely the Tree model, k-nearest neighbors (KNN), and Artificial Neural Networks (ANN), to develop predictive models for rock strength. The models were trained on a dataset of rock properties and corresponding mechanical strength values. The study’s results revealed that the M5 tree model is the most suitable method for predicting UCS. It demonstrates robust performance across a spectrum of metrics and boasts low prediction errors. Following the M5 tree model are the KNN, ANN, and regression methods in descending order of performance.

1. Introduction

Rocks, as natural resources, have played a pivotal role in human civilization for millennia, and their significance in various applications continues to expand in modern times. In the assessment of engineering projects, the essential properties of rocks, both in their physical and mechanical aspects, assume a pivotal role in establishing feasibility and ensuring safety. The behavior of rocks under varying environmental conditions, including temperature and moisture fluctuations, becomes

discernible through key physical attributes like density, porosity, permeability, and durability.

Recent advancements in rock engineering material property prediction have been fueled by technological progress and enhanced comprehension of geological processes. These progressions play a pivotal role in enhancing the safety and efficacy of projects within sec- tors like civil engineering, mining, and geotechnical engineering.

Notable developments include Machine Learning, Digital Rock Physics, Advanced Sensor Technologies, Geological Information Systems (GIS),

* Corresponding author.

E-mail address: [email protected] (S.R. Rabab’ah).

Contents lists available at ScienceDirect

Results in Engineering

journal homepage: www.sciencedirect.com/journal/results-in-engineering

https://doi.org/10.1016/j.rineng.2023.101593

Received 19 August 2023; Received in revised form 7 November 2023; Accepted 12 November 2023

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Remote Sensing, Geological Data Fusion, and Numerical Modeling.

These innovative breakthroughs are reshaping the landscape of rock engineering, facilitating the ability to predict material properties with unprecedented precision and efficiency. Their significance is paramount in enhancing the safety and success of infrastructure projects and opti- mizing the extraction of natural resources.

The mechanical properties comprise parameters such as strength, deformation, and fracture toughness, which are vital in establishing the load-bearing capability and overall stability of the rock formation. A profound comprehension of these engineering characteristics is imper- ative for designing and realizing significant infrastructure projects like dams, tunnels, and underground mines. The precise and meticulous characterization of rock engineering properties also plays a pivotal role in evaluating the vulnerabilities linked to natural occurrences and human-induced dangers, such as earthquakes and landslides. One widely utilized method to glean insights into diverse rock engineering properties is the UCS test [1].

Two main methodologies are commonly employed in estimating UCS values for rocks. The first method, called the “Direct” approach, directly assesses rock specimens in a laboratory setting. The second approach, the “Indirect” approach, takes a different route by relying on empirical equations in earlier studies [2]. For testing the UCS, the American So- ciety for Testing and Materials (ASTM) and the International Society for Rock Mechanics (ISRM) introduced standardized testing procedures.

These procedures require the extraction of high-quality core specimens, which does not apply to all rock types, as some rocks are weak, highly fracturable, weathered, or thinly bedded. Besides, these procedures are considered time-consuming, costly, and involve destructive tests [2,3].

Therefore, non-destructive, portable, fast, and cost-effective tests that can reasonably estimate UCS is recommended.

Indirect methods for predicting material properties in rock engineering are vital for assessing elastic mechanical properties. These properties, including UCS, Young’s Modulus, Poisson’s Ratio, and more, are integral for comprehending how rocks and geological formations respond to stress and deformation. Elastic mechanical properties are linked to a material’s capacity to deform elastically under the influence of loads and to revert to its original state upon load removal. Engineers and geologists can predict how rocks will deform when exposed to various stresses, whether from natural forces or engineering activities, by estimating these elastic mechanical properties using indirect methods. This knowledge is critical for designing structures and infrastructure that can withstand elastic deformations without harm or structural failure. Such predictive insights are critical for making well- informed decisions during infrastructure project conception and execution and maintaining the safety and stability of geological formations across a wide range of applications.

UCS tests are extensively utilized in geomechanics to evaluate the mechanical characteristics of rocks. Nonetheless, the destructive nature, high cost, time-intensive procedures, and demanding sample re- quirements associated with UCS tests often impede their practical application. Consequently, there is a growing interest in developing predictive models that can indirectly estimate the UCS of geological materials. The utilization of diverse machine learning techniques and the integration of various rock properties have proven effective in indirectly estimating the UCS of rocks. Prior research has extensively investigated the application of different machine learning algorithms, including artificial neural networks (ANN) [2,4–7], support vector regression (SVR) [8–11], decision trees (DT) [12–14], Adaptive Neuro-Fuzzy Inference System (ANFIS) [15–17], and many other techniques [18–21], not only in rock mechanics but also in numerous civil engineering applications.

In this study, parametric and non-parametric techniques are utilized to predict the UCS of travertine rocks depending on cost-effective non- destructive techniques through the rock’s physical and mechanical properties (e.g., Dry density, Absorption, Pulse velocity, Schmidt hammer, Porosity, Leeb rebound hardness, and Point Load). This study’s

predictive models can be utilized to assess rock strength in geotechnical engineering applications such as rock slope stability analysis, rock tunneling, and building rehabilitation. These models can provide important insights into the mechanical behavior of rocks, which can help to ensure the safety and efficiency of geotechnical projects.

2. Materials and methods 2.1. Materials

Travertine is a sedimentary limestone used for centuries for several construction uses. Because of its excellent durability, this rock is a popular choice for cladding, paving, and flooring. Jordanian travertine rocks’ physical and mechanical engineering features have previously been researched, finding that they have high compressive strength and low porosity, making them a perfect material for construction projects requiring a durable and weather-resistant material. Travertine has been utilized as a construction stone in Jordan since Ancient Rome, with most travertine outcrops located in the eastern Jordan Valley and the Dead Sea Transform Fault System [22]. As shown in Fig. 1, samples for this investigation were obtained from several travertine outcrops along the Dead Sea’s borders.

Travertine is characterized as a freshwater deposit of carbonates that precipitates from streams and spring waters, exhibiting a wide range of conditions. The standard color ranges from creamy to white; however, the presence of impurities can result in shades of yellow, brown, red, and pink. Deposits formed from ambient-temperature water that are similar but softer and highly porous are referred to as tufa [23]. A diverse range of other rock types can serve as substrates and sources of the elements necessary for the formation of these carbonates. Igneous rocks such as basalts, rhyolites, carbonatites, syenites, and granites, as well as sedimentary rocks like dolomites and marls, may contribute as source materials for calcium and other essential elements involved in the development of travertine and tufa [24].

To collect samples for this experiment, stone blocks were collected from various sites within the study region. According to the ASTM standard, 61 cores with a diameter of 10 cm and a height of 20 cm were prepared [25]. After the samples were cut, the gridding machine ground and polished the end surfaces until all surface irregularities were eliminated for the most uniform load distribution possible.

2.2. Experimental tests

The USC tests were carried out to determine the compressive strength of the core specimens based on the ASTM standards [26]. The specimens had flat, smooth, and parallel ends, which are important for ensuring consistent test results. Furthermore, the specimens had a length-to-diameters ratio of 2:1, which helps to ensure that the compressive force is applied uniformly to the specimen and prevents any buckling or failure during testing. A computerized MTS (Material Testing System) compression machine (Fig. 2) applied compressive force to the specimens. The applied load and deformation during the test are typically measured by using a load cell and a pair of strain gauges simultaneously, which are wired to a logging system. Computers are used to log the stress-strain continuously, and the UCS of specimens is considered the stress that will be recorded at failure (Peak stress level), which is generally defined by the major deformation or fracture of the sample.

To determine the dry density of each core specimen, the volume was calculated by taking measurements of its diameter and length using calipers. Subsequently, the specimen was subjected to a 24-h drying process in an oven set at 105 ^◦C. After the drying period, the dry mass of the specimen was determined.

The ultrasonic pulse velocity (UPV) test is an in-situ, non-destructive test to check the quality of natural rocks. In this test, the strength and quality of the rock are assessed by measuring the velocity of an

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ultrasonic pulse passing through a natural rock formation. This test was conducted on cylindrical rock core specimens with a diameter of 100 mm according to the recommendations of ISRM (2007) [27] and ASTM D2845 [28] by using PUNDIT Pulse (Portable Ultrasonic Non-Destructive Digital Indicating Tester).

The Schmidt Hammer Rebound (SHR) test is one of the most frequently used to assess the relative compressive strength of rock based on the hardness at or near its exposed surface without causing damage.

This test is non-destructive, quick, easy to apply, and cheap in either site or laboratory to provide preliminary information on the strength of rock materials. The test was conducted on the core samples according to the ISRM [27] recommendations, which recommended that the samples be intact (free of visible cracks), air-dried or saturated before testing, and have a diameter at least greater or equal to 54.7 mm.

Since hardness is an important physical index of rocks, it shows the

resistance of rocks to permanent deformation [29]. The Leeb Rebound Hardness (LH) test was selected because it is a non-destructive, simple, low-cost portable device, low energy, suitable for soft and hard rock, and was adopted by many researchers [29–32]. A standardized testing procedure for utilizing the LH on rock materials has not been established.

Therefore, the assumption Corkum et al. (2018) [32] mentioned was considered. Core samples with a diameter of 100 mm and 200 mm length were prepared according to ASTM D4543 [25] and ISRM (2007) [33]. The specimens had flat, smooth surfaces free from micro-fractures, fissures, and anisotropic features. The approach involved subjecting the core specimens to a single impact method involving 12 impacts [34].

The highest and lowest measurements were disregarded, and the average of the remaining ten readings was utilized as recommended by Refs. [35,36].

The Point Load (PL) index test presents an alternative Fig. 1. Location map showing the travertine outcrops in Jordan.

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straightforward and efficient approach for assessing the strength of rocks. This method is deemed cost-effective for testing rock samples on- site or in a controlled laboratory environment. The procedure involved assessing irregular fragments originating from fractured cores, following the guidelines outlined in the ASTM D 5731 Test Method [37].

3. Results and discussion 3.1. Laboratory tests results

The summary of the results of the experiments conducted on the studied samples of travertine rocks collected from the study area is presented in Table 1. As shown in Table 1, the results revealed that the ultrasonic wave velocity ranged from 4609 to 5416, averaging 5037.

According to the International Association for Engineering Geology (IAEG) [38], the travertine samples can be classified as very high ultrasonic wave velocities of class 5. Additionally, this variation of the obtained velocities reflects that the results have been affected by porosity and micro-cracks in some samples. Moreover, according to Onur et al. [39], ultrasonic traveling velocities are high in homogenous rock masses with good mechanical properties and can be used to identify the quality of the rock structure. This is consistent with the ultrasonic test results and indicates that the travertine rock from the study area has high-quality and desirable properties. Based on rock densities and po- rosities, (IAEG) proposed a classification of rocks of five classes.

Accordingly, the average density value of the tested travertine is 2.3 g/cm³, which classified as moderate dense rock, and the average porosity value is 9.7%, which is classified as medium porous rock. Based on strength values, the average value of UCS is 54.5 MPa, ranging from (25 MPa–85 MPa), which is classified as moderately hard rock according to the International Society for Rock Mechanics (ISRM) [33]. Also, the

dominant failure mode of travertine rocks was observed under uniaxial compression: axial splitting (multiple extensional planes). The dispersion of the UCS test results may be due to internal spatial variability in terms of mineralogical composition, anisotropy, and grain size that affect the yielding process. The Schmidt hammer rebound number and Leeb rebound hardness show an average value of (35, 583), respectively, with a limited range of variation. These values indicate that the travertine rock in the study area has a high hardness.

3.2. Data analysis

In data analysis, exploratory analysis is critical for identifying the underlying relationships between variables in a dataset. The exploratory analysis allows researchers to identify patterns, trends, and outliers in data, which can provide crucial insights into the topic under investigation. By exploring their relationships, researchers can better understand how these variables influence one another and contribute to overall data patterns. This research aims to predict the UCS of Jordanian travertine rocks and to analyze the links between the UCS and other physical and mechanical parameters.

The selection of seven specific rock indices for analysis in this study was chosen primarily because they collectively provide a comprehensive view of a rock’s physical and mechanical properties. They encompass density, porosity, stiffness, hardness, and resistance to deformation and loading. When analyzed together, they enable a complete understanding of a rock’s behavior under different stress conditions and can be used to develop predictive models for properties like UCS. Moreover, these indices are relatively easy to measure and applicable in laboratory and field settings, making them practical choices for a study that predicts material properties. Their inclusion allows for a robust analysis that can lead to more accurate predictions of rock strength and behavior, which, in turn, have significant implications for engineering and geological applications.

To propose significant relationships for real field applications, approaches capable of successfully recording the dataset’s complicated patterns and trends must be used. In this context, using both parametric and non-parametric approaches aids in modeling the correlations between the UCS of travertine rocks and other relevant variables. For example, Simple and multiple linear regression assumes a predefined functional form for the relationship between variables and estimates the model parameters based on the data. Non-parametric techniques, on the other hand, such as tree models, KNN, and ANN, make no assumptions about the underlying functional form and instead employ a data-driven approach to characterize variable associations. Researchers can utilize these strategies to create reliable models that can be used to anticipate Fig. 2. (a)The MTS machine was used to conduct a uniaxial compression test, (b) specimen failure during the UCS test.

Table 1

Descriptive Statistics for the conducted physical and mechanical tests.

Predictor Range Mean ST. DV CV (%)

UCS (MPa) 25.07–84.88 54.5 12.28 22.52

DD: (g/cm³) 2.2–2.4 2.3 0.059 2.55

n (%) 6–13 9.75 1.56 15.98

UPV (m/s) 4609–5416 5037 194.98 3.87

A (%) 1.18–3.34 2.57 0.48 18.78

RH 32–37.7 35 1.048 2.99

LH 550–624 583 13.8 2.37

PL(MPa) 1.75–3.31 2.57 0.35 13.62

ST. DV: Standard deviation.

CV: Coefficient of variation.

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system behavior under diverse conditions and influence field-level decision-making.3.2 Parametric Analysis.

3.2.1. Predicting UCS using simple regression analysis

The prediction of UCS using simple linear regression from non- destructive simple tests is widely used by many researchers [4,40–43].

Developing relationships between UCS and other index properties is critical in geotechnicals.

Statistical parameter values of maximum, minimum, mean, coefficient of variation, and standard deviation of the results are shown in Table 1, which summarizes all the tests conducted on the studied samples.

In this study, the UCS is used as the dependent variable (response variable), whereas Dry density (DD), Absorption (A), Pulse Velocity (P), Schmidt Hammer (RH), Porosity (n), Leeb Rebound Hardness (LH), and Point Load (PL) were used as independent variables (predictors).

Various regression relationships, including linear, power, exponential, and logarithmic, were performed between UCS and each one of the index parameters (independent variables). Moreover, to evaluate the

predicted simple regression equations and try to select the best equation to estimate the UCS, the evaluation was conducted based on statistical performance indices, Coefficient of determination (R²), Root Mean Square Error (RMSE), and mean absolute error (MEA). The equations of these indices are shown below:

R²=1−

⎛

⎜⎜

⎝

∑ⁿ

i=1

(ym− yp

)₂

∑ⁿ

i=1

(ym− ya)²

⎞

⎟⎟

⎠ (1)

RMSE=

̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅

1 n

∑ⁿ

i=1

(ym− yp

)₂

√

(2)

MAE=1

n

∑ⁿ

i=1

⃒⃒ym− yp

⃒⃒ (3)

Where y_m: is the measured value, y_p: is the predicted value, y_a: is the average of the measured value, and n: is the total number of data.

Fig. 3.The developed equations for estimating UCS using (a) Point load, (b) Schmidt hammer rebound number, (c) Leeb rebound hardness, (d) P-wave velocity, (e) Dry density, (f) porosity, and (g) Absorption.

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R²measures the goodness of fit between the independent and dependent variables in the regression model. A higher R²value indicates a better fit. The MAE calculates the average percentage difference between the observed and predicted values. A lower MAE indicates better predictive accuracy. On the other hand, RMSE calculates the average deviation between the observed and predicted values. A lower RMSE indicates better predictive accuracy. By using R², MAE, and RMSE, the researchers aimed to thoroughly evaluate the performance of different regression relationships and identify the most suitable model for UCS prediction based on the rock indices.

Fig. 3 depicts a scatter plot of the data points, with the x-axis representing the independent variable and the y-axis representing the dependent variable (UCS). The dispersion and clustering of the data points can provide valuable insights into the nature and strength of the relationship between the variables.

The results show that all of these statistical models have high reli- ability in estimating UCS since the majority are scattered around the diagonal line (y =x), and the linear, logarithmic functions were the more reliable in estimating the UCS with a greater coefficient of correlation. Table 2 displays the chosen equations for predicting UCS using those with the highest R²values and lowest MAE and RMSE values in comparison to other equation types. The simple regression analyses exhibit favorable correlation coefficients between UCS and most independent variables used in the predictions.

Table 2 shows each predictor’s fit functions (mathematical functions) with the UCS using the Social Sciences Software (SPSS) estimation tool. These functions were selected based on the R²from the suggested functions. Based on the R²values, some predictors (RH, LH, and PL) have a strong relationship with UCS, while others have a weak or weak relationship (DD and n). For instance, the RH model has the highest R² values, which indicates a strong relationship with UCS. Models such as Dry unit weight, porosity, and absorption have low R²values, indicating no clear evidence of the effect of these parameters on the UCS values in this study. The plot of ultrasonic pulse velocity and UCS shows a trend of increasing UCS with increasing ultrasonic pulse velocity. However, a wide scatter indicates a lower and upper boundary. This is probably due to the effect of factors related to the fabric and composition, and the effect of porosity influences the response of the selected samples to the UPV.

3.2.2. Multiple linear regression (MLR)

This study used MLR analysis to investigate the relationships between UCS and various independent variables. MLR is a statistical technique commonly used to model the relationship between a dependent variable and multiple independent variables.

Simple bivariate correlation is one of the statistical techniques that can investigate how strong the relationships between two variables are.

For this study, the Pearson Correlation coefficient was used to determine if there is a relation between UCS and the other independent variables.

Table 3 shows the Pearson Correlation coefficient values for the UCS with the independent variables. As seen, UCS has a relatively strong relationship with the predictors. For example, UCS has a weak positive correlation with DD, a medium positive correlation with the UPV, and a strong positive correlation with PL, RH, and LH. As a result, all independent variables were considered in the analysis.

The MLR opted to halt after including six independent variables since it could no longer uncover statistically significant predictors of the UCS, as seen in Table 4. Additionally, Table 5 shows the R²ranging from Model 1’s R²of 0.899 to Model 6’s R²of 0.967, which contained additional predictors (RH, PL, LH, A, n, and UPV). The Durbin-Watson (DW) statistical method examines the possibility that the data may be serially correlated. The DW value of 1.933, between 1.2 and 2.5, indicates no significant serial correlations.

The process of deciding which independent variables to include and exclude from a regression equation is known as model specification.

Table 6 shows the models’ coefficients; Model 6 has two variables with p-values more than 0.05, whereas Model 4 has all variables with p- values less than 0.05.

Multicollinearity is a scenario in which two independent variables are highly correlated. Using the SPSS software, the Variance Inflation Factor (VIF), which is one of the collinearity diagnostic methods, is used to investigate if one of the predictors has a strong linear relationship with others (VIF <1 or >11). Therefore, based on Table 6, Model 4 is selected, with variables eliminated (DD, UPV, & n).

The homoscedasticity test was used to check if the residuals were equally distributed. Fig. 4 illustrates that residuals are approximately normal as they are equally distributed around Zero.

3.3. Non-parametric analysis

The selection of machine learning techniques in a study for predicting rock material properties is typically based on a combination of factors. Decision trees are well-known for their interpretability, result- ing in an easy-to-understand and visualize decision tree structure. This is critical for understanding the relationships between input, such as rock indices, and the target variable, UCS, thus assisting in identifying key predictive features. Decision trees excel at modeling intricate, non-linear relationships within data, which is critical when dealing with rocks’ complex and frequently non-linear properties. Furthermore, decision

Table 2

Best fit function for the independent variables.

Variable Correlation Equation R² RMSE MAE

PL UCS =32.93PI − 30.25 0.885 4.1391 3.0019

RH UCS =11.10(SHR) − 334.34 0.899 3.8796 2.9367 LH UCS =480.01lnLRH − 3002.76 0.856 4.6545 3.6992

UPV UCS =0.035UPV − 123.9 0.317 10.154 7.891

DD UCS =114.13*ln (DD) − 40.254 0.056 11.934 9.486 n UCS = −2.5649(n) +79.532 0.106 11.612 9.070

A UCS =64.337(A)⁻^0.21 0.048 12.01 9.56

Table 3

Correlation values of independent variables with UCS.

Independent variables Pearson correlation

DD 0.233

A −0.209

UPV 0.563

RH 0.948

n 0.338

LH 0.923

PL 0.940

Table 4

Variables Entered/Removed to predict UCS.

Model Variables

entered Variables

removed Method

1 RH . Stepwise (Criteria: Probability-of-F-to-

enter ≤.050, Probability-of-F-to- remove ≥.100).

2 PL . Stepwise (Criteria: Probability-of-F-to-

3 LH . Stepwise (Criteria: Probability-of-F-to-

4 A . Stepwise (Criteria: Probability-of-F-to-

5 n . Stepwise (Criteria: Probability-of-F-to-

6 UPV . Stepwise (Criteria: Probability-of-F-to- enter ≤.050, Probability-of-F-to- remove ≥.100).

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trees are resistant to the influence of outliers and adept at dealing with noisy data. This robustness is extremely useful in rock engineering, characterized by data with inherent variability.

KNN is a simple and intuitive algorithm that classifies data points based on the majority class of their nearby neighbors in feature space. Its user-friendly nature makes it accessible even to those with limited machine-learning experience. KNN is non-parametric, adapting effectively to the data’s underlying distribution. This adaptability is particularly valuable when predicting rock properties, given the often complex and irregular data distributions in geology. Moreover, KNN does not impose specific assumptions on the data, which is advanta- geous when dealing with diverse geological data that may deviate from standard statistical expectations.

ANNs have a remarkable capacity for capturing intricate patterns and relationships in data, making them an ideal choice for tasks characterized by non-linear and complex dependencies, as is often encoun- tered in rock engineering. ANNs are well-equipped to handle extensive datasets and exhibit high scalability, a valuable trait when dealing with geological data from diverse sources and locations. Notably, ANNs possess strong generalization capabilities, making accurate predictions even when confronted with new, previously unseen data. This adaptability is essential in rock engineering, where conditions can widely differ from one geographical location to another.

3.3.1. SimpleKMeans clustering

SimpleKMeans is a fast and efficient clustering algorithm that can discover patterns in large datasets and identify groups of similar data points. Clustering in machine learning is simply the process of creating different dataset groups that consists of similar elements (features). The k-mean algorithm is an exclusive clustering machine learning algorithm to group or cluster observations (u) based on their features or attributes into k-groups or partitions, where k <u (k represents the number of clusters); therefore, this algorithm is approximately capable of being scaled so it is efficient in handling large datasets. In general, clusters are established by assuming that each observation has the least distance from the centroid. To measure the distance between observations, this study used the Waikato Environment for Knowledge Analysis (WEKA) software [44] by utilizing its Euclidean distance function (Equation (4)).

du,v=

̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅

(u1− v1)²+ (u2− v2)²+…+( uq− vq

)₂

√

(4) where, Observations u =(u₁, u₂, …, u_q) and v =(v₁, v₂, …, v_q) each comprise measurements of q variables.

In this study, clustering analysis was used to identify the rock’s UCS clusters by grouping the rocks based on their strengths in the hope of better understanding how the strength of a mineral depends on its properties. It is possible to identify natural groupings or clusters of rocks with similar UCS values by applying the SimpleKMeans clustering

method to a dataset of rock properties. This method can aid in a better understanding of the relationships between various rock properties and UCS and shed light on the mechanisms governing granite strength.

Furthermore, clustering can be useful for identifying outliers or anomalous samples that may require further investigation and for identifying trends and patterns in the data that may not be immediately apparent. Overall, using the SimpleKMeans clustering method to identify UCS clusters in rocks can provide a valuable tool for rock mechanics studies and enable more efficient and accurate estimation of UCS. Fig. 5 demonstrates that the UCS data can be categorized into two clusters.

Each group has unique components, indicating that the UCS data are distributed uniformly.

3.3.2. Tree model

Tree models are characterized by their ease of interpretation and ability to offer valuable insights into the relative significance of various input variables in predicting UCS. This analysis treats the data as if they are a series of decisions. This procedure results in a comparison between each of the different possible outcomes. This supervised learning algorithm predicts a target variable given a set of independent variables.

Using the eight independent variables, this study used the M5 tree model to predict the USC. Generally, the M5 model starts from the tree’s top node, trying to find an attribute that will reduce the variability in the target variable. At the end of the tree, the M5 model concludes with a constant value, the average of the samples reaching the node. To reduce overfitting, 20 instances for each node are required [45].

This study used the WEKA software to build the M5 tree model. This WEKA software provides different test modes for the M5 tree modeling.

These modes are 1- All data was used for training, 2- Split mode, where data is split to training and testing, and three- Cross Validation mode.

This study used the split mode (80% for Training and 20% for Testing) to predict the UCS using the previous eight independent variables. After that, a supplied dataset was used to test and validate the M5 tree model.

Besides, Equation (5) illustrates the rule for this tree model.

UCS =8.3482 ×A +0.0061 ×UPV +3.5776 ×RH +0.297 ×n +0.2298

×LH +14.1174 ×PL − 304.9694 (5) As seen, the UCS value is directly proportional to all input variables.

The M5 tree model excluded one variable from the equation (DD) to avoid multicollinearity. Besides, Point Load (PL) has more effect than the other variables. Additionally, Table 7 shows the M5 tree model performance. As seen, Kendall’s tau and Spearman’s rho values show a perfect positive relationship between the variables, which means the M5 model performs well in ranking the data points. In addition, the Root Mean Square Logarithmic Error (RMSLE) has a value close to zero, which measures the ratio between the predicted and actual UCS values.

Moreover, Fig. 6 shows the goodness of fit for the developed M5 tree model. The performance metrics suggest that the M5 tree model is reliable in predicting UCS, as it effectively captures the relationship between the input features and the target variable. For instance, a linearity behavior is seen between the predicted and actual UCS values;

this suggests that the model is performing well in predicting the UCS values.

3.3.3. 3K-nearest neighbors (KNN)

This study uses the KNN algorithm to predict the UCS value based on several independent variables. KNN is a non-parametric method that finds the k-nearest data points to a given sample based on the distance metric. Once the k-nearest neighbors are identified, the algorithm uses their corresponding UCS values to calculate a predicted value for the new sample. The WEKA’s supervised machine learning algorithm was utilized to build this prediction model using the lazy IBK. The model parameters used to build this model are shown in Fig. 7. This study used several K values and selected the cross-validation as “True.” Therefore, the best K value is included in the prediction model.

Table 5

The UCS model summary.

Model R R

Square Adjusted R

Square sStd. Error of the

Estimate Durbin-

Watson

1 0.948^a 0.899 0.897 3.93180

2 0.973^b 0.946 0.944 2.90894

3 0.976^c 0.952 0.950 2.74760

4 0.980^d 0.960 0.957 2.54709

5 0.981^e 0.963 0.960 2.46694

6 0.984^f 0.967 0.964 2.34127 1.933

g. Dependent Variable: UCS.

aPredictors: (Constant), RH.

b Predictors: (Constant), RH, PL.

cPredictors: (Constant), RH, PL, LH.

dPredictors: (Constant), RH, PL, LH, A.

ePredictors: (Constant), RH, PL, LH, A, n.

fPredictors: (Constant), RH, PL, LH, A, n, UPV.

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The distance function is a key component of the KNN algorithm, as it controls the prediction model’s performance. This study chooses the Euclidean with distance weighting as the distance function. This function is one of the most natural distance functions for numeric attributes to solve the outlier and noise sensitivity problem. This process utilizes the “1/the distance” weight to provide the selected closest neighbor with the most effective weight. However, this distance function requires that the data parameters have homogeneous features. Particularly, the scale of the included parameters should be the same. As a result, all numeric attributes included in this case were normalized before estimating the distance between neighbor instances.

Fig. 8 shows graphically the goodness of fit using the developed KNN model for the predicted UCS vs. actual UCS plot. Besides, Table 8 depicts

the KNN model performance metrics for predicting the UCS values. The values of Kendall’s tau and Spearman’s rho show a positive relationship between the variables. In addition, the low RMSLE value shows the strength of the developed model. Based on the performance metrics, the KNN model performs reasonably well predicting the UCS. For instance, the mean absolute logarithmic error is (0.0685), and the RMSLE is (0.099); these values measure the average logarithmic differences between the predicted and actual values. Generally, lower values indicate better model performance. These metrics are higher than those for the M5 tree model, suggesting that the KNN model has a higher prediction error on a logarithmic scale. Similarly, lower values indicate better model performance for the Mean absolute percentage error (0.073) and root mean square percentage error (0.11). These values are higher than Table 6

The MLR models summary.

Model Unstandardized Coefficients Standardized Coefficients t Sig. 95.0% Confidence Interval for B

B Std. Error Beta Lower Bound

1 (Constant) − 334.341 16.958 −19.716 <.001 − 368.274

RH 11.109 .484 .948 22.941 <.001 10.140

2 (Constant) − 206.852 21.997 −9.404 <.001 − 250.883

RH 6.266 .774 .535 8.093 <.001 4.716

PL 16.331 2.315 .466 7.056 <.001 11.698

3 (Constant) − 243.136 24.413 −9.959 <.001 − 292.023

RH 3.912 1.108 .334 3.532 <.001 1.694

PL 15.706 2.197 .448 7.148 <.001 11.306

LH .206 .073 .232 2.830 .006 .060

4 (Constant) − 264.887 23.622 −11.213 <.001 − 312.208

RH 3.706 1.029 .316 3.603 <.001 1.646

PL 15.807 2.037 .451 7.759 <.001 11.726

LH .245 .069 .276 3.573 <.001 .108

A 2.324 .723 .091 3.214 .002 .875

5 (Constant) − 272.950 23.179 −11.776 <.001 − 319.403

RH 3.156 1.028 .269 3.070 .003 1.096

PL 16.156 1.980 .461 8.161 <.001 12.189

LH .257 .067 .288 3.848 <.001 .123

A 6.387 2.001 .251 3.192 .002 2.377

n .241 .111 .177 2.167 .035 .018

6 (Constant) − 304.969 25.082 −12.159 <.001 − 355.255

RH 3.578 .989 .305 3.619 <.001 1.596

PL 14.117 2.029 .403 6.956 <.001 10.049

LH .230 .064 .258 3.585 <.001 .101

A 8.348 2.037 .328 4.098 <.001 4.263

n .297 .107 .219 2.763 .008 .081

UPV .006 .002 .098 2.658 .010 .002

Model 95.0% Confidence Interval for B Correlations Collinearity Statistics

Upper Bound Zero-order Partial Part Tolerance VIF

1 (Constant) − 300.408

RH 12.078 .948 .948 .948 1.000 1.000

2 (Constant) − 162.820

RH 7.816 .948 .728 .247 .214 4.670

PL 20.964 .940 .680 .216 .214 4.670

3 (Constant) − 194.249

RH 6.130 .948 .424 .102 .093 10.712

PL 20.106 .940 .688 .206 .212 4.718

LH .352 .923 .351 .082 .124 8.042

4 (Constant) − 217.566

RH 5.767 .948 .434 .096 .093 10.754

PL 19.888 .940 .720 .208 .212 4.719

LH .383 .923 .431 .096 .120 8.301

A 3.773 −.209 .395 .086 .889 1.124

5 (Constant) − 226.497

RH 5.217 .948 .382 .080 .087 11.452

PL 20.124 .940 .740 .212 .210 4.751

LH .390 .923 .461 .100 .120 8.353

A 10.397 −.209 .395 .083 .109 9.174

n .463 .338 .281 .056 .101 9.939

6 (Constant) − 254.683

RH 5.560 .948 .442 .089 .085 11.754

PL 18.186 .940 .687 .171 .180 5.543

LH .358 .923 .438 .088 .117 8.566

A 12.433 −.209 .487 .101 .095 10.560

n .512 .338 .352 .068 .097 10.341

UPV .011 .563 .340 .065 .449 2.226

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those for the M5 tree model, indicating that the KNN model’s predictions are less accurate on a percentage scale.

3.3.4. Artificial neural networks (ANN)

One of the most prominent machine learning algorithms, neural

networks, encompasses many concepts and methodologies. This study utilizes the Multilayer Perceptron (MLP), one of the ANN techniques used to classify and develop prediction models. MLP is a feedforward neural network with multiple layers of interconnected nodes, or neurons, that can understand complex relationships between input variables and their corresponding UCS values. The perceptron is inspired by the Fig. 4. Frequency and P–P plots of regressions standardized residual for the

dependent variable.

Fig. 5.UCS clustering using the SimpleKMeans method.

Table 7

M5 tree model performance.

Model performance Value

Kendall’s tau 0.884

Mean absolute logarithmic error 0.035

Mean absolute percentage error 0.034

Root mean square logarithmic error 0.046

Root mean square percentage error 0.045

Spearman’s rho 0.958

Fig. 6. UCS (predicted) versus UCS (actual) using the M5 tree model.

Fig. 7. The WEKA’s window for lazy IBK classifier.

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Neuron, the human brain’s most fundamental unit of thought. The MLP method can, therefore, both solve and predict complex cases.

Consequently, it has the potential to circumvent several limitations, such as linear separability. The MLP is trained using a backpropagation algorithm that modifies the weights of the connections between neurons to minimize the deviation between the predicted and actual UCS values.

The MLP was implemented using the Neural Designer software, which enables users to build a prediction model manually or through a simple heuristic interface and monitor and modify/edit the ANN parameters.

In this study, the learning (training) strategy was carried out. This strategy is utilized in the neural network to achieve the best possible loss. In ANNs, the type of learning is determined by how the adjustment of the parameters in the neural network occurs. The Levenberg- Marquardt algorithm is a popular method for training machine learning models to balance the first-order and second-order learning rates. In this study, the algorithm was employed to improve the training process without the need to compute the Hessian matrix, which can be computationally expensive. By utilizing the Levenberg-Marquardt algorithm, the convergence speed of the model was improved, and therefore, more accurate results were obtained.

The data was divided into three distinct datasets: a training set, a validation set, and a test set to train and evaluate the performance of the developed model. The training set was utilized to fit the model parameters by trial and error, enabling the model to learn from the data and improve its accuracy over time. The validation set was utilized to tune the model’s hyperparameters and prevent overfitting when a model performs well on training data but poorly on unseen data. The test set was then utilized to evaluate the model’s performance on unseen data and estimate its generalization capability. Separating the data into these three subsets developed a robust and dependable model that accurately predicts outcomes with new, unobserved data.

A trial-and-error method was used to determine the appropriate percentage of data to divide between the three datasets: training, validation, and testing. The software Neural Designer analyzed the data and recommended a split percentage for optimal model performance. The

software recommended a split of 60% for training, 20% for validation, and 20% for testing based on the dataset’s characteristics utilized in this study. However, the exact percentages were also determined through iterative experimentation, as split ratios can significantly affect model performance. The final split percentages were determined by balancing the need for adequate training data, the importance of avoiding overfitting, and the desire to accurately evaluate model performance on unseen data. By selecting the split percentages with care, the researchers could ensure that the developed model was robust and reliable for the provided dataset. Fig. 9 shows the graphical representation of the developed model architecture.

The Levenberg-Marquardt algorithm is applicable only when the loss index has the form of a sum of squares (as the mean squared error or the normalized squared error). Fig. 10 depicts the training and selection errors for each iteration of the developed model. The blue line indicates a training error, while the orange line indicates a selection error.

The efficacy of the developed ANN model can be evaluated based on the training and selection error values. In this study, the initial training error value was 2.3397, and after 104 epochs, it decreased to 0.0106, indicating that the model could learn from the data and increase its accuracy over time. Similarly, the initial selection error value was 4.4436, and after 104 iterations, it decreased to 0.143, indicating that the model could generalize well to new data. These results indicate that the developed ANN model predicted the target variable accurately and performed well on the dataset. Table 9 displays the training results of the Levenberg-Marquardt algorithm.

In this study, the neural network was designed with eight inputs, and the size of the scaling layer was also set to eight, using the Scaler:

MeanStandardDeviation algorithm. Table 10 provides an overview of the inputs included in the neural network, highlighting the variables considered for predicting the UCS. For the perceptron layer, two layers were used for the neural network. Table 11 shows the size of each layer and its corresponding activation function. In addition, the size of the unscaling layer is 1, which is the number of outputs (UCS).

Model selection algorithms in neural networks provide the created neural network model with a topology that maximizes the error on the new dataset. There are two types of model selection algorithms: order selection algorithms and input selection algorithms. For example, order selection methods identify the optimal number of hidden neurons in a neural network. In contrast, input selection algorithms are in charge of identifying the optimal subset of input variables.

The expanding neurons algorithm was utilized in this study to determine the ideal number of neurons for the constructed neural network. This algorithm starts with a few neurons and adds a fixed number in each iteration, thus increasing the network over time. The algorithm calculates the ideal number of neurons by evaluating the network’s performance on a validation set and determining the point at which adding additional neurons no longer enhances performance. The growing neurons technique is very helpful in avoiding overfitting since it allows the network to add neurons only where needed based on the Fig. 8. UCS (predicted) versus UCS (actual) using the KNN model.

Table 8

The KNN model performance for predicting the UCS values.

Model performance Value

Kendall’s tau 0.7716

Mean absolute logarithmic error 0.0685

Mean absolute percentage error 0.073

Root mean square logarithmic error 0.099

Root mean square percentage error 0.11

Spearman’s rho 0.923

Fig. 9.Graphical representation of the ANN model architecture.

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correlations between the inputs and the desired output. This method can result in a more efficient and accurate neural network by removing unnecessary parameters and reducing computational complexity. Using the growing neurons algorithm to determine the optimal number of neurons for the developed neural network, a model was developed that was robust, reliable, and capable of accurately predicting the target variable on new, unseen data.

The loss index measures the quality of any neural network model.

When using the loss index, users must consider error and regularization.

The model’s numerical fit to the dataset is assessed in the error term. It depends on the application to choose from multiple error methods. The Normalized Squared Error (MSE) has been selected in this study.

ANN model accuracy is often measured using the Normalized Squared Error (MSE). This study calculated the MSE for the training, validation, and testing datasets to determine how well the model predicts the target variable. A lower MSE suggests that the model can better capture data patterns, indicating better performance. In this study, the normalized MSE for training data was 0.047, validation data 0.035, and testing data 0.086. The MSE for all study datasets (Training, Validation, and Testing) is shown in Table 11. The low MSE values indicate that the model performed well on the three datasets.

In addition, the error histogram is used to investigate the error

distribution of the developed model for the testing samples. Generally, a model with high predictive performance would have an approximately normal distribution. Fig. 11 shows the relative error distribution for UCS (output/Target).

Based on the previously mentioned performance metrics, the M5 tree model is the most suitable for predicting UCS, as it shows strong performance across various metrics and has low prediction errors. The KNN model performs reasonably well but has higher prediction errors than the M5 tree model. While performing well on the training data, the ANN model performs less on the validation and testing datasets.

In this study, the proposed techniques can be integrated into the preliminary design stages of projects by providing critical data and insights regarding rock properties. This thorough understanding of rock properties, which includes critical factors such as compressive strength, porosity, and elastic behavior, enables engineers to make accurate and informed decisions in foundation design. Because of this informed decision-making process, the foundation is accurately designed to support the structure. It improves the foundation’s load-bearing capacity while significantly lowering the risk of problems like settlement or instability. Engineers and geologists use this information for foundation design and tunnel stability, material selection, support systems, and safety precautions. Incorporating these techniques during the preliminary design stage is the foundation for a successful project. It enables professionals to make data-driven decisions that improve project safety, efficiency, and overall effectiveness while ensuring that structures are designed to align with geological conditions seamlessly.

4. Conclusions

The findings of this study make substantial contributions to the advancement of the fields of rock and geotechnical engineering. Accu- rate material property prediction is fundamental for designing safe and resilient infrastructure. The study’s findings enhance the ability of geotechnical engineers to assess geological conditions, which, in turn, leads to the design of foundations, tunnels, and support systems that can withstand stress and deformation. By understanding the material properties of rock formations, engineers can optimize the allocation of resources and reduce over-engineering. This translates to substantial cost savings. The study incorporates advanced techniques into geotechnical engineering, including machine learning and data analysis. This integration fosters innovation by encouraging the exploration of new technologies and methodologies. Geotechnical engineers can leverage cutting-edge tools to address complex problems, adapt to evolving challenges, and continuously improve industry practices.

This study was conducted to provide engineers and practitioners with more cost-effective and time-saving methods to determine travertine rocks’ UCS. Several laboratory tests were performed to develop prediction models that can predict the rocks’ UCS (e.g., Schmidt Fig. 10. The training and selection errors in the developed model for

each iteration.

Table 9

The training results of the Levenberg-Marquardt algorithm.

Levenberg-Marquardt algorithm Results Value

Training Error 0.0106

Selection Error 0.143

Epoch Number 104

Stopping Criterion Minimum loss decrease

Table 10

Developed Models layer size and activation functions.

Layer Number Input Number Neurons Number Activation Function

1 8 12 HyperbolicTangent

2 12 1 Linear

Table 11

The normalized Squared Error (MSE) for the Model datasets.

Error Training Validation Testing

Normalized Squared Error 0.047 0.035 0.086

Fig. 11.The distribution of the ANN Model’s relative errors.