Machine Learning Algorithms in Rock Strength Prediction: A Novel Method for Evaluating Dynamic Compressive Strength of Rocks Under Freeze-Thaw Cycles

The combined impact of freeze-thaw cycles and dynamic loads significantly influences the long-term durability of rock engineering in high-cold regions. Consequently, investigating the dynamic compressive strength (DCS) of rocks subjected to freeze-thaw cycles has emerged as a crucial area of scientific research to advance rock engineering construction in cold regions. Presently, the determination of the DCS of rocks under freeze-thaw cycles primarily relies on indoor experiments. However, this approach has faced criticism due to its drawbacks, including prolonged duration, high costs, and reliance on rock samples. To address these limitations, the exploration of using artificial intelligence technology to develop more accurate and convenient DCS prediction models for rocks under freeze-thaw cycles is a promising attempt. In this context, this paper introduces a DCS prediction model for rocks under freeze-thaw cycles, which integrates the Sparrow Search Algorithm (SSA) with Random Forest (RF). Firstly, employing a dataset of 216 samples, Principal Component Analysis (PCA) is utilized to reduce the dimensionality of ten influential factors. Subsequently, five optimization algorithms are employed to optimize the hyperparameters of both the BP and RF algorithms. Finally, a comprehensive evaluation and comparative analysis are carried out to assess the predictive performance of the optimized model, using evaluation metrics such as Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Coefficient of Determination (R2).The research findings demonstrate that the SSA-RF model exhibits the best predictive performance, surpassing the other nine models in terms of generalization. The prediction model proposed in this study has good applicability for predicting DCS of freeze-thaw rock in cold regions, and also provides new ideas for the combination of machine learning and rock mass engineering in cold regions.


Introduction
China's permafrost area is widely distributed.Permafrost and seasonal frozen soil areas account for approximately 22% and 53.5% of the land area, respectively, ranking third in the world [1,2] .Guided

Machine Learning Algorithms in Rock Strength Prediction: A Novel Method for Evaluating Dynamic Compressive Strength of Rocks Under Freeze-Thaw Cycles
. Consequently, predicting the dynamic compressive strength (DCS) of rock in alpine regions has become a significant research focus aimed at facilitating the sustainable development of engineering construction in cold regions [20][21][22] .
At present, in order to establish a high-precision and universally applicable method for predicting the DCS of rock masses in high-altitude cold regions, researchers have made a lot of research practices, mainly including: (1) exploring the main factors affecting the DCS of rocks under freeze-thaw cycles.For example, Meng et al. [23] have found that freeze-thaw cycles can reduce the mechanical properties of sandstone, while confining pressure and strain rate can enhance the mechanical properties of sandstone.Gholamreza et al. [24] assessed the long-term durability of sandstone during freeze-thaw cycles using a decay function model, revealing that an increased number of freeze-thaw cycles reduces uniaxial compressive strength, longitudinal wave velocity, and increases effective porosity.Pu et al. [25] delved into the damage process of sandstone under freeze-thaw cycles and impact loads, discovering that the DCS of sandstone under different air pressures was influenced by freeze-thaw cycles.Xu et al. [26] conducted a systematic study on the dynamic compression characteristics of green sandstone under various hydrostatic confining stresses after freeze-thaw cycles.Their findings indicated that hydrostatic pressure restricts the opening and propagation of microcracks through constraint, thereby improving the DCS of the rock.In summary, the DCS of freeze-thaw rocks is related to the number of freeze-thaw cycles (FTC), confining pressure (CP), impact force (IP), dry density (DD), natural water content (NWC), water absorption (WA), average dry weight (ADW), porosity (P), P-wave velocity (PWV), strain rate (SR) etc; (2) Construction of deterioration models for DCS of rocks under freezethaw cycles.For example, Ke et al.'s [27] derivation of deterioration model parameters through conventional uniaxial compression tests on samples subjected to different freeze-thaw cycles, allowing for the development of a comprehensive dynamic increase factor model for different rock types and sample sizes.Luo et al. [28] observed that the DCS of sandstone samples decreases exponentially with an increase in the number of freeze-thaw cycles when the strain rate remains constant.When the number of freeze-thaw cycles is constant, the DCS of the sandstone sample increases linearly with an increase in the strain rate.This data informed the formulation of a DCS formula for sandstone that considers both the number of freeze-thaw cycles and the strain rate, leading to the construction of a deterioration model.Xu et al. [29] utilized a decay model based on longitudinal wave velocity to describe and measure rock disintegration rates.This model assumes a proportional relationship between the loss of rock integrity during accelerated freezing and thawing and the change in longitudinal wave velocity.It can be employed to predict the mechanical degradation and durability of tested rocks.Wang et al. [30] established a mechanical property decay model reflecting the strain rate effect, using uniaxial compressive strength and deformation modulus as integrity indices for rock.Regression analysis of experimental results yielded the variation of attenuation constants and halflives with strain rates.Despite the diversity of deterioration models for DCS of rocks under freezethaw cycles, the complexity of variables affecting DCS has hindered the establishment of an accurate and applicable empirical equation for estimation.Consequently, DCS prediction for rocks under freeze-thaw cycles still necessitates time-consuming indoor experiments.Currently, the advent of artificial intelligence has revolutionized the data analysis for DCS of rocks under freeze-thaw cycles.With its formidable self-learning and data processing capabilities, it effectively compensates for the inherent limitations of traditional technologies when solving rock mechanics problems and employing numerical methods [31,32] .Machine learning has found widespread application in rock mechanics engineering [33,34] , encompassing rock mechanics property prediction [35] , surrounding rock classification [36] , flying rock prediction [37] , rock burst prediction [38] , rock fracture extraction [39] , and more.Unfortunately, the above-mentioned intelligent technologies for predicting rock mechanics are mostly based on static compressive strength, and the research on intelligent prediction of DCS of freeze-thaw rocks has not received sufficient attention from scholars.

Figure 1. Flow chart for predicting the DCS of rocks under freeze-thaw cycles
In summary, based on the intelligent prediction technology of DCS of freeze-thaw rocks in cold regions, firstly, it is necessary to comprehensively consider various relevant indicators and obtain corresponding original database; At the same time, compress and extract characteristic information from the database of DCS impact indicators of freeze-thaw rocks, in order to reflect sufficient variable information with as few variables as possible; Finally, the hybrid algorithm is trained and tested based on the dimensionality reduced database, and its predictive performance was evaluated.Based on this, this article focuses on the scientific problem of intelligent prediction of DCS of freeze-thaw rocks and attempts to carry out the following work: (1) Based on the primary influencing factors of DCS in rocks under freeze-thaw cycles, a set of 10 indicators, namely FTC, CP, IP, DD, NWC, WA, ADW, P, PWV, and SR, were chosen to create a sample database.The correlation coefficients between each indicator were analyzed, and the original data was dimensionally reduced using Principal Component Analysis (PCA).( 2) By combining BP neural network and Random Forest (RF) algorithms with 5 optimization algorithms, including Sparrow Search Algorithm (SSA), Grey Wolf Optimization Algorithm (GWO), Bacterial Foraging Optimization Algorithm (BFO), Particle Swarm Optimization Algorithm (PSO), and Whale Optimization Algorithm (WOA), a total of 10 hybrid algorithms were developed for predicting the DCS of rocks under freeze-thaw cycles.(3) Based on the prediction results and various evaluation indicators, a comprehensive model evaluation was conducted to identify the best prediction model.The results indicate that the proposed prediction model has good generalization performance and practical value, which can improve the prediction accuracy of DCS of rocks in cold regions.The research results can provide new ideas for predicting the mechanical strength of rocks in high cold regions in the future.

Methodology
In this study, PCA was employed for the dimensionality reduction of high-dimensional data [40,41] , and the fundamental models used in this research were the BP and RF models.To enhance the prediction accuracy of the models, both the model samples and hyperparameters were optimized.The dataset underwent a 10-fold cross-validation method.Regarding model hyperparameters, this paper utilized five optimization algorithms to automatically search for the optimal solution through the learning of data samples, thereby improving the predictive performance of the model.

Regression model 2.1.1. BP
The BP neural network involves the forward propagation of data and the backward propagation of error signals.The forward propagation direction is from the input layer to the hidden layer and then to the output layer.The state of each layer of neurons only affects the neurons in the subsequent layer [29] .If the desired output is not achieved, the error signal processing is converted into reverse propagation.The output vector of the hidden layer is represented by Equation (1), and the output vector of the output layer is represented by Equation (2): In the formula: i x represents the input value of the i-th node in the input layer; j b denotes the output value of the j-th node in the hidden layer; k y signifies the output value of the k-th node in the output layer; ij w represents the connection weight between the nodes of the input layer and the hidden layer; jk w is the connection weight between the nodes of the hidden layer and the output layer; j  represents the threshold of the nodes in the hidden layer; k  is the threshold of the node in the output layer.
The error function is represented by Equation (3): In the formula: t represents the expected output of the network 2.1.2.RF RF is a method that integrates multiple decision trees, combining the results of each individual tree to obtain the final outcome.The training samples for each base learner in RF are obtained through bootstrap sampling.In essence, a subset is randomly chosen from all features, and the remaining samples that are not included in this subset are referred to as out-of-bag samples (OOB).
The randomness in RF is evident in two aspects: the training samples for each tree are randomly Ultimately, the forest is formed, and the results are averaged according to a specific formula, leading to a final fused model with higher accuracy [35] .
In the formula: f is the average value; B represents a tree, and the output of each tree can be represented as  

SSA The update equation for producer location in SSA is:
In the formula: , t i j X represents the value of the j-th dimension of the i-th sparrow during the t-th iteration; T is the current number of iterations; max iter is the maximum number of iterations; a is a random number, where all elements are 1; Q is a random number that follows a normal distribution.
The update equation for the position of beggars is: In the formula: where P X is the optimal location for the producer; worst X is the current global worst position; A is a 1 × A d-dimensional matrix in which all elements are randomly set to 1 or -1, and . When 2 n i＞ , the i-th beggar needs to search for food elsewhere.

The update equation for reconnaissance position is:
In the formula: Best X is the current global optimal position,  is a step size control parameter; K is the direction of movement of the sparrow , f is the worst-case fitness value, and g f is the global optimal fitness value.

GWO
A wolf pack can be divided into four parts, including  wolves,  wolves,  wolves, and  wolves.The mathematical model for wolf pack encirclement is as follows: In the formula:   P X t represents the position of the prey represented by t;   X t represents the position of Generation t gray wolf; 1 r 、 2 r is a random vector within the interval (0,1); The random number C determines the distance between the gray wolf and its prey.
The positions of  Wolf,  Wolf, and  Wolf are 1 X , 2 X , and 3 X .During the iteration process, these three types of wolves will guide  Wolf to move: When 1 A  , it has good global search ability, therefore: In the formula: t is the current number of iterations; T is the maximum number of iterations;   0, 2   .when 1 A  , the wolf pack performs global optimization; When 1 A  , the wolf pack performs local optimization.
is the unit vector of values at random angles within the interval (-1, 1);  is a random variable.
After the chemotaxis operation, the bacterial population survives according to its health level, and the health function of bacterial i is: calculate the fitness values of each example during the iteration process, and update pBest and gBest .The d-th dimensional velocity and position of particle i are shown in the following equation: In the formula:  is the inertia weight; 1 c and 2 c are acceleration coefficients;  are coefficient vectors, and a is a linearly decreasing vector from 2 to 0; r is a random vector with a range of intervals   0,1 , and the position can be updated based on the current optimal position.When whales prey, they first calculate the distance between the individual whale and the target prey, and then prey by simulating the spiral motion of the whale: In the equation, b1 is the coefficient that controls the shape of the spiral path, and l is the random number in   0,1 .
When whales search for prey, they randomly select prey based on the mutual position of each whale.When ＜ , the search behavior can be defined as:

Raw data processing
This study employed a dataset of rock samples obtained from Tongchuan City, Shaanxi Province, China [23] .The sampling area belongs to a seasonally cold area (with temperatures below 0 ℃ for at least 3 months each year).Prior to undergoing freeze-thaw weathering, all rock samples were treated as standard samples following the methodology recommended by ISRM.The temperature range for freeze-thaw testing was set between -20 and 20 ℃, with the number of freeze-thaw cycles varying at 0, 20, 40, 60, 80, and 100 cycles.DCS of rocks was determined using Hopkinson separated compression rods, with confining pressure gradients of 0, 2, 5, and 10 MPa, and an impact pressure range of 0.2 to 0.6 MPa.This paper conducted intelligent prediction research based on ten influencing factors affecting the DCS of rocks under freeze-thaw cycles [28,[47][48][49][50][51] , including FTC, CP, IP, DD, NWC, WA, ADW, P, PWV, and SR.After thorough investigation, this study amassed a total of 216 sets of data for the training and testing sets.The ratio between the training and testing sets was established at 8:2, with the initial 173 groups designated as training samples and the final 43 groups allocated for testing.The rock sample dataset is visually presented in Figure 10.   Figure 13 shows the characteristic values, variance contribution rates, and cumulative contribution rates of each component sample data.As depicted, the variance contribution rates of the first four principal components are 25.12%, 19.88%, 11.10%, and 10.42%, all exceeding 10%, with a cumulative contribution rate of 66.53%.In contrast, the variance contribution rates of the last six principal components are 9.92%, 7.89%, 7.53%, 5.13%, 2.83%, and 0.18%, respectively, resulting in a cumulative contribution rate of only 33.47%.Consequently, the first four principal components were chosen for model construction, and Figure 14 displays the principal component coefficient matrix.PCA dimensionality reduction was applied to the influencing factors of DCS in 216 groups of rocks under freeze-thaw cycles, and the processed data is illustrated in Figure 15.The dimensionality reduction data were employed for subsequent model training and testing.

Evaluating indicator
To comprehensively assess the prediction model's performance for the DCS of rocks under freezethaw cycles, this study employs RMSE, MAE, MAPE, and R 2 as evaluation indicators for the generalization ability of various models.The calculation equations for the aforementioned indicators are as follows: In the formula: N is the number of data, i y is the measured value, i f is the predicted value, and R is to 1, the better the fitting effect is.The closer it is to 0, the worse the fitting effect is.

Optimization algorithm construction 4.2.1. Model construction process
This section employs optimization algorithms such as SSA, GWO, BFO, PSO, and WOA to optimize hyperparameters for BP and RF, respectively.For the dataset, a 10fold cross-validation is utilized to automatically search for the optimal solution by learning data samples, thereby optimizing the predictive performance of the model.The iterative optimization process employs mean square error as the fitness function for iteration.When the number of iterations meets the termination condition, the iteration can be terminated, and the optimal parameters can be output.Among them, the three important parameters of BP are: hidden_layer_sizes, learning_rate_init, and alpha.The four important parameters of RF are: n_estimators, max_depth, min_samples_split, and min_samples_leaf.The optimal parameters after iteration are substituted into the BP or RF models to obtain the trained optimal prediction model.Then, 43 sets of test set data are input into the above 10 models for testing, and the test results are compared with the DCS of rocks under freeze-thaw cycles in the original data.The two are fitted and error analyzed to obtain the optimal model.

Hyperparameter optimization
Machine learning models involve numerous hyperparameters, and varying combinations of these hyperparameters significantly impact the computational efficiency and generalization performance of the model.Therefore, to achieve a model with optimal computational efficiency and generalization performance, hyperparameter optimization is essential.This paper employs five optimization algorithms to optimize hyperparameters for existing BP and RF models, defining the parameter space to search for specified hyperparameters and determining the optimal hyperparameter combination.The optimal parameters are detailed in Table 1.Apart from the hyperparameters discussed in this paper, the other model hyperparameters are kept at default values.Substituting the optimized hyperparameters into the model yields the trained DCS prediction model.
Table 1.Optimal hyperparameters of BP model     From the graph, it can be seen that the five prediction models exhibit similar patterns during the testing phase.However, upon closer observation, it is apparent that the prediction curve of the SSA-RF model is closer to the actual measurement curve, followed by the remaining models in the order of GWO-RF, WOA-RF, PSO-RF, and BFO-RF.This indicates that the prediction error of the SSA-RF model during the testing phase is superior to other prediction models, making it more suitable for predicting the DCS of rocks under freeze-thaw cycles.

Comprehensive evaluation of predictive models
The above evaluation of the model's prediction performance provides insights into the differences between the models during the training and testing stages.To further assess the performance and stability of the 10 hybrid models, this study calculated evaluation indicators for each hybrid model during the training and testing stages.A scoring system was implemented for each evaluation indicator to comprehensively evaluate the predictive models.The study normalized the individual evaluation indicators of each model, using the results as scores for a comprehensive evaluation.This approach effectively avoids situations where there is a significant difference in scores between similar evaluation parameters [52,53] .Normalization of various evaluation indicators in the prediction model is shown in equation ( 33

Conclusion
This study focuses on the swift and precise prediction of DCS in rocks subjected to freeze-thaw cycles in cold regions.Initially, principal component analysis is used to reduce the dimensions of influencing factor indicators in the original database, ensuring that the minimum number of variables reflects sufficient information.Subsequently, five optimization algorithms were applied to perform hyperparameter optimization on the BP and RF algorithms.Finally, the predictive performance of the hybrid model was comprehensively evaluated.The main conclusions are summarized as follows: (1) The impact of diverse influencing factors and conditions on the DCS of rocks varies.The study encounters an issue of overlapping parameter information among the various indicators affecting DCS.Hence, the principal component analysis method is employed to substitute the original indicators with four new indicators, aiming to retain the information of the original variables to the maximum extent and avoid the impact of information overlap on the accuracy of model establishment and prediction. (

Figure 5 .
Figure 5. Flow chart of GWO algorithm 2.2.3.BFO The mathematical model of bacterial i in chemotaxis is:

Figure 6 .
Figure 6.Flow chart of BFO algorithm 2.2.4.PSO Assign values to the velocity and position of the particle swarm, set the historical optimal pBest of the individual to the current position, set the optimal individual in the swarm to gBest , random numbers on the interval   0,1 .

Figure 7 .
Figure 7. Flow chart of PSO algorithm 2.2.5.WOA The two behaviors of whales in determining the location of prey and surrounding prey can be expressed as:

Figure 8 .
Figure 8. Flow chart of WOA algorithm2.3.10-fold cross-validationWhen developing machine learning models for regression prediction and aiming to eliminate randomness in the selection of training and testing sets, cross-validation becomes imperative.10-fold cross-validation stands out as a widely adopted method for evaluating model stability.In the course of 10-fold cross-validation, the data in the training set gets divided into ten mutually exclusive subsets.One of these subsets is allocated as the test set, while the remaining nine serve as the training set.Following multiple calculations of the model's accuracy, the average accuracy is then assessed, providing a more robust measure of the model's effectiveness.Given the potential interference of training and testing set divisions with model results, relying on the average of 10 cross-validation results proves to be a superior evaluation approach.Taking into account the dataset size and code runtime considerations, this study opted for 10-fold cross-validation, dividing the training set, which comprises 173 datasets, into 10 parts for cross-validation purposes.Using root mean square error (RMSE) as a performance evaluation indicator, Figure9illustrates the fundamental process of 10-fold cross-validation.

Figure 10 .
Figure 10.DCS dataset of rocks under freeze-thaw cycles Figure11illustrates the violin chart for each feature parameter.This chart reveals that the distribution of each feature parameter in the dataset is broad and uniform.It is noteworthy that the violin chart, incorporating kernel density plots and box plots, not only showcases the overall data distribution but also furnishes statistical insights through the inclusion of boxes.Furthermore, Figure11presents the frequency distribution histogram and frequency accumulation curve for each characteristic parameter.Descriptive statistics such as mean, variance, maximum, median, minimum,

Figure 11 .
Figure 11.Distribution characteristics of raw data for each indicator Conducting a correlation analysis on the 10 indicators, including FTC, CP, IP, DD, NWC, WA, ADW, P, PWV, SR, resulted in the correlation coefficients displayed in Figure 12.Through analysis, it is evident that various indicators exhibit a certain degree of correlation, with notable instances such as the high correlation coefficient of 0.98 between SR and IP, and -0.64 between WA and DD.This implies significant information overlap among the samples.Directly utilizing these 10 indicators for training and testing would inevitably impact the model establishment and prediction accuracy.Therefore, employing principal component analysis (PCA) is deemed necessary to reduce the dimensionality of the original data.

Figure 12 .
Figure 12.Pearson correlation coefficient of each indicator Principal component analysis was conducted on the original data of DCS of 216 groups of rocks.Figure13shows the characteristic values, variance contribution rates, and cumulative contribution rates of each component sample data.As depicted, the variance contribution rates of the first four principal components are 25.12%, 19.88%, 11.10%, and 10.42%, all exceeding 10%, with a cumulative contribution rate of 66.53%.In contrast, the variance contribution rates of the last six principal components are 9.92%, 7.89%, 7.53%, 5.13%, 2.83%, and 0.18%, respectively, resulting in a cumulative contribution rate of only 33.47%.Consequently, the first four principal components were

Figure 13 .Figure 15 .
Figure 13.Principal component analysis results chart Figure 14.Principal component coefficient matrix diagram

iy 2 R
is the average of the measured values.RMSE can well reflect the deviation between predicted and measured values.  0, RMSE   , MAE represents the average absolute error between predicted and measured values,   0, MAE   , MAPE represents the average percentage error of predicted values deviating from the true value, and   0, MAPE   .The closer RMSE , MAE , and MAPE approach 0, the better the regression effect.reflects the fitness of the regression model and represents the degree to which the regression model explains the changes in the dependent variable.closer the value of 2

4. 2 . 3 .Figure 16 .Figure 17 .
Figure 17.Fitting analysis between experimental values of DCS of rocks under freeze-thaw cycles and predicted values of RF optimization model

Figure 18 .
Figure 18.Taylor plots for each prediction model 5. Discussion on the superiority of the model5.1.Evaluation of model prediction effectTo assess the predictive performance of the 10 models, this study computed the errors during both the training and testing stages and illustrated the error histograms for each prediction model in Figures19 and 20.In these figures, the horizontal axis represents the error between the predicted and actual values, while the vertical axis depicts the percentage of error.These graphs provide a clear visualization of the cumulative error values within different error intervals for each model, facilitating an evaluation of their predictive performance.

Figure 19 .
Figure 19.Error histogram of BP optimization algorithm In accordance with the principles of error histogram representation, a more accurate prediction model is indicated by lower error bars and a concentration of high error bars near the origin.

Figure 20 .
Figure 20.Error Histogram of RF Optimization Algorithm To further assess the predictive performance of each model, this study analyzed the error curves of the models.Due to space limitations, this section only displays error curves of five RF optimization algorithms during the testing phase, as shown in Figure 21.In the figure, the black lines represent the measured values of DCS, while the colored lines represent the predicted values.From the graph, it can be seen that the five prediction models exhibit similar patterns during the testing phase.However, upon closer observation, it is apparent that the prediction curve of the SSA-RF model is closer to the actual measurement curve, followed by the remaining models in the order of GWO-RF, WOA-RF, PSO-RF, and BFO-RF.This indicates that the prediction error of the SSA-RF model during the testing phase is superior to other prediction models, making it more suitable for predicting the DCS of rocks under freeze-thaw cycles.

Figure 21 .
Figure 21.RF optimization algorithm prediction curve ) ： In the formula: i y is the normalized value; i x is the evaluation indicator of a certain model; 0 max is the maximum value of the evaluation indicator in all models; 0 min is the minimum value of the evaluation indicator in all models.
signifies a better fitting effect, whereas smaller values of MSE, RMSE, MAE, and MAPE indicate a superior predictive effect for the model.Therefore, in the process of assigning scores to these five evaluation indicators, the R 2 value score is equal to its normalized value, and the scores of the other four evaluation indicators are equal to the difference between 1 and their normalized values.Upon observing Table3, it can be inferred that the scoring parameters for each model in the training set and testing set are relatively close, indicating the stability of the algorithm.Additionally, plot the cumulative scores of evaluation indicators for each model's training and testing sets, as illustrated in Figure22.According to the evaluation results, SSA-RF emerges as the superior prediction model, earning a perfect score of 10 points.In contrast, the other nine hybrid algorithms have comparatively lower scores, with BFO-BP, PSO-BP, and WOA-BP scoring only between 0.670 and 1.485.On the other hand, GWO-BP, SSA-BP, BFO-RF, PSO-RF, WOA-RF, and GWO-RF achieved scores ranging from 4.389 to 8.087.

Figure 22 .
Figure 22.Accumulative graph of evaluation scores for each model

2 )
The ranking of prediction accuracy among the 10 hybrid models, from high to low, is as follows: SSA-RF model, GWO-RF model, WOA-RF model, PSO-RF model, BFO-RF model, SSA-BP model, GWO-BP model, WOA-BP model, PSO-BP model, and BFO-BP model.The SSA-RF hybrid model exhibits the best overall performance among the 10 hybrid models.When predicting the DCS of rocks in cold regions, the SSA-RF model should be considered as the first choice.

Table 2 .
Optimal hyperparameters of RF model

Table 3 .
Evaluation indicators of each prediction modelNormalize the evaluation indicators of the training and testing sets for each model using equation(33), and the results are presented in Table3.According to the calculation results, the score range of each evaluation indicator falls within the range of [0, 1].As discussed earlier, a larger R 2 value