Slope stability prediction method based on Gaussian process regression

Abstract

The invention provides a side slope stability prediction method based on Gaussian process regression. The side slope stability prediction method comprises the following steps of S1, performing normalization processing on collected side slope observation data; s2, selecting slope observation data characteristics by adopting a particle swarm optimization algorithm; s3, constructing a covariance matrix in Gaussian process regression; s4, calculating posterior probability distribution of stability corresponding to newly input slope observation data; and S5, carrying out normalization processing on a prediction result by adopting a sigmoid function. According to the method, the Gaussian process regression method is adopted to effectively solve the problems of insufficient precision and improper processing of a nonlinear relation in slope stability prediction processing of a traditional regression analysis method. Compared with other machine learning prediction models, the method provided by the invention not only can provide a stability prediction result, but also can quantify the uncertainty of prediction and improve the credibility of the prediction result.

Description

Technical Field

[0001] The present invention belongs to the field of slope stability prediction, and in particular relates to a slope stability prediction method based on Gaussian process regression. Background Art

[0002] Slope stability analysis is a key technology in geological engineering, particularly in mountainous and hilly areas and large-scale infrastructure projects. Natural disasters such as landslides and collapses caused by unstable slopes often result in significant property losses and casualties. Their stability directly impacts project safety and the safety of the surrounding environment. Therefore, accurately assessing slope stability and promptly predicting potential geological hazards are crucial for disaster prevention and mitigation.

[0003] Traditional slope stability analysis methods primarily include empirical and physical methods. Empirical methods assess slope stability qualitatively or quantitatively, drawing on historical disaster cases and empirical formulas. Commonly used methods, such as the limit equilibrium method and sliding analysis, can provide effective predictions under simple conditions but often overlook the complex interrelationships between multiple variables, such as soil mechanical properties and the external environment. Consequently, empirical methods have limited applicability and, in particular, cannot provide accurate assessment results under complex geological environments and varying climatic conditions. Physical methods, such as the finite element method and discrete element method, simulate the deformation and mechanical behavior of slopes by establishing soil mechanical models, enabling more accurate stability analysis. However, physical methods are computationally complex, requiring significant computational resources and time, and require extremely high accuracy of input data. Especially for large-scale slope areas, the high computational cost and strong reliance on accurate data limit their applicability in practical applications.

[0004] In recent years, machine learning technology has provided new solutions for slope stability analysis. Machine learning can learn the complex relationships between slope stability and multiple influencing factors from large amounts of historical data, enabling efficient and accurate predictions. Machine learning methods (regression models, such as linear regression, support vector machine regression, decision tree regression, and random forest regression) are capable of processing large amounts of data and exploring nonlinear relationships between multidimensional variables. These methods, to a certain extent, overcome the limitations of traditional empirical and physical methods in processing complex and nonlinear data, particularly by automatically adjusting model parameters to improve prediction accuracy. However, classic regression models still face challenges when dealing with high-dimensional data and complex geological environments. For example, traditional linear regression methods often assume a linear relationship between input variables and outputs, failing to effectively capture the complex nonlinear characteristics between variables. Neural network and support vector machine regression models have high data requirements, requiring a large amount of slope observation data to train appropriate prediction models. Prediction models based on decision trees or random forests require manual intervention in both tree depth and branching criteria. Gaussian Process Regression (GPR) is a regression method based on a probability model. It assumes that the mapping relationship between slope stability prediction results and observed characteristics obeys a Gaussian distribution. Based on the slope observation data, it designs a kernel function and calculates the Gaussian distribution covariance to overcome the shortcomings of traditional regression models in dealing with nonlinear relationships. In addition, when establishing a data distribution model through the Gaussian process, it can provide a confidence interval for each predicted value and quantify the uncertainty of the prediction.

[0005] In summary, although traditional slope stability analysis methods can provide effective assessments under certain conditions, their applicability and accuracy are significantly limited in complex environments. Machine learning, especially the Gaussian process regression method, provides a new approach for slope stability prediction. The Gaussian process regression model can automatically learn the complex nonlinear relationships between multidimensional features from data through the construction of kernel functions, thereby improving prediction accuracy and efficiency. At the same time, Gaussian process regression can provide confidence in the prediction results, quantify uncertainty, and further enhance the reliability of the model. Compared with traditional methods, the slope stability prediction method based on Gaussian process regression has higher prediction accuracy, stronger adaptability, and lower computational cost, providing a more effective tool for large-scale slope stability assessment and disaster warning. Summary of the Invention

[0006] To address the challenges of the prior art, the present invention proposes a slope stability prediction method based on Gaussian process regression. This method addresses the difficulties of existing slope stability prediction models in accurately modeling the inherent relationships between slope characteristics and their poor generalization capabilities. This method uses a Gaussian process regression model to establish a mapping relationship between slope stability and observed slope characteristics (such as slope height, slope angle, soil particle density, internal friction angle, water content, and porosity). Specifically, the mapping relationship between slope stability and observed slope characteristics follows a Gaussian distribution. By mathematically modeling this mapping relationship, accurate slope stability prediction is achieved. Compared to traditional methods, the present invention not only truly and comprehensively expresses the inherent relationships between observed characteristics through kernel function design, but also quantifies the uncertainty of the prediction results. The core technical solution of the present invention is a slope stability prediction method based on Gaussian process regression, which employs a Gaussian process regression model to fit the nonlinear relationship between input features and stability. By solving the posterior distribution of the regression model, the present invention can effectively predict slope stability. The predicted results are then converted into numerical values between 0 and 1 using a sigmoid function to facilitate the determination of slope stability.

[0007] The specific technical solutions adopted by the present invention to solve the problems existing in the prior art are as follows:

[0008] A slope stability prediction method based on Gaussian process regression includes the following steps:

[0009] Step S1: Preprocessing of slope observation data: For the collected slope observation data (such as slope height, slope angle, soil particle density, internal friction angle, water content, porosity, etc.), the data are normalized to ensure the uniformity of the scale of each factor;

[0010] Step S2: Slope observation data feature selection: Slope observation data usually contains many features. Directly inputting these features into the Gaussian regression model will greatly increase the computational complexity of stability prediction. At the same time, some irrelevant and redundant features will also affect the prediction accuracy. Therefore, the present invention will use the particle swarm optimization algorithm to select features;

[0011] Step S3: Constructing the covariance matrix in Gaussian process regression: Based on the features in the slope observation data after feature selection, a kernel function with linear and nonlinear components is used to model the observation data to obtain the covariance matrix obtained from the slope observation data;

[0012] Step S4: Calculate the posterior probability distribution of stability corresponding to the newly input slope observation data: Based on the covariance matrix calculated in step S3, calculate the correlation between the newly input slope characteristics and the original observation data to obtain an updated covariance matrix, and calculate the mean and variance of the predicted stability based on the covariance matrix;

[0013] Step S5: Normalization: The stability mean obtained in step S4 is difficult to guarantee to be between [0, 1]. In order to convert the prediction result into a stability judgment, the prediction result is normalized using the sigmoid function, and the result is mapped between 0 and 1, so that the prediction result is probabilistically interpretable. If the normalized result is close to 0, the slope is judged to be unstable; if the normalized result is close to 1, the slope is judged to be stable.

[0014] The step S2 specifically includes:

[0015] Step S2.1: Initial positions of particles in particle swarm optimization algorithm And the parameters {ω, c1, c2} are initialized, where α i represents the eigenvalue set of the i-th particle, represents the Nth feature of the i-th particle in the particle swarm. When , it means selecting the Nth feature of the i-th particle in the particle swarm. It means that the feature is not selected, ω represents the inertia weight, c1 and c2 are acceleration constants;

[0016] Step S2.2: Estimate the fitness of each particle according to the following formula (1):

[0017]

[0018] In the above formula, represents the fitness function, ErrorRate represents the accuracy on the validation set, υ is the weight that controls the final prediction performance and the size of the feature subset, and the index number of the best position experienced by all particles in the group is represented by the symbol g, that is, P g , the velocity of particle i is expressed as V i =(v i1 ,v i2 ,…,v iN ) indicates that the strategy of the particle swarm moving in the solution space is as follows:

[0019]

[0020] In formula (2), rand() represents a random number between [0,1], t represents the current number of iterations, represents the best feature subset found by the i-th particle, P g Represents the best feature subset found by all particles. It should be noted that the bold α i Represents a vector, and α represents an element in the vector. Repeat the above steps iteratively, and within a certain number of iterations, let the fitness function The largest slope feature subset α is the feature selection result.

[0021] The step S3 specifically includes:

[0022] After feature selection, some redundant and irrelevant features are screened out; the kernel function is used to model the relationship between slope observation data. This kernel function can not only represent linear relationships, but also describe nonlinear relationships between data. Its expression is:

[0023]

[0024] In formula (3), k(x n ,x m ) represents the kernel function, where x n ,x m is the slope characteristic vector represented by the nth observation data and the mth observation data, Represents x n The transpose of this vector, θ0, θ1, θ2, θ3 are the hyperparameters of the Gaussian regression model. θ0 represents the amplitude of the Gaussian kernel, which controls the overall scale of the function change. θ1 controls the nonlinear relationship in the observed data and determines the decay rate of the correlation between features (the larger θ1 is, the faster the similarity decays with distance). θ2 is a constant term, which represents the global offset in the data. θ3 is the coefficient of the linear relationship of the observed data, which captures the linear correlation of the input features. These four parameters are obtained through grid search in the 5-fold cross-validation step. The calculation results in formula (3) are the corresponding elements in the covariance matrix.

[0025] The step S4 specifically includes:

[0026] Step S4.1: Construct the joint probability distribution, the observation data {x1,...,x N} and the corresponding stability result y, where x i Corresponding to a certain observation data, which contains several observation characteristics of the slope, such as slope height, slope angle, pore water pressure, etc., the observation result y obeys the following distribution:

[0027]

[0028] In formula (4), p(y) represents the probability density function that the observation result y obeys, Represents a Gaussian distribution with a mean of 0 and a covariance of K, where K represents the Gram matrix and the element K nm Represents the input point x n and x m covariance of

[0029] Step S4.2: Model the noisy observation data, the actual observation value t n =y n +ε n, t n Indicates the actual stability result of the nth observation data, y n is the true stability result of the nth slope observation data, ε n Represents the noise in the nth data, noise The marginal distribution of the actual stability result t taking into account the influence of noise is:

[0030]

[0031] Among them, β represents the noise precision, which controls the credibility of the observation value, C is the covariance matrix, I N is the N×N identity matrix.

[0032] The updated covariance matrix is:

[0033]

[0034] Among them, C N+1 represents the N+1×N+1 covariance matrix, k is an N-dimensional vector, k n =k(x n ,x N+1 ) represents the covariance between the newly observed data and the original data points, c is c=k(x N+1 ,x N+1 )+β -1 represents the autocovariance of the test point plus the noise variance;

[0035] Step S4.3: Using Gaussian inequality, we can get the mean value m(x N+1 ) and covariance σ 2 (x N+1 ):

[0036]

[0037] Among them, k is the covariance matrix of the new input observation data and the existing observation data, c = k (x N+1 ,x N+1 )+β -1 represents the autocovariance of the test point plus the noise variance, C N -1 t is the weighted contribution of existing observation data to the prediction, reflecting the influence between similar samples.

[0038] The step S5 specifically includes:

[0039] Step S5.1: For any new input slope observation data x N+1 , the predicted mean m(x N+1), then, the sigmoid function is mapped to the output of the regression model to limit the output value range to between 0 and 1. The specific form is:

[0040]

[0041] In formula (9), f(m) is the predicted value of the model, and m represents m(x N+1 ), which is the mean of the final Gaussian process regression result;

[0042] Step S5.2: Since f(m)∈[0,1], the model regression results have a probabilistic interpretation. If the result is close to 0, it means that the predicted slope is unstable; if the result is close to 1, it means that the predicted slope is stable.

[0043] The present invention has the following advantages:

[0044] The present invention employs a Gaussian process regression method to effectively address the issues of insufficient precision and improper handling of nonlinear relationships encountered by traditional regression analysis methods in slope stability prediction. Compared with other machine learning prediction models, the present invention not only provides stability prediction results but also quantifies the uncertainty of the prediction, thereby enhancing the credibility of the prediction results. The introduction of the sigmoid function further enhances the interpretability of the prediction results. The method of the present invention can be applied to slope stability assessment, providing reliable technical support for the early warning and prevention of geological disasters. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 1 is a flow chart of a slope stability prediction method based on Gaussian process regression in an embodiment of the present invention;

[0046] Figure 2 1 is a schematic diagram of slope feature selection based on particle swarm algorithm in an embodiment of the present invention;

[0047] Figure 3 This is a confusion matrix result diagram of the stability prediction method in the present invention and the benchmark method. DETAILED DESCRIPTION

[0048] The technical solution of the present invention will be further described in detail below through embodiments and in conjunction with the accompanying drawings. Figure 1 As shown, the embodiment of the present invention provides a slope stability prediction method based on Gaussian process regression, such as Figure 2 As shown, the embodiment of the present invention uses the particle swarm algorithm to select the optimal features of the slope. Figure 2The table in the upper left corner shows the corresponding slope observation data features. The table includes 221 slope observation features, including 14 slope features such as water content, pore water pressure, slope height, slope angle, internal friction angle, soil particle density, cohesion, liquid limit, plastic limit, and saturation. These features cover common features of slope stability monitoring. The specific steps of the slope stability prediction method are as follows:

[0049] Step 1: Experimental data collection and preprocessing.

[0050] Observation data related to slope stability are collected. The observation data consists of two parts: one is the slope characteristics (slope height, slope angle, soil particle density, internal friction angle, pore water pressure ratio, and cohesion), and the other is the stability judgment y corresponding to the slope characteristics. y = 1 indicates stability, and 0 indicates instability. The data is then preprocessed to quantify and normalize all features to between [0, 1].

[0051] Step 2: Slope feature selection based on particle swarm algorithm.

[0052] Step S2.1: Determine the size of the particle swarm, and initialize the initial positions and parameters of the particles. Figure 2 As shown, for the characteristics of N-dimensional slope observation data (N=14 at this time), the initial position of the particle needs to be Initialization, where Indicates that all slope features are selected during initialization. Indicates that the feature of the Nth dimension is selected, and the initialization parameters {ω, c1, c2} = {1.2, 2, 2};

[0053] Step S2.2: Calculate the fitness of each particle according to formula (1),

[0054]

[0055] Then update the best position of the individual particle, and then update the best position found by all particles, and use formula (2) to update the speed and position:

[0056]

[0057] Iterate and repeat the above fitness calculation and speed update. After 5000 iterations, the feature subset α corresponding to the maximum fitness i This is the result after feature selection. At this time, α i =[1,1,0,0,...,1], in this vector of length 14, 6 elements are 1, indicating that the characteristics of these positions are used for subsequent stability prediction. These 6 characteristics are slope height, slope angle, internal friction angle, gravity, pore water pressure ratio and cohesion.

[0058] Step 3: Determine the kernel function in Gaussian process regression.

[0059] After feature selection, some redundant and irrelevant features were removed. The 221 slope observations were divided into training and test data in an 8:2 ratio, with 177 data points used for Gaussian process training and 44 slope data points used to test the performance of the proposed method. Throughout the training and testing process, a kernel function was used to model the relationships between the slope observations. This kernel function can represent not only linear relationships but also nonlinear relationships between data. Its expression is:

[0060]

[0061] Among them, θ0, θ1, θ2, θ3 are the hyperparameters of the Gaussian regression model. θ0 represents the amplitude of the Gaussian kernel, which controls the overall scale of the function change. θ1 controls the nonlinear relationship in the observed data and determines the decay rate of the correlation between features (the larger θ1 is, the faster the similarity decays with distance). θ2 is a constant term, which represents the global offset in the data. θ3 is the coefficient of the linear relationship of the observed data, which captures the linear correlation of the input features. These four parameters are obtained through grid search during the 5-fold cross-validation process. The calculation result in formula (3) is the corresponding element in the covariance matrix, where x n and x m are slope characteristic vectors of length 6, corresponding to the six observation indicators mentioned above: slope height, slope angle, internal friction, pore water pressure ratio, gravity, and cohesion. n -x m || 2 represents the square of the Euclidean distance between these two vectors, represents the inner product of these two vectors.

[0062] Step 4: Calculate the mean and variance of the slope stability prediction distribution.

[0063] Step S4.1: Construct joint probability distribution. The observation data {x1,...,x N} and the corresponding stability result y, where x i For a certain observation data, which contains several observation characteristics of the slope (slope height, slope angle, pore water pressure ratio, internal friction angle, gravity, cohesion), the observation result y obeys the following distribution:

[0064] p(y)=N(y|0,K), K nm =k(x n ,x m ) (4)

[0065] Among them, K represents the Gram matrix, element K nmRepresents the input point x n and x m Since there are 177 slope observations in the training data, K is a 177×177 matrix.

[0066] Step S4.2: Model the noisy observation data. Actual observation value t n =y n +ε n , noise ε n ~N(0,β -1 ), the marginal distribution of the actual stability result t taking into account the influence of noise is:

[0067] p(t)=N(t|0,C), C=K+β -1 I N (5)

[0068] Where β represents the noise accuracy, I N is the N×N identity matrix.

[0069] The updated covariance matrix is:

[0070]

[0071] Among them, C N+1 represents a 178×178 covariance matrix, which is the covariance matrix formed after adding a slope feature of the test data in formula (4); k is a 177-dimensional vector, k n =k(x n ,x N+1 ) represents the covariance between the newly observed data and the original data points, c=k(x N+1 ,x N+1 )+β -1 is a scalar representing the autocovariance of the test point plus the noise variance.

[0072] Step S4.3: Using Gaussian inequality, we can get the mean value m(x N+1 ) and covariance σ 2 (x N+1 ):

[0073]

[0074]

[0075] Among them, k is the covariance column vector of the new input observation data and the existing observation data, c = k (x N+1 ,x N+1 )+β -1 represents the autocovariance of the test point plus the noise variance, CN -1 t is the weighted contribution of existing observation data to the prediction, reflecting the influence between similar samples. Finally, m(x N+1 ) This scalar represents the new input observation data x N+1 The mean of all possible outcomes after Gaussian process regression.

[0076] Step 5: Prediction and normalization processing.

[0077] Step S5.1: For any new input slope observation data x N+1 , the predicted mean m(x N+1 ), then input the regression model results into sigmoid to limit the output value range to between 0 and 1. The specific form is:

[0078]

[0079] Among them, f(m) is the prediction result of the final model regarding stability;

[0080] Step S5.2: Since f(m)∈[0,1], the model regression results have a probabilistic interpretation. If the result is close to 0, it means that the predicted slope is unstable; if the result is close to 1, it means that the predicted slope is stable. The comparison between the prediction results of the method proposed in this patent and other classic machine learning slope stability prediction results is shown in Figure 3 middle, Figure 3 Shown is the confusion matrix of each method on 44 test datasets.

[0081] The scope of protection of the present invention is not limited to the above-described embodiments. Obviously, those skilled in the art may make various modifications and variations to the present invention without departing from the scope and spirit of the present invention. If such modifications and variations fall within the scope of the claims of the present invention and their equivalents, the present invention is intended to include such modifications and variations.

Claims

1. A slope stability prediction method based on Gaussian process regression, characterized in that: The following steps are involved: Step S1: Preprocessing of slope observation data: normalizing the collected slope observation data; Step S2: Slope observation data feature selection: particle swarm optimization algorithm is used to select features; Step S3: Constructing the covariance matrix in Gaussian process regression: Based on the features in the slope observation data after feature selection, a kernel function with linear and nonlinear components is used to model the observation data to obtain the covariance matrix obtained from the slope observation data; Step S4: Calculate the posterior probability distribution of stability corresponding to the newly input slope observation data: Based on the covariance matrix calculated in step S3, calculate the correlation between the newly input slope characteristics and the original observation data to obtain an updated covariance matrix, and calculate the mean and variance of the predicted stability based on the covariance matrix; Step S5: Normalization: The prediction results are normalized using the sigmoid function and mapped to a value between 0 and 1. If the normalized result is close to 0, the slope is determined to be unstable. If the normalized result is close to 1, the slope is judged to be stable.

2. The slope stability prediction method based on Gaussian process regression according to claim 1, characterized in that: The data collected in step S1 include slope height, slope angle, soil particle density, internal friction angle, water content, and porosity.

3. The slope stability prediction method based on Gaussian process regression according to claim 1, characterized in that: The step S2 specifically includes: Step S2.1: Initial positions of particles in particle swarm optimization algorithm And the parameters {ω, c1, c2} are initialized, where α i represents the eigenvalue set of the i-th particle, represents the Nth feature of the i-th particle in the particle swarm. When , it means selecting the Nth feature of the i-th particle in the particle swarm. It means that the feature is not selected, ω represents the inertia weight, c1 and c2 are acceleration constants; Step S2.2: Estimate the fitness of each particle according to the following formula (1): In formula (1), represents the fitness function, ErrorRate represents the accuracy on the validation set, υ is the weight that controls the final prediction performance and the size of the feature subset, and the index number of the best position experienced by all particles in the group is represented by the symbol g, that is, P g , the velocity of particle i is expressed as V i =(v i1 ,v i2 ,…,v iN ) indicates that the strategy of the particle swarm moving in the solution space is as follows: In formula (2), rand() represents a random number between [0,1], t represents the current number of iterations, represents the best feature subset found by the i-th particle, P g Represents the best feature subset found by all particles. It should be noted that the bold α i Represents a vector, α represents an element in the vector, iterate and repeat the above steps, within a certain number of iterations, let the fitness function The largest slope feature subset α is the feature selection result.

4. The slope stability prediction method based on Gaussian process regression according to claim 1, characterized in that: The step S3 specifically includes: After feature selection, some redundant and irrelevant features are screened out; the kernel function is used to model the relationship between slope observation data. This kernel function can not only represent linear relationships, but also describe nonlinear relationships between data. Its expression is: In formula (3), k(x n ,x m ) represents the kernel function, where x n ,x m is the slope characteristic vector represented by the nth observation data and the mth observation data, Represents x n The transpose of this vector, θ0, θ1, θ2, θ3 are the hyperparameters of the Gaussian regression model. θ0 represents the amplitude of the Gaussian kernel, which controls the overall scale of the function change. θ1 controls the nonlinear relationship in the observed data and determines the decay rate of the correlation between features. The larger θ1 is, the faster the similarity decays with distance. θ2 is a constant term, which represents the global offset in the data. θ3 is the coefficient of the linear relationship of the observed data, which captures the linear correlation of the input features. These four parameters are obtained through grid search in the 5-fold cross-validation step. The result of calculation of formula (3) is the corresponding element in the covariance matrix.

5. The slope stability prediction method based on Gaussian process regression according to claim 1, characterized in that: The step S4 specifically includes: Step S4.1: Construct the joint probability distribution, the observed data {x1,...,x N } and the corresponding stability result y, where x i Corresponding to a certain observation data, which contains several observation characteristics of the slope, such as slope height, slope angle, and pore water pressure, the observation result y obeys the following distribution: In formula (4), p(y) represents the probability density function that the observation result y obeys, Represents a Gaussian distribution with a mean of 0 and a covariance of K, where K represents the Gram matrix and the element K nm Represents the input point x n and x m covariance of Step S4.2: Model the noisy observation data, the actual observation value t n =y n +ε n , t n Indicates the actual stability result of the nth observation data, y n is the true stability result of the nth slope observation data, ε n Represents the noise in the nth data, noise The marginal distribution of the actual stability result t taking into account the influence of noise is: Among them, β represents the noise precision, which controls the credibility of the observation value, C is the covariance matrix, I N is the N×N identity matrix; The updated covariance matrix is: Among them, C N+1 represents the N+1×N+1 covariance matrix, k is an N-dimensional vector, k n =k(x n ,x N+1 ) represents the covariance between the newly observed data and the original data points, c is c=k(x N+1 ,x N+1 )+β -1 represents the autocovariance of the test point plus the noise variance; Step S4.3: Using Gaussian inequality, obtain the mean value m(x N+1 ) and covariance σ 2 (x N+1 ): Among them, k is the covariance matrix of the new input observation data and the existing observation data, c = k (x N+1 ,x N+1 )+β -1 represents the autocovariance of the test point plus the noise variance, C N -1 t is the weighted contribution of existing observation data to the prediction, reflecting the influence between similar samples.

6. The slope stability prediction method based on Gaussian process regression according to claim 1, characterized in that: The step S5 specifically includes: Step S5.1: For any new input slope observation data x N+1 , the predicted mean m(x N+1 ), then, the sigmoid function is mapped to the output of the regression model to limit the output value range to between 0 and 1. The specific form is: In formula (9), f(m) is the predicted value of the model, and m represents m(x N+1 ), which is the mean of the final Gaussian process regression result; Step S5.2: Since f(m)∈[0,1], the model regression results have a probabilistic interpretation. If the result is close to 0, it means that the predicted slope is unstable; if the result is close to 1, it means that the predicted slope is stable.

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