A method for predicting slope stability based on Gaussian process regression

Abstract

The application provides a slope stability prediction method based on Gaussian process regression, comprising the following steps: step S1: normalizing collected slope observation data; step S2: selecting slope observation data features by using a particle swarm optimization algorithm; step S3: constructing a covariance matrix in Gaussian process regression; step S4: calculating a posterior probability distribution of the stability corresponding to new input slope observation data; and step S5: normalizing a prediction result by using a sigmoid function. The application effectively solves the problems of insufficient precision and improper nonlinear relationship processing in traditional regression analysis methods in the processing of slope stability prediction by using the Gaussian process regression method. Compared with other machine learning prediction models, the application can not only provide a stability prediction result, but also quantify the uncertainty of prediction, and improve the credibility of the prediction result.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of slope stability prediction, and particularly relates to a slope stability prediction method based on Gaussian process regression. BACKGROUND

[0002] Slope stability analysis is a key technology in geotechnical engineering, especially in mountainous areas, hilly areas and large infrastructure construction projects. Landslides, collapses and other natural disasters caused by unstable slopes often cause significant property damage and casualties, and the stability of the slope directly affects the safety of the project and the surrounding environment. Therefore, accurately assessing the stability of the slope and timely predicting potential geological disasters are of great significance for disaster prevention and mitigation.

[0003] Traditional slope stability analysis methods mainly include empirical methods and physical methods. Empirical methods assess the stability of the slope through qualitative or quantitative methods based on historical disaster cases and empirical formulas. Common methods such as limit equilibrium method and sliding analysis method can effectively predict under simple conditions, but often ignore the complex relationships between soil mechanical properties, external environment and other multivariate factors. Therefore, the applicability of empirical methods is relatively narrow, especially in the face of complex geological environments and different climate conditions, which cannot provide accurate assessment results. Physical methods simulate the deformation process and mechanical behavior of the slope by establishing soil mechanical models such as finite element method and discrete element method, which can accurately analyze the stability. However, physical methods are complex to calculate, requiring a large amount of computing resources and time, and the model requires high accuracy of input data, especially when facing large-scale slope areas. The high computational cost and strong dependence on accurate data of physical methods limit their applicability in practical applications.

[0004] In recent years, machine learning techniques have provided new solutions for slope stability analysis. Machine learning can learn the complex relationship between slope stability and multiple influencing factors from a large amount of historical data, thereby making efficient and accurate predictions. Machine learning methods (regression models such as linear regression, support vector machine regression, decision tree regression, random forest regression, etc.) can handle large-scale data and mine the non-linear relationships between multi-dimensional variables. These methods overcome the limitations of traditional empirical methods and physical methods in dealing with complex and non-linear data to some extent, especially in automatically adjusting model parameters and improving prediction accuracy. However, classical regression models still have some problems when facing high-dimensional data and complex geological environments. For example, traditional linear regression methods often assume a linear relationship between input variables and output results, which cannot capture the complex non-linear characteristics between variables; neural networks and support vector machine regression models require a large amount of slope observation data to train suitable prediction models; decision tree or random forest-based prediction models require manual intervention in tree depth and branching criteria. Gaussian Process Regression (Gaussian Process Regression) is a regression method based on a probabilistic model that assumes the mapping relationship between slope stability prediction results and observed features follows a Gaussian distribution. Based on slope observation data, kernel functions are designed and Gaussian distribution covariance is calculated to overcome the shortcomings of traditional regression models in dealing with non-linear relationships. In addition, when establishing a data distribution model using Gaussian Process, a confidence interval can be provided for each prediction value to quantify the uncertainty of the prediction.

[0005] In summary, although traditional slope stability analysis methods can provide effective evaluation under certain conditions, their applicability and accuracy have significant limitations in complex environments. Machine learning, especially the Gaussian Process Regression method, provides a new approach to slope stability prediction. The Gaussian Process Regression model can automatically learn the complex non-linear relationships between multi-dimensional features from data through the construction of kernel functions, improving prediction accuracy and efficiency. At the same time, Gaussian Process Regression can provide the confidence of the prediction results, quantifying the uncertainty and further enhancing 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 evaluation and disaster warning. SUMMARY

[0006] In view of the problems in the prior art, the present application provides a slope stability prediction method based on Gaussian process regression, which solves the problems that the existing slope stability prediction model is difficult to accurately establish a mathematical model for the internal relationship of slope characteristics, and the prediction model has poor generalization ability. The method establishes a mapping relationship between the slope stability and the slope observation characteristics (such as slope height, slope angle, soil particle specific gravity, internal friction angle, water content, and pore ratio) by a Gaussian process regression model. Specifically, the mapping relationship between the slope stability and the slope observation characteristics is subject to Gaussian distribution, and the accurate prediction of the slope stability is realized by modeling the mapping relationship. Compared with the traditional method, the present application not only truly and comprehensively expresses the internal relationship between the observation characteristics through the kernel function, but also quantifies the uncertainty of the prediction result. The core technical scheme of the present application is a slope stability prediction method based on Gaussian process regression, which uses a Gaussian process regression model to fit the nonlinear relationship between the input characteristics and the stability. By solving the posterior distribution of the regression model, the present application can effectively predict the slope stability, and convert the prediction result into a value between 0 and 1 by a Sigmoid function, so as to facilitate the judgment of the stability of the slope.

[0007] The present application adopts the following specific technical scheme to solve the problems in the prior art:

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

[0009] Step S1: slope observation data preprocessing: for the collected slope observation data (such as slope height, slope angle, soil particle specific gravity, internal friction angle, water content, and pore ratio), the data is normalized to ensure the scale of each factor is uniform;

[0010] Step S2: slope observation data feature selection: the slope observation data usually contains many features, and directly inputting these features into the Gaussian regression model will greatly increase the calculation complexity of the stability prediction, and some irrelevant and redundant features will also affect the prediction accuracy, therefore, the present application will select the features by using a particle swarm optimization algorithm;

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

[0012] Step S4: calculating the posterior probability distribution of the stability corresponding to the newly input slope observation data: according to the covariance matrix calculated in step S3, the correlation between the newly input slope characteristics and the original observation data is calculated, an updated covariance matrix is obtained, and the mean and variance of the predicted stability are calculated according to the covariance matrix;

[0013] Step S5: normalization processing: the stability mean value obtained in step S4 is difficult to guarantee in the value between [0, 1], in order to convert the prediction result into stability judgment, the sigmoid function is used to normalize the prediction result, and the result is mapped to 0-1, so that the prediction result has interpretability in probability, if the normalized result is close to 0, it is determined that the slope is unstable; if the normalized result is close to 1, it is determined that the slope is stable.

[0014] The step S2 specifically comprises:

[0015] Step S2.1: initial position of particle in particle swarm optimization algorithm And the parameters {ω, c1, c2} are initialized, where α i represents the characteristic value set of the i-th particle, represents the Nth feature of the i-th particle in the particle swarm, when represents the Nth feature of the i-th particle in the particle swarm, when represents that the feature is not selected, ω represents the inertia weight, and 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 for controlling the last 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 speed of particle i is represented by V i =(v i1 ,v i2 ,…,v iN ), and the strategy for moving the particle swarm in the solution space is as follows:

[0019]

[0020] In formula (2), rand() represents a random number between 0 and 1, t represents the current iteration number, represents the best feature subset found by the i-th particle, P g represents the best feature subset found by all particles, and it should be noted that the bold α i represents a vector, and α represents the elements in the vector. Repeat the above steps in iteration, and within a certain number of iterations, the feature selection result is the feature subset α with the maximum fitness function ​

[0021] The step S3 specifically comprises:

[0022] After feature selection, some redundant and irrelevant features are filtered out. A kernel function is used to model the relationship between the slope observation data. The kernel function can not only represent the linear relationship, but also describe the nonlinear relationship between the data, and 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 feature vector represented by the nth observation data and the mth observation data, represents the transpose of the vector x n , θ0, θ1, θ2, θ3 are the hyperparameters of the Gaussian regression model, θ0 represents the amplitude of the Gaussian kernel, controls the overall scale of the function change, θ1 controls the nonlinear relationship among the observation data, determines the decay rate of the correlation between features (the greater θ1, the faster the similarity decays with distance), θ2 is the constant term, indicating the global offset in the data, and θ3 is the coefficient of the linear relationship of the observation data, capturing the linear correlation of the input features. The four parameters are obtained by grid search in the 5-fold cross-validation step. The calculation result in formula (3) is the corresponding element in the covariance matrix.

[0025] The step S4 specifically comprises:

[0026] Step S4.1: Construct a joint probability distribution. The obtained observation data {x1,...,x N} and the corresponding stability results y are obtained, where x i corresponds to an observation data, which contains observation features of several slopes, such as slope height, slope angle, pore water pressure, etc. The observation results y obey the following distribution:

[0027]

[0028] In formula (4), p(y) represents the probability density function to which the observation results y are subjected, represents a Gaussian distribution with mean 0 and covariance K, K represents the Gram matrix, and the element K nm represents the covariance of the input points x n and x m ;

[0029] Step S4.2: Model the observation data containing noise. The actual observation value t n =y n +ε n, t n represents 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 edge distribution of the actual stability result t considering the influence of noise is:

[0030]

[0031] wherein β represents the noise precision, controlling the credibility of the observation value, C is the covariance matrix, I N is the N×N unit matrix.

[0032] The updated covariance matrix is:

[0033]

[0034] wherein 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 of 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 the Gaussian inequality, the mean m(x N+1 ) and the covariance σ 2 (x N+1 ) of the stability prediction can be obtained:

[0036]

[0037] wherein k is the covariance matrix composed of the newly 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 the existing observation data to the prediction, reflecting the influence between similar samples.

[0038] The step S5 specifically comprises:

[0039] Step S5.1: for any newly input slope observation data x N+1 , the prediction mean m(x N+1), and then the regression model output is mapped by a sigmoid function to limit the output value range to between 0 and 1, and the specific form is:

[0040]

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

[0042] Step S5.2: Since f(m) is between 0 and 1, the model regression result has a probabilistic interpretation, and the result close to 0 indicates that the predicted slope is unstable, and the result close to 1 indicates that the predicted slope is stable.

[0043] The present application has the following advantages:

[0044] The present application effectively solves the problems of insufficient precision and improper handling of nonlinear relationships in traditional regression analysis methods in the prediction of slope stability by using the Gaussian process regression method. Compared with other machine learning prediction models, the present application not only provides a stability prediction result, but also quantifies the uncertainty of the prediction, thereby improving the credibility of the prediction result. By introducing the sigmoid function, the interpretability of the prediction result is further improved. The method of the present application can be applied to slope stability evaluation, and provides reliable technical support for the early warning and prevention of geological disasters. BRIEF DESCRIPTION OF DRAWINGS

[0045] Figure 1 is a flowchart of the slope stability prediction method based on Gaussian process regression in the embodiment of the present application;

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

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

[0048] The technical solutions of the present application will be further specifically described below by means of embodiments and in conjunction with the drawings, such as Figure 1 As shown in the drawings, the embodiment of the present application provides a slope stability prediction method based on Gaussian process regression, as shown in the drawings, Figure 2 As shown in the drawings, the embodiment of the present application uses a particle swarm algorithm to select the optimal slope features, Figure 2The table in the upper left corner represents the corresponding slope observation data characteristics. This table includes 221 slope observation characteristics, covering 14 aspects 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 characteristics encompass common features for slope stability monitoring. The specific steps of this slope stability prediction method are as follows:

[0049] Step 1: Data collection and preprocessing.

[0050] Observational data related to slope stability were collected. The observational data consisted of two parts: one part was slope characteristics (slope height, slope angle, soil particle density, internal friction angle, pore water pressure ratio, and cohesion), and the other part was the stability judgment y corresponding to the slope characteristics. y = 1 indicates stability, and 0 indicates instability. The data was then preprocessed to quantify and normalize all characteristics to the range [0,1].

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

[0052] Step S2.1: Determine the particle swarm size and initialize the initial positions and parameters of the particles. For example... Figure 2 As shown, for the characteristics of N-dimensional slope observation data (where N=14), it is necessary to determine the initial position of the particles. Initialization, where This indicates that all slope features are selected during initialization. This indicates that the feature in the Nth dimension has been selected, and the initial parameters are {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 optimal position of the individual particles, and then update the optimal positions found by all particles, using formula (2) to update the velocity and position:

[0056]

[0057] Repeat the fitness calculation and speed update iteratively 5000 times. The feature subset α corresponding to the maximum fitness is then determined. i This is the result after feature selection, where α i =[1,1,0,0,...,1], this vector of length 14 has 6 elements that are 1, indicating that the features at these locations are used for subsequent stability prediction. These 6 features are slope height, slope angle, internal friction angle, unit weight, pore water pressure ratio, and cohesion.

[0058] Step 3: Kernel function determination in Gaussian process regression.

[0059] After feature selection, some redundant and irrelevant features are filtered out. For the 221 slope observation data, they are divided into training data and test data according to the ratio of 8:2, in which 177 data are used for Gaussian process training, and 44 slope data are used for testing the performance of the proposed method. In the whole training and testing process, the kernel function is used to model the relationship between the slope observation data, which can not only represent the linear relationship but also describe the nonlinear relationship between the data, and its expression is:

[0060]

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

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

[0063] Step S4.1: Construct the joint probability distribution. The observation data {x1,...,x N} and the corresponding stability results y have been obtained, where x i corresponds to a certain observation data, which contains the observation features (slope height, slope angle, pore water pressure ratio, internal friction angle, unit weight, and cohesion) of several slopes, and the observation result y follows the following distribution:

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

[0065] where K represents the Gram matrix, and the element K nmdenotes the input point x n and x m The covariance matrix K is a 177x177 matrix since there are 177 slope observations in the training data.

[0066] Step S4.2: Model the noisy observations. The actual observation t n = y n + ε n , where the noise ε n ~ N(0, β -1 ) is Gaussian. The marginal distribution of the actual stability outcome t, taking into account the noise, is:

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

[0068] where β represents the noise precision and I N is the N x N identity matrix.

[0069] The updated covariance matrix is:

[0070]

[0071] where C N+1 represents the 178x178 covariance matrix, k is a 177-dimensional vector, and k n = k(x n , x N+1 ) represents the covariance of the new observation with the original data points, and c = k(x N+1 , x N+1 ) + β -1 is a scalar representing the self-covariance of the test point plus the noise variance.

[0072] Step S4.3: Using the Gaussian inequality, the mean m(x N+1 ) and covariance σ 2 (x N+1 ) of the stability prediction can be obtained:

[0073]

[0074]

[0075] where k is the covariance column vector of the new input observation with the existing observations, c = k(x N+1 , x N+1 ) + β -1 represents the self-covariance of the test point plus the noise variance, and CN -1 t is the weighted contribution of existing observation data to the prediction, which reflects the influence between similar samples, and finally m(x N+1 ) is a scalar representing the new input observation data x N+1 The mean of all possible results 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 ) is obtained by the Gaussian process regression model, then the regression model result is input into the sigmoid, so that the output value range is limited between 0 and 1, and the specific form is:

[0078]

[0079] Wherein, f(m) is the final model prediction result about stability;

[0080] Step S5.2: since f(m) is between 0 and 1, the model regression result has a probability explanation, and the result close to 0 indicates that the predicted slope is unstable; if the result is close to 1, it indicates that the predicted slope is stable. The comparison between the prediction results of the method proposed in the patent and other classical machine learning slope stability prediction results is shown in Figure 3 , and Figure 3 shows the confusion matrix of each method on the 44 test data sets.

[0081] The protection scope of the present application is not limited to the above-mentioned embodiments, and it is obvious that those skilled in the art can make various modifications and changes to the present application without departing from the scope and spirit of the present application. If these modifications and changes belong to the scope of the claims of the present application and its equivalent technologies, the intention of the present application also includes these modifications and changes.

Claims

1. A method for predicting slope stability based on Gaussian process regression, characterized in that, The method comprises the following steps: Step S1: slope observation data preprocessing: normalizing the collected slope observation data; Step S2: slope observation data feature selection: selecting features by using a particle swarm optimization algorithm; Step S3: constructing a covariance matrix in Gaussian process regression: modeling the observation data by using a kernel function with linear and nonlinear components for the features selected from the slope observation data, to obtain a covariance matrix derived from the slope observation data; Step S4: calculating the posterior probability distribution of the stability corresponding to the new input slope observation data: calculating the correlation between the new input slope features and the original observation data according to the covariance matrix obtained in step S3, obtaining an updated covariance matrix, and calculating the mean and variance of the predicted stability according to the covariance matrix; Step S5: normalization: normalizing the prediction results by using a sigmoid function, mapping the results to between 0 and 1, if the normalized result is close to 0, it is determined that the slope is unstable; if the normalized result is close to 1, it is determined that the slope is stable. 2.The slope stability prediction method based on Gaussian process regression of claim 1, wherein: The data collected in step S1 includes slope height, slope angle, soil particle specific gravity, internal friction angle, water content, and pore ratio. 3.The method of claim 1, wherein, The step S2 specifically comprises: Step S2.1: initial position of particles in particle swarm optimization algorithm and parameters {ω, c1, c2} are initialized, where α i denotes the feature value set of the i-th particle, denotes the N-th feature of the i-th particle in the particle swarm, when denotes the selection of the N-th feature of the i-th particle in the particle swarm, when denotes the non-selection of the feature, and ω denotes the inertia weight, and 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, and υ is a weight controlling the final prediction performance and the size of the feature subset. The index number of the best position experienced by all particles in the population is denoted by the symbol g, that is, P g The velocity of particle i is denoted by V i =(v i1 ,v i2 ,…,v iN ). The strategy for moving the particle swarm in the solution space is as follows: In formula (2), rand() represents a random number between [0, 1], t represents the current iteration number, represents the best feature subset found by the i th particle, P g represents the best feature subset found by all particles, it is noted that the bold α i represents a vector, α represents an element in the vector, the above steps are repeated iteratively, and within a certain number of iterations, the fitness function The side slope feature subset α with the maximum fitness function is the feature selection result.

4. The method of claim 1, wherein the method further comprises: The step S3 specifically comprises: After feature selection, some redundant and irrelevant features are screened out; the relationship between the slope observation data is modeled by using a kernel function, which can not only represent the linear relationship but also describe the nonlinear relationship between the data, and its expression is: In formula (3), k(x n ,x m ) represents a kernel function, where x n ,x m are the slope feature vectors represented by the nth observation data and the mth observation data, represents the transpose of the vector x n , θ0, θ1, θ2, θ3 are hyperparameters of the Gaussian regression model, θ0 represents the amplitude of the Gaussian kernel, controls the overall scale of the function variation, θ1 controls the nonlinear relationship among the observation data and determines the decay rate of the correlation between the features, the greater θ1, the faster the similarity decays with the distance, θ2 is a constant term, representing the global offset in the data, and θ3 is the coefficient of the linear relationship of the observation data, capturing the linear correlation of the input features. The four parameters are obtained by grid search in the 5-fold cross-validation step, and the calculation result of formula (3) is the corresponding element in the covariance matrix.

5. The method of claim 1, wherein the method further comprises: The step S4 specifically comprises: Step S4.1: Constructing the joint probability distribution, the observed data {x1,..., x N} has been obtained and the corresponding stability results y, where x i corresponds to a certain observation data, which contains the observed characteristics of several slopes, such as slope height, slope angle, pore water pressure, and the observation results y obey the following distribution: In Equation (4), p(y) represents a probability density function to which the observation result y is subjected, represents a Gaussian distribution with a mean of 0 and a covariance of K, K represents a Gram matrix, and elements K nm represent a covariance of input points x n and x m ; Step S4.2: Model the observed data with noise, actual observation t n = y n + ε n , t n represents 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 considering the noise effect is: where β represents the noise precision, controlling the reliability of the observations, C is the covariance matrix, I N is the N x N identity matrix; The updated covariance matrix is: where C N+1 represents the N+1 x N+1 covariance matrix, k is an N-dimensional vector, k n = k(x n , x N+1 ) represents the covariance of the newly observed data with 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 the Gaussian inequality, obtain the mean m(x) and covariance σ(x) of the stability prediction: N+1 2 N+1 ​​​ where 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 the existing observation data to the prediction, reflecting the influence between similar samples.

6. The method of claim 1, wherein the method further comprises: The step S5 specifically comprises: Step S5.1: For any new input slope observation data x N+1 , get the prediction mean m(x N+1 ) by Gaussian process regression model, then map the regression model output by sigmoid function to limit the output value range between 0 and 1, the specific form is: In Equation (9), f(m) is the predicted value of the model, and m represents m(x N+1 ), i.e., the mean of the final Gaussian process regression result; Step S5.2: since f(m) is between 0 and 1, the model regression result has a probabilistic explanation, the result close to 0 indicates that the predicted slope is unstable; if the result is close to 1, it indicates that the predicted slope is stable.

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