Foundation pit horizontal displacement probability prediction method based on sparse Bayesian extreme learning machine

Abstract

The invention discloses a foundation pit horizontal displacement probability prediction method and system based on a sparse Bayesian extreme learning machine, and belongs to the technical field of civil engineering monitoring and artificial intelligence crossing. According to the method, a probability model containing input and output noise is constructed, feature selection and uncertainty quantification are automatically carried out by using a sparse Bayesian framework, and probability prediction of horizontal displacement at the position where a sensor is not arranged is realized. The method can output the prediction mean value and the confidence interval, effectively solves the problems of data sparsity and noise, and improves the prediction reliability and the engineering decision support capability. The method has the advantages of being high in automation degree, high in anti-interference capacity, suitable for actual engineering monitoring and the like.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of civil engineering monitoring and artificial intelligence, and particularly relates to a horizontal displacement probability prediction method based on sparse sensor data in deep foundation pit engineering. BACKGROUND

[0002] Deep foundation pit engineering is a key link in urban underground space development, and the safety of its supporting structure is directly related to the safety of the surrounding environment and the lives and property of construction personnel. Traditional foundation pit displacement monitoring mainly relies on manual measurement methods such as inclinometer and total station. Although these methods are mature, they have problems such as low monitoring frequency, strong data lag, high labor cost and susceptibility to environmental interference, and are difficult to meet the real-time and high-precision monitoring needs of modern complex deep foundation pit engineering.

[0003] With the development of sensing technology and automation technology, automatic monitoring systems such as automatic inclinometers have been gradually applied in foundation pit engineering, which can realize nearly continuous data acquisition and remote transmission, significantly improving the timeliness and data density of monitoring. However, due to the cost of equipment, sensors can only be sparsely arranged (such as shown in the figure, the sensor spacing is up to 5.0m), resulting in discontinuous displacement data in space, which cannot fully reflect the real deformation pattern of the supporting structure along the depth. This data sparsity makes it impossible to directly obtain the horizontal displacement at the depth where no sensors are arranged, thereby bringing great challenges to the overall stability evaluation and risk warning of the foundation pit. Figure 1

[0004] To infer the displacement at the unmonitored depth from sparse measurement data, existing methods mainly include physical models and data-driven models. Physical models are based on soil mechanics and structural mechanics principles, and have strong mechanism explainability. However, their application is severely dependent on accurate soil parameters, supporting structure attributes and construction process data, and the model construction is complex, the calculation cost is high, and it is difficult to apply to real-time online prediction scenarios. Data-driven models (such as support vector machines, random forests, artificial neural networks, etc.) avoid explicit physical modeling, and directly learn the input-output mapping relationship from historical data through machine learning algorithms, with the advantages of high computational efficiency and strong adaptability, and have been successfully applied in many geotechnical engineering prediction tasks. However, existing data-driven models are mostly deterministic models, which can only provide point prediction results and cannot quantify the uncertainty of the prediction results. In the context of sparse sensor arrangement, the incompleteness of the data itself introduces significant cognitive uncertainty, and the deterministic prediction result is often too confident and cannot reflect the possible displacement fluctuation range in the real situation, making it difficult to support risk-based engineering decisions.

[0005] ​Although there are studies (such as CN120541946A) that attempt to integrate BIM with Bayesian methods for parameter updating and deformation prediction in foundation pit construction, it is still subject to three major bottlenecks: first, the method relies on a pre-constructed high-precision BIM-finite element coupling model, which requires high modeling details and computing power, making it difficult to quickly start real-time prediction on site with only sensor data; second, each parameter update requires re-running time-consuming numerical simulation and Bayesian inversion, which is computationally heavy and cannot achieve real-time or near real-time response; finally, the entire framework focuses on model parameter inversion and does not address the actual scenario of sparse sensor deployment and missing monitoring points, providing a lightweight solution that can directly interpolate and extrapolate the displacement field. SUMMARY

[0006] The present application aims to provide a foundation pit horizontal displacement probability prediction method and system based on sparse Bayesian extreme learning machine, which can infer the displacement at unmonitored depth from sparse sensor data and output a prediction interval with uncertainty quantification, thereby improving the reliability and decision support capability of the monitoring system.

[0007] The technical solution of the present application includes the following steps:

[0008] S1: Based on the measured results in the engineering case, the horizontal displacement measured at the position of the embedded automatic inclinometer sensor is taken as the input, and the horizontal displacement measured by manual measurement at the position without the embedded automatic inclinometer sensor is taken as the output to construct the data set required for model training.

[0009] S2: Construct a sparse Bayesian extreme learning machine mathematical model containing input noise and output noise;

[0010] S3: Based on the mathematical model of S2, give the posterior probability analytical expression of the weight parameter;

[0011] S4: Based on the mathematical model of S2, give the iteration formula of the hyperparameter in the posterior probability analytical expression of the weight parameter;

[0012] S5: Construct the analytical expression of the horizontal displacement prediction model;

[0013] S6: Based on the training set constructed in step S1, start training the mathematical model in step 2, and obtain the model parameters through the expressions in steps 3 and 4;

[0014] S7: Take the horizontal displacement monitored by the automatic inclinometer sensor in the target engineering at any site as the input of the model, and based on the analytical expression of the horizontal displacement prediction model constructed in step S5 and the model parameters obtained in step S6, give the prediction result of the horizontal displacement at the position without the automatic inclinometer sensor.

[0015] Furthermore, the specific operation method of step S1 is as follows:

[0016] Obtain the distribution of the horizontal displacement measured at different times in the engineering case along the depth direction. Extract the horizontal displacement at the location where the inclinometer sensor is buried as the input of the model, and the horizontal displacement at the location where there is no buried sensor (manually measured) as the output of the model.

[0017] Furthermore, the mathematical model for predicting the probability of the foundation pit horizontal displacement in step S2 is:

[0018]

[0019] Y = H0B + E

[0020] In the formula: is the model input in the training set, x i (i = 1, 2,..., N) is the i-th row in X, T represents the matrix or vector transpose operation; N represents the number of rows in X, which is also the number of training samples; d is the number of sensors in the inclinometer; is the unobservable noise-free model input, is the i-th row in; is the input noise matrix, δ i (i = 1, 2,..., N) is the i-th row in Δ; is the model output matrix, y i (i = 1, 2,..., N) is the i-th row in Y; is the output matrix of the hidden layer under noise-free input, h k (·) is the activation function of the k-th hidden node, are the input weight and bias of this node respectively, L is the number of hidden nodes, k represents the number of the hidden node, k = 1, 2,..., L; is the output weight matrix, β j (j = 1, 2,..., m) is the j-th column in B, m is the number of positions to be predicted by the model; is the output noise matrix, ε i (i = 1, 2,..., N) is the i-th row in E; δ i obeys the normal distribution with a mean of zero and a covariance matrix of I d is the d×d identity matrix, is the variance of the input noise. ε i obeys the normal distribution with a mean of zero and a precision matrix of Γ = diag(γ1…γ m ), where is the variance of the output noise in the jth dimension, and diag(·) represents a diagonal matrix composed of the elements in the parentheses.

[0021] Further, the horizontal displacement probability prediction mathematical model of the foundation pit in step S3 is:

[0022] Under the Bayesian framework, given the data set The posterior distribution of the output weight B is represented by integrating the latent true input :

[0023]

[0024] where p(B) is the prior distribution of B. According to the established sparse Bayesian extreme learning machine framework, in order to realize automatic feature selection, we impose an automatic relevance determination (ARD) prior on the output weight B:

[0025]

[0026] where α = [α1α2,...α L ] and α k is the prior precision parameter of the kth hidden node. β (k) is the vector composed of the kth row of the matrix B. Considering , the posterior distribution of the output weight B is calculated by substituting the mathematical model in step S2 and the prior distribution p(B|α) of the output weight B into : j The posterior distribution of the relevant weight vector β for the jth dimension output is given by the following formula:

[0027]

[0028] It can be found that the posterior distribution of the weight vector β j is a Gaussian distribution:

[0029]

[0030] where represents the training data set, represents the normal distribution, and S j can be expressed as:

[0031]

[0032] where represents the gradient is calculated in ; x i (i = 1, 2,...,N) is the ith row in the model input X in the training set, It is the unobservable, noise-free model input in the training set. The i-th row in;

[0033] h k (·) is the activation function of the k-th hidden node. These are the input weights and biases of the node, respectively; L is the number of hidden nodes; k represents the hidden node number, k = 1, 2, ..., L; T represents the matrix or vector transpose operation; A = diag(α1, α2, ..., α... L ), α k (k = 1, 2, ..., L) is the prior precision parameter of the k-th hidden node, and diag(·) represents a diagonal matrix composed of the elements in parentheses; y i (i = 1, 2, ..., N) is the i-th row in the model output matrix Y.

[0034] Furthermore, the iterative formula for the hyperparameter in the posterior probability analytical expression of the weight parameters in step S4 is obtained using the following method:

[0035] hyperparameter α k and γ j Estimation is typically performed using evidence approximation or the Type II maximum likelihood method. First, we construct the marginal likelihood function (evidence function) as follows:

[0036]

[0037] To estimate α k We Regarding α k Differentiate α k And set it to zero. Solving this equation will give α. k Update rules:

[0038]

[0039] Similarly, to estimate γ j We Regarding γ j Differentiate and set it to zero. Solving the equation will yield γ. j Update rules:

[0040]

[0041] Where, α k new It is α k The updated value, S j,kk It is S j The kth diagonal element, yes k elements, S j and It is the output weight β j The covariance matrix and mean are given, and the output weights β are calculated. j It is the j-th column of the output weight matrix B; m is the number of positions that the model needs to predict; γ j new It is γ j The updated value, where N represents the number of training samples; y i (i = 1, 2, ..., N) is the i-th row in the model output matrix Y; h k (·) is the activation function of the k-th hidden node. These are the input weights and biases of the node, respectively; L is the number of hidden nodes; k represents the hidden node number, k = 1, 2, ..., L; tr() represents calculating the rank of the matrix. yes The updated value, Representative to exist Find the gradient; x i (i = 1, 2, ..., N) is the i-th row of the model input X in the training set. It is the unobservable, noise-free model input in the training set. The i-th row in; I d It is a d×d identity matrix. It is the variance of the input noise.

[0042] Furthermore, the analytical expression formula for the horizontal displacement prediction model described in step S5 is as follows:

[0043] For the j-th dimension of the output, the prediction distribution becomes a Gaussian distribution:

[0044]

[0045] Where h(·) represents the activation function; S j and It is the output weight β j The covariance matrix and mean are given, and the output weights β are calculated. j It is the j-th column of the output weight matrix B; It represents the variance of the output noise in the j-th dimension; T represents the matrix or vector transpose operation; Representative to exist Find the gradient; It is the unobservable, noise-free model input in the prediction set; I dis a dxd identity matrix, is the variance of the input noise.

[0046] Given the mean test,j and the standard deviation The confidence interval can be computed by the following formula:

[0047]

[0048] where is the critical value under the standard normal distribution corresponding to 1 - a CI , a CI is the significance level. This value controls the width of the confidence interval, which is usually used to represent the probability of the data point falling within the interval.

[0049] Further, the step S6 comprises:

[0050] (6-1) Dividing the training set into a training set and a validation set

[0051] (6-2) Define the hyperparameter search space: specify the candidate number of hidden nodes, and the type of activation function.

[0052] (6-3) Select a hyperparameter combination, based on the iterative formula of and step S4, to obtain the parameters a k and Then, substitute the obtained parameters a k and and the input in into the prediction model described in step S5, compare the prediction results of the model with the output in , and obtain the prediction performance of the model.

[0053] (6-4) Repeat step (6-3) until every hyperparameter combination in the hyperparameter search space is exhausted, and select the hyperparameter with the best prediction performance on .

[0054] Further, the accuracy of the prediction results of the model described in step S7 is evaluated, and the evaluation index used is:

[0055]

[0056] where y i,j is the measured displacement of the i-th sample at the j-th position, is the predicted mean displacement of the i-th sample at the j-th position.

[0057] In addition, the application also provides application of the foundation pit horizontal displacement probability prediction method based on the sparse Bayesian extreme learning machine in foundation pit horizontal displacement measurement.

[0058] Compared with the prior art, the application has the following beneficial effects:

[0059] (1) The probabilistic prediction of fusing data sparsity and input noise is realized, and the reliability of the prediction result is significantly improved. Unlike the traditional deterministic machine learning model which can only provide a single displacement point prediction value, the model proposed in the application can consider the cognitive uncertainty caused by sparse sensor arrangement and the measurement noise in the input data under the Bayesian framework, and output a prediction interval (such as a 95% confidence interval) with a probability significance. This provides important uncertainty quantification information for engineering decision-making, enabling engineers to assess the credibility and potential risks of the prediction value, thereby overcoming the misjudgment risk that may be caused by the deterministic model due to the neglect of uncertainty.

[0060] (2) The model has automatic feature selection capability, and the robustness and generalization of the model are enhanced. The application introduces an automatic relevance determination (ARD) prior, which can automatically identify and prune the hidden layer nodes that have little contribution to prediction by imposing sparsity constraints on the output weights. This process is equivalent to an internal feature selection, avoiding overfitting and improving the generalization ability of the model for different foundation pit projects.

[0061] (3) The risk-based engineering decision-making provides direct and quantitative scientific basis. The probabilistic prediction result (prediction interval) provided by the application can directly serve the risk control. Engineers can develop differentiated intervention strategies based on the probability of displacement value exceeding the safety threshold (rather than only whether the point estimate value exceeds the threshold), to realize real "risk warning".

[0062] (4) The input data noise is effectively processed, and the practicability and anti-interference ability of the model are improved. Existing models usually assume that the input data is accurate and error-free, which is seriously inconsistent with the engineering practice. The application constructs a likelihood function containing input noise, and introduces first-order Taylor expansion for linearization approximation in the derivation, effectively including the measurement error of the input data into the entire probability prediction framework. This makes the prediction result of the model not sensitive to data quality, and has stronger anti-interference ability and engineering practicability. BRIEF DESCRIPTION OF DRAWINGS

[0063] Figure 1 is a schematic diagram of automatic monitoring results of soil horizontal displacement caused by foundation pit excavation in the technical background of the application.

[0064] Figure 2 is a geological condition and foundation pit support structure diagram of an engineering case used when constructing a training set in the embodiment of the application.

[0065] Figure 3 The geological conditions and foundation pit support structure of the engineering case used in the implementation case of this invention to test the trained model.

[0066] Figure 4 This is a comparison chart of the model's prediction results and actual test results in an embodiment of the present invention.

[0067] Figure 5 This is a MAE display of the prediction results of the automated inclinometer for a total of 328 days in this embodiment of the invention. Detailed Implementation

[0068] The following will take the calculation of the breakdown time of a double-layer composite gasket in a wrinkled state as an example, and further elaborate and explain the present invention with reference to the accompanying drawings and tables. The technical features of each embodiment of the present invention can be combined accordingly without conflict with each other.

[0069] Step S1: Selected project case (e.g.) Figure 2 The geological conditions (as shown) are as follows: the soil is mainly composed of silty clay and silty clay, with silty clay dominating within the excavation area. The excavation depth is 28.4m, and a retaining structure of diaphragm wall + five-layer support is adopted. Based on the measured results in this engineering case, the horizontal displacement measured at locations where automated inclinometer sensors are installed (every 5 meters) is used as input, and the horizontal displacement measured manually at locations where automated inclinometer sensors are not installed is used as output, to construct the dataset required for model training.

[0070] Step S2: Construct a mathematical model for predicting the probability of horizontal displacement of the foundation pit based on a sparse Bayesian extreme learning machine:

[0071]

[0072] Y = H0B + E

[0073] In the formula: It is the model input in the training set, x i (i = 1, 2, ..., N) is the i-th row in X, T represents the matrix or vector transpose operation; N represents the number of rows in X, which is also the number of training samples; d is the number of sensors in the automatic inclinometer. It is an unobservable, noise-free model input. yes The i-th row in; It is the input noise matrix, δ i (i = 1, 2, ..., N) is the i-th row in Δ; It is the model output matrix, y iis the i-th row of Y; is the hidden layer output matrix under noiseless input, h k is the activation function of the k-th hidden node, are the input weight and bias of the node, respectively, L is the number of hidden nodes, and k represents the number of hidden nodes, k = 1, 2,..., L; is the output weight matrix, β j is the j-th column of B, and m is the number of positions that need to be predicted by the model; is the output noise matrix, ε i is the i-th row of E; δ i obeys a normal distribution with a mean of zero and a covariance matrix I d is a dxd identity matrix, is the variance of the input noise. ε i obeys a normal distribution with a mean of zero and a precision matrix Γ = diag(γ1…γ m ), where is the variance of the output noise in the j-th dimension, and diag(·) represents a diagonal matrix composed of the elements in the parentheses. N = 20185, d = 9, m = 69.

[0074] Step S3: Based on the foundation pit horizontal displacement probability prediction mathematical model, the posterior probability analytical expression of the weight parameter is given:

[0075] For the j-th dimensional output, the posterior distribution of the related weight vector β j is given by the following formula:

[0076]

[0077] The posterior distribution of the weight vector β j is a Gaussian distribution:

[0078]

[0079] where, and S j can be represented as:

[0080]

[0081] where, represents the gradient of with respect to ; x i is the i-th row of the model input X in the training set, It is the unobservable, noise-free model input in the training set. The i-th row in;

[0082] h k (·) is the activation function of the k-th hidden node. These are the input weights and biases of the node, respectively; L is the number of hidden nodes; k represents the hidden node number, k = 1, 2, ..., L; T represents the matrix or vector transpose operation; A = diag(α1, α2, ..., α... L ), α k (k = 1, 2, ..., L) is the prior precision parameter of the k-th hidden node, and diag(·) represents a diagonal matrix composed of the elements in parentheses; y i (i = 1, 2, ..., N) is the i-th row in the model output matrix Y.

[0083] Step S4: Based on the aforementioned mathematical model for predicting the horizontal displacement of the foundation pit, the iterative formula for the hyperparameters in the posterior probability analytical expression of the weight parameters is given:

[0084]

[0085]

[0086] Where α k new It is α k The updated value, S j,kk It is S j The kth diagonal element, yes k elements, S j and It is the output weight β j The covariance matrix and mean are given, and the output weights β are calculated. j It is the j-th column of the output weight matrix B; m is the number of positions that the model needs to predict; γ j new It is γ j The updated value, where N represents the number of training samples; y i (i = 1, 2, ..., N) is the i-th row in the model output matrix Y; h k (·) is the activation function of the k-th hidden node. These are the input weights and biases of the node, respectively; L is the number of hidden nodes; k represents the hidden node number, k = 1, 2, ..., L; tr() represents calculating the rank of the matrix. yes The updated value, Representative to In Gradient is sought; x i is the i-th row in the model input X in the training set, is the unobserved noise-free model input in the training set is the t-th row in the training set. I d is a dxd identity matrix, is the variance of the input noise.

[0087] Step S5: Construct the analytical expression of the horizontal displacement prediction model:

[0088] For the j-th dimensional output, the prediction distribution becomes a Gaussian distribution:

[0089]

[0090] where x test is a sample of the input of the test set (or a measured result of the sensor at one time in actual engineering application), y test,j is the j-th dimensional output corresponding to x test ; h(·) represents the activation function; S j and are the covariance matrix and mean of the output weight β j ; β j is the j-th column of the output weight matrix B. is the variance of the output noise in the j-th dimension; T represents the matrix or vector transpose operation; represents the gradient of in ; is the unobserved noise-free model input in the prediction set; I d is a dxd identity matrix, is the variance of the input noise. Given the mean μ test,j and the standard deviation , the confidence interval can be calculated by the following formula:

[0091]

[0092] where is the critical value corresponding to 1-α CI in the standard normal distribution. This value controls the width of the confidence interval, which is usually used to represent the probability of the data point falling within the interval.

[0093] Step S6: Based on the training set constructed in step S1 Train the mathematical model in step 2 to obtain the model parameters in the expressions in step 3 and step 4:

[0094] (6-1) Divide 80% of the data in the training set into a training set and 20% of the data into a validation set

[0095] (6-2) Define the hyperparameter search space: specify the candidate hidden node number L ∈ {50, 100, 150}, and the activation function type h k (·) ∈ {sigmoid, tanh}.

[0096] (6-3) Select a hyperparameter combination, based on and the iterative formula in step S4, obtain the parameters a k and Then, substitute the obtained parameters a k and and into the prediction model described in step S5, compare the prediction results of the model with the outputs in to obtain the prediction performance of the model.

[0097] (6-4) Repeat step (6-3) until every hyperparameter combination in the hyperparameter search space is exhausted, select the hyperparameter with the best prediction performance on , that is, L = 100, h k (·) ∈ {sigmoid}.

[0098] Step S7: The target project (as shown in Figure 3 ) mainly has silty sand and silty clay, and the silty sand is mainly in the foundation pit excavation range. The excavation depth of the foundation pit is 18.8m, and the enclosure structure adopts underground continuous wall + three supports. The automatic inclinometer sensor is arranged every 5 meters, and the horizontal displacement obtained by monitoring is used as the input (x test ) of the model. Based on the analytical expression of the horizontal displacement prediction model constructed in step S5 and the model parameters obtained in step S6, the prediction results of the horizontal displacement at the positions without automatic inclinometer sensors are given. Figure 4 The monitoring results and prediction results (mean and confidence interval) of the 100th day of one automatic inclinometer in the target project are shown in . The error index MAE = 1.09, which shows very high prediction accuracy. Figure 5 The MAE of the prediction results of this automatic inclinometer for a total of 328 days is shown in . The maximum value of MAE is 1.89mm, indicating that the model has shown very high prediction accuracy throughout the monitoring period.

[0099] The embodiments of the present application are not limited to the above-mentioned preferred examples, all combinations and improvements combined with the above technical features shall be considered as the protection scope of the present application. For those skilled in the art, several improvements and refinements can be made without departing from the principles of the present application, which shall also be considered as the protection scope of the present application.

Claims

1. A method for probability prediction of horizontal displacement of foundation pit based on sparse Bayesian extreme learning machine, characterized in that, The method comprises the following steps: S1: constructing a training data set, taking the horizontal displacement measured at the position of the automatic inclinometer sensor as input data, and taking the manually measured horizontal displacement at the position without the sensor as output data, to obtain the training data set; S2: establishing a sparse Bayesian extreme learning machine mathematical model containing input noise and output noise; S3: based on the mathematical model in S2, an analytical expression of the posterior probability of the weight parameter is given; S4: based on the mathematical model in S2, an iterative formula of the hyperparameter in the analytical expression of the posterior probability of the weight parameter is given; S5: constructing an analytical expression of the horizontal displacement prediction model; S6: based on the training data set in S1, and S2, S3 and S4, the model is trained, and the optimal model parameters are determined through the hyperparameters; S7: for any site, taking the horizontal displacement monitored by the automatic inclinometer sensor in the target project as the input of the model, based on the analytical expression of the horizontal displacement prediction model constructed in step S5 and the model parameters obtained in step S6, the prediction result of the horizontal displacement at the position without the automatic inclinometer sensor is given.

2. The method of claim 1, wherein, The construction method of the training data set in step S1 is: taking the horizontal displacement at the position of the automatic sensor as input, and taking the manually measured displacement at the position without the sensor as output.

3. The method of claim 1, wherein, The mathematical model in step S2 is: Y=H0B+E where: is the model input in the training set, x i is the ith row in X, T represents the matrix or vector transpose operation; N represents the number of rows of X, which is also the number of samples of the training samples; d is the number of sensors in the automatic inclinometer; is the unobservable noise-free model input, is the ith row in ; is the input noise matrix, δ i is the ith row in Δ; is the model output matrix, y i is the ith row in Y; is the hidden layer output matrix under noise-free input, h k (·) is the activation function of the kth hidden node, are the input weight and bias of the node, respectively, L is the number of hidden nodes, k represents the number of hidden nodes, k = 1, 2, …, L; is the output weight matrix, β j is the jth column of B, m is the number of positions that need to be predicted by the model; is the output noise matrix, ε i is the ith row in E; δ i is subject to a normal distribution with a mean of zero and a covariance matrix I d is a dxd identity matrix, is the variance of the input noise; ε i Under the assumption of a normal distribution with zero mean, the precision matrix is Γ = diag(γ1…γ m ), where is the variance of the output noise in the j-th dimension, and diag(·) denotes a diagonal matrix composed of the elements in the parentheses.

4. The method of claim 1, wherein, The input noise Δ and the output noise E both obey the normal distribution with zero mean.

5. The method of claim 1, wherein, The weight parameter is an output weight matrix β j (j = 1, 2, …, m) is the jth column of B, m is the number of positions to be predicted by the model, and the output weight β j The posterior distribution of β is a Gaussian distribution, whose covariance matrix S j and mean are given by the following equations: in, Representative to exist Find the gradient; x i (i = 1, 2, ..., N) is the i-th row of the model input X in the training set. It is the unobservable, noise-free model input in the training set. The i-th row in; h k (·) is the activation function of the k-th hidden node. These are the input weights and biases of the node, respectively; L is the number of hidden nodes; k represents the hidden node number, k = 1, 2, ..., L; T represents the matrix or vector transpose operation; A = diag(α1, α2, ..., α... L ), α k (k = 1, 2, ..., L) is the prior precision parameter of the k-th hidden node, and diag(·) represents a diagonal matrix composed of the elements in parentheses; y i (i = 1, 2, ..., N) is the i-th row in the model output matrix Y.

6. The method of claim 1, wherein, The super parameters include super parameters a k and g j The iterative formula of super parameters a k and g j in step S4 is: where α k new is α k the updated value, S j,kk is S j the kth diagonal element, is the k elements of S j and is the covariance matrix and mean of the output weights β j j is the jth column of the output weight matrix B;​ m is the number of locations that need to be predicted by the model; γ j new is γ j updated value, N represents the number of samples of the training samples; y i (i = 1, 2,..., N) is the i-th row in the model output matrix Y; h k (·) is the activation function of the k-th hidden node, and b k respectively the input weight and bias of the node, L is the number of hidden nodes, k represents the number of hidden nodes, k = 1, 2,..., L; tr() represents the rank of the matrix; is updated value, represents the prediction of In x i derivative; x i (i = 1, 2,..., N) is the i-th row in the model input X in the training set, is the i-th row in the unobservable noise-free model input in the training set; I d is a dxd identity matrix, is the variance of the input noise.

7. The method of claim 1, wherein, The analytical expression of the horizontal displacement prediction model in step S5 is: for the test sample x test The prediction distribution of the jthdimensional output is: wherein, representing a training data set, representing a normal distribution, mean μ test,j and variance is given by: where h(·) represents the activation function; S j and is the covariance matrix and mean of the output weights β j is the jth column of the output weight matrix B; j is the variance of the output noise in the jth dimension; T represents the matrix or vector transpose operation; represents taking the gradient with respect to in ; and is the unobserved noise-free model input in the prediction set; I d is the dxd identity matrix, is the variance of the input noise.​ 8. The method of claim 1, wherein, Step S6 selects the optimal hidden node number L and the activation function type on the validation set through grid search.

9. The method of claim 1, wherein, Step S7 uses the mean absolute error (MAE) as the evaluation index of the prediction accuracy of the model.

10. The application of a foundation horizontal displacement probability prediction method based on a sparse Bayesian extreme learning machine in the measurement of foundation horizontal displacement according to any one of claims 1-9.

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