Copula-based bridge response and load joint probability modeling method

Abstract

The invention relates to the technical field of bridge monitoring, and provides a Copula-based bridge response and load joint probability modeling method, which comprises the following steps: S1, acquiring a structure response index and a load parameter through a structure health monitoring system, and carrying out probability density modeling on the structure response index and the load parameter through a Gaussian mixture model to obtain a probability density model; screening and fitting an optimal probability distribution model of each structure response index and load parameter through a Bayesian information criterion; s2, analyzing the correlation between variable pairs formed by the structure response indexes and the load parameters by adopting a Pearson correlation coefficient, and determining the strength and direction of the correlation; s3, constructing a joint probability density function between variables through a Copula probability density modeling method in combination with the fitted variable probability distribution marginal model; and S4, verifying the constructed joint probability density function by adopting a Monte Carlo sampling method. According to the method, the multivariable joint probability model between the key load and the response is generated, and the uncertainty is quantified.

Description

Technical Field

[0001] The present application relates to the technical field of bridge monitoring, and in particular to a copula-based joint probability modeling method for bridge response and load. Background Art

[0002] Structural health monitoring (SHM) systems have become an important method for real-time and continuous monitoring of bridge conditions. By collecting data on a variety of structural response variables, such as displacement, strain, and vibration, SHM systems can detect anomalies and implement timely maintenance. To effectively utilize the large amounts of data generated by SHM systems to support decision-making, sophisticated probabilistic modeling methods are required that account for the inherent uncertainties and correlations between response variables. In civil engineering applications such as bridges, statistical probability models play a key role in characterizing the field characteristics of loads and responses. Common distributions such as Gaussian, Gamma, Gumbel, Weibull, and Pareto provide support for the probabilistic behavior of structural elements and environmental forces.

[0003] However, relying solely on univariate PDFs ignores critical correlations and dependencies between different aspects of structural behavior. Furthermore, real civil engineering structures consist of numerous interconnected components whose structural behaviors exhibit correlations that reflect their interdependencies and interactions. Extracting correlations between different types of responses can help managers better understand the relationships between various structural elements, as these correlations may propagate throughout the structure, potentially leading to joint failure of structural components or system-wide cascading failures. Consequently, current probabilistic modeling cannot accurately quantify and characterize the inherent uncertainties in structural behavior and cannot effectively handle the uncertainties introduced by fluctuations in multiple factors, such as material properties, loading conditions, and environmental factors. Summary of the Invention

[0004] To help solve the above technical problems, this application provides a copula-based joint probabilistic modeling method for bridge response and load, which adopts the following technical solutions: A copula-based joint probabilistic modeling method for bridge response and load, wherein the method comprises: Step S1: Structural response indicators and load parameters are collected through the structural health monitoring system, and probability density modeling is performed on the structural response indicators and load parameters through the Gaussian mixture model. The optimal probability distribution model of each structural response indicator and load parameter is selected and fitted through the Bayesian information criterion. The structural response indicators include the bridge tower offset. , pier offset , main beam strain and expansion joint displacement The load parameters include temperature TMP, wind speed ,wind direction and traffic load ; Step S2: using the Pearson correlation coefficient to analyze the correlation between the variable pairs consisting of the structural response index and the load parameter, and determining the strength and direction of the correlation; Step S3: constructing a joint probability density function between the variable pairs by combining the fitted variable probability distribution marginal model with the Copula probability density modeling method; Step S4: Use the Monte Carlo sampling method to verify the constructed joint probability density function.

[0005] Preferably, the step S1 includes: Step S101: continuously collecting the structural response index and the load parameter through a structural health monitoring system installed on the bridge structure, and performing data preprocessing on the collected structural response index and load parameter; Step S102: Probability density modeling is performed on the structural response index and the load parameter using a Gaussian mixture model. The probability density function of the Gaussian mixture model is: ,in, represents the mixing weight of the kth component; The probability density function corresponding to the Gaussian mixture model of the kth component is represented by its mean. And the covariance is .

[0006] Preferably, the step S1 further includes: Step S103: The optimal probability distribution model of each structural response index and load parameter is selected and fitted by the Bayesian information criterion as follows: , where L represents the likelihood of the model given the data, k represents the number of parameters in the model, and n represents the number of data points; Step S104: For each of the structural response indicators and load parameters, fitting is performed based on the monitoring data, the corresponding Bayesian information criterion value is calculated, the Bayesian information criterion values ​​of the candidate models are compared, and the distribution model with the smallest Bayesian information criterion value is selected as the optimal distribution of the corresponding variable parameters. The candidate models include Gaussian distribution model, exponential distribution model and Weibull distribution model.

[0007] Preferably, the step S2 includes: Step S201: Quantitatively analyze the correlation between the variable pairs consisting of the selected structural response index and the load parameter using the Pearson correlation coefficient. The calculation formula of the Pearson correlation coefficient is as follows: ,in, and are the observed values ​​of the two variables, and is the mean of the variable; Step S202: Screening out variable pairs with significant correlation based on the calculated correlation coefficients, and identifying variable pairs with high positive correlations whose Pearson correlation coefficients are greater than a Pearson correlation coefficient threshold or high negative correlations whose Pearson correlation coefficients are less than a Pearson correlation coefficient threshold as key variables that require joint modeling; Step S203: Draw a correlation matrix diagram to display the correlation distribution between variables, and use the selected key variables as the basic input for joint probability modeling.

[0008] Preferably, step S3 includes: Step S301: The Gaussian Copula density function is expressed as: , where R represents the correlation coefficient matrix; represents the inverse cumulative distribution function of the standard normal; represents the joint cumulative distribution function of the multivariate normal distribution with zero mean and covariance matrix equal to the correlation coefficient matrix; Step S302: Capture the joint probability density function as follows: ; Step S303: Construct the joint probability density function between variables as follows: , where u represents the normal distribution vector .

[0009] Preferably, the step S4 includes: Step S401: For each variable, randomly generate samples according to the corresponding marginal distribution function; combine the marginal samples through the selected Copula function to generate a joint distribution sample, and set the number of samples; Step S402: Visualize the generated joint distribution samples, using a scatter plot to display the joint distribution characteristics of each pair of variables, and check whether the sampling points closely follow the theoretical distribution pattern of the joint probability density function. If the scatter plot shows that the sampling points are uniform and consistent with the theoretical distribution, it means that the fitting effect is good; Step S403: verifying the joint probability density function by calculating the deviation between the distribution of the sampling points and the distribution of the observed data and the joint probability values ​​under different variable combinations.

[0010] In summary, this application provides a copula-based joint probabilistic modeling method for bridge responses and loads. Based on a bridge health monitoring system and copula functions, a multivariate joint probabilistic model of key loads and responses is generated to reveal corresponding probabilistic relationships and quantify uncertainty. Furthermore, copula functions can be used to predict loads and responses, conduct structural system reliability analysis, and perform joint failure identification. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Figure 1 A schematic block diagram of an embodiment of a copula-based joint probabilistic modeling method for bridge response and load in the present application; Figure 2 A flowchart of an embodiment of a copula-based joint probability modeling method for bridge response and load in the present application; Figure 3 Schematic diagram of the probability distribution model of bridge tower displacement in the structural response index; Figure 4 Schematic diagram of the probability distribution model of pier displacement in the structural response index; Figure 5 Schematic diagram of the probability distribution model of the main beam strain in the structural response index; Figure 6 Schematic diagram of the probability distribution model of expansion joint displacement in the structural response index; Figure 7 Schematic diagram of the correlation between key loads and responses; Figure 8 Schematic diagram of the joint probability density function of the main beam strain and the pier displacement in the structural response index; Figure 9 Schematic diagram of the joint probability density function of expansion joint displacement and bridge tower offset in the structural response index; Figure 10 Schematic diagram of the joint probability density function of expansion joint displacement and pier offset in the structural response index; Figure 11 Schematic diagram of the joint probability density function of expansion joint displacement and main beam strain in the structural response index; Figure 12 Schematic diagram of sampling verification of pier displacement and main beam strain in structural response indicators based on Copula joint probability density function; Figure 13 Schematic diagram of sampling verification of bridge tower offset and expansion joint displacement in structural response indicators based on Copula joint probability density function; Figure 14 Schematic diagram of sampling verification of pier offset and expansion joint displacement in structural response indicators based on Copula joint probability density function; Figure 15 Schematic diagram of sampling verification of expansion joint displacement and main beam strain in structural response indicators based on Copula joint probability density function. DETAILED DESCRIPTION

[0012] The present invention will be further described below with reference to the accompanying drawings. The structure and principle of the present invention will be very clear to those skilled in the art. It should be understood that the specific embodiments described herein are only intended to explain the present invention and are not intended to limit the present invention.

[0013] Figure 1 A schematic block diagram of an embodiment of a copula-based joint probabilistic modeling method for bridge response and load in the present application; Figure 2 This is a flow chart of an embodiment of a copula-based joint probability modeling method for bridge response and load in this application. Figure 3 Schematic diagram of the probability distribution model of bridge tower displacement in the structural response index; Figure 4 Schematic diagram of the probability distribution model of pier displacement in the structural response index; Figure 5 Schematic diagram of the probability distribution model of the main beam strain in the structural response index; Figure 6 Schematic diagram of the probability distribution model of expansion joint displacement in the structural response index; Figure 7 Schematic diagram of the correlation between key loads and responses; Figure 8 Schematic diagram of the joint probability density function of the main beam strain and the pier displacement in the structural response index; Figure 9 Schematic diagram of the joint probability density function of expansion joint displacement and bridge tower offset in the structural response index; Figure 10 Schematic diagram of the joint probability density function of expansion joint displacement and pier offset in the structural response index; Figure 11 Schematic diagram of the joint probability density function of expansion joint displacement and main beam strain in the structural response index; Figure 12 Schematic diagram of sampling verification of pier displacement and main beam strain in structural response indicators based on Copula joint probability density function; Figure 13 Schematic diagram of sampling verification of bridge tower offset and expansion joint displacement in structural response indicators based on Copula joint probability density function; Figure 14 Schematic diagram of sampling verification of pier offset and expansion joint displacement in structural response indicators based on Copula joint probability density function; Figure 15 Schematic diagram of sampling verification of expansion joint displacement and main beam strain in structural response indicators based on Copula joint probability density function.

[0014] Figures 3 to 6 In the equation, Density represents the probability density and Value represents the value. Figure 7 The value in is the correlation coefficient, Figures 8 to 11The Copula Density in represents the joint probability density function, Figures 12 to 15 Measured represents the measured distribution, and Simulated represents the estimated distribution.

[0015] Combine Figures 1 to 15 It can be understood that the method of the present application may include the following steps.

[0016] Step S1: Structural response indicators and load parameters are collected through the structural health monitoring system, and probability density modeling of the structural response indicators and load parameters is performed through the Gaussian mixture model. The optimal probability distribution model of each structural response indicator and load parameter is selected and fitted through the Bayesian information criterion. The structural response indicators include the bridge tower offset. , pier offset , main beam strain and expansion joint displacement , load parameters include temperature TMP, wind speed ,wind direction and traffic load .

[0017] It should be noted that the structural response indicators and load parameters used in this application are exclusive to the bridge field and are specifically set to solve the technical problem that "relying solely on single-variable PDFs will ignore the key correlations and dependencies between different aspects of structural behavior."

[0018] Step S2: Use the Pearson correlation coefficient to analyze the correlation between the variable pairs consisting of the structural response index and the load parameter, and determine the strength and direction of the correlation.

[0019] Step S3: The joint probability density function between the variables is constructed by combining the fitted variable probability distribution marginal model with the Copula probability density modeling method.

[0020] Step S4: Use the Monte Carlo sampling method to verify the constructed joint probability density function.

[0021] The following is a detailed description of each step.

[0022] Step S1 includes: Step S101: A structural health monitoring (SHM) system installed on the bridge structure continuously collects load and response data, including parameters such as temperature, wind load, traffic load, displacement, strain, and tilt. The collected raw data is systematically preprocessed to ensure data integrity and reliability.

[0023] Step S102: In the process of constructing the marginal distribution of the multivariate parameter copula function, the present invention preferably uses the Gaussian distribution as the basis function. The Gaussian mixture model (GMM) can adapt to various data distribution forms and can accurately fit the actual data through parameter adjustment. The probability density function (PDF) of the GMM is usually expressed as ,in, represents the mixing weight of the kth component; The probability density function corresponding to the Gaussian mixture model of the kth component is represented by its mean. And the covariance is .

[0024] Step S103: To determine the optimal distribution, the present invention uses the Bayesian Information Criterion (BIC) to select marginal distribution fits. BIC is a statistical evaluation criterion used to measure the similarity between the fitted distribution and the original data distribution, while taking into account the fitting accuracy and complexity of the model, thereby effectively avoiding overfitting. Generally, models with smaller BIC values ​​have higher priority among candidate models. The calculation formula for BIC is as follows: , Here, L represents the likelihood of the model given the data, k represents the number of parameters in the model, and n represents the number of data points.

[0025] Step S104: For each structural response index and load parameter, fitting is performed based on the monitoring data, the corresponding Bayesian information criterion value is calculated, the Bayesian information criterion values ​​of each candidate model are compared, and the distribution model with the smallest Bayesian information criterion value is selected as the optimal distribution of the corresponding variable parameters. The candidate models include Gaussian distribution model, exponential distribution model and Weibull distribution model.

[0026] Step S2 includes: Step S201: Use the Pearson correlation coefficient to quantitatively analyze the relationship between the selected variables and calculate their correlation. The calculation formula of the Pearson correlation coefficient is as follows: , in, and are the observed values ​​of the two variables, and is the mean of the variable.

[0027] Step S202: Based on the calculated correlation coefficient, select variable pairs with significant correlation. Variable pairs with high positive correlation (e.g., r > 0.7) or high negative correlation (e.g., r < -0.7) are identified as key variables requiring joint modeling. 0.7 is the correlation coefficient threshold. That is, based on the calculated correlation coefficient, select variable pairs with significant correlation. Variable pairs with high positive correlation (Pearson correlation coefficients greater than the Pearson correlation coefficient threshold) or high negative correlation (Pearson correlation coefficients less than the Pearson correlation coefficient threshold) are identified as key variables requiring joint modeling.

[0028] Step S203: Draw a correlation matrix to visually display the correlation distribution between variables and further verify the accuracy and rationality of the data analysis results. The selected set of highly correlated variables is used as the basic input for joint probability modeling, providing a basis for subsequent Copula function modeling to ensure the rationality and effectiveness of the joint modeling.

[0029] Step S3 includes: Step S301: In the framework of Gaussian Copula, the Copula density function It can be derived by joint multivariate normal distribution. When considered as independent standard normal random variables, the Gaussian Copula density function is given by the determinant of the correlation matrix The function is defined by the power of the function. This function is designed to capture the interdependence between variables to ensure that the resulting copula can accurately reflect the inherent correlation structure in the multivariate normal distribution. Specifically, the Gaussian copula density function is expressed as: , Where R represents the correlation coefficient matrix; represents the inverse cumulative distribution function of the standard normal; Represents the joint cumulative distribution function of the multivariate normal distribution with mean zero and a covariance matrix equal to the correlation matrix.

[0030] Step S302: Based on the marginal distribution of multiple random variables, the dependency relationship between variables is established through the Copula function and the joint probability density function. The Gaussian Copula density function is used to evaluate the normalized variables under the given correlation conditions. The probability of jointly taking a specific value provides a quantitative tool. When using the copula function to capture the interdependence between variables in the joint probability density function (JPDF), especially in the presence of Gaussian characteristics, this involves integrating the copula function with the marginal probability density function of each variable. From a mathematical point of view, the multivariate JPDF based on the copula can be expressed as: .

[0031] Step S303: Using the properties of the multivariate normal distribution, the Gaussian Copula density function can model complex dependency patterns between variables in a manner consistent with the Gaussian properties of each variable. By combining the Copula function with the Gaussian marginal distribution, this method can effectively represent the joint distribution of variables and take into account the dependency structure between them. The corresponding density function can be expressed as: , Where u represents the normal distribution vector .

[0032] Step S4 includes: Step S401: Generate random samples from each joint distribution model using the constructed joint probability density function. The specific method is as follows: For each variable, randomly generate samples according to its marginal distribution function; combine the marginal samples using the selected copula function to generate a joint distribution sample; and set the number of samples (e.g., 2000 points) to ensure that the sample size is sufficient to reflect the overall characteristics of the model.

[0033] Step S402: Visualize the sampling results. Visualize the generated joint distribution samples: Use a scatter plot to display the joint distribution characteristics of each pair of variables. Check whether the sampling points closely follow the theoretical distribution pattern of the joint probability density function. If the scatter plot shows that the sampling points are uniform and consistent with the theoretical distribution, the fit is good. Compare the sampling results with the observed data to verify whether the model can accurately capture the dependencies between the variables.

[0034] Step S403: Model suitability and predictive performance evaluation. The accuracy of the quantitative model is evaluated through goodness-of-fit testing, such as calculating the deviation between the distribution of sampling points and the distribution of observed data. The model's predictive performance is then evaluated by calculating the joint probability values ​​for different variable combinations. The model is then verified to accurately predict the variable relationships and uncertainty characteristics in real-world scenarios. If the results indicate a good model fit and excellent predictive performance, the model is confirmed to meet project requirements. If deviations exist, the copula function or marginal distribution model needs to be readjusted.

Claims

1. A joint probability modeling method for bridge response and load based on Copula, characterized in that: The method comprises: Step S1: Structural response indicators and load parameters are collected through the structural health monitoring system, and probability density modeling is performed on the structural response indicators and load parameters through the Gaussian mixture model. The optimal probability distribution model of each structural response indicator and load parameter is selected and fitted through the Bayesian information criterion. The structural response indicators include bridge tower offset , Pier offset , beam strain and expansion joint displacement The load parameters include temperature TMP, wind speed ,wind direction and traffic load ; Step S2: using the Pearson correlation coefficient to analyze the correlation between the variable pairs consisting of the structural response index and the load parameter, and determining the strength and direction of the correlation; Step S3: constructing a joint probability density function between the variable pairs by combining the fitted variable probability distribution marginal model through the Copula probability density modeling method; Step S4: Using the Monte Carlo sampling method to verify the constructed joint probability density function.

2. The method for joint probability modeling of bridge response and load based on Copula according to claim 1, characterized in that: The step S1 comprises: Step S101: continuously collecting the structural response index and the load parameter through a structural health monitoring system installed on the bridge structure, and performing data preprocessing on the collected structural response index and load parameter; Step S102: Probability density modeling is performed on the structural response index and the load parameter by using a Gaussian mixture model. The probability density function of the Gaussian mixture model is: ,in, represents the mixing weight of the kth component; The probability density function corresponding to the Gaussian mixture model of the kth component is represented by its mean value. And the covariance is .

3. The method for joint probability modeling of bridge response and load based on Copula according to claim 2 is characterized in that: The step S1 further comprises: Step S103: The optimal probability distribution model of each structural response index and load parameter is selected and fitted by the Bayesian information criterion as follows: , where L represents the likelihood of the model given the data, k represents the number of parameters in the model, and n represents the number of data points; Step S104: For each of the structural response indicators and load parameters, fitting is performed based on the monitoring data, the corresponding Bayesian information criterion value is calculated, the Bayesian information criterion values ​​of the candidate models are compared, and the distribution model with the smallest Bayesian information criterion value is selected as the optimal distribution of the corresponding variable parameters. The candidate models include Gaussian distribution model, exponential distribution model and Weibull distribution model.

4. The method for joint probability modeling of bridge response and load based on Copula according to claim 2 is characterized in that: The step S2 comprises: Step S201: Use the Pearson correlation coefficient to quantitatively analyze the correlation between the variable pairs consisting of the selected structural response index and the load parameter. The calculation formula of the Pearson correlation coefficient is as follows: ,in, and are the observed values ​​of the two variables, and is the mean of the variable; Step S202: According to the calculated correlation coefficient, select variable pairs with significant correlation, and identify variable pairs with high positive correlation whose Pearson correlation coefficient is greater than the Pearson correlation coefficient threshold or high negative correlation whose Pearson correlation coefficient is less than the Pearson correlation coefficient threshold as key variables that need joint modeling; Step S203: Draw a correlation matrix diagram to display the correlation distribution between variables, and use the selected key variables as the basic input for joint probability modeling.

5. The method for joint probability modeling of bridge response and load based on Copula according to claim 4 is characterized in that: The step S3 comprises: Step S301: Gaussian Copula density function is expressed as: , where R represents the correlation coefficient matrix; represents the inverse cumulative distribution function of the standard normal; represents the joint cumulative distribution function of the multivariate normal distribution with zero mean and covariance matrix equal to the correlation coefficient matrix; Step S302: Capture the joint probability density function as follows: ； Step S303: construct the joint probability density function between variables as follows: , where u represents the normal distribution vector .

6. The method for joint probability modeling of bridge response and load based on Copula according to claim 5 is characterized in that: The step S4 comprises: Step S401: for each variable, randomly generate samples according to the corresponding marginal distribution function; combine the marginal samples through the selected Copula function to generate joint distribution samples, and set the sampling quantity; Step S402: Visualize the generated joint distribution samples, use a scatter plot to display the joint distribution characteristics of each pair of variables, and check whether the sampling points closely follow the theoretical distribution pattern of the joint probability density function. If the scatter plot shows that the sampling points are uniform and consistent with the theoretical distribution, it means that the fitting effect is good; Step S403: verifying the joint probability density function by calculating the deviation between the distribution of the sampling points and the distribution of the observed data and the joint probability values ​​under different variable combinations.

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