A Copula-based joint probabilistic modeling method for bridge response and load

Abstract

The present application relates to the technical field of bridge monitoring, and provides a joint probability modeling method of bridge response and load based on Copula, including step S1: collecting structural response indicators and load parameters through a structural health monitoring system, performing probability density modeling on the structural response indicators and load parameters through a Gaussian mixture model, and screening and fitting the optimal probability distribution model of each structural response indicator and load parameter through the Bayesian information criterion; step S2: using the Pearson correlation coefficient to analyze the correlation between the variable pairs consisting of the structural response indicators and load parameters, and determining the strength and direction of the correlation; step S3: constructing a joint probability density function between variables by combining the Copula probability density modeling method with the fitted variable probability distribution marginal model; step S4: using the Monte Carlo sampling method to verify the constructed joint probability density function. The present application generates a multivariate joint probability model between key loads and responses, and quantifies uncertainty.

Description

Technical Field

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

[0002] At present, structural health monitoring (SHM) systems have become an important method for real-time and continuous monitoring of bridge conditions. By collecting data on multiple response variables such as displacement, strain, vibration, etc. of the structure, it can detect anomalies and implement timely maintenance. In order to effectively use the large amount of data generated by SHM systems to support decision-making, complex probabilistic modeling methods are needed that can consider the inherent uncertainty and correlation 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 distribution, gamma distribution, Gumbel distribution, Weibull distribution and Pareto distribution provide support for the probabilistic behavior of structural elements and environmental forces.

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

[0004] In order to help solve the above technical problems, this application provides a joint probability modeling method of bridge response and load based on Copula, which adopts the following technical solutions:

[0005] A joint probability modeling method for bridge response and load based on Copula, wherein the method comprises:

[0006] 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 ;

[0007] 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;

[0008] 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;

[0009] Step S4: Using the Monte Carlo sampling method to verify the constructed joint probability density function.

[0010] Preferably, the step S1 comprises:

[0011] 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;

[0012] 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:

[0013] ,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 .

[0014] Preferably, the step S1 further comprises:

[0015] 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:

[0016] , 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;

[0017] 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.

[0018] Preferably, step S2 comprises:

[0019] 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:

[0020] ,in, and are the observed values ​​of the two variables, and is the mean of the variable;

[0021] 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;

[0022] 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.

[0023] Preferably, step S3 comprises:

[0024] Step S301: Gaussian Copula density function is expressed as:

[0025] , 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;

[0026] Step S302: Capture the joint probability density function as follows:

[0027] ;

[0028] Step S303: construct the joint probability density function between variables as follows:

[0029] , where u represents the normal distribution vector .

[0030] Preferably, step S4 comprises:

[0031] 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;

[0032] 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;

[0033] 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.

[0034] In summary, this application provides a joint probability modeling method for bridge response and load based on Copula. Based on the bridge health monitoring system and Copula function, a multivariate joint probability model between key loads and responses is generated to reveal the corresponding probability relationship and quantify uncertainty. In addition, the Copula function can be used to predict loads and responses, and to perform structural system reliability analysis and joint failure identification. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 A schematic block diagram of an embodiment of a copula-based joint probability modeling method for bridge response and load of the present application;

[0036] Figure 2 A schematic flow chart of an embodiment of a joint probability modeling method of bridge response and load based on Copula of the present application;

[0037] Figure 3 Schematic diagram of the probability distribution model of bridge tower displacement in the structural response index;

[0038] Figure 4 It is a schematic diagram of the probability distribution model of the pier displacement in the structural response index;

[0039] Figure 5 It is a schematic diagram of the probability distribution model of the main beam strain in the structural response index;

[0040] Figure 6 It is a schematic diagram of the probability distribution model of expansion joint displacement in the structural response index;

[0041] Figure 7Schematic diagram of the correlation between key loads and responses;

[0042] Figure 8 It is a schematic diagram of the joint probability density function of the main beam strain and the pier displacement in the structural response index;

[0043] Fig. 9 It is a schematic diagram of the joint probability density function of expansion joint displacement and bridge tower displacement in the structural response index;

[0044] Fig.10 It is a schematic diagram of the joint probability density function of expansion joint displacement and pier displacement in the structural response index;

[0045] Fig.11 It is a schematic diagram of the joint probability density function of the expansion joint displacement and the main beam strain in the structural response index;

[0046] Fig.12 It is a schematic diagram of sampling verification of pier displacement and main beam strain in structural response indicators based on Copula joint probability density function;

[0047] Fig.13 The schematic diagram of sampling verification of bridge tower offset and expansion joint displacement in structural response indicators based on Copula joint probability density function;

[0048] Fig.14 The schematic diagram of sampling verification of pier offset and expansion joint displacement in structural response indicators based on Copula joint probability density function;

[0049] Fig.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

[0050] The present application is further described below in conjunction with the accompanying drawings, and the structure and principle of the present application are very clear to people in the field. It should be understood that the specific embodiments described herein are only used to explain the present application and are not used to limit the present application.

[0051] Figure 1 A schematic block diagram of an embodiment of a copula-based joint probability modeling method for bridge response and load of the present application; Figure 2 This is a flow chart of an embodiment of a joint probability modeling method of bridge response and load based on Copula of the present application. Figure 3 Schematic diagram of the probability distribution model of bridge tower displacement in the structural response index; Figure 4 It is a schematic diagram of the probability distribution model of the pier displacement in the structural response index; Figure 5 It is a schematic diagram of the probability distribution model of the main beam strain in the structural response index; Figure 6 It is a 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 It is a schematic diagram of the joint probability density function of the main beam strain and the pier displacement in the structural response index; Fig. 9 It is a schematic diagram of the joint probability density function of expansion joint displacement and bridge tower displacement in the structural response index; Fig.10 It is a schematic diagram of the joint probability density function of expansion joint displacement and pier displacement in the structural response index; Fig.11 It is a schematic diagram of the joint probability density function of the expansion joint displacement and the main beam strain in the structural response index; Fig.12 It is a schematic diagram of sampling verification of pier displacement and main beam strain in structural response indicators based on Copula joint probability density function; Fig.13 The schematic diagram of sampling verification of bridge tower offset and expansion joint displacement in structural response indicators based on Copula joint probability density function; Fig.14 The schematic diagram of sampling verification of pier offset and expansion joint displacement in structural response indicators based on Copula joint probability density function; Fig.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.

[0052] 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 11 The Copula Density in represents the joint probability density function. Figures 12 to 15 Measured represents the measured distribution, and Simulated represents the estimated distribution.

[0053] Combination Figures 1 to 15 It can be understood that the method of the present application may include the following steps.

[0054] Step S1: Collect structural response indicators and load parameters through the structural health monitoring system, perform probability density modeling on the structural response indicators and load parameters through the Gaussian mixture model, and screen and fit the optimal probability distribution model of each structural response indicator and load parameter through the Bayesian information criterion. The structural response indicators include the bridge tower offset , Pier offset , beam strain and expansion joint displacement , load parameters include temperature TMP, wind speed ,wind direction and traffic load .

[0055] It should be noted here 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 only on single variable PDFs will ignore the key correlations and dependencies between different aspects of structural behavior."

[0056] Step S2: The Pearson correlation coefficient is used to analyze the correlation between the variable pairs consisting of the structural response index and the load parameter, and to determine the strength and direction of the correlation.

[0057] Step S3: The joint probability density function between variables is constructed by combining the fitted variable probability distribution marginal model through the Copula probability density modeling method.

[0058] Step S4: Using the Monte Carlo sampling method to verify the constructed joint probability density function.

[0059] Next, each step is described in detail.

[0060] Step S1 includes:

[0061] Step S101: The 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 the integrity and reliability of the data.

[0062] 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 basic function. The Gaussian mixture model (GMM) can adapt to a variety of 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 value. And the covariance is .

[0063] Step S103: In order to determine the optimal distribution, the present invention uses the Bayesian Information Criterion (BIC) to select the marginal distribution fit. 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 the overfitting problem. In general, models with smaller BIC values ​​have higher priority among candidate models. The calculation formula of BIC is as follows:

[0064] ,

[0065] 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.

[0066] 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 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.

[0067] Step S2 includes:

[0068] 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:

[0069] ,

[0070] in, and are the observed values ​​of the two variables, and is the mean of the variable.

[0071] Step S202: According to the calculated correlation coefficient, select the variable pairs with significant correlation. Variable pairs with high positive correlation (such as r > 0.7) or high negative correlation (such as r <-0.7) are identified as key variables that need to be jointly modeled. 0.7 is the correlation coefficient threshold, that is, according to the calculated correlation coefficient, select the variable pairs with significant correlation, and identify the 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 to be jointly modeled.

[0072] Step S203: By drawing a correlation matrix diagram, the correlation distribution between variables is intuitively displayed to further verify the accuracy and rationality of the data analysis results. The screened high-correlation variable set is used as the basic input for joint probability modeling to provide a basis for subsequent Copula function modeling to ensure the rationality and effectiveness of joint modeling.

[0073] Step S3 includes:

[0074] Step S301: In the framework of Gaussian Copula, 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:

[0075] ,

[0076] 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 a covariance matrix equal to the correlation matrix.

[0077] 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 normalization of variables under the given correlation between variables. The probability of jointly taking a particular 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:

[0078] .

[0079] Step S303: Using the properties of the multivariate normal distribution, the Gaussian Copula density function can model the complex dependency patterns between variables in a way that is 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:

[0080] ,

[0081] Where u represents the normal distribution vector .

[0082] Step S4 includes:

[0083] 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 through the selected Copula function to generate joint distribution samples; set the number of samples (such as 2000 points) to ensure that the number of samples is sufficient to reflect the overall characteristics of the model.

[0084] Step S402: Visualization of sampling results. Visualize the generated joint distribution samples: Use scatter plots 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, it means that the fitting effect is good. Compare the sampling results with the observed data to verify whether the model can accurately capture the dependency relationship between variables.

[0085] Step S403: Model suitability and prediction performance evaluation. The accuracy of the quantitative model is tested by goodness of fit, such as calculating the deviation between the distribution of sampling points and the distribution of observed data; the prediction performance of the model is evaluated by calculating the joint probability value under different variable combinations; verifying whether the model can accurately predict the variable relationship and uncertainty characteristics in the actual scenario; if the results show that the model fits well and has excellent prediction performance, it is confirmed that the model meets the engineering requirements; if there is a deviation, 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, and the step S3 includes: 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 ; 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 is 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 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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