Index construction method for representing sensitivity of multi-dimensional joint probability model

Abstract

The invention relates to an index construction method for representing sensitivity of a multi-dimensional joint probability model. The index construction method comprises the following steps: step 1, carrying out generalized extreme value distribution edge fitting on each single-variable sample; according to the edge distribution, T-year once-encounter extreme values in different return periods are calculated; converting the sample into a [0, 1] standard probability space; 2, constructing a three-dimensional joint probability model; 3, based on each three-dimensional joint probability model, generating three-dimensional environment contour surfaces in different return periods by adopting an inverse first-order reliability method to obtain a joint extreme load combination; and 4, constructing a composite disaster index based on a univariate extreme value and a combined load combination. According to the method, the problem of significant uncertainty of an existing multi-dimensional joint probability model in extreme marine environment load inference can be solved, and the defect that different models are lack of unified sensitivity evaluation indexes is overcome. And a stable and reliable basis is provided for environmental load selection of structures such as offshore wind power and offshore oil and gas platforms.

Description

Technical Field

[0001] This invention belongs to the field of marine disaster prevention and mitigation technology, specifically relating to a method for constructing an index to characterize the sensitivity of a multidimensional joint probability model. Background Technology

[0002] Typhoons are among the most destructive natural disasters affecting marine and nearshore areas. Their occurrence is typically accompanied by a combination of extreme environmental factors, including strong winds, high waves, storm surges, and strong currents. These wind-wave-current multi-hazard environments pose a significant threat to marine engineering structures such as cross-sea bridges, offshore oil and gas platforms, and offshore wind farms, whose structural responses are highly sensitive to the combined loads of these multiple factors. Therefore, accurately characterizing the interdependence between typhoon-induced wind speed, significant wave height, and current velocity is a crucial foundation for achieving structural reliability design and risk assessment.

[0003] Currently, several mature multivariate co-modeling methods exist. For example, Gaussian Copulas can conveniently describe correlation structures, but they are weak in characterizing heavy-tailed or asymmetric dependencies; Multivariate Extreme Value Theory (MEVT) can be used to describe tail dependencies in extreme regions; Nested Archimedesian Copulas are suitable for hierarchical dependency structures; and Vine Copulas can flexibly construct heterogeneous correlations through paired Copulas. Although the above methods have been applied in fields such as wind waves, water levels, and floods, research on modeling extreme events involving wind, waves, and current induced by typhoons remains limited.

[0004] Existing research mainly focuses on bivariate modeling of wind and waves or wave-related factors, while systematic evaluation of joint modeling of multiple hazards in 3D or even high-dimensional typhoons is significantly lacking. Relevant literature indicates that different joint probability models may produce significant differences in extreme value inference, joint return period assessment, and environmental isosurface construction, which is one of the biggest sources of uncertainty affecting marine engineering design values.

[0005] To address the aforementioned shortcomings, existing technologies lack a systematic, multi-model comparative sensitivity evaluation framework, failing to quantitatively characterize the impact of different multidimensional joint probability models on extreme environmental load estimation results. Therefore, it is necessary to construct a multidimensional joint probability model sensitivity index that integrates marginal distribution extremes, joint probability models, and environmental isosurfaces to guide the safety design and risk management of marine engineering structures in typhoon-prone environments. Summary of the Invention

[0006] The purpose of this invention is to address the significant uncertainties inherent in existing multidimensional joint probability models for inferring loads in extreme marine environments and to overcome the lack of a unified sensitivity evaluation index among different models. To this end, this invention proposes a method for constructing an index characterizing the sensitivity of multidimensional joint probability models. This method quantifies the differences in design values ​​derived from typhoon-induced wind-wave-current factors under different joint models, providing a robust and reliable basis for selecting environmental loads for structures such as offshore wind power plants and offshore oil and gas platforms.

[0007] This invention provides a method for constructing an index to characterize the sensitivity of a multidimensional joint probability model, comprising the following steps: The first step is to perform a generalized extreme value distribution marginal fit on each univariate sample to obtain the parameters of the marginal distribution of each univariate; then, based on the marginal distribution, calculate the parameters for different return periods. T The annual extreme value; and based on the CDF function of the fitted marginal distribution, the sample is transformed to the standard probability space of [0,1]. The second step is to construct a three-dimensional joint probability model for the samples transformed into the probability space; and to evaluate the fitting effect of the model by calculating its log-likelihood value, Akaike information criterion, and Bayesian information criterion. The third step is to use the constructed three-dimensional joint probability model to generate three-dimensional environmental isosurfaces under different return periods using the inverse first-order reliability method, thereby obtaining the joint extreme load combination. The fourth step involves constructing a Compound Hazard Index (CHI) based on the univariate extreme values ​​obtained in the first step and the combined load combination obtained in the third step, thereby quantifying the sensitivity of different combined probability models in describing extreme events.

[0008] Furthermore, the three-dimensional joint probability model is selected from any one of the following: Gaussian Copula model, symmetric Logistic multivariate extreme value distribution model, Vine Copula model, and nested Archimedesian Copula model.

[0009] Furthermore, the construction of the nested Archimedes Copula model involves the following steps: (1) Inner Copula Build exist , , Two-dimensional copulas are selected for pairwise variables, and the optimal model is determined using the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC), thereby obtaining the inner copulas. and its parameters ; (2) Outer Copula Build Will = and Constructing a new two-dimensional Copula (i.e., the outer Copula) ), capture and The dependency characteristics between them, where These are the parameters of the outer Copula model; (3) The final three-dimensional nested Copula model is composed of inner and outer Copulas, which fully describes the joint distribution structure of the three-dimensional variables, and is expressed as: .

[0010] For a three-dimensional nested Copula, the selection of the variable C2 in the inner Copula function adopts the maximum correlation criterion.

[0011] Furthermore, the construction of the Vine copula model involves the following steps: (1) Selecting variables As intermediate variables, establish respectively and Two-dimensional Copulas are used to characterize their dependencies; (2) Through the conditional distribution function and Calculate condition variables and Thus, samples of the second-level conditional Copula are obtained; (3) Fitting conditions Copula , to characterize in a given under conditions and Dependency structure.

[0012] More preferably, when the method for establishing the three-dimensional joint distribution model employs nested Copulas and symmetric Logistic multivariate extreme value distributions, the joint probability distribution function has a definite analytical form. (Conditional distribution function) and It can be directly determined by the following formula:

[0013] When the Vine Copula method is used to establish the three-dimensional joint distribution model, the conditional distribution function can be obtained by integrating the probability density function. When the method for establishing the three-dimensional joint distribution model adopts multivariate Gaussian Copula, the standard normal distribution variables are mapped to the original physical space using the Nataf transform; random vectors in the standard normal space Environmental variables in physical space The relationship between them is: ; ; .

[0014] Furthermore, in the third step, the establishment of the environmental isosurface includes the following steps: (1) IFORM method for determining the radius of a sphere in normal space Reliability index The relationship with the return period T is as follows: ; In the formula, It follows a standard normal distribution; (2) Generate normal space samples Reliability index With normal space samples The relationship is as follows: ; Polar angle / co-latitude angle; It is the azimuth angle; (3) Nataf transform or Rosenblatt transform to realize random vectors in standard normal space Environmental variables in physical space The conversion; Random vectors in standard normal space Environmental variables in physical space The relationship between them is: ; ; .

[0015] More preferably, the calculation of the composite disaster index CHI value in the fourth step includes the following two steps: (1) Calculate the hazard index HI as shown in the following formula: ; in, and Let represent the value of the i-th variable at the k-th point on the environmental isosurface, and represent the value obtained from the marginal probability distribution, respectively. TA value that occurs only once a year; (2) Calculate the Composite Hazard Index (CHI) using the following formula: ; ; In the formula, K and I These represent the total number of points considered on the isosurface and the total number of environmental variables, respectively. Let be the weight of the i-th variable.

[0016] This invention addresses the significant uncertainties inherent in existing multidimensional joint probability models for inferring loads in extreme marine environments, overcoming the lack of a unified sensitivity evaluation index among different models. This invention proposes a method for constructing an index characterizing the sensitivity of multidimensional joint probability models, used to quantify the differences in design values ​​derived from typhoon-induced wind-wave-current factors under different joint models, providing a robust and reliable basis for selecting environmental loads for structures such as offshore wind power and offshore oil and gas platforms. Attached Figure Description

[0017] Figure 1 This is a flowchart of the method proposed in this invention; Figure 2 To perform edge fitting of the generalized extreme value distribution (GEV) on the extracted extreme value sample dataset; Figure 3 This is a diagram illustrating the method of nesting Archimedean Copula and Vine Copula models; Figure 4 To evaluate the fitting performance of the four joint models; Figure 5 Environmental isosurfaces were established for different joint probability models; Figure 6 A comparison of the Composite Hazard Index (CHI) calculated by different models. Detailed Implementation

[0018] The present invention will be further described below with reference to a specific embodiment, but the invention is not limited to these specific embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0019] This invention provides a method for constructing an index to characterize the sensitivity of a multidimensional joint probability model, specifically including the following steps: The first step is to perform a generalized extreme value distribution (GEV) marginal fit on each univariate sample to obtain the parameters of the marginal distribution for each univariate. Then, based on the marginal distribution, the parameters for different return periods are calculated. TThe extreme value occurs once a year. Based on the CDF function of the fitted marginal distribution, the sample is transformed to the standard probability space of [0,1].

[0020] The second step involves constructing four three-dimensional joint probability models for the samples transformed into the probability space: the Gaussian Copula model, the symmetric Logistic Multivariate Extreme Value Distribution (MEVD) model, the Vine Copula model, and the nested Archimedes Copula model. The fitting performance of the four joint models is then evaluated by calculating their log-likelihood values, the Akaike Information Criterion (AIC), and the Bayesian Information Criterion (BIC).

[0021] The construction of the nested Archimedean Copula model involves the following steps: (1) Inner Copula Build exist , , By selecting appropriate two-dimensional copulas for each pair of variables and determining the optimal model using the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC), the inner copulas are obtained. and its parameters .

[0022] (2) Outer Copula Build Will = and Constructing a new two-dimensional Copula (i.e., the outer Copula) ), capture and The dependency characteristics between them, where These are the parameters of the outer Copula model; (3) The final three-dimensional nested Copula model is composed of inner and outer Copulas, which fully describes the joint distribution structure of the three-dimensional variables, and can be expressed as: .

[0023] For a three-dimensional nested Copula, the selection of the variable C2 in the inner Copula function adopts the maximum correlation criterion.

[0024] The construction of a Vine copula model involves the following steps: (1) Selecting variables As intermediate variables, establish respectively and Two-dimensional Copulas are used to characterize their dependencies; (2) Through the conditional distribution function and Calculate condition variables and Thus, samples of the second-level conditional Copula are obtained; (3) Fitting conditions Copula , to characterize in a given under conditions and The dependency structure. Through this decomposition method, the three-dimensional joint distribution can be accurately and flexibly represented as a combination of several low-dimensional copulas.

[0025] The third step involves generating three-dimensional environmental isosurfaces with different return periods based on the three-dimensional joint probability models, using the inverse first-order reliability method (IFORM), to obtain the joint extreme load combination.

[0026] The establishment of environmental isosurfaces involves the following two steps: (1) IFORM method for determining the radius of a sphere in normal space Reliability index The relationship with the return period T is as follows: In the formula, It follows a standard normal distribution; (2) Generate normal space samples Reliability index With normal space samples The relationship is as follows: ; (3) Nataf transform or Rosenblatt transform to realize random vectors in standard normal space Environmental variables in physical space The conversion.

[0027] Random vectors in standard normal space Environmental variables in physical space The relationship between them is, ; ; .

[0028] The fourth step involves constructing a Compound Hazard Index (CHI) based on the univariate extreme values ​​obtained in the second step and the combined load combination obtained in the third step, thereby quantifying the sensitivity of different combined probability models in describing extreme events.

[0029] The calculation of the CHI (Combined Hazard Index) value involves the following two steps: (1) Calculate the risk index HI The concept of the hazard index HI is defined as follows: ; in, and Let represent the value of the i-th variable at the k-th point on the environmental isosurface, and represent the value obtained from the marginal probability distribution, respectively. T A value that occurs only once a year.

[0030] (2) Calculate the Composite Hazard Index (CHI) using the following formula: ; ; In the formula, K and I These represent the total number of points considered on the isosurface and the total number of environmental variables, respectively. Let be the weight of the i-th variable.

[0031] It should be noted that when the chosen method for establishing the three-dimensional joint distribution is a nested Copula and a symmetric Logistic multivariate extreme value distribution, the conditional distribution function has a clear analytical form because its joint probability distribution function has a definite analytical form. and It can be directly determined by the following formula: ; .

[0032] When the chosen method for establishing the three-dimensional joint distribution is Vine Copula, the conditional distribution function can be obtained by integrating the probability density function.

[0033] When the three-dimensional joint distribution model used is a multivariate Gaussian Copula, since its dependency structure is characterized by the correlation coefficient matrix, the standard normal distribution variables can be mapped to the original physical space using the Nataf transformation. In this case, random vectors in the standard normal space... Environmental variables in physical space The relationship between them is: ; ; ; in, , , These are random variables in the standard normal space. and The linear correlation coefficient between them.

[0034] See Figure 1 The present invention provides a method for constructing an index to characterize the sensitivity of a multidimensional joint probability model. A specific application example includes the following steps: Step 1: First, select the sea area to be analyzed. In this implementation, the selected area is the offshore wind farm sites along the Chinese coast, from which 6 sites were selected. These sites are approximately 20-100 kilometers from the coastline, and their specific locations are shown in Table 1.

[0035] Table 1 Location information of the selected site .

[0036] To obtain the aforementioned physical quantities under typhoon conditions, this invention employs multiple datasets. Wind and wave data are derived from the ERA5 dataset. Since the ERA5 dataset does not include ocean current data, this invention uses a product from the Copernicus Marine Environment Monitoring Service (CMEMS), numbered MULTIOBS_GLO_PHY_MYNRT_015_003. Typhoon information is obtained from the China Meteorological Administration (CMA) optimal path dataset. For each site, typhoons passing through a circular area with a radius of 250 km centered on that site are selected, and the corresponding typhoon impact periods are extracted. Based on the same impact period for each typhoon event, time-series data of surface wind speed and significant wave height are obtained from the ERA5 dataset, and time-series data of ocean current velocity are obtained from the CMEMS dataset. The maximum values ​​of wind speed, significant wave height, and ocean current velocity corresponding to each typhoon event are calculated. The above steps are repeated for all typhoons in the CMA optimal path dataset and for all marine sites selected in this invention. The extreme values ​​of wind speed, significant wave height, and ocean current velocity corresponding to all typhoon events are recorded, forming an extreme value sample dataset.

[0037] Next, the extracted extreme value sample dataset is subjected to generalized extreme value distribution (GEV) marginal fitting to obtain the parameters of each univariate marginal distribution. For example... Figure 2 As shown in Table 2, the extreme values ​​of the T-year return period are calculated based on the marginal distribution. Then, the samples are transformed into the standard probability space [0,1] based on the CDF function of the fitted marginal distribution.

[0038] Table 2. Extreme values ​​of the T-year return period under different return periods .

[0039] Step 2: For the samples transformed into the probability space, construct four three-dimensional joint probability models: Gaussian Copula model, symmetric Logistic Multivariate Extreme Value Distribution (MEVD) model, nested Archimedes Copula model, and Vine Copula model. The methods for the nested Archimedes Copula model and the Vine Copula model are illustrated below. Figure 3 As shown. The fitting performance of the four joint models was evaluated, as follows: Figure 4 As shown in Table 3, the log-likelihood value, Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC) were calculated.

[0040] Table 3. Log-likelihood values, Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC) for four three-dimensional joint probability models. .

[0041] Step 3: Based on the established joint probability model, the inverse first-order reliability method (IFORM) is used to generate three-dimensional environmental isosurfaces under different return periods, thus obtaining the joint extreme load combination; such as Figure 5 As shown.

[0042] Step 4: Then, based on the single-variable extreme values ​​obtained above and the combined load combinations obtained on the environmental isosurface, a Compound Hazard Index (CHI) is constructed to quantify the sensitivity of different joint probability models in describing extreme events.

[0043] The calculation of the CHI (Combined Hazard Index) value involves the following two steps: (1) Calculate the risk index HI The concept of the hazard index HI is defined as follows: ; in, and Let represent the value of the i-th variable at the k-th point on the environmental isosurface, and represent the T-year return value obtained from the marginal probability distribution, respectively.

[0044] (2) Calculate the Composite Hazard Index (CHI) using the following formula: ; ; In the formula, K and I These represent the total number of points considered on the isosurface and the total number of environmental variables, respectively. Let be the weight of the i-th variable.

[0045] In this implementation, K=3, which considers the positions where the three variables on the environmental isosurface are at their maximum values.

[0046] Figure 6 A comparison of the calculated CHI (Combined Hazard Index) values ​​when using different models to establish a joint probability model is presented.

[0047] The above specific embodiments illustrate the principles and implementation methods of the present invention. These descriptions are merely for the purpose of helping to understand the method and core ideas of the present invention; furthermore, those skilled in the art will recognize that, based on the ideas of the present invention, there will be changes in specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A method for constructing an index to characterize the sensitivity of a multidimensional joint probability model, characterized in that, Includes the following steps: The first step is to perform a generalized extreme value distribution marginal fit on each univariate sample to obtain the parameters of the marginal distribution of each univariate; then, based on the marginal distribution, calculate the parameters for different return periods. T The annual extreme value; and based on the CDF function of the fitted marginal distribution, the sample is transformed to the standard probability space of [0,1]. The second step is to construct a three-dimensional joint probability model for the samples transformed into the probability space, and evaluate the fitting effect of the model by calculating its log-likelihood value, Akaike information criterion, and Bayesian information criterion. The third step is to use the constructed three-dimensional joint probability model to generate three-dimensional environmental isosurfaces under different return periods using the inverse first-order reliability method, thereby obtaining the joint extreme load combination. The fourth step involves constructing a composite disaster index based on the univariate extreme values ​​obtained in the first step and the combined load combination obtained in the third step, thereby quantifying the sensitivity of different joint probability models in describing extreme events.

2. The method for constructing an index to characterize the sensitivity of a multidimensional joint probability model according to claim 1, characterized in that, The three-dimensional joint probability model is selected from any one of the following: Gaussian Copula model, symmetric Logistic multivariate extreme value distribution model, Vine Copula model, and nested Archimedes Copula model.

3. The method for constructing an index to characterize the sensitivity of a multidimensional joint probability model according to claim 2, characterized in that, The construction of a nested Archimedes Copula model involves the following steps: (1) Inner Copula Build: exist , , Two-dimensional copulas are selected for pairwise variables, and the optimal model is determined using the Akaike information criterion or the Bayesian information criterion, thereby obtaining the inner copula. and its parameters ; (2) Outer Copula Build: Will = and Construct a new two-dimensional Copula, namely the outer Copula Capture and The dependency characteristics between them, where These are the parameters of the outer Copula model; (3) The final three-dimensional nested Copula model is composed of inner and outer Copulas, and is represented as: .

4. The method for constructing an index to characterize the sensitivity of a multidimensional joint probability model according to claim 3, characterized in that: For a three-dimensional nested Copula, the selection of the variable C2 in the inner Copula function adopts the maximum correlation criterion.

5. The method for constructing an index to characterize the sensitivity of a multidimensional joint probability model according to claim 2, characterized in that, The construction of a Vine Copula model involves the following steps: (1) Selecting variables As intermediate variables, establish respectively and Two-dimensional Copulas are used to characterize their dependencies; (2) Through the conditional distribution function and Calculate condition variables and Thus, samples of the second-level conditional Copula are obtained; (3) Fitting conditions Copula , to characterize in a given under conditions and Dependency structure.

6. The method for constructing an index to characterize the sensitivity of a multidimensional joint probability model according to claim 2, characterized in that, When establishing a three-dimensional joint distribution using nested Copulas and symmetric Logistic multivariate extreme value distributions, the conditional distribution function... and Determined by the following formula: ； ； When using Vine Copula to establish the three-dimensional joint distribution, the conditional distribution function is obtained by integrating the probability density function. When establishing a three-dimensional joint distribution model using multivariate Gaussian Copula, the standard normal distribution variables are mapped to the original physical space using the Nataf transformation. Random vectors in standard normal space Environmental variables in physical space The relationship between them is: ； ； 。 7. The method for constructing an index to characterize the sensitivity of a multidimensional joint probability model according to claim 1, characterized in that, In the third step, the establishment of the environmental isosurface includes the following steps: (1) IFORM method for determining the radius of a sphere in normal space Reliability index With recurrence period T The relationship is: ； In the formula, It follows a standard normal distribution; (2) Generate normal space samples Reliability index With normal space samples The relationship is as follows: ； Random vectors in the standard normal space can be realized using the Nataf transform or the Rosenblatt transform. Environmental variables in physical space The conversion; Random vectors in standard normal space Environmental variables in physical space The relationship between them is: ； ； 。 8. The method for constructing an index to characterize the sensitivity of a multidimensional joint probability model according to claim 1, characterized in that, The calculation of the composite disaster index value in the fourth step includes the following two steps: (1) Calculate the risk index HI The concept of the hazard index HI is defined as follows: ； in, and Let represent the value of the i-th variable at the k-th point on the environmental isosurface, and represent the value obtained from the marginal probability distribution, respectively. T A value that occurs only once a year; (2) Calculate the Composite Hazard Index (CHI) using the following formula: ； ； In the formula, K and I These represent the total number of points considered on the environmental isosurface and the total number of environmental variables, respectively. Let be the weight of the i-th variable.

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