Expansion circle line type Copula method for wind speed and wind direction joint probability modeling
By introducing an additional degree-of-freedom parameter β into the traditional circular Copula model, an extended QS Copula function is constructed, which solves the problem of low accuracy in wind speed and wind direction modeling, achieves higher accuracy in wind load assessment, and improves the safety of engineering structures.
Patent Information
- Application Number
- CN202511479838.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-10-16
- Publication Date
- 2026-01-23
AI Technical Summary
The existing circular QS Copula model has too few degrees of freedom, resulting in limited applicability and low accuracy, making it difficult to effectively characterize the complex correlation between wind speed and wind direction angle.
An additional degree-of-freedom parameter β is introduced into the traditional circular Copula model to construct an extended QS Copula function, thereby improving the modeling accuracy of the joint probability distribution of wind speed and wind direction angle.
It significantly expands the applicability and modeling accuracy of the model, and can more effectively capture the complex periodic correlation characteristics between wind speed and wind direction angle, thereby improving the reliability of wind load assessment and the safety of engineering structures.
Smart Images
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Abstract
Description
Technical Field
[0001] This invention belongs to the field of structural wind engineering technology, specifically relating to an extended circular Copula method for joint probabilistic modeling of wind speed and direction. Background Technology
[0002] Wind load is a major environmental load on wind-sensitive structures such as high-rise buildings, long-span bridges, wind turbines, and photovoltaic supports, significantly impacting their long-term service life and safety performance. To accurately assess the wind load on a structure, various wind conditions it may experience during its service life must be considered. Since wind speed and direction significantly influence the wind load effect on a structure, wind direction and wind speed are typically used to describe wind conditions during service. During long-term service, wind conditions exhibit significant randomness; therefore, wind speed and wind direction are two random variables. Furthermore, due to climatic conditions, there is a complex correlation between wind speed and wind direction at the site. To reasonably describe the long-term wind conditions at the site, a joint probability distribution model of wind speed and wind direction needs to be constructed based on long-term site observation data.
[0003] There is considerable research on the joint probability distribution modeling of wind speed and wind direction angle, using methods such as conditional probability modeling and Copula models to obtain the joint probability distribution of these two parameters. While conditional probability modeling can capture the dependencies between variables to some extent, it cannot characterize the periodic variation correlation between wind direction angle, a circular variable, and wind speed. Currently widely used Copula models are typically line-to-line Copulas, such as elliptical Copulas and Archimedesian Copulas, which also cannot effectively describe the probabilistic correlation structure involving circular variables like wind direction angle. For modeling the probabilistic correlation between circular-to-line variables like wind direction angle and wind speed, there are relatively few usable circular-to-line Copula models, mainly QS Copula, Johnson & Wehrly Copula, and Bernstein Copula models. Among them, QS Copula has the simplest form, making it relatively easy and widely used. However, the QS Copula function contains only one variable parameter, limiting its applicability and accuracy, and in some cases, it is difficult to accurately characterize the correlation between wind speed and wind direction angle. Therefore, the core technical problem to be solved by this invention is how to expand the applicability of QSCopula by introducing additional degrees of freedom parameters while ensuring the ability to characterize periodic correlation features, thereby improving the modeling accuracy of the joint probability distribution of wind speed and wind direction angle. Summary of the Invention
[0004] To address the aforementioned problems, the purpose of this invention is to provide an extended circular Copula method for modeling the joint probability distribution of wind speed and wind direction angle, solving the issues of limited applicability and low accuracy in existing circular QS Copula models due to insufficient degrees of freedom. Based on the traditional circular Copula joint probability model, additional degrees of freedom parameters are introduced to expand the applicability of the traditional circular Copula function and improve the accuracy of modeling the joint probability distribution between circular variables (wind direction angle) and linear variables (wind speed).
[0005] To achieve the above objectives, the present invention provides the following technical solution:
[0006] An extended circular Copula method for joint probabilistic modeling of wind speed and direction specifically includes the following steps:
[0007] (1) Estimate the marginal probability distribution of wind direction angle and wind speed
[0008] Based on long-term wind condition observation data at the site, observed sample values of wind speed X and wind direction angle Θ are extracted. Then, based on the long-term site observation data, marginal probability distribution models for wind speed and wind direction angle are established respectively. The marginal probability distribution function of wind speed is denoted as F. X The marginal probability distribution function of the wind direction angle (x) is denoted as F. Θ (θ); x is the sample value of wind speed, and θ is the sample value of wind direction angle;
[0009] (2) A probabilistic correlation model of wind speed and wind direction angle is constructed using the extended QS Copula function.
[0010] Based on the traditional single-parameter QS Copula function, an additional degree of freedom parameter β is introduced to construct an extended circular Copula function, the expression of which is: Where u = F Θ (θ) and v = F X (x) represents the marginal probability distribution functions of wind direction angle and wind speed, respectively.
[0011] The corresponding Copula density function is:
[0012] Subsequently, based on n observation samples of wind speed and wind direction angle... Calculate the empirical probability distribution function value of the sample Then, the maximum likelihood estimation method is used to determine the two undetermined parameters λ and β in the model, so as to accurately characterize the probabilistic correlation structure between wind speed and wind direction angle;
[0013] (3) Obtain the joint probability distribution model and implement random sampling.
[0014] Based on Sklar's theorem, and combined with the marginal probability distribution model F of wind speed and wind direction angle obtained in step (1), X (x) and F Θ (θ), and the extended QS Copula model determined in step (2), the joint probability density function of wind speed and wind direction angle is:
[0015] f ΘX (θ,x)=[1+2πλcos(2πu)(1―vβ―βvβ)]f Θ (θ)f X (x); where f Θ (θ) and f X (x) are the marginal probability density functions for wind direction angle and wind speed, respectively;
[0016] To perform joint random sampling of wind speed and wind direction angle, the conditional distribution function of wind direction angle at a given wind speed needs to be obtained: F Θ|X (θ|x)=u+λsin(2πu)(1―v β ―βv β Let θ,x be the sampled data of Θ,X, and let u,v be the corresponding probability distribution functions. Then the specific sampling process is as follows:
[0017] (a) Generate two independent random numbers z1 and z2 that are uniformly distributed in the interval [0,1]. z1 and z2 are considered as random variables Z1 = F. X (x), Z2=F Θ|X For samples of (θ|x), then F X (x)=v=z1、F Θ|X (θ|x)=z2=u+λsin(2πu)(1―v β ―βv β );
[0018] (b) Substituting v = z1 into the conditional distribution function equation z2 = u + λsin(2πu)(1 - v β ―βv β In the original text, u is solved numerically.
[0019] (c) Finally, using the inverse function of the marginal distribution, the sample values of wind speed and wind direction angle are calculated: in, and F respectively Θ and F X The inverse function;
[0020] Repeat this process to obtain the required number of wind condition samples.
[0021] Preferably, in step (1), the wind speed variable is selected as a Weibull distribution, Gamma distribution, or Lognormal distribution parameter model as an alternative model, and the optimal distribution type is determined by parameter estimation and combined with the AIC criterion.
[0022] Preferably, in step (1), the circular variable wind direction angle is fitted using a mixed von Mises distribution model, and the parameters of different order models are obtained through maximum likelihood estimation, and the optimal model is selected according to the AIC criterion.
[0023] Preferably, the marginal probability density function of the wind direction In the formula, n is the order of the mixed von Mises distribution, usually taken as 1, 2, or 3; a i As weights, all a i The sum is 1; I0 is the zeroth-order modified Bessel function of the first kind; μ i k is the mean direction parameter. i This is a concentration parameter.
[0024] Compared with the prior art, the beneficial effects of the present invention are:
[0025] This invention significantly expands the applicability and modeling accuracy of the traditional circular Copula by introducing additional degrees of freedom parameters. Compared with existing methods, this invention can more effectively capture the complex periodic correlation between circular variables (wind direction angle) and linear variables (wind speed), overcoming the limitations of traditional models such as QS Copula, which struggle to accurately describe nonlinear correlation structures due to their single parameter. The constructed joint probability model not only significantly improves the accuracy of joint distribution modeling of wind speed and wind direction angle but also possesses greater applicability, covering traditional models as special cases. This method provides a more reliable theoretical basis for the probabilistic assessment of wind loads on wind-sensitive structures, contributing to the refinement and improvement of wind-resistant design and safety in engineering structures. Attached Figure Description
[0026] The accompanying drawings, which are included to provide a further understanding of embodiments of the invention and form part of this application, do not constitute a limitation thereof. In the drawings:
[0027] Figure 1 This is a flowchart illustrating the implementation of the extended circular Copula method for joint probability modeling of wind speed and direction proposed in this invention.
[0028] Figure 2 This is the fitting result of the bimodal von Mises distribution of the wind direction angle in this invention.
[0029] Figure 3 This is the Weibull distribution fitting result for wind speed according to the present invention.
[0030] Figure 4 This is a comparison of the joint distribution scatter plots of the original observation data and the generated data in this invention.
[0031] Figure 5 This invention extends the comparison of AIC values between the QS Copula model and the traditional QS Copula model.
[0032] Figure 6 This invention relates to the cumulative distribution function (CDF) curve of the bimodal von Mises distribution and the inverse transformation interpolation process. Detailed Implementation
[0033] To further illustrate the technical means and effects of the present invention in achieving its intended purpose, the following detailed description of the specific implementation methods, structures, features, and effects of the present invention, in conjunction with the accompanying drawings and preferred embodiments, is provided below.
[0034] like Figure 1 As shown, the extended circular Copula method for joint probabilistic modeling of wind speed and direction according to the present invention specifically includes the following steps:
[0035] (1) Estimate the marginal probability distribution of wind direction angle and wind speed
[0036] The extended circular Copula method proposed in this invention still falls within the Copula theoretical framework. Therefore, to establish a joint probability distribution model of wind speed and wind direction angle, it is necessary to first estimate the marginal probability distribution models of wind speed and wind direction angle. For wind speed (denoted as X), commonly used parameterized probability distribution models such as the Weibull distribution, Gamma distribution, and log-normal distribution can be used as candidate marginal probability distribution models. Then, parameter estimation is performed, and the optimal distribution type is selected according to the AIC criterion.
[0037] For the wind direction angle (denoted as Θ), a mixture von Mises distribution can be used for estimation, and the expression for the probability density function is as follows: In the formula, θ is the sample value of the wind direction angle, n is the order of the mixed von Mises distribution, which is usually taken as 1, 2, or 3; a i As weights, all a i The sum is 1; I0 is the zeroth-order modified Bessel function of the first kind; μ i k is the mean direction parameter. i Let F be the concentration parameter. Parameters of von Mises distribution models of different orders are obtained through maximum likelihood estimation, and the optimal model is selected according to the AIC criterion. For ease of calculation, the marginal probability distribution functions for wind speed and heading angle are denoted as Fi. X (x) and F Θ (θ).
[0038] (2) The extended QS Copula model is used to model the correlation structure between wind direction angle and wind speed.
[0039] In current research and applications, the QS Copula function is often used to describe the probabilistic correlation structure between circular and linear random variables. The QS Copula function is denoted as CS. ΘX (u,v), where u=F Θ (θ) and v = F X (x) are the marginal probability distribution functions of two random variables, circular and linear, respectively. The expression for the QS Copula function is C. Θx (u,v)=uv+λsin(2πu)·v(1―v), and the corresponding expression for the Copula density function is c ΘX (y,v)=1+2πλcos(2πu)(1―2v), where λ is the Copula parameter to be determined.
[0040] From the expression for the QS Copula density function above, it can be seen that an important property of the circular Copula function is... This is the key to its ability to describe periodic correlation characteristics. However, the QS Copula function has only one undetermined parameter λ in its expression, resulting in low degrees of freedom in its functional form, which limits its applicability and, in some cases, its accuracy. Therefore, this invention, referencing the construction method of asymmetric Copula functions, proposes an extended QS Copula function model. The construction form is α and β are additional parameters. This is to ensure that the extended QSCopula still satisfies the basic properties of the circular-linear Copula function, i.e. Through derivation and analysis, it is found that the constructed extended QS Copula only satisfies the basic properties of the circular Copula function when α = 1. Therefore, the expression for the proposed extended QS Copula function is:
[0041] The corresponding extended QS Copula density function is expressed as follows: As can be seen, the extended QS Copula function has two undetermined parameters, namely λ and β, thus having more degrees of freedom and a wider range of applications.
[0042] Based on the extended QS Copula function form obtained above, and combined with the empirical probability distribution function sample values of wind speed-wind direction angle observation data, the parameters λ and β in the model can be estimated using the maximum likelihood estimation method, thereby obtaining the probability correlation structure model between wind direction angle and wind speed.
[0043] (3) Obtain the joint probability distribution model of wind speed and wind direction angle and perform random sampling.
[0044] Based on the fundamental theory of Copula, after obtaining the marginal probability distributions of wind speed and wind direction angle and the extended QS Copula model, their joint probability density function is: Where f Θ (θ) and f X (x) are the marginal probability density functions for wind direction angle and wind speed, respectively.
[0045] After obtaining the joint probability distribution model of wind speed and wind direction angle, a corresponding joint sampling method needs to be provided. In joint sampling of two-dimensional random variables based on the Copula method, the conditional probability distribution function of the secondary variable needs to be obtained. In structural wind load assessment, wind speed is usually taken as the dominant variable and wind direction angle as the secondary variable; therefore, the conditional probability distribution function of the wind direction angle needs to be obtained. According to the basic theory of Copula, the conditional probability distribution function can be expressed as... Based on the expression of the obtained extended QS Copula model F can be derived Θ|X (θ|x)=u+λsin(2πu)(1―v β ―βv β Therefore, the random sampling steps for wind speed and wind direction angle based on the extended QS Copula model are as follows:
[0046] (a) Let the random variable Z1 = F X (x), Z2=F Θ|X If (θ|x), then Z1 and Z2 are independent random variables uniformly distributed in the interval [0,1]. Therefore, it is easy to obtain sample points (Z1,Z2), denoted as (z1,z2). At this point, v=z1=F X (x), z2=F Θ|X (θ|x)=u+λsin(2πu)(1―v β ―βv β ).
[0047] (b) According to v = Z1 and Z2 = u + λsin(2πu)(1 - v) β ―βv β u can be obtained by reverse calculation.
[0048] (c) Since the marginal probability distribution model F for wind speed and wind direction angle has been obtained Θ (θ) and F X (x), according to u=F Θ (θ) and v = F X (x) can be used to calculate the sampling results of wind direction angle and wind speed, i.e. in, and F respectively Θ and F X The inverse function of .
[0049] When multiple samples need to be randomly selected, each time (z1, z2) is randomly selected independently and uniformly in the interval [0, 1]. Then, by following the three steps (a), (b), and (c), the corresponding sample points (θ, x) of wind direction and wind speed can be obtained.
[0050] The above describes the joint probability distribution modeling method for wind speed and direction based on the extended circular Copula method proposed in this invention. It expands the degrees of freedom of the existing QS Copula method and has higher modeling accuracy.
[0051] Implementation Case:
[0052] Taking observational data from the European Centre for Mesoscale Weather Prediction's reanalysis database as an example, wind speed and wind direction angle data for a certain site in the South my country Sea from 1979 to 2016 (a total of 38 years) were downloaded from this database. This data represents the average of 30-minute observations with a time resolution of 6 hours, meaning there are four data points per day. According to existing research, when performing probabilistic modeling of environmental parameters, classification modeling should be performed based on the different meteorological conditions under which the data occurred. Previous research indicates that the environmental data for this site can be divided into winter data (October 3rd to March 26th of the following year) and summer data. Furthermore, since the time series of environmental parameter observation data has short-term correlations, which affect the probabilistic modeling results, one data point is usually randomly selected from the weekly data, and then probabilistic analysis is performed based on the selected data. Therefore, this example yields 950 observational data points for wind speed and wind direction angle for subsequent analysis, as shown below. Figure 2 As shown.
[0053] According to the implementation steps of this invention, firstly, based on the above-mentioned observation data, the marginal probability distribution models for wind speed and wind direction angle are estimated respectively. The modeling results of the marginal probability distribution of wind speed using Weibull distribution, Gamma distribution, and Lognormal distribution are as follows: Figure 3 As shown in (a), the optimal marginal probability distribution for wind speed is determined to be the Weibull distribution model according to the AIC criterion, and the KS test proves the effectiveness of the modeling results. Marginal probability distribution models for wind direction angle are then performed using both first-order von Mises distribution and second-order mixed von Mises distribution, with the results shown in (a). Figure 3 As shown in (b), the marginal probability distribution of the optimal wind direction angle is determined according to the AIC criterion as a second-order mixed von Mises distribution, which can also be determined by the KS test.
[0054] Subsequently, the probabilistic correlation structure of wind speed and wind direction angle was estimated using both the traditional QS Copula model and the QS Copula model with extended degrees of freedom proposed in this invention, and the AIC values for the two cases were calculated to be -1.08 and -11.76, respectively. This demonstrates that establishing the probabilistic correlation structure of wind speed and wind direction angle using the QS Copula model with extended degrees of freedom proposed in this invention has higher accuracy.
[0055] Based on the obtained marginal probability distribution models of wind speed and wind direction angle, and the extended-degree-of-freedom QS Copula model, the sampling method presented in this invention can be used to generate samples of wind speed and wind direction. A comparison of scatter plots of the generated samples and the original data is provided below. Figure 4 As shown in the figure, the probability distribution characteristics of the wind speed and wind direction angle samples obtained by the method of this invention match the original data well, which initially demonstrates the effectiveness of this invention. To further clarify the comparison of the probability characteristics of the generated samples and the original data, Figure 5 Statistical histograms for two types of data are given. Figure 6 Empirical probability distribution functions for the two types of data are presented. It is evident that the probability distribution characteristics of the samples obtained using the method of this invention agree well with the original data, further validating the effectiveness of the method.
[0056] The above results demonstrate that the extended circular Copula method proposed in this invention for joint probability modeling of wind speed and wind direction angle can effectively characterize the complex correlation between the two variables. Compared with the traditional QS Copula method, it has higher accuracy and has important engineering application value for evaluating the stochastic wind load on wind-sensitive structures.
[0057] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.
Claims
1. An extended circular Copula method for joint probabilistic modeling of wind speed and direction, characterized in that, Specifically, the following steps are included: (1) Estimate the marginal probability distribution of wind direction angle and wind speed Based on long-term wind condition observation data at the site, observed sample values of wind speed X and wind direction angle Θ are extracted. Then, based on the long-term site observation data, marginal probability distribution models for wind speed and wind direction angle are established respectively. The marginal probability distribution function of wind speed is denoted as F. X The marginal probability distribution function of the wind direction angle (x) is denoted as F. Θ (θ); x is the sample value of wind speed, and θ is the sample value of wind direction angle; (2) A probabilistic correlation model of wind speed and wind direction angle is constructed using the extended QS Copula function. Based on the traditional single-parameter QS Copula function, an additional degree of freedom parameter β is introduced to construct an extended circular Copula function, the expression of which is: Where u = F Θ (θ) and v = F X (x) represent the marginal probability distribution functions of wind direction angle and wind speed, respectively, and the corresponding Copula density function is: Subsequently, based on n observation samples of wind speed and wind direction angle... Calculate the empirical probability distribution function value of the sample Then, the maximum likelihood estimation method is used to determine the two undetermined parameters λ and β in the model, so as to accurately characterize the probabilistic correlation structure between wind speed and wind direction angle; (3) Obtain the joint probability distribution model and implement random sampling. Based on Sklar's theorem, and combined with the marginal probability distribution model F of wind speed and wind direction angle obtained in step (1), X (x) and F Θ (θ), and the extended QS Copula model determined in step (2), the joint probability density function of wind speed and wind direction angle is: f ΘX (θ,x)=[1+2πλcos(2πu)(1―v β ―βv β )]f Θ (θ)f X (x); where f Θ (θ) and f X (x) are the marginal probability density functions for wind direction angle and wind speed, respectively; To perform joint random sampling of wind speed and wind direction angle, the conditional distribution function of wind direction angle at a given wind speed needs to be obtained: F Θ|x (θ|x)=u+λsin(2πu)(1―v β ―βv β Let θ,x be the sampled data of Θ,X, and let u,v be the corresponding probability distribution functions. Then the specific sampling process is as follows: (a) Generate two independent random numbers z1 and z2 that are uniformly distributed in the interval [0,1]. z1 and z2 are considered as random variables Z1 = F. X (x), Z2=F Θ|X For samples of (θ|x), then F X (x)=v=z1、F Θ|x (θ|x)=z2=u+λsin(2πu)(1―v β ―βv β ); (b) Substituting v = z1 into the conditional distribution function equation z2 = u + λsin(2πu)(1 - v β ―βv β In the original text, u is solved using numerical methods. (c) Finally, using the inverse function of the marginal distribution, the sample values of wind speed and wind direction angle are calculated: in, and F respectively Θ and F X The inverse function; Repeat this process to obtain the required number of wind condition samples.
2. The extended circular Copula method for joint probabilistic modeling of wind speed and direction as described in claim 1, characterized in that, In step (1), the wind speed variable is selected from Weibull distribution, Gamma distribution or Lognormal distribution parameter model as the candidate model, and the optimal distribution type is determined by parameter estimation and combined with the AIC criterion.
3. The extended circular Copula method for joint probabilistic modeling of wind speed and direction as described in claim 1, characterized in that, In step (1), the circular variable wind direction angle is fitted using a mixed von Mises distribution model. The parameters of different order models are obtained through maximum likelihood estimation, and the optimal model is selected according to the AIC criterion.
4. The extended circular Copula method for joint probabilistic modeling of wind speed and direction as described in claim 3, characterized in that, The marginal probability density function of the wind direction In the formula, n is the order of the mixed von Mises distribution, usually taken as 1, 2, or 3; a i As weights, all a i The sum is 1; I0 is the zeroth-order modified Bessel function of the first kind; μ i k is the mean direction parameter. i This is a concentration parameter.