A Copula Framework for Joint Probability Density of Wind Speed, Wind Direction, and Wind Attack Angle Based on Dirichlet Process Mixture Model

Abstract

The structural safety and performance of long-span bridges in coastal areas are significantly influenced by the wind field. Traditional univariate or simplified multivariate probabilistic models often fail to capture the multimodal and nonlinear dependencies among wind parameters, leading to inaccuracies in wind-induced risk assessment. This study proposes a novel joint probability density function framework for wind speed, wind direction, and wind attack angle, integrating a Dirichlet process mixture model (DPMM) for marginal distributions and a Regular Vine (R-Vine) copula for dependency modeling. The DPMM adaptively identifies the number of mixture components without presetting, effectively capturing multimodal and periodic characteristics of the wind field. The R-Vine copula flexibly models complex nonlinear dependencies among the three variables. A case study using field data from the Beikou Bridge demonstrates the capability to reveal seasonal wind patterns of the proposed model. By explicitly parametrizing the underlying wind regimes, the DPMM-based framework enables physically interpretable wind field decomposition. This provides direct support for probabilistic wind field modeling, supporting enhanced wind-resistant design and operational management of coastal long-span bridges.

1. Introduction

The rapid expansion of coastal transportation infrastructure has led to a growing number of long-span bridges in recent years [1]. The structural integrity and flexible performance inherent in the long-span bridges are sensitive to wind-induced hazards and the statistical characteristics of the wind field [2,3]. As span lengths increase, nonlinear structural behavior becomes more outstanding, and the interaction between the wind field and the structure grows significantly more influential [4]. Therefore, a comprehensive probability density function (PDF) model of wind field forms the basis for a robust probabilistic assessment, enables accurate evaluation of structural safety, and directly supports the long-term management strategies.

Owing to the highly fluctuating and nonlinear nature of measured wind field data, the accurate characterization of the wind field in various terrains and climate regions necessitates the application of multiple statistical probability models for fitting analysis [5]. Ozay and Celiktas applied the maximum likelihood (ML) to estimate the two parameters of the Weibull distribution for modeling the wind speed distribution in the Alaçatı region [6]. Wang et al. utilized several classic parametric probability distributions, including the Lognormal, Gamma, and Loglogistic, to fit the wind field [7]. Ding et al. adopted three parametric distribution functions for modeling wind speed on the long-span bridge: the Gumbel distribution, Weibull distribution, and Rayleigh distribution [8]. Despite advantages such as computational efficiency and parameter transparency, the single parametric models are limited by their predefined forms, which often result in systematic errors and an inability to resolve the multimodal structures of the wind field. Hence, for complex wind fields or high-precision applications, these simple models should be superseded by more flexible mixture models or nonparametric models [9]. Kollu et al. conducted a comparative study of three mixture distributions (Weibull-GEV, Weibull-Lognormal, GEV-Lognormal) against single parametric models and other mixture models for characterizing the wind field [10]. Mazzeo et al. proposed the Mixture of Two Truncated Normal Distributions (MTTNDs) based on a linear combination of two normal distributions for modeling bimodality or asymmetry of the wind speed distribution [11]. Wang et al. assessed the applicability of the mixture Weibull distribution (2W2W) in modeling the probability distribution of wind fields in engineering and compared it with the traditional single-component Weibull distribution [12]. Khamees et al. pointed out that the two-component mixed distribution is not the best choice in wind field modeling, and then introduced the three-component mixture Weibull distribution and the meta-heuristic optimization method to construct a more optimal wind speed probability model [13]. Wang et al. extended the limited-components of mixture model for estimating wind field PDF to the Bayesian infinite Gaussian mixture model, but still needed to set an upper limit for the number of components [14]. Wu et al. developed a truncated Gaussian mixture model for wind speed probability distribution by using the continuous ranked probability score (CRPS) loss function and the False Discovery Rate (FDR) algorithm [15]. Furthermore, the nonparametric density estimation methods (such as the maximum entropy theory and kernel density estimation) have the ability to adaptively fit complex wind speed distributions without any prior assumptions about the parameter model; thus, the advantages have been extensively studied by researchers [16,17,18].

A comprehensive analysis of statistical properties of the wind field requires moving beyond a single-parameter approach. The joint probability density function (JPDF) is a key statistical characteristic model for characterizing the dependence between random variables, with broad applications in wind engineering [19,20]. Carta et al. employed an Angular-linear (AL) model to construct a joint probability distribution for wind speed and direction by combining a Normal–Weibull mixture for wind speed with a finite mixture of von Mises distributions for wind direction [21]. Han et al. adopted the classic Johnson-Wehrly (JW) model structure to estimate the JPDF of wind speed and wind direction for characterizing their dependence in wind field analyses [22]. While the JW model can overcome the symmetry restriction that limits the AL method in complex scenarios, its reliance on subjective judgment for model selection and its limited capacity for explaining complex dependencies present significant drawbacks [23].

In the wind-resistant design of large-span bridges, the wind attack angle is a critical parameter influencing structural behavior [24]. The Copula method offers both flexible marginal fitting and accurate simulation of the complex interdependencies among all wind field parameters. Chen et al. proposed two Copula-based strategies to estimate the JPDF of wind speed, wind direction and wind attack angle at typical mountain bridge sites: Strategy I employed the Vine Copula framework in combination with the modified binary Bernstein Copula for modeling; Strategy II directly utilized the modified ternary Bernstein Copula for estimation [25]. Ding et al. characterized the marginal distribution (univariate PDF) of each variable by the finite mixture (FM) model, and used the copula model to integrate wind speed, direction, and attack angle into a unified trivariate JPDF at the bridge site [26]. Zhang et al. (2022, 2024) developed trivariate joint probability models for wind speed, direction, and angle of attack using on-site measurement data from a deep-cut gorge bridge site, employing distinct Copula structures in each study [27,28]. The 2022 model utilized a D-Vine structure with a Frank Copula, while the 2024 model was based on a C-Vine structure with Gumbel, mixed von Mises, and Logistic marginal distributions [27,28].

In general, the above studies have revealed different advantages of frameworks for JPDF of wind field, such as “more suitable”, “better fit”, “better performance”, or “is proposed”. However, for large-span bridges located in coastal areas with distinct seasonal climates, there are relatively few joint probability modeling systems for wind speed, wind direction, and wind attack angle. Most of the research pays more attention to the JPDF of wind speed with other factors in the coastal areas of bridges [29,30,31], but the wind horizontal and vertical components should be further considered. Consequently, the joint probability modeling of wind fields for large-span bridges in coastal areas demands a framework capable of both elucidating the intrinsic physical structure of the wind and fulfilling the requirements for computational efficiency in engineering. The development of methods specifically towards real engineering applications has been a shared emphasis in other structural engineering fields [32,33].

In this study, a JPDF modeling framework based on Copula theory is proposed for statistical characteristics analysis of wind fields. For marginal distribution modeling, the Dirichlet process mixture model (DPMM) is adopted: a Gaussian mixture model is used for wind speed and wind attack angle to capture their continuous multimodal characteristics, while a von Mises mixture distribution is employed for wind direction to strictly preserve the periodic nature of circular variables. In terms of dependency structure modeling, a parametric Vine Copula method is applied to effectively capture complex nonlinear dependencies among multiple variables. The framework achieves a balance between data-driven physical mechanism representation and computational efficiency optimization, avoiding both the oversimplified assumptions of purely physical models and the high computational costs of fully nonparametric methods, while strictly respecting the physical nature of each variable, which includes the periodicity of wind direction and the non-negativity of wind speed. Compared with existing Copula-based models that predominantly adopt single parametric distributions or finite mixtures with prespecified component counts, the DPMM adopted here adaptively infers the number of wind regimes directly from the data. Each resulting component corresponds to a physically interpretable meteorological state, enabling data-driven regime discovery without prior assumptions on the underlying wind climate structure. The research outcome provides a comprehensive solution for wind field characteristic modeling in coastal bridge engineering, combining physical rigor with computational feasibility.

2. Dirichlet Process Mixture Model

2.1. The DPMM in the Stick-Breaking Construction

The Dirichlet process (DP) was introduced by Ferguson in 1973 [34], which is widely used in Bayesian nonparametric inference. As a prior on the space of probability measures, the DP addresses both uncertainty in the number of clusters and data-adaptive density estimation [35].

Let $X=(x_1,x_2,\cdots ,x_n)$ be a sample space, $G_0$ a base distribution on $X$, $\alpha >0$ a positive scaling parameter. A random measure $G$ follows a DP, denoted:

$$G|\left\{\alpha ,G_0\right\}\sim \mathrm{DP}\left(\alpha ,G_0\right)$$

(1)

Thus, for any finite partition $(x_1,x_2,\cdots ,x_n)$, the random vector $(G(x_1),G(x_2),\cdots ,G(x_n))$ follows a Dirichlet distribution:

$$\left(G\left(x_1\right),G\left(x_2\right),\cdots ,G\left(x_N\right)\right)\sim Dirichlet\left(\alpha G_0\left(x_1\right),\alpha G_0\left(x_2\right),\cdots ,\alpha G_0\left(x_N\right)\right)$$

(2)

where $Dirichlet(•)$ is the Dirichlet distribution. Moreover, the DPMM under stick-breaking construction admits an infinite number of components with random weights [36], the random measure $G$ is expressed as follows:

$$G={\displaystyle \sum _{k=1}^{\infty }{\pi }_k{\delta }_{{\theta }_k}}$$

(3)

where ${\delta }_{{\theta }_k}$ is the indicator function located at the atom ${\theta }_k$ and ${\theta }_k$ is drawn independently from the base distribution $G_0$; ${\pi }_k={\beta }_k{\displaystyle {\prod }_{j=1}^{k-1}(1-{\beta }_j)}$ is the weight constructed by the stick-breaking process: ${\beta }_k|\alpha \sim \mathrm{Beta}\left(1,\alpha \right)$ and ${\displaystyle {\sum }_{k=1}^{\infty }{\pi }_k}=1$. Each observation $x_i$ is associated with a latent variable $z_i$ that indicates its component assignment. The DPMM is defined by the following generative process: (1) the collection of components ${\left\{{\beta }_k\right\}}_{k=1}^{\infty }$ independently drawn from a Beta distribution with parameters 1 and $\alpha$. (2) the collection of components ${\left\{{\theta }_k\right\}}_{k=1}^{\infty }$ corresponding to each mixture component are independently drawn from $G_0$. (3) For each $x_i$, the latent variables $z_i$ is drawn from the prior distribution of weights. Conditioned on $z_i$ and the component parameters, the $x_i$ is generated from an observation model $p(\left.x_i\right|{\theta }_{z_i})$.

2.2. Accelerated Variational Bayesian Inference Algorithm

The accelerated variational Bayesian algorithm extends standard variational inference by introducing a soft truncation strategy for the DP model. While conventional methods select a simple parametric family for the variational distribution, the soft truncation allows the variational posterior to theoretically possess infinitely many components. However, for components beyond a prespecified level $T$ (i.e., $i>T$), the variational distributions are fixed to their priors and excluded from optimization. This ensures the variational family remains nested as $T$ increases, enabling adaptive model complexity selection. Accordingly, the intractable posterior of the DP mixture is approximated by a factorized variational distribution of the form:

$$q\left(\theta ,\beta ,z;\varphi \right)=\left[{\displaystyle \prod _{i=1}^L{q}_{{\beta }_i}\left({\beta }_i;{\varphi }_i^{\beta }\right)q_{{\theta }_i}\left({\theta }_i;{\varphi }_i^{\theta }\right)}\right]\left[{\displaystyle \prod _{i=1}^n{q}_{z_i}\left(z_i\right)}\right]$$

(4)

where $q_{{\beta }_i}({\beta }_i;{\varphi }_i^{\beta })$ and $q_{{\theta }_i}({\theta }_i;{\varphi }_i^{\theta })$ are parametric models with parameters ${\varphi }_i^{\beta }$ and ${\varphi }_i^{\theta }$, and $q_{z_i}(z_i)$ are discrete distributions over the component labels. To handle large-scale data efficiently, the dataset is divided among the leaf nodes of a KD-tree. The same responsibility distribution is assigned to all samples that belong to the same leaf, thereby avoiding point-wise updates. Complexity is further reduced by caching sufficient statistics at the node level.

The objective of the variational inference process would be to minimize the Kullback–Leibler divergence $KL\left[q\left(\theta ,\beta ,z;\varphi \right)‖q\left(\left.\theta ,\beta ,z\right|x,\lambda ,\alpha \right)\right]$, or equivalently, that minimize the free energy [37] in this paper, the form of free energy can be written as follows:

$$F={\displaystyle {\displaystyle \sum _{i=1}^K}\left\{E_{q_{{\beta }_i}}\left[\log \frac{q_{{\beta }_i}\left({\beta }_i;{\varphi }_i^{\beta }\right)}{p_{\beta }\left(\left.{\beta }_i\right|\alpha \right)}\right]+E_{q_{{\beta }_i}}\left[\log \frac{q_{{\theta }_i}\left({\theta }_i;{\varphi }_i^{\theta }\right)}{p\left(\left.{\theta }_i\right|\lambda \right)}\right]\right\}}-{\displaystyle \sum _A\left|n_A\right|\log {\displaystyle \sum _{i=1}^{\infty }\exp \left(S_{A,i}\right)}}$$

(5)

where $n_A$ is the number of data in the node outer node $A$ of KD-tree. Based on the stick-breaking construction, $p_{{\beta }_i}(\left.{\beta }_i\right|\alpha )$ and $q_{{\beta }_i}({\beta }_i;{\varphi }_i^{\beta })$ can be assumed as follows:

$$p_{\beta }\left(\left.{\beta }_i\right|\alpha \right)\sim \mathrm{Beta}\left({\alpha }_1,{\alpha }_2\right)$$

(6)

$$q_{{\beta }_i}\left({\beta }_i;{\varphi }_i^{\beta }\right)\sim \mathrm{Beta}\left({\varphi }_{i,1}^{\beta },{\varphi }_{i,2}^{\beta }\right)$$

(7)

To obtain an analytical solution for the variational update equations, this algorithm assumes conjugate exponential family distributions for both the prior and the variational posterior. Thus, the analytical solution for prior $p(\left.{\theta }_i\right|\lambda )$ and variational posterior $q_{{\theta }_i}({\theta }_i;{\varphi }_i^{\theta })$ in the exponential family are given by:

$$p\left(\left.{\theta }_i\right|\lambda \right)=h\left({\theta }_i\right)\exp \left\{{\lambda }_1{\theta }_i+{\lambda }_2\left(-a\left({\theta }_i\right)-a\left(\lambda \right)\right)\right\}$$

(8)

$$q_{{\theta }_i}\left({\theta }_i;{\varphi }_i^{\theta }\right)=h\left({\theta }_i\right)\exp \left\{{\varphi }_{i,1}^{\theta }{\theta }_i+{\varphi }_{i,2}^{\theta }\left(-a\left({\theta }_i\right)-a\left({\varphi }_i^{\theta }\right)\right)\right\}$$

(9)

where $h(•)$ is the base measure; $a(•)$ is the log-partition function. The form of $S_{A,i}$ can be written as follows:

$$S_{A,i}=E_{q_{{\theta }_i}}{\left[{\theta }_i\right]}^T{〈x〉}_A-E_{q_{{\theta }_i}}\left[a\left({\theta }_i\right)\right]$$

(10)

where ${〈x〉}_A$ denotes the average over all data contained in the node $A$. The optimal rules for different parameters with the KD-tree are as follows:

$${\varphi }_{i,1}^{\beta }={\alpha }_1+{\displaystyle \sum _A\left|n_A\right|}{q}_{z_A}\left(z_A=i\right)$$

(11)

$${\varphi }_{i,2}^{\beta }={\alpha }_2+{\displaystyle \sum _A\left|n_A\right|}{\displaystyle {\displaystyle \sum _{j=i+1}^{\infty }}{q}_{z_A}\left(z_A=j\right)}$$

(12)

$${\varphi }_{i,1}^{\theta }={\lambda }_1+{\displaystyle \sum _A\left|n_A\right|}{q}_{z_A}\left(z_A=i\right){〈x〉}_A$$

(13)

$${\varphi }_{i,2}^{\theta }={\lambda }_2+{\displaystyle \sum _A\left|n_A\right|}{q}_{z_A}\left(z_A=i\right)$$

(14)

where $q_{z_A}(z_A)$ is given by:

$$q_{z_A}\left(z_A=i\right)={\displaystyle \frac{\exp \left(S_{A,i}\right)}{{\displaystyle \sum _{j=1}^{\infty }\exp \left(S_{A,j}\right)}}}$$

(15)

3. Copula Framework

3.1. Pair-Copula Constructions

A Copula $C(•)$: ${[0,1]}^N\to [0,1]$ is a multivariate cumulative distribution function (CDF) with uniform marginal distributions on $[0,1]$. According to Sklar’s theorem [38], for any joint CDF $F$ with marginal distribution functions $F_1,F_2,\cdots ,F_N$ can be expressed as follows:

$$F\left(X_1,X_2,\cdots ,X_N\right)=C\left\{F_1(X_1),F_2(X_2),\cdots ,F_N(X_N)\right\}$$

(16)

If the marginal distributions are continuous, the copula $C(•)$ is unique. The joint probability density function $f$ can be expressed as the product of the copula density $c(•)$ and the marginal densities

$$f\left(X_1,X_2,\cdots ,X_N\right)=\mathrm{c}\left\{F\left(X_1\right),F\left(X_2\right),\cdots ,F\left(X_N\right)\right\}{\displaystyle \prod _{i=1}^N{f}_i\left(X_i\right)}$$

(17)

where $c(•)$ is the partial derivative of $C(•)$. Sklar’s theorem decouples marginal modeling from dependence modeling, which allows the $c(•)$ and $C(•)$ to be modeled independently. To address the issue of dimension caused by high-dimensional joint distributions, Joe (1997) [39] introduced the Pair-Copula Construction (PCC) framework, which decomposes a high-dimensional joint distribution into a product of conditional bivariate copulas. The general formula of conditional marginal density can be written as follows:

$$f(X\mid v)=c_{X{v}_j\mid v_{-j}}\left\{F(X\mid v_{-j}),F(v_j\mid v_{-j})\right\}\cdot f(X\mid v_{-j})$$

(18)

where $v$ is the n-dimensional vector; $v_j$ is the $j$ component of $v$; $v_{-j}$ is vector $v$ without its $j$ component. for every $j$, the formula of $F(X\mid v)$ can be written as follows:

$$F(X\mid v)={\displaystyle \frac{\mathsf{\partial }{C}_{X{v}_j\mid v_{-j}}\left\{F(x\mid v_{-j}),F(v_j\mid v_{-j})\right\}}{\mathsf{\partial }F(v_j\mid v_{-j})}},$$

(19)

For the special case where $v$ is univariate and $X$, $v$ are uniform, the conditional distribution of $F(X\mid v)$ is obtained by the h-function. That is

$$h(X,v)=F(X\mid v)={\displaystyle \frac{\mathsf{\partial }{C}_{X,v}\left(X,v\right)}{\mathsf{\partial }v}}$$

(20)

where the second parameter of $h(•)$ always denotes the conditioning variable. While the $h(•)$ provides the analytical mechanism to evaluate conditional distributions in a pair-copula construction, the sequential order in which these pair copulas are cascaded is not predetermined.

3.2. Vine Copula

Bedford & Cooke (2001, 2002) [40,41] introduced the Regular Vine (R-Vine) as a systematic framework for organizing all valid pair-copula decompositions. A $N$ dimensional R-Vine is defined by an ordered sequence of ($N-1$) trees. Each tree $T_i$ consists of a set of nodes $N_i$ and a set of edges $E_i$. The sequence of trees forms a valid vine when [42]:

(1)

Tree $T_1$ has a node set $N_1$ representing the $N$ random variables.

(2)

Tree $T_i$ (for $i\ge 2$) has a node set $N_i$ formed by the edges $E_{i-1}$ of the previous tree $T_{i-1}$.

(3)

Proximity Condition: For any edge $e=\left\{a,b\right\}\in E_i$ (where $i\ge 2$), it must hold that $\left|a\cap b\right|=1$.

As the most general vine copula structure, R-vine subsumes C-vine and D-vine as special cases. Its distribution-free properties and configurable architecture enable flexible adaptation to complex dependence patterns without parametric constraints [43]. For a model specified by the R-Vine structure, a set of copula families, and a set of parameters, the joint density function is given by the product of marginal densities and all pair-copula densities [44]:

$$f\left(X_1,X_2,\cdots ,X_N\right)={\displaystyle {\displaystyle \prod _{n=1}^N}{f}_n\left(X_n\right)}{\displaystyle {\displaystyle \prod _{i=1}^{N-1}}{\displaystyle \prod _{e\in E_i}{c}_{C_{e,a},C_{e,b}|D_e}\left\{F_{C_{e,a}|D_e}\left(X_{C_{e,a}}|X_{D_e}\right),F_{C_{e,b}|D_e}\left(X_{C_{e,b}}|X_{D_e}\right)\right\}}}$$

(21)

where $e=\left\{a,b\right\}\in E_i$ is an edge in tree $T_i$, connecting the random variables $X_{C_{e,a}}$ and $X_{C_{e,b}}$ conditioned on the set $X_{D_e}$. $C_{e,a}$,$C_{e,b}$ is the conditioned set of $e$, defined as follows:

$$C_{e,a}=a\backslash D_e$$

(22)

$$C_{e,b}=b\backslash D_e$$

(23)

where $D_e$ is the conditioning set of $e$, defined as follows:

$$D_e=a\cap b$$

(24)

Building an R-Vine copula model requires solving three intertwined tasks: (1) selecting the vine structure; (2) selecting a suitable bivariate copula family for each pair-copula in the structure; (3) estimating the parameters for all selected copulas. As the number of possible R-Vine structures grows super-exponentially with dimension, a manual approach is infeasible. Therefore, the Dißmann algorithm is adapted in this paper. Figure 1 shows the structure of an exemplary three-dimensional R-vine copula.

3.3. Regular Vine Model Selection and Estimation

The construction of an R-Vine copula model involves three steps: structure selection, copula family selection, and parameter estimation. As the number of possible R-Vine structures grows exponentially with dimension, the efficient sequential algorithm of Dißmann et al. (2013) [45] is adopted in this paper. This algorithm proceeds by capturing the strongest conditional dependencies at each level of the vine hierarchy through the following steps:

(1)

Starting from uniform pseudo-observations, the empirical Kendall’s τ matrix is computed for all variable pairs to serve as a nonparametric weight for dependence strength.

(2)

For each tree level ($T_1$ to $T_{N-1}$), select a Maximum Spanning Tree using absolute τ values as weights. For each edge, choose the best-fitting bivariate copula family by Bayesian Information Criterion ($BIC$) and estimate its parameters by maximum likelihood. The preliminary independence test based on τ is performed for each candidate pair. The independence copula is selected if independence is not rejected. The h-function is then applied to obtain the transformed variables for the next tree.

(3)

The algorithm outputs the full model specification in three lower triangular arrays: vine structure, copula families, and parameters.

This methodology yields a flexible characterization of high-dimensional dependence by automating the identification of prominent associations and the individual tailoring of pair-copula tail properties.

4. Goodness-of-Fit Test

This study employs a two-stage modeling framework comprising marginal PDF fitting for wind field variables, followed by the construction of the JPDF to capture their interdependencies [46]. To ensure rigorous evaluation across distinct modeling objectives and error sources, a hierarchical assessment scheme aligned with the DPMM-Copula structure is adopted, spanning three layers: univariate margins, bivariate copula dependence, and the JPDF.

At the univariate marginal distribution layer, Root Mean Square Error ($RMSE$), Coefficient of Determination ($R^2$), and Mean Absolute Error ($MAE$) are adopted to evaluate the fitting accuracy of the PDF and CDF. Their mathematical formulations are given below:

$$RMSE=\sqrt{{\displaystyle \frac{1}{L}}{\displaystyle {\sum }_{i=1}^L{\left(x_i-{\widehat{x}}_i\right)}^2}}$$

(25)

$$R^2=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^L{\left(x_i-{\widehat{x}}_i\right)}^2}}{{\displaystyle {\sum }_{i=1}^L{\left(x_i-\overline{x}\right)}^2}}}$$

(26)

$$MAE={\displaystyle \frac{1}{L}}{\displaystyle {\sum }_{i=1}^L\left|x_i-{\widehat{x}}_i\right|}$$

(27)

where $L$ is the number of data samples; $x_i$ is the $i$-th measured value of the data sample; ${\widehat{x}}_i$ is the $i$-th output value of the model; $\overline{x}$ is the sample mean of the measured value.

At the bivariate copula dependence structure layer, the $BIC$ is employed to evaluate the statistical validity and parsimony of the selected copula structure. The $BIC$ is defined as follows:

$$BIC=-2·\ln \left(\mathcal{L}\right)+k·\ln \left(L\right)$$

(28)

where $\mathcal{L}$ is the model likelihood, and $k$ is the number of model parameters.

At the JPDF layer, $RMSE$, $MAE$, and the Index of Agreement ($IA$) are utilized to quantify Copula model accuracy. The $IA$ formula is as follows:

$$IA=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^L{\left(x_i-{\widehat{x}}_i\right)}^2}}{{\displaystyle {\sum }_{i=1}^L{\left(\left|x_i-\overline{x}\right|+\left|x_i-\overline{x}\right|\right)}^2}}}$$

(29)

5. Case Study

5.1. Instrument and Data

This study is bases on the Beikou Bridge at the Oujiang River in Wenzhou, Zhejiang, China—a three-tower four-span double-layer continuous steel truss girder suspension bridge with a 2178 m main cable span (230 + 800 + 800 + 348). Its main cable is 1/10, with north and south spans measuring 213.6 m and 273.6 m, respectively. Wind speed, direction, and attack angle were recorded by sensors installed at four positions on the main girder. Refer to Figure 2 for further details.

It should be noted that this bridge is located in a coastal area with predominantly flat estuarine terrain and gentle hills to the north and south, resulting in relatively unobstructed wind flow from seaward directions. The region experiences strong seasonal influences, with wind field variations significantly affecting the structural behavior of the bridge.

In this study, the JPDF is constructed using 2023 wind data collected at 10 Hz from four 3-axis ultrasonic anemometers (F-06, F-08, F-14, and F-16). The main technical parameters of anemometers are presented in

Table 1. The main technical parameters of the anemometer.

Equipment	Drawing	Technical Parameters	Value
HD2003		Wind speed	0~65 m/s ± 1
		Wind direction	0~360° ± 1
		Wind attack angle	−90°~90° ± 1

.

The wind speed, direction, and angle of attack are analyzed based on 10 min mean values, a standard averaging interval in engineering applications. Prior to JPDF modeling, the raw data (sample size 10 Hz) is preprocessed to address missing values and outliers. Outliers are identified and removed using the Pauta criterion (3σ rule), which assumes that residual errors follow a normal distribution and defines a safety interval of ±3σ, beyond which the probability of occurrence is less than 0.3% [47]. Following outlier elimination, missing and excised data points are imputed by cubic spline interpolation [48], a method selected for its high smoothness and continuity, which are particularly well suited to the fluctuating components of wind field data. While this preprocessing procedure enhances the reliability and temporal consistency of the dataset, it does not distort the tails or the dependence structure.

5.2. Marginal PDFs for Wind Field

In this study, the DPMM is adopted for marginal density estimation of each variable (wind speed, wind direction, and wind attack angle). This step constitutes the first stage in the construction of the JPDF.

Specifically, a Gaussian mixture model is employed for wind speed and wind attack angle to capture their continuous multimodal characteristics, while a von Mises mixture model is applied to wind direction to strictly preserve the periodicity inherent in circular variables. This mixture model within the DPMM framework can adaptively determine the number of mixture components without a predefined structure. This advantage can avoid the functional form constraints and selection biases typical of conventional parametric models. As a benchmark for comparison, kernel density estimation (KDE) is also implemented and evaluated.

For the quantitative assessment of goodness-of-fit metrics, the continuous probability density estimates are compared against the empirical probabilities derived from the discretized bins. Accordingly, the domain of each variable is partitioned into intervals (bins), a procedure that balances statistical resolution with computational efficiency. The wind speed at the observation points is discretized in 0.5 m/s bins between site-specific minimum and maximum values. The wind direction is discretized into 36 bins (0° to 360°, 10° interval, measured clockwise from true north) [49]. For wind attack angle, the theoretical range spans from −90° to 90°; however, practical observations are typically more concentrated. To preserve statistical fidelity while maintaining valid bin coverage, a 2° bin spacing is adopted throughout this interval.

As wind direction belongs to circular data, the application of linear mixture models ignores periodicity. Therefore, the wind direction observations should be transformed onto the two-dimensional Euclidean plane, and a DP mixture of von Mises models (DPMM-vM) is adopted as the marginal model. For wind speed and wind attack angle, DP Gaussian mixture models (DPGMMs) are employed. The fitted PDFs and CDFs are compared with the actual bin values, enabling a quantitative assessment of the goodness-of-fit between the DPMM-based marginals (DPGMM and DPMvMM) and the KDE benchmark.

5.2.1. Marginal Model for Wind Speed

The marginal model of wind speed at the four positions (F-06, F-08, F-14, and F-16) is fitted respectively using DPGMM and KDE. The corresponding PDF and CDF estimates are displayed in Figure 3 and Figure 4.

Table 2. The goodness-of-fit results for wind speed PDFs.

Positions	Method	RMSE	MAE	R2
F-06	DPGMM	0.0015	0.0009	0.9998
	KDE	0.0011	0.0005	0.9993
F-08	DPGMM	0.0023	0.0012	0.9975
	KDE	0.0005	0.0003	0.9999
F-14	DPGMM	0.0016	0.0008	0.9978
	KDE	0.0011	0.0004	0.9990
F-16	DPGMM	0.0023	0.0013	0.9970
	KDE	0.0011	0.0005	0.9994

and

Table 3. The goodness-of-fit results for wind speed CDFs.

Positions	Method	RMSE	MAE	R2
F-06	DPGMM	0.0024	0.0022	0.9999
	KDE	0.0012	0.0005	0.9999
F-08	DPGMM	0.0026	0.0022	0.9999
	KDE	0.0015	0.0015	0.9999
F-14	DPGMM	0.0082	0.0080	0.9991
	KDE	0.0011	0.0005	0.9999
F-16	DPGMM	0.0017	0.0015	0.9999
	KDE	0.0012	0.0010	0.9999

show the goodness-of-fit results for marginal models of wind speed.

As indicated by the goodness-of-fit metrics presented in Table 2 and Table 3, both methods yield low $RMSE$ and $MAE$ values, with $R^2$ values close to 1, confirming satisfactory overall fit. It is worth mentioning that while KDE achieves slightly lower $RMSE$ and $MAE$ in some PDF fitting cases, the DPMM is still retained as the marginal model. Unlike KDE, which yields a nonparametric estimate without structural interpretability, the DPMM provides explicit parametric components that correspond to physically meaningful wind regimes and form the basis for the analysis of seasonal variation.

Table 4. The model parameters for marginal PDFs of wind speed.

Item	Position	Parameter	Value
Wind speed	F-06	μF-06	$\left[\begin{array}{ccccc}3.77& 2.36& 1.31& 5.22& 0.75\end{array}\right]$
		σF-06	$\left[\begin{array}{ccccc}1.10& 0.71& 0.42& 1.86& 0.22\end{array}\right]$
		πF-06	$\left[\begin{array}{ccccc}0.4068& 0.2899& 0.1275& 0.1191& 0.0567\end{array}\right]$
	F-08	μF-08	$\left[\begin{array}{ccccc}3.88& 2.78& 1.74& 1.04& 5.79\end{array}\right]$
		σF-08	$\left[\begin{array}{ccccc}1.10& 0.73& 0.46& 0.24& 2.07\end{array}\right]$
		πF-08	$\left[\begin{array}{ccccc}0.3396& 0.3143& 0.1828& 0.0946& 0.0687\end{array}\right]$
	F-14	μF-14	$\left[\begin{array}{ccccccc}5.56& 2.29& 3.43& 1.47& 3.69& 7.72& 0.99\end{array}\right]$
		σF-14	$\left[\begin{array}{ccccccc}1.56& 0.64& 0.95& 0.36& 1.01& 2.92& 0.19\end{array}\right]$
		πF-14	$\begin{array}{l}\left[\begin{array}{cccc}0.2689& 0.2195& 0.1526& \cdots \end{array}\right.\\ \left.\begin{array}{ccccc}\cdots & 0.1337& 0.0813& 0.0805& 0.0635\end{array}\right]\end{array}$
	F-16	μF-16	$\left[\begin{array}{ccccc}2.68& 4.04& 1.50& 6.13& 0.78\end{array}\right]$
		σF-16	$\left[\begin{array}{ccccc}0.81& 1.24& 0.49& 2.21& 0.24\end{array}\right]$
		πF-16	$\left[\begin{array}{ccccc}0.3403& 0.3400& 0.1697& 0.0790& 0.0710\end{array}\right]$

shows the model parameters for the marginal PDFs of wind speed. The variation in the number of DPGMM components along the bridge span, seven at F-14 versus four or five at the remaining positions, reflects pronounced spatial heterogeneity in the wind speed distribution. F-14 is located at the mid-span of the main girder (Figure 2). While the wind field at F-14 is subject to aerodynamic interference from the bridge towers and cables, it is largely unaffected by the surrounding terrain due to its distance from the shoreline. This observation confirms the effectiveness of the DPGMM in adaptively determining the required model complexity and providing high-quality marginal foundations for the subsequent Vine copula modeling.

5.2.2. Marginal Model for Wind Direction

Since the wind direction is circular data, the application of linear mixture models would ignore its inherent periodicity. Therefore, the DPMM-vM is adopted as the marginal model, with the number of components adaptively determined from the data. Each von Mises component is parameterized by a mean direction and a concentration parameter, naturally satisfying the periodic boundary condition $F(0)=F(2\pi )$. For the integration purpose in the copula framework, the fitted DPMM-vM cumulative distribution function is used to transform wind direction to uniform pseudo-observations. Since the CDF satisfies $F(\theta +2\pi )=F(\theta )+1$, observations near 0° and 360° map consistently to values near 0 and 1. The corresponding PDF and CDF estimates are displayed in Figure 5 and Figure 6.

Table 5. The goodness-of-fit results for wind direction PDFs.

Positions	Method	RMSE	MAE	R2
F-06	DPMM-vM	0.0079	0.0048	0.8912
	von Mises KDE	0.0047	0.0023	0.9610
F-08	DPMM-vM	0.0040	0.0031	0.9656
	von Mises KDE	0.0024	0.0014	0.9874
F-14	DPMM-vM	0.0035	0.0029	0.9640
	von Mises KDE	0.0016	0.0011	0.9929
F-16	DPMM-vM	0.0067	0.0051	0.9113
	von Mises KDE	0.0026	0.0017	0.9870

and

Table 6. The goodness-of-fit results for wind direction CDFs.

Positions	Method	RMSE	MAE	R2
F-06	DPMM-vM	0.0220	0.0197	0.9942
	von Mises KDE	0.0221	0.0203	0.9942
F-08	DPMM-vM	0.0261	0.0232	0.9924
	von Mises KDE	0.0262	0.0236	0.9923
F-14	DPMM-vM	0.0329	0.0249	0.9899
	von Mises KDE	0.0326	0.0244	0.9901
F-16	DPMM-vM	0.0213	0.0180	0.9941
	von Mises KDE	0.0229	0.0211	0.9931

show the goodness-of-fit results for marginal models of wind direction.

As summarized in Table 5 and Table 6, the goodness-of-fit metrics indicate that the DPMM-vM preserves the inherent periodicity of wind direction while maintaining a robust characterization of the global distributional form.

Table 7. The model parameters for marginal PDFs of wind direction.

Item	Position	Parameters	Value
Wind direction	F-06	μF-06	$\left[\begin{array}{ccccc}0.84& 2.15& 3.37& 5.22& 1.22\end{array}\right]$
		κF-06	$\left[\begin{array}{ccccc}20.65& 6.22& 7.59& 5.55& 6.34\end{array}\right]$
		πF-06	$\left[\begin{array}{ccccc}0.2908& 0.1818& 0.1695& 0.2247& 0.1332\end{array}\right]$
	F-08	μF-08	$\left[\begin{array}{ccccc}0.79& 1.81& 2.90& 4.94& 1.25\end{array}\right]$
		κF-08	$\left[\begin{array}{ccccc}29.42& 6.42& 5.99& 5.39& 6.54\end{array}\right]$
		πF-08	$\left[\begin{array}{ccccc}0.2504& 0.1832& 0.1991& 0.2523& 0.1150\end{array}\right]$
	F-14	μF-14	$\left[\begin{array}{ccccc}1.06& 2.00& 3.44& 4.96& 1.50\end{array}\right]$
		κF-14	$\left[\begin{array}{ccccc}21.99& 5.67& 4.40& 11.50& 4.78\end{array}\right]$
		πF-14	$\left[\begin{array}{ccccc}0.2018& 0.2263& 0.2638& 0.1879& 0.1202\end{array}\right]$
	F-16	μF-16	$\left[\begin{array}{cccc}0.70& 3.06& 5.15& 1.37\end{array}\right]$
		κF-16	$\left[\begin{array}{cccc}20.82& 4.02& 6.55& 3.14\end{array}\right]$
		πF-16	$\left[\begin{array}{cccc}0.2553& 0.2016& 0.2454& 0.2977\end{array}\right]$

provides the estimated mean direction, concentration parameter $\kappa$, and mixture weight for each component of the DPMM-vM. The prevailing wind direction intervals and their corresponding directional concentration are distinctly observable at each measurement site within the bridge area. For example, the dominant directional components at sites F-06 and F-08 are centered at approximately 0.84 rad and 0.79 rad, respectively, and are associated with notably high $\kappa$. These estimates indicate the presence of stable, well-defined prevailing wind directions within the study region.

5.2.3. Marginal Model for Wind Attack Angle

The marginal models of wind attack angle are modeled using the DPGMM. The corresponding PDF and CDF estimates are displayed in Figure 7 and Figure 8.

Table 8. The goodness-of-fit results for wind attack angle PDFs.

Positions	Method	RMSE	MAE	R2
F-06	DPGMM	0.0018	0.0006	0.9982
	KDE	0.0029	0.0007	0.9950
F-08	DPGMM	0.0020	0.0007	0.9956
	KDE	0.0004	0.0001	0.9998
F-14	DPGMM	0.0026	0.0009	0.9939
	KDE	0.0009	0.0002	0.9992
F-16	DPGMM	0.0044	0.0012	0.9875
	KDE	0.0038	0.0005	0.9906

and

Table 9. The goodness-of-fit results for wind attack angle CDFs.

Positions	Method	RMSE	MAE	R2
F-06	DPGMM	0.0014	0.0006	0.9999
	KDE	0.0123	0.0090	0.9993
F-08	DPGMM	0.0041	0.0029	0.9999
	KDE	0.0014	0.0009	0.9999
F-14	DPGMM	0.0030	0.0020	0.9999
	KDE	0.0007	0.0002	0.9999
F-16	DPGMM	0.0109	0.0079	0.9995
	KDE	0.0237	0.0170	0.9975

show the goodness-of-fit results for marginal models of wind attack angle.

Figure 8 illustrates the pronounced central tendency evident in the measured wind attack angle data.

As shown in

Table 10. The model parameters for marginal PDFs of wind attack angle.

Item	Position	Parameters	Value
Wind attack angle	F-06	μF-06	$\left[\begin{array}{cccc}-2.89& 11.91& 2.06& 12.15\end{array}\right]$
		σF-06	$\left[\begin{array}{cccc}1.28& 6.70& 1.74& 3.01\end{array}\right]$
		πF-06	$\left[\begin{array}{cccc}0.5536& 0.1895& 0.1522& 0.1047\end{array}\right]$
	F-08	μF-08	$\left[\begin{array}{ccccc}12.27& -0.96& 5.76& 7.91& 3.96\end{array}\right]$
		σF-08	$\left[\begin{array}{ccccc}5.14& 3.43& 1.95& 1.11& 0.87\end{array}\right]$
		πF-08	$\left[\begin{array}{ccccc}0.4346& 0.2167& 0.1261& 0.1163& 0.1063\end{array}\right]$
	F-14	μF-14	$\left[\begin{array}{cccc}8.56& 1.57& 3.69& 0.59\end{array}\right]$
		σF-14	$\left[\begin{array}{cccc}5.37& 0.98& 1.68& 0.60\end{array}\right]$
		πF-14	$\left[\begin{array}{cccc}0.7594& 0.1367& 0.0669& 0.0370\end{array}\right]$
	F-16	μF-16	$\left[\begin{array}{cccc}-1.05& 0.26& 15.57& -1.27\end{array}\right]$
		σF-16	$\left[\begin{array}{cccc}1.80& 4.84& 3.78& 0.73\end{array}\right]$
		πF-16	$\left[\begin{array}{cccc}0.3185& 0.2853& 0.2446& 0.1516\end{array}\right]$

, the relatively small estimated means and standard deviations of the individual mixture components indicate that the observed wind attack angle exhibits a limited range of fluctuation, concentrated within a considerably narrower interval than the theoretical bounds of −90° to 90°.

5.3. Seasonal Variation in the Wind Field

Bayesian inference in the DPMM framework provides the posterior component-assignment probabilities for each observation. These quantities serve as the basis for a seasonal analysis of the wind field. The monthly occurrence probabilities of the mixture components for wind speed, wind direction, and wind attack angle are shown in Figure 9, Figure 10 and Figure 11, respectively. This probabilistic decomposition reveals the seasonal variability of the wind climate at the bridge site and illustrates the annual progression of the dominant regional weather regimes.

As shown in Figure 3a and Figure 10a, at position F-06, the fourth Gaussian component of wind speed (characterized by the highest mean value of approximately 5.22 m/s) reaches its peak monthly probability of 39.75% in July, indicating that winds of this intensity dominate during this month.

Located within the subtropical monsoon climate of southeastern China, the bridge site exhibits a pronounced seasonal shift in wind direction components, as shown in Figure 10a. Specifically, the third wind direction component (mean direction: 194.01°, concentration parameter: 7.95) emerges as the dominant regime in July, accounting for a notably high occurrence probability of 60.35%. Concurrently, the third Gaussian component of the wind attack angle (mean: 2.06°, standard deviation: 1.74°) attains its maximum probability of 65.11% during the same month.

It is noteworthy that the highest mean wind speeds also have markedly elevated probabilities at the other locations in July: the fifth component at F-08 (mean: 5.79 m/s, 42.95%), the sixth at F-14 (mean: 7.72 m/s, 49.81%), and the fourth at F-16 (mean: 6.13 m/s, 30.48%). During the same month, the prevailing wind direction and wind attack angle components exhibit strong concentration, each dominated by a single regime.

The probabilistic decomposition identifies seasonal wind regimes and their annual progression, which are meaningful for the vulnerability assessment in the construction phase during the dominant wind period, maintenance scheduling during low-wind months, reliability assessment with regime-conditional limit states, and wind-traffic control decision-making.

5.4. Construction of JPDF Based on Vine Copula

Based on Sklar’s theorem, the JPDF can be decomposed into the marginal distributions of each variable and a Copula function that describes the dependency structure among the variables. Section 5.2 has already obtained the precise marginal models of wind speed, wind direction, and wind attack angle using a mixture model. Thus, the JPDF of the three variables is constructed by selection and fitting of the optimal R-vine structure and bivariate copulas, following the sequential algorithm of Dißman. The bivariate copula families employed in this study are listed in

Table 11. The bivariate copula families.

Copula	Formulation	Parameter Range
Gauss	$C_{\rho }(u,v)={\Phi }_2\left({\Phi }^{-1}(u),{\Phi }^{-1}(v);\rho \right)$	$\rho \in (-1,1)$
t	$C_{\rho ,\nu }(u,v)=t_{2,\nu }\left(t_{\nu }^{-1}(u),t_{\nu }^{-1}(v);\rho ,\nu \right)$	$\begin{array}{l}\rho \in (-1,1),\\ v\in [1,1000000]\end{array}$
Clayton	$C_{\theta }(u,v)={\left(u^{-\theta }+v^{-\theta }-1\right)}^{-{\displaystyle \frac{1}{\theta }}},$	$\rho \in [0.00001,150]$
Gumbel	$C_{\theta }(u,v)=\exp \left\{-{\left[{(-\log (u))}^{\theta }+{(-\log (v))}^{\theta }\right]}^{{\displaystyle \frac{1}{\theta }}}\right\},$	$\theta \in [1,120]$
Frank	$C_{\theta }(u,v)=-{\displaystyle \frac{1}{\theta }}\log \left({\displaystyle \frac{1-e^{-\theta }-(1-e^{-\theta u})(1-e^{-\theta v})}{1-e^{-\theta }}}\right)$	$\theta \in [-700,700]$
Ali–Mikhail–Haq	$C_{\theta }(u,v)={\displaystyle \frac{uv}{1-\theta (1-u)(1-v)}}$	$\theta \in [-1,1]$
Farlie–Gumbel–Morgenstern	$C_{\theta }(u,v)=uv(1+\theta (1-u)(1-v))$	$\theta \in [-1,1]$
Plackett	$\begin{array}{l}{C}_{\theta }(u,v)={\displaystyle \frac{1}{2(\theta -1)}}\times \cdots \\ \cdots \left(1+(\theta -1)(u+v)-{\left({(1+(\theta -1)(u+v))}^2-4\theta (\theta -1)uv\right)}^{{\displaystyle \frac{1}{2}}}\right)\end{array}$	$\theta \in [0,1000000]$
Joe	$C_{\theta }(u,v)=1-{\left({(1-u)}^{\theta }+{(1-v)}^{\theta }-{(1-u)}^{\theta }{(1-v)}^{\theta }\right)}^{{\displaystyle \frac{1}{\theta }}}$	$\theta \in [1,150]$
Survival Clayton	$C_{\theta }(u,v)={\left(u^{-\theta }+v^{-\theta }-1\right)}^{-{\displaystyle \frac{1}{\theta }}},$	$\theta \in [0.00001,150]$
Survival Gumbel	$C_{\theta }(u,v)=\exp \left\{-{\left[{(-\log (u))}^{\theta }+{(-\log (v))}^{\theta }\right]}^{{\displaystyle \frac{1}{\theta }}}\right\},$	$\theta \in [1,120]$
Survival Joe	$C_{\theta }(u,v)=1-{\left({(1-u)}^{\theta }+{(1-v)}^{\theta }-{(1-u)}^{\theta }{(1-v)}^{\theta }\right)}^{{\displaystyle \frac{1}{\theta }}}$	$\theta \in [1,150]$

.

Table 12. The optimal Copula model and R-Vine structure.

Position	Tree	Edge	Copula	Parameter	BIC
F-06	T1	23	Plackett	27.136	−5.8561 × 104
		13	Plackett	0.447	−3.6962 × 103
	T2	12|3	Ali–Mikhail–Haq	0.533	−1.9048 × 103
F-08	T1	23	Joe	1.632	−1.3913 × 104
		12	Farlie–Gumbel–Morgenstern	−0.363	−7.6686 × 102
	T2	13|2	t	[−0.131, 28.538]	−1.0302 × 103
F-14	T1	23	Ali–Mikhail–Haq	−1.000	−8.8872 × 103
		13	Frank	1.9616	−5.3369 × 103
	T2	12|3	Farlie–Gumbel–Morgenstern	−0.8206	−3.2166 × 103
F-16	T1	23	Survival Clayton	1.0189	−2.1613 × 104
		13	Survival Gumbel	1.1427	−2.9634 × 103
	T2	12|3	Farlie–Gumbel–Morgenstern	−0.4763	−1.2521 × 103

Note: Variables are coded as 1 = wind speed, 2 = wind direction, and 3 = wind attack angle; the edge label “ab|c” indicates that variables a and b are conditioned on variable c.

presents the optimal R-vine structure, the selected bivariate copula families, their corresponding parameter estimates, and the associated $BIC$ values for the JPDF of wind parameters at the four positions (F-06, F-08, F-14, and F-16).

The bivariate copula selection results show that there are differences in the optimal Copula types at each position, which reflects pronounced spatial heterogeneity in the wind field dependence structure at the bridge site. All selected optimal bivariate copulas return low $BIC$ values, which indicates a favorable balance between goodness-of-fit and model complexity. These models provide a reliable dependence skeleton for JPDF construction.

Table 13. The goodness-of-fit test for JPDF of wind field.

Position	BIC	RMSE	MAE	IA
F-06	−6.4136 × 104	1.2299 × 10−6	8.2202 × 10−8	0.9999
F-08	−1.5675 × 104	3.2344 × 10−6	4.0045 × 10−7	0.9994
F-14	−1.7514 × 104	9.8721 × 10−8	1.1222 × 10−8	0.9999
F-16	−2.5802 × 104	1.8430 × 10−6	2.2527 × 10−7	0.9998

and

Table 14. The goodness-of-fit test for JCDF of wind field.

Position	RMSE	MAE	IA
F-06	1.8112 × 10−4	8.3158 × 10−5	0.9999
F-08	5.6988 × 10−4	1.6102 × 10−4	0.9999
F-14	4.2763 × 10−5	1.4163 × 10−5	0.9999
F-16	5.4428 × 10−4	2.1227 × 10−4	0.9999

The lower $BIC$ value indicates that the model achieves the balance between goodness-of-fit and model parsimony. For the JPDF, $RMSE$ and $MAE$ values are on the order of 10⁻⁸ to 10⁻⁶; for the JCDF, both metrics are on the order of 10⁻⁵ to 10⁻⁴. The $IA$ consistently exceeds 0.999 across all positions. These results collectively confirm the high accuracy of the proposed framework.

report the goodness-of-fit test for the JPDF and the joint CDF derived from the fitted R-vine copula models.

6. Conclusions

This study develops a comprehensive JPDF modeling framework for wind speed, wind direction, and wind attack angle. The proposed approach integrates the flexibility of the DPMM for adaptive marginal model estimation with the structural adaptability of R-vine copulas for dependence modeling. The framework is established to simultaneously accommodate the multimodal and non-Gaussian features of the wind parameters and the complex and nonlinear interdependencies among them, and is subsequently validated against measured data to confirm its availability and robustness. The following findings from discussions are noteworthy:

(1)

The DPMM offers great advantages in marginal distribution fitting. It autonomously identifies the intrinsic clustering structure of the wind field data and exhibits strong performance and stability in capturing multimodal, non-Gaussian characteristics as well as the tail behavior of the distribution.

(2)

The posterior component-assignment probabilities from the Bayesian framework enable a monthly decomposition of the wind field. This analysis reveals a distinct seasonal pattern, with a dominant summer regime of elevated wind speeds, stable direction, and narrow attack-angle range. These results provide direct support for season-specific wind-resistant design and reliability assessment.

(3)

The diversity of adaptively selected bivariate copula families reflects pronounced spatial heterogeneity in the wind field characteristics along the bridge span, thereby affirming the framework’s flexibility in capturing complex nonlinear dependence.

(4)

The proposed DPMM-Copula framework yields excellent goodness-of-fit performance, demonstrating high-fidelity characterization of the marginal distributions, the JPDF, and the JCDF.

(5)

It is worth mentioning that the framework is validated using one year of measurements from a single coastal bridge site. While it is sufficient for establishing seasonal wind regime characteristics, the record length may not fully represent interannual variability. Moreover, typhoon events are not separated from the overall dataset, and their distinct wind structures may need specific further study in future work.