Extension of REBMIX algorithm to von Mises parametric family for modeling joint distribution of wind speed and direction

Highlights

• An improved mixture parameter estimation algorithm is proposed.

• The joint mixture distribution model of wind speed and direction is established.

• The wind direction observations are represented by von Mises mixed distributions.

• The proposed algorithm exhibits a better performance with less computing time.

Abstract

A statistical analysis of the wind speed and wind direction serves as a solid foundation for the wind-induced vibration analysis. The probabilistic modeling of wind speed and direction can effectively characterize the stochastic properties of wind field. The joint distribution model of wind speed and direction involves a circular distribution and has a multimodal characteristic. In this paper, the finite mixture distribution model is introduced and used to represent the joint distribution model that is comprised of the mixture Weibull distributions and von Mises distributions. An extended parameters estimation algorithm for multivariate and multimodal circular distributions is proposed to construct the joint distribution model. The proposed algorithm estimates the component parameters, mixture weight of each component and the number of components successively by an iterative process. The major improvement is accomplished by adding a circular distribution model. The effectiveness of the proposed algorithm is verified with numerical simulations and one-year field monitoring data and compared with the expectation maximization algorithm-based angular-linear approach in terms of the Akaike’s information criterion and computing time. The results indicate that the finite mixture model represents the joint distribution model of wind speed and direction well and that the proposed algorithm has a good and time-saving performance in parameter estimation for multivariate and multimodal models.

Introduction

For the long-span bridges, the environmental wind load will cause relatively large vibration and deformation with the increase of flexibility and span. As stress cycles would occur under frequent wind load induced by wind speed, critical components of long-span steel bridges may suffer an enormous amount of stress cycles, which will lead to fatigue damage accumulation and finally structural failure may appear without extreme climate condition. In addition to wind speed, wind direction is another important factor when considering the effect of wind load because wind in the along and transverse directions of a bridge will cause a completely different effect [23]. Due to the wind-induced fatigue damage of long-span steel bridges depending on wind speed and wind direction, the statistical model of field monitoring of wind data near the bridge site will lay a solid foundation for the accurate fatigue evaluation. A suitable model of joint distribution and an efficient parameter estimation method are major challenges for constructing the statistical model of wind speed and direction.

Nowadays, structural health monitoring systems have been installed on structures to measure environmental load and actual structural response of the existing structures. Field monitoring of wind data includes wind speed variable and wind direction variable. For the wind speed variable, the two-parameter Weibull distribution has been commonly used and accepted to model wind speed distribution [2]. Compared with the literature on wind speed, the literature on wind direction modeling is more limited because the distribution of the wind direction variable is a circular distribution [8]. Masseran et al. [12] used the circular distribution based on nonnegative trigonometric sums as the probability density function of wind direction. Carta et al. [5] proposed a finite mixture model comprised of a finite mixture of von Mises distribution to represent the distribution of directional wind speed. Based on the distribution functions of wind speed and direction, there are some approaches to construct the joint distribution model. Feng et al. [9] constructed the joint distribution of wind speed and direction based on the parameters of the sector-wise Weibull distribution and interpolations between direction sectors. Additionally, the angular-linear distribution models are used to construct joint distributions with specified marginal distributions [6]. While the models described above are all comprised of one-dimensional marginal distributions that will lead to the deviation between the predictive model and the distribution of observations, very little has been written in the literature about the finite mixture distribution models that are multidimensional.

Many optimization algorithms have been proposed to optimize the joint distribution models by estimating the parameters in models and selecting the optimal joint distribution model, such as the least square method [4], [5], the expectation maximization (EM) algorithm[21], [3] and the genetic algorithm [10]. Carta et al. [5] used the least squares method to estimate the parameters in a mixture of von Mises distributions. Masseran [13] separated wind direction data into eight groups based on geographical directions and used Markov chain models to model the distribution. Nasr et al. [20] applied three maximum likelihood technics based on Newton Raphson algorithm, Levenberg Marquardt algorithm and Trust Region reflective algorithm to the parameter estimation of the Weibull distribution. Calderara et al. [3] proposed a new specific EM algorithm for estimating the parameters in a mixture of von Mises distributions. In addition, the EM algorithm has been used widely to estimate parameters for the Weibull distribution [22], [15]. However, the EM algorithm is prone to converge to the local maximum of the likelihood function and has a limitation for the parameter estimation of multidimensional models.

The basic purpose of this paper is to develop an effective and efficient parameter estimation algorithm for the multivariate and multimodal circular distribution. The finite mixture of von Mises distributions is selected as the predictive model of the circular distribution. The unknown parameters in the predictive models are estimated by the extended REBMIX algorithm using an iterative process to assign the component densities one after another to the empirical mixture density. The optimal model will be selected from the models of different component numbers by judging the value of Akaike’s information criterion (AIC). This paper is organized as follows: the basic finite mixture model is introduced firstly in Section 2. Moreover, the mixture models of wind direction and the joint mixture models of wind speed and direction are formulated by the mixture of von Mises distributions and the mixture of Weibull-von Mises distributions, respectively. Then, Section 3 presents an extended REBMIX algorithm for estimating the unknown parameters in the predictive models and a detailed description of the flow of the algorithm. In Section 4, the extended algorithm is verified by being compared to the simulated data. Meanwhile, its performance of modeling with the field monitoring data, as compared to the performance calculated by the EM algorithm, is presented in Section 5. Finally, the conclusions are listed in Section 6.