Joint segmentation of the wind speed and direction

Abstract

We present in this article a Bayesian estimation method for the joint segmentation of a set of piecewise stationary processes. The estimate we propose is based on the maximization of the posterior distribution of the change instants conditionally to the process parameter estimation. It is defined as a penalized contrast function with a first term related to the fit to the observation and a second term of penalty. The expression of the contrast function is deduced from the log-likelihood of the parametric distribution that models the statistic evolution of processes in the stationary segments. In the case of joint segmentation the penalty term is deduced from the prior law of change instants. It is composed of parameters that guide the number and the position of changes and of parameters that will bring prior information on the joint behavior of processes. This work is applied to the estimation of wind statistics parameters. We use data available from a cup anemometer and a wind vane, supposed to be piecewise stationary. The contrast function is deduced from the circular Von Mises distribution for the wind direction and from the log-normal distribution for the speed. The feasibility and the contribution of our method are shown on synthetic and real data.

Introduction

The problem of change-point detection has been studied considerably over the years. We found in the literature sequential or on-line segmentation methods [1] (in which analysis is performed as each new observation is gathered) and off-line methods for which it is considered that all the data are available at the same time and perform a global segmentation. One interesting area of study is the extension of change-point methods in the problem of the joint segmentation of piecewise stationary processes. In this paper we present a non-sequential Bayesian approach for the joint segmentation of wind direction and wind velocity supposed to be piecewise stationary.

The classical change-point problem supposes that the data are normally distributed with constant (but unknown) variance and the mean is suspected to undergo an abrupt change in value. The first works on this problem are due to Page [2] and Hinkley [3] for the detection of one change in the process. When the number of changes is unknown, the problem of change-point detection can be seen as a problem of model selection. In this case, each change configuration is a model and the number of changes is the dimension of the model. A classical solution to solve this problem is the penalized maximum likelihood. Let M be the model set, $m\in M$ a model and the empirical penalized criterion:

$$\widehat{m}=\underset{m\in M}{\underbrace{\mathrm{argmin}}}\{f({\widehat{s}}_m)+\mathrm{pen}(m)\}\text{,}$$

where ${\widehat{s}}_m$ are the estimated parameters of the model, and $\mathrm{pen}(m)$ a penalty function (in our presentation, we choose $f(t)=-$log-likelihood of the noise distribution $f(..)$ of the model observations t). The first work about such criteria is due to Mallows [4] and Akaike [5]. Both approaches are based on some unbiased estimation of the quadratic risk and aim at choosing a model which minimizes this risk: this is the efficiency point of view. Recent and more detailed works about this kind of criteria can be found in [6]. Another point of view about model selection consists in assuming the existence of a true model of minimal size and aiming at finding it. This is the case of change-point detection where all possible configurations of changes m are in M. The criteria described in [7], [8] have been designed to find the model with a probability tending to 1 when the variance of the process goes to 0: this is the consistency point of view. For those criteria the penalty term is a linear function of the model dimension. In this paper we propose a criterion deduced from the MAP estimate of the posterior distribution of the change instants conditionally to the process parameter estimation of a set of processes. We suppose the configuration of changes to be a sequence of Bernoulli variables [9] and we obtain for the penalty term a linear function of the model dimension. In the case of joint segmentation the penalty term is deduced from the prior law of the change instants. It is composed of parameters that guide the number and the position of changes and parameters that will bring prior information on the joint behavior of the process.

This work is applied to the statistical estimation of horizontal wind vector parameters. For modelling the transport and the diffusion of airborne pollutants, knowledge of the wind characteristics plays a crucial role in the numerical models of air quality [10], [11]. The dispersion of airborne material is mainly due to turbulent diffusion inherent in atmospheric motion. For realistic estimates of dispersion, it is therefore of primary importance to have an accurate description of atmospheric turbulence and wind vector parameters. The wind vector is described, in an Euclidean co-ordinate system, by three typical parameters: the vector of mean wind speed, the vector of mean wind direction and the angular standard deviation. The angular standard deviation is an indicator of atmospheric stability (Pasquill classes [12]). It allows appreciating the atmospheric turbulence taking into account the local topographical and climatic conditions. In our approach the wind velocity is expressed by two components, the wind speed and the wind direction, supposed to be piecewise stationary. The statistical properties of the wind speed are described by a positive linear random variable: the variable follows a log-normal distribution [13]. The wind direction is a circular random variable: it follows a Von Mises distribution [11]. In the preceding works the wind vector parameters were estimated on data supposed to be stationary. In this case the measurements were inaccurate and needed manual human intervention [10].

The problem, then, consists in recovering the instants of change in the recorded wind data. The wind velocity vector is described by an angle and a module. The value of these two parameters is available from a cup anemometer and a wind vane placed at the seafront. For this application we can define prior information on the behavior of the process. First, there is a correlation of the change instants with the tide hours. Second, the modifications of climatic conditions induce simultaneous changes in the two processes [14]. The goal of the joint estimation method we propose in this article is to integrate in the segmentation the prior information we have on the joint behavior of the process.

The paper is organized as follows. Section 2 describes the Bayesian technique. We detail, in this section, the joint estimation method used for estimating the posterior distribution of the change points. The parametric model and the MAP estimate of wind vector parameters are described in Section 3. In Section 4 we present numerical experiments on synthetic and real data.