Spatiotemporal probabilistic wind vector forecasting over Saudi Arabia
Motivated by a data set with observations from 28 wind stations in Saudi Arabia, the authors propose statistical models for probabilistic forecasting of wind vectors. In the proposed univariate model the value \(Y(s,t)\) denotes the square-root transformed wind speed at station \(s\) and time \(t\). There is assumed that \(Y(s,t)\) has a normal distribution, \(Y(s,t)\sim N(\mu(s,t),\sigma^2)\), where \(\sigma^2\) is the variance of a Gaussian white noise and \(\mu(s,t)\) is the mean function which includes the fixed and random effects. It is of the form \[ Y(s,t) \mu(s,t) = \beta_0 + \beta_1 \mathrm{Temp}(s,t) + \beta_2 \mathrm{Elev}(s) + \beta_3 \sin\{\theta(s,t)\} + \beta_4 \cos\{\theta(s,t)\} + f (H(t)) + W(s,t), \] where \(\beta_0\) is the intercept, \(\text{Temp}(s,t) \) is the covariance temperature, \(\mathrm{Elev}(s)\) is the elevation that is constant in time, \(\theta(s,t)\) represents the wind direction for each location and time, \(f (H(t))\) is a smooth function known as the ``time-of-day'' effect, \(W(s,t)\) is a spatiotemporal model with first order autoregressive dependence \[ W(s,t) = aW (s,t - 1) + Z(s,t),\, t = 2,\dots,T, \,|a| < 1. \] Here \(Z(s,t)\) is a realization of a Gaussian random field (GRF) with mean zero and the covariance function \[ \mathrm{cov}\{Z(s,t),Z(s',t')\}=\frac{\sigma^2_z}{\Gamma(v)2^{v-1}}(kh)^vK_v(kh)\cdot\delta_{t}^{t'}, h=\|s-s'\|, \] where \(K_v(\cdot)\) is the modified Bessel function of the second kind of order \(v\), \(\delta_{t}^{t'}\) is the Kronecker function. In the developed bivariate model a wind vector is determined by west-east (horizontal), denoted by \(U(s,t)\), and north-south (vertical), denoted by \(U(s,t)\), components wich are assumed to be Gaussian \(U(s,t)\sim N(\mu_u(s,t),\sigma_u^2)\) and \(V(s,t)\sim N(\mu_v(s,t),\sigma_v^2)\), where \(\sigma_u^2\) and \(\sigma_v^2\) are variances of the white noise and \(\mu_u(s,t)\) and \(\mu_v(s,t)\) are means of the random processes of the form \(Y(s,t)\) with the corresponding (different) parameters. The authors develop a model for the wind vector in a hierarchical framework, where at the first stage \(U(s,t)\) and \(V (s,t)\) are considered to be conditionally independent with their own set of parameters. At the next stage the bivariate spatiotemporal process is modeled as a function of covariates. The cross-covariance function is based on a linear combination of independent univariate spatiotemporal processes, also known as linear model of coregionalization (LMC). This approach provides covariance models that are richer and more flexible than the separable structures because different parameters represent each component of the bivariate process. The Bayesian inference and prediction are performed by means of a stochastic partial differential equation (SPDE) approach and the integrated nested Laplace approximation (INLA). Besides the originality of considering the wind components jointly, the main novelty of this work lies in the formulation of a spatiotemporal linear model of coregionalization as part of a more general model which utilizes the state-of-the-art INLA-SPDE techniques to construct the Gaussian random fields that encode the dependence structures. For more results and references see [\textit{E. T. Krainski} et al., Advanced spatial modeling with stochastic partial differential equations using R and INLA. Boca Raton, FL: CRC Press (2019; Zbl 1418.62011)].
coregionalization spatial and spatio-temporal models
scientific article; zbMATH DE number 7378431
scientific article