A Gaussian state-space model for wind fields in the North-East Atlantic

A Gaussian state-space model for wind fields in the North-East Atlantic Julie BESSAC - Université de Rennes 1 with Pierre AILLIOT and Valï 1 rie MONBET 2 Juillet 2013

Plan Motivations 1 Motivations 2 Context and goals 3 Wind data 4 Model Gaussian Linear State-Space Model Parameter Estimation Procedure 5 Validation of the model Validation by simulation Validation by prediction Improvement of prediction? 6 A more parsimonious model 7 Conclusion and extension of the model

Motivations Motivations for the use of weather generators Many natural phenomena and human activities depend on wind conditions Production of electricity by wind turbines Evolution of a coast line Maritime transport Drift of objects in the ocean... Wind data generally available on short periods of time 50 years of data maximum Not enough to compute reliable estimates of the probability of complex events Stochastic model used to simulate artificial wind conditions Monte-Carlo methods

Plan Context and goals 1 Motivations 2 Context and goals 3 Wind data 4 Model Gaussian Linear State-Space Model Parameter Estimation Procedure 5 Validation of the model Validation by simulation Validation by prediction Improvement of prediction? 6 A more parsimonious model 7 Conclusion and extension of the model

Context and goals latitude 46 48 50 52 Goals: 350 352 354 356 358 360 longitude 12 10 8 6 4 wind speed 0 5 10 15 20 Available data: 0 5 10 15 20 25 30 day Reanalysis data from ECMWF: wind speed to propose a stochastic state-space model for wind fields to generate realistic wind conditions

Context and goals One of the main difficulties and goals is: reproducing space-time motions of meteorological systems e.g. propagation of storms. 358 360 52 50 48 46 350 352 52 longitude 354 356 360 350 352 354 longitude 20 20 15 15 latitude latitude 50 358 longitude 52 356 5 50 354 10 5 46 46 5 10 48 48 10 5 46 5 46 352 15 10 48 48 10 350 20 latitude 50 15 latitude 50 15 latitude 20 52 52 20 350 352 354 longitude 356 358 360 350 352 354 longitude 356 358 360 356 358 360

Plan Context and goals 1 Motivations 2 Context and goals 3 Wind data 4 Model Gaussian Linear State-Space Model Parameter Estimation Procedure 5 Validation of the model Validation by simulation Validation by prediction Improvement of prediction? 6 A more parsimonious model 7 Conclusion and extension of the model

Plan Wind data 1 Motivations 2 Context and goals 3 Wind data 4 Model Gaussian Linear State-Space Model Parameter Estimation Procedure 5 Validation of the model Validation by simulation Validation by prediction Improvement of prediction? 6 A more parsimonious model 7 Conclusion and extension of the model

Wind data Data under study: reanalysis data ERA Interim from ECMWF Available with regular sampling: x = 0.75 o, t = 6h To eliminate seasonality: study of months of January from 1979 to 2011 Wind speed are transformed according to Box-Cox method to get approximately a Gaussian distribution (Box-Cox transformation = wind speedλ 1 λ ) 18 gridded points under study in the North-East Atlantic channel 12 46 48 50 52 13 14 15 16 17 18 7 8 9 10 11 12 1 2 3 4 5 6 10 8 6 4 350 352 354 356 358 360

Some time series Wind data Site 13 Site 18 vitesse du vent 5 10 15 vitesse du vent 5 10 15 20 0 5 10 15 20 25 30 jour 0 5 10 15 20 25 30 jour Time shifts are observable on time series

Wind data Some statistics computed on data Cross-correlations highlight the temporal advance of western locations on eastern locations Cross correlation between locations 13 & 18 cross correlation 0.0 0.2 0.4 0.6 0.8 1.0 6 4 2 0 2 4 6 lag (day)

Plan Model 1 Motivations 2 Context and goals 3 Wind data 4 Model Gaussian Linear State-Space Model Parameter Estimation Procedure 5 Validation of the model Validation by simulation Validation by prediction Improvement of prediction? 6 A more parsimonious model 7 Conclusion and extension of the model

Model Gaussian Linear State-Space Model Modeling by a Gaussian state-space model [DK01], [CMR] X t = ρx t 1 + σɛ t Y t (1) = α1 1X t+1 + α0 1X t + α 1 1 X t 1 + η t (1).. Y t (K) = α1 K X t+1 + α0 K X t + α 1 K X t 1 + η t (K) pour t 1, - Y t R K : vector of observations, K: number of locations studied - X t R: scalar hidden state, - ɛ and η are serially independent Gaussian white noises, η t N K (0, Γ) Parameters and equations interpretation: X : mean transformed wind conditions at regional scale, temporal dynamic is contained in the state equation, Y : mean corrected transformed wind speed at local scale, (α 1, α 0, α 1 ) : links regional and local scales, Γ : contains a part of the spatial structure of the error between the two scales.

Model Stationarity and identifiability Gaussian Linear State-Space Model Stationarity: Under the assumption ρ < 1, the process Y is stationary [BD]. Identifiability: based on the second order structure of the Gaussian process Y. In order to ensure identifiability of the model: σ - fixing the variance of the hidden variable X (we choose 2 = 1) 1 ρ 2 - constraining the vectors α 1, α 0 et α 1 to be linearly independent

Model Parameter Estimation Procedure Parameter Estimation Procedure The following estimation procedure is performed: First initialization: - ρ initialized as the mean of all available ratios: cov(y t, Y t+3 ) cov(y t, Y t+2 ), - Λ = (α 1 α 0 α 1 ) and Γ initialized by a least square estimation between the observed and theoretical quantities defined in the identifiability procedure, Least square estimation on the auto-covariance (GMM): initialized with the previous parameters EM algorithm: initialized with the parameters estimated by GMM

Model Maximum Likelihood Estimation Parameter Estimation Procedure Expectation-Maximization algorithm is used [CMR]: E-step: Linearity and Gaussianity of the model enable to use Kalman filter and smoother to derive the incomplete likelihood, M-step: Explicit forms of the parameters: ρ, Λ and Γ. Starting point: Least square estimation on second order structure.

Plan Validation of the model 1 Motivations 2 Context and goals 3 Wind data 4 Model Gaussian Linear State-Space Model Parameter Estimation Procedure 5 Validation of the model Validation by simulation Validation by prediction Improvement of prediction? 6 A more parsimonious model 7 Conclusion and extension of the model

Marginal distributions Validation of the model Validation by simulation quantile of simulation 10 5 0 5 10 ++ ++ +++++ + + + + 5 0 5 10 quantile of data Figure : Quantile-Quantile plot at location 18 with 90% confidence intervals (red), parameters estimated by ML

Temporal dynamics Validation of the model Validation by simulation 13 & 18 Autocorrelation 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Observed GMM ML 4 2 0 2 4 lag(day) Figure : Cross-correlation of Y between locations 13 and 18

Spatial structure Validation of the model Validation by simulation correlation 0.65 0.75 0.85 0.95 Observed Theoretical correlation 0.65 0.75 0.85 0.95 Observed Theoretical 4 2 0 2 4 1.5 1.0 0.5 0.0 0.5 1.0 1.5 dlong dlat Figure : Correlation of Y at lag 0 differences of longitude (left) and latitude (right)

Spatial structure Validation of the model Validation by simulation correlation 0.5 0.6 0.7 0.8 Observed Theoretical correlation 0.5 0.6 0.7 0.8 Observed Theoretical 4 2 0 2 4 1.5 1.0 0.5 0.0 0.5 1.0 1.5 dlong dlat Figure : Correlation of Y at lag 1 differences of longitude (left) and latitude (right)

Prediction Validation of the model Validation by prediction RMSE 1.4 1.6 1.8 2.0 2.2 2.4 2.6 Persistence X~AR1 (ML) Y~VAR(1) (ML) 5 10 15 site Figure : Root mean square error between observed wind speed and predicted wind speed at each location

Validation of the model Validation by prediction 52 20 52 20 52 20 352 354 356 358 360 46 5 46 5 46 5 350 350 352 longitude 354 356 358 360 350 52 50 latitude latitude 50 15 46 5 46 360 350 352 354 356 358 360 350 360 Model 52 latitude 50 15 50 latitude 10 48 48 10 5 46 5 46 358 360 20 52 52 356 358 15 50 latitude 48 46 longitude 356 20 5 354 354 longitude 15 352 352 longitude 20 350 10 48 48 10 longitude 10 Observation 20 52 52 50 latitude 358 360 5 46 356 358 15 5 354 356 20 48 10 354 longitude 15 352 352 longitude 20 350 10 48 48 10 latitude latitude latitude 48 10 50 15 50 15 50 15 350 352 354 longitude 356 358 360 350 352 354 longitude 356 358 360

Validation of the model Improvement of prediction? Improvement of prediction? The state equation is: X t = ρ 1 X t 1 + ρ 2 X t 2 + ɛ t RMSE 1.8 2.0 2.2 2.4 2.6 Persistence AR1 (ML) AR2 (ML) 5 10 15 site Figure : Root mean square error between observed wind speed and predicted wind speed at each location No improvement in prediction...

Plan A more parsimonious model 1 Motivations 2 Context and goals 3 Wind data 4 Model Gaussian Linear State-Space Model Parameter Estimation Procedure 5 Validation of the model Validation by simulation Validation by prediction Improvement of prediction? 6 A more parsimonious model 7 Conclusion and extension of the model

A more parsimonious model A more parsimonious model (Observation equation: Y t = α 1 X t+1 + α 0 X t + α 1 X t 1 + η t, η t N K (0, Γ)) Gamma 0.5 0.0 0.5 1.0 1.5 2.0 dlat = 1.5 dlat = 0.75 dlat = 0 dlat = 0.75 dlat = 1.5 Gamma 0.5 0.0 0.5 1.0 1.5 2.0 dlat = 3.75 dlat = 3 dlat = 2.25 dlat = 1.5 dlat = 0.75 dlat = 0 4 2 0 2 4 1.5 1.0 0.5 0.0 0.5 1.0 1.5 dlong dlat Figure : Estimated Γ against differences of longitude (left) and latitude (right), solid lines : mean value

A more parsimonious model A more parsimonious model Spatial structure of covariance ([Cre], [HR89], [Abr]): Γ is fitted to parametric spatial covariance: ( )) Γ i,j = (σ i σ j exp( λ 1 di,j) 2 + λ 2 δ i,j or Γ i,j = ( σ i σ j ( sin(λ1 d i,j ) λ 1 d i,j + λ 2 δ i,j )) for for with (σ 1,..., σ K, λ 1, λ 2 ) strictly positive to be estimated. i, j {1,..., K}, i, j {1,..., K}, d: anisotropic distance defined as: d i,j = θ 1 Lat(i, j) 2 + θ 2 Long(i, j) 2 + θ 3 Lat(i, j) Long(i, j) with (θ 1, θ 2, θ 3 ) to be estimated for each spatial structure.

A more parsimonious model A more parsimonious model Gamma 0.5 0.0 0.5 1.0 1.5 2.0 dlat = 1.5 dlat = 0.75 dlat = 0 Observed Estimated Gamma 0.5 0.0 0.5 1.0 1.5 2.0 dlat = 1.5 dlat = 0.75 dlat = 0 Observed Estimated 4 2 0 2 4 4 2 0 2 4 dlong dlong Figure : Estimated Γ against differences of longitude, left: Gaussian structure, right: sinus structure, solid lines: mean value, dashed line: estimated mean

A more parsimonious model How to select a model? Structure of Γ Nb of parameters Log-likelihood BIC Pas de structure 208-24781 51894 Gaussienne 78-29264 59402 Sinus 78-35784 72442 X AR(2) 209-24790 51922 VAR(1) 495-20707 46961 Average RMSE (Standard deviation) Structure sur Γ GMM ML Pas de structure 2.161 (0.19) 2.181 (0.2) Gaussienne 2.215 (0.16) 2.199 (0.16) Sinus 2.312 (0.16) 2.225 (0.17) X AR(2) 2.257 (0.18) 2.292 (0.221) VAR(1) 2.064 (0.17)

Plan Conclusion and extension of the model 1 Motivations 2 Context and goals 3 Wind data 4 Model Gaussian Linear State-Space Model Parameter Estimation Procedure 5 Validation of the model Validation by simulation Validation by prediction Improvement of prediction? 6 A more parsimonious model 7 Conclusion and extension of the model

Conclusion and extension of the model Conclusion Advantages of the model: Ease of interpretation of the parameters Linearity and Gaussian properties: ease of implementation of estimation procedure Some statistics are well reproduced by the model Drawbacks: Lack of flexibility of the model: it only catches average behaviors of wind speed in the area, does not account for the weather type Linearity of the model may not be appropriate to catch possible local effects Extensions: Improve the parameterization of the parameters Take into account wind direction

Conclusion and extension of the model P. Abrahamsen. A review of gaussian random fields and correlation functions. Peter J. Brockwell and Richard A. Davis. Introduction to time series and forecasting. Springer Texts in Statistics. Second edition. Olivier Cappé, Eric Moulines, and Tobias Rydén. Inference in hidden Markov models. Springer Series in Statistics. Noel A. C. Cressie. Statistics for spatial data. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. J. Durbin and S. J. Koopman. Time series analysis by state space methods, volume 24 of Oxford Statistical Science Series. Oxford University Press, Oxford, 2001.

Conclusion and extension of the model J. Haslett and A.E. Raftery. Space-time modelling with long-memory dependence: Assessing ireland s wind power ressource. Applied Statistics, 1989.

Conclusion and extension of the model correlation 0.65 0.75 0.85 0.95 Observed Theoretical correlation 0.5 0.6 0.7 0.8 4 2 0 2 4 4 2 0 2 4 dlat dlat Figure : Estimated Γ against differences of longitude. Solid lines: mean value