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Introduction         Theory           RMAWGEN content          Example          Conclusions   References               RMA...
Introduction         Theory           RMAWGEN content          Example          Conclusions   ReferencesOutline       Intr...
Introduction         Theory           RMAWGEN content          Example          Conclusions   ReferencesMotivationMotivati...
Introduction         Theory               RMAWGEN content               Example         Conclusions   ReferencesMotivation...
Introduction           Theory            RMAWGEN content                Example        Conclusions   ReferencesMotivationH...
Introduction          Theory          RMAWGEN content          Example          Conclusions   ReferencesA Daily Weather Ge...
Introduction          Theory          RMAWGEN content          Example          Conclusions   ReferencesVector Auto-Regres...
Introduction          Theory          RMAWGEN content          Example          Conclusions   ReferencesVector Auto-Regres...
Introduction          Theory          RMAWGEN content          Example          Conclusions   ReferencesVector Auto-Regres...
Introduction                                   Theory                              RMAWGEN content                        ...
Introduction               Theory                   RMAWGEN content          Example          Conclusions         Referenc...
Introduction          Theory          RMAWGEN content          Example          Conclusions   ReferencesGaussianizationThe...
Introduction         Theory           RMAWGEN content            Example          Conclusions    ReferencesGaussianization...
Introduction          Theory            RMAWGEN content          Example          Conclusions    ReferencesWeather Variabl...
Introduction              Theory             RMAWGEN content             Example                 Conclusions   ReferencesW...
Introduction             Theory                RMAWGEN content                   Example                 Conclusions   Ref...
Introduction         Theory           RMAWGEN content          Example          Conclusions   ReferencesDatasetDaily Tempe...
Introduction           Theory         RMAWGEN content          Example          Conclusions   ReferencesPrecipitation Gene...
Introduction           Theory         RMAWGEN content          Example          Conclusions   ReferencesPrecipitation Gene...
Introduction           Theory         RMAWGEN content          Example          Conclusions   ReferencesPrecipitation Gene...
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
A seminar about RMAWGEN
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A seminar about RMAWGEN

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RMAWGEN: A software project for a daily Multi-Site Weather Generator with R

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A seminar about RMAWGEN

  1. 1. Introduction Theory RMAWGEN content Example Conclusions References RMAWGEN: A software project for a daily Multi-Site Weather Generator with R RMAWGEN Daily Weather Generator Emanuele Cordano, Emanuele Eccel CRI - Fondazione Edmund Mach San Michele all’Adige (TN), Italy September 26, 2012 Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  2. 2. Introduction Theory RMAWGEN content Example Conclusions ReferencesOutline Introduction Motivation A Daily Weather Generator Theory Vector Auto-Regressive Models Gaussianization RMAWGEN content Weather Variables and Generation Flowcharts Example Dataset Precipitation Generation Temperature Generation with Precipitation as an exogenous predictor Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  3. 3. Introduction Theory RMAWGEN content Example Conclusions ReferencesMotivationMotivation Understanding the effects of climate change on phenological and agricultural processes in Trentino, an alpine region, Italy. Need of daily time-series of temperature and precipitation corresponding to climate projections at several sites! Development of generation algorithms with R platform Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  4. 4. Introduction Theory RMAWGEN content Example Conclusions ReferencesMotivationMotivation #+,-,.& 78,9:8;&<828;,90;& /01234,.526 !=>$7<"?* !"#$%&())* Courtesy of Riccardo De Filippi Downscaling climate predictions from Global Climate Models (GCM), mean monthly climatology is obtained for several sites in the Trentino region and its neighborhood. RMAWGEN aims to generate randomly daily time series from the obtained monthly climatology in each site. Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  5. 5. Introduction Theory RMAWGEN content Example Conclusions ReferencesMotivationHydrologic Modelling (water cycle), on going ... Hydrologic models need daily or hourly weather time series In collaboration with Fabio Zottele, Daniele Andreis and www.geotop.org Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  6. 6. Introduction Theory RMAWGEN content Example Conclusions ReferencesA Daily Weather GeneratorA Daily Weather Generator A weather generator (WG) aims to generate weather time series with the same statistical patterns of the observed ones. This presentation shows in detail how to calibrate a WG and generate series with RMAWGEN . Currently RMAWGEN generates daily maximum and minimum temperature and precipitation. Major details are explained going on with the presentation! Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  7. 7. Introduction Theory RMAWGEN content Example Conclusions ReferencesVector Auto-Regressive ModelsTheory: Vector Auto-Regressive Model (VAR) A VAR(K ,p)) is: xt = A1 · xt−1 + ... + Ap · xt−p + ut (1) where xt is a K -dimensional vector representing the set of weather variables generated at day t by the model, called ”endogenous” variables, Ai is a coefficient matrix K × K for i = 1, ..., p and ut is a K -dimensional stochastic process. xt and ut are usually normalized to have a null mean. ut is a Standard White Noise [Luetkepohl, 2007,def. 3.1]. Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  8. 8. Introduction Theory RMAWGEN content Example Conclusions ReferencesVector Auto-Regressive ModelsTheory: VAR with exogenous variable xt = A1 · xt−1 + ... + Ap · xt−p + ut ut = C · dt + B · t (2) Here, ut is a process described by (2) where dt is a set of known K -dimensional processes, whose components are called ”exogenous” variables, and t is a K -dimensional uncorrelated standard white noise, i.e. E[ t · T ] = Σ t = IK , C and B are the t respective coefficient matrices and IK is the K -dimensional identity matrix. The process dt is a generic, previously generated, known meteorological variable that works as a predictor vector for the stochastic generation of xt . Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  9. 9. Introduction Theory RMAWGEN content Example Conclusions ReferencesVector Auto-Regressive ModelsVAR Coefficient Estimation Model Calibration: Method of Moments (based on Yule-Walker Equations); Method of Maximum Likelihood; Method of Least Squares (OLS) (vars and RMAWGEN use this!! Model Diagnostic, once calibrated VAR: Multivariate Portmanteau and Breusch-Godfrey (Lagrange Multiplier) test, seriality tests on residuals; Jarque-Bera multivariate skewness and kurtosis test, normaility tests on residuals Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  10. 10. Introduction Theory RMAWGEN content Example Conclusions ReferencesGaussianizationTheory: 1D Marginal Gaussianization It is based on quantile transformation: library(RMAWGEN) set.seed(123456) z <- rexp(10000,rate=0.5) −1 x = FG (Fz (z)) (3) x <- normalizeGaussian(x=z,data=z) z1 <- normalizeGaussian(x=x,data=z,inverse=TRUE) #-- Plot probability histogram of z, x where Fz (z) is cumulate distribution and z1 zhist <- probability of z, FG (x) is the Gaussian hist(z,probability=TRUE,breaks=50) xhist <- cumulative function with zero mean and hist(x,probability=TRUE,breaks=50) z1hist <- standard deviation equal to 1. hist(z1,probability=TRUE,breaks=50) Histogram of z Histogram of x Histogram of z1 zhist$counts <- zhist$density xhist$counts <- xhist$density 0.4 z1hist$counts <- z1hist$density 0.4 0.4 def.par <- par(no.readonly = TRUE) # 0.3 0.3 0.3 save default, for resetting... Frequency Frequency Frequency 0.2 par(mfrow=c(1,3),oma=c(15.0,0.0,15.0,0.0)) 0.2 0.2 plot(zhist) 0.1 0.1 0.1 plot(xhist) plot(z1hist) 0.0 0.0 0.0 0 5 10 15 20 -4 -2 0 2 4 0 5 10 15 20 z x z1 par(def.par) Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  11. 11. Introduction Theory RMAWGEN content Example Conclusions ReferencesGaussianizationTheory: Monthly 1D Marginal Gaussianization The gaussianization Jan Feb transformation can be 3 Mar Apr May different for each month. Jun 2 Jul Aug Sep Oct 1 Nov Dec library(RMAWGEN) data(trentino) 0 x col <- rainbow(n=12,start=0.1,end=0.9) col[6:1] <- col[1:6] -1 col[7:12] <- col[12:7] plot sample(x=TEMPERATURE MIN$T0090, sample="monthly", -2 origin="1958-1-1",axes=FALSE,xlab="Tn [degC]",ylab="x",abline=NULL,col=col) -3 -10 0 10 20 Tn [degC] Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  12. 12. Introduction Theory RMAWGEN content Example Conclusions ReferencesGaussianizationTheory: Multi-Dimensional Gaussianization (1) Similarly to the 1D case, it is intuitive that an invertible function exists between any multi-dimensional random variable and a Gaussian variable with the same dimensions [Chen and Gopinath, 2000]. Recently, Laparra et al. [2009] proposed an iterative method based on Principal Component Analysis (PCA): One-dimensional Gaussianization of each marginal variable (component of x), i.e. Marginal Gaussianization; Orthonormal transformation of the coordinates based on the eigenvector matrix of the covariance matrix according to PCA analysis. Details about convergence of the method are given in Laparra et al. [2009]. Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  13. 13. Introduction Theory RMAWGEN content Example Conclusions ReferencesGaussianizationTheory: Multi-Dimensional Gaussianization (2) The PCA Gaussianization (G-PCA) is described by the following mathematical formulation (Laparra et al. [2009]). x(k+1) = B(k) · Ψ(k) (x(k) ) (4) where x(k) and x(k+1) are the vector random variable at the k-th and the k + 1-th iteration level, Ψ(k) (x(k) ) is the marginal Gaussianization of each component of random variable x(k) performed by applying (3), B(k) is the PCA transform matrix, i.e. the orthonormal eigenvector matrix of the covariance matrix of Ψ(k) (x(k) ). Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  14. 14. Introduction Theory RMAWGEN content Example Conclusions ReferencesWeather Variables and Generation FlowchartsRepresentation of a Weather Variable: defining zt Temperature Generation: → preliminary deseasonalization Txt + Tnt Txst + Tnst zt = − |Txt − Tnt (5) 2 2 Txt : daily maximum temperature Tnt : daily minimum temperature Txst : daily maximum spline-interpolated temperature (from ”monthly climatology”) Tnst : daily minimum spline-interpolated temperature (from ”monthly climatology”) Precipitation Generation: zt is equal to daily precipitation Marginal Gaussianization for both temperature or precipitation case =⇒ xt = Gm (zt ) Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  15. 15. Introduction Theory RMAWGEN content Example Conclusions ReferencesWeather Variables and Generation FlowchartsCalibration of a VAR for weather generationfrom observed data A VAR model is calibrated from observed weather data according to the following algoritm daily time series (observed) of daily weather variable time series (observed) exogenous variables (e. g.another weather variable) monthly sampling and Marginal Gaussianization monthly sampling and Marginal Gaussianization PCA Gaussianization (multi-variate) of data VAR calibration PCA Gaussianization (multi-variate) of residuals Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  16. 16. Introduction Theory RMAWGEN content Example Conclusions ReferencesWeather Variables and Generation FlowchartsStochastic weather generation by a previously created VAR Inversely, once calibrated the VAR model, weather data are generated as follows: daily time series of random generation of N(0,1) normally distributed values GPCA parameters exogenous variables for residuals Inverse PCA Gaussianization of residuals monthly sampling and Marginal Gaussianization Execution of the VAR model: generation of "x" VAR model Inverse PCA Gaussianization of "x" GPCA parameters for "x" Marginal gaussianization and deseasonalization: "z" monthly mean value Daily weather variable time-series of daily weather variable Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  17. 17. Introduction Theory RMAWGEN content Example Conclusions ReferencesDatasetDaily Temperature and Precipitation in trentino dataset[Eccel et al., 2012] See file trentino map.R in the doc/example directory of RMAWGEN package source code. Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  18. 18. Introduction Theory RMAWGEN content Example Conclusions ReferencesPrecipitation GenerationLet’s start with a precipitation generation 4 different models with different auto-regression order and with or without GPCA are considered. rm(list=ls()) library(RMAWGEN) # Call RMAWGEN set.seed(1222) # Set the seed for random generation data(trentino) # Load the dataset # Two stations where to work! station <- c("T0090","T0083") Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  19. 19. Introduction Theory RMAWGEN content Example Conclusions ReferencesPrecipitation Generation4 VAR models for Precipitation Generation Precipitation Generation with auto-regression order p = 3 and PCA Gaussianization (P03GPCA) Precipitation Generation with auto-regression order p = 1 and PCA Gaussianization (P01GPCA) Precipitation Generation with auto-regression order p = 3 and without PCA Gaussianization (P03) Precipitation Generation with auto-regression order p = 1 and without PCA Gaussianization (P03) Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator
  20. 20. Introduction Theory RMAWGEN content Example Conclusions ReferencesPrecipitation GenerationPrecipitation VAR Model Diagnostic R> serial test(generationP03GPCA prec$var) R> normality test(generationP03GPCA prec$var) Normality Test Serial Test P01 Unsuccessful Unsuccessful P03 Unsuccessful Successful P01GPCA Successful Successful P03GPCA Successful Successful Table 1: Test results on VAR residuals (precipitation generation). Emanuele Cordano, Emanuele Eccel RMAWGEN Daily Weather Generator

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