A seminar about RMAWGEN

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


Outline
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



Motivation

Motivation

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



Motivation

Motivation

#+,-,.& 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.


Motivation

Hydrologic 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





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!




Theory: 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].




Theory: 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 .



VAR 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



Gaussianization

Theory: 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)



Gaussianization

Theory: 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]



Gaussianization

Theory: 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].



Gaussianization

Theory: 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) ).




Representation 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 )



Calibration of a VAR for weather generation
from 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




Stochastic 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



Dataset

Daily 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.



Let’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")




4 VAR models for Precipitation Generation

Precipitation Generation with auto-regression order p = 3 and
PCA Gaussianization (P03GPCA)
PCA Gaussianization (P01GPCA)
without PCA Gaussianization (P03)
without PCA Gaussianization (P03)




Precipitation 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).


A seminar about RMAWGEN

Empfohlen

Empfohlen

Weitere ähnliche Inhalte

Was ist angesagt?

Was ist angesagt? (20)

Ähnlich wie A seminar about RMAWGEN

Ähnlich wie A seminar about RMAWGEN (20)

Mehr von Emanuele Cordano

Mehr von Emanuele Cordano (6)

A seminar about RMAWGEN