Transcript part 2
Maximum energy of hurricane tracks
Series of integrated PDI of hurricane tracks : 1880-2005
▼
Series of annual maximum PDI
Corresponding potential predictors : NAO, SOI, AMO, Tropical
Atlantic SST (beware : reduced SST), Global mean temperature
Vector Additive Modelling of the GEV
WHY?
Joint variations of non-independent parameters →structural trend
models are often difficult to formulate
Model LOCATION (µ), SCALE () parameters of the GEV distribution as
smooth functions of covariates. For computation reasons, the SHAPE
parametrer () remains constant in our study.
µ=µo+f1(X1)+f2(X2)
o+g3(X3)+g4(X4)
= o
Data driven approach rather than model driven
approach
Vector Additive Modelling of the GEV
HOW?
Vector Generalized Additive Modelling technique (Yee & Wild, 1996)
provides flexible smoothing via modified vector backfitting algorithm
Implementation and vector spline: VGAM package in R (Yee, 2006).
WARNINGS
– few predictors should be used in additive models
– Only pointwise standard error estimates are provided, not the full
covariance matrix: full inferences should be obtained using linear
techniques
– Convergence may be hard to achieve – use link functions : log()
Additive effects estimation
Vector linear modelling of extremes of PDI
Parametric models based on previous VGAM results
µ modelled as a linear function of SOI and SST
modelled as a linear function of SST + change-point model in SOI
Deviance tests
– Gumbel approximation is valid
(p value : 0.36)
– Change-point model in position -0.55hPa
(same number of parameters, but better fit than linear trend in SOI)
Modelled parameters of the Gumbel distribution
90th Quantile of the PDI distribution
Standard error (DELTA method)
Model fit
Prediction
Reliability plots
Learning
Cross validation
Example 2
Evolution of GCM maxima of
temperatures
DATA
Annual maxima of air temperature
Period 1860-2099
IPSL GCM (5th IPCC Report Assessment)
Concentrations of the GHG and aerosols are prescribed during
the whole simulations using observations prior to 2000 and
according to a SRES-A2 IPCC scenario for 2000-2100.
EVOLUTION OF TEMPERATURE EXTREMES FOR
ONE GRIDPOINT OVER FRANCE
CO2 concentration plays a major role in extremes rise. This
evolution is modulated by time.
µ=f1(CO2)+f2(YEAR)
=g1(CO2)+g2(YEAR)
=constant
EVOLUTION OF TEMPERATURE EXTREMES
FOR ONE GRIDPOINT OVER FRANCE
VGAM exhibits linear dependency in CO2 for µ parameter.
Computation of GEV 90th quantile (VGAM and VGLM)
GRIDPOINTS GEV PARAMETERS
GRIDPOINTS GEV PARAMETERS
GRIDPOINTS GEV PARAMETERS
µ:2100-2000 difference
GRIDPOINTS GEV PARAMETERS
GRIDPOINTS GEV PARAMETERS
:2100-2000 difference
SHAPE PARAMETER
90th percentile
90th percentile
90th percentile
90th quantile:2100-2000 difference
CONCLUSION
GAM & VGAM
Data driven approaches
Flexible exploratory tools
Less precise inferences : full covariance matrix of
parameters not computed
Few predictors only
Computational problems may occur (extremes)
BIBLIOGRAPHY
GAM
Hastie & Tibshirani, 1990
Generalized Additive Models, Monographs on statistics and applied
probability 43, Chapman & Hall/CRC, 335 p.
VGAM
Yee & Wild, 1996
Vector Generalized Additive Models
JRSS series B, Vol. 58, n°3, pp. 481-493
VGAM and extremes
BIBLIOGRAPHY
VGAM and extremes
Yee & Stephenson, 2007
Vector Generalized and Additive Extreme Value Models.
To appear in Extremes.
Chavez-Demoulin & Davison, 2005
Generalized Additive Modelling of sample extremes
Applied Statistics 54, 207-222.
COMPUTATION
R useful packages
« gam »
Hastie
« VGAM »
Yee, 2006