Maximum energy of hurricane tracks

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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

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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.
• µµof1(X1)f2(X2) ???og3(X3)g4(X4) ? ?o
• Data driven approach rather than model driven
  approach

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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(?)

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Additive effects estimation
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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)

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Modelled parameters of the Gumbel distribution
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90th Quantile of the PDI distribution
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Standard error (DELTA method)
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Model fit

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Prediction

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Reliability plots

• Learning

• Cross validation

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Example 2

• Evolution of GCM maxima of temperatures

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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.

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EVOLUTION OF TEMPERATURE EXTREMES FOR ONE
GRIDPOINT OVER FRANCE

• CO2 concentration plays a major role in extremes
  rise. This evolution is modulated by time.
• µf1(CO2)f2(YEAR) ?g1(CO2)g2(YEAR)
  ?constant

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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)

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GRIDPOINTS GEV PARAMETERS
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GRIDPOINTS GEV PARAMETERS
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GRIDPOINTS GEV PARAMETERS
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µ2100-2000 difference
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GRIDPOINTS GEV PARAMETERS
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GRIDPOINTS GEV PARAMETERS
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?2100-2000 difference
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SHAPE PARAMETER

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90th percentile
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90th percentile
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90th percentile
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90th quantile2100-2000 difference
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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) ?

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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, n3, pp. 481-493
• VGAM and extremes

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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.

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COMPUTATION

• R useful packages
•  gam  Hastie
•  VGAM  Yee, 2006