Maximum energy of hurricane tracks

1. 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

2. 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

3. 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()

4. Additive effects estimation

5. 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)

6. Modelled parameters of the Gumbel distribution

7. 90th Quantile of the PDI distribution

8. Standard error (DELTA method)

9. Model fit

10. Prediction

11. Learning Cross validation Reliability plots

12. Example 2 Evolution of GCM maxima of temperatures

13. 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.

14. 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

15. 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)

16. GRIDPOINTS GEV PARAMETERS

17. GRIDPOINTS GEV PARAMETERS

18. GRIDPOINTS GEV PARAMETERS

19. µ:2100-2000 difference

20. GRIDPOINTS GEV PARAMETERS

21. GRIDPOINTS GEV PARAMETERS

22. :2100-2000 difference

23. SHAPE PARAMETER

24. 90th percentile

25. 90th percentile

26. 90th percentile

27. 90th quantile:2100-2000 difference

28. 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) 

29. 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

30. 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.

31. COMPUTATION • R useful packages « gam » Hastie « VGAM » Yee, 2006