Some advanced methods in extreme value analysis

1. Some advanced methods in extreme value analysis Peter Guttorp NR and UW

2. Outline • Nonstationary models • Extreme dependence (Cooley) • When a POT approach is better than a block max approach (Wehner and Paciorek) • A Bayesian space-time model • An extreme climate event

3. Trends

4. What do we mean by trends in extreme values?

5. Time-dependent location estimates Stockholm data (Guttorp and Xu, Environmetrics 2011)

6. Simple model Model Estimate -LLR Fixed μ,all -16.6 706.0 Fixed μ, early -18.1 336.8 Fixed μ, late -14.2 350.2 Early + late 687.0 Linear model in μ (-18.9,-14.0) 687.9 A linear change in mean value for annual minima seems a good model. Modal prediction for 2050: -11.5°C 2100: -10.5°C

7. Extreme dependence

8. Measuring dependence

9. Storm surges and wave heights Risk region Most of these data are not extreme!

11. Describing “tail dependence”

12. Estimating H • Transform margins to Frechet; keep largest 150 obs (95th %ile)

13. Density of H

14. Probability of risk region

15. Block extremes vs peaks over threshold

16. Climate model output • Daily precip from 450 year control run of climate model (long stationary series) • Fit GEV to seasonal max

17. POT analysis • 99th percentile

18. Why are the two analyses so different? • Desert regions: large amount of lack of precipitation. GEV therefore will include many zeros, while GPD only uses data where high values are actually recorded.

19. Space-time data

20. Some temperature data • SMHI synoptic stations in south central Sweden, 1961-2008

21. Annual minimum temperatures and rough trends

22. Location slope vs latitude

23. Dependence between stations Common coldest day in 48 years 5 common to all 4 northern stations

24. Max stable processes • Independent processes Yi(x) • (e.g. space-time processes with weak temporal dependence) • A difficulty is that we cannot compute the joint distribution of more than 2-3 locations. So no likelihood.

25. Spatial model • where • Allows borrowing estimation strength from other sites • Can include more sites in analysis

26. Trend estimates Posterior probability of slope ≤ 0 is very small everywhere

27. Spatial structure of parameters

28. Location slope vs latitude

29. Prediction Borlänge

30. A different kindof extreme event

31. What made Gudrun so destructive? • Hurricane Gudrun, Sweden, January 2005. 15 deaths, 340 000 households without power up to 4 weeks.

32. Consequences • 75 million cubic feet of forest fell (normal annual production) • Wind speeds up to 40 m/s • Forest damage due to large amounts of precipitation, temperatures around 0°C, high winds • Much work on multivariate extreme asymptotics needs all components to be extreme–here temperature is not. • Want (limiting) conditional distribution of wind given temperature and previous precipitation

33. The Heffernan-Tawn approach • Assume we can find normalizing factors a-i(yi), b-i(yi) so that • where G-i has non-degenerate margins. • Pick aji(yi) so that • and • where hji is the conditional hazard function of Yj given Yi=yi.

34. Asymptotic properties • Let . • Then given Yi>u, • Yi - u and Z-i are asymptotically independent as . Furthermore • Fitting using observed data; forecasting using regional model output

35. Some R software • ExtRemes • ismev • evlr • SpatialExtremes