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 ModelEstimate-LLR Fixed  all -16.6706.0 Fixed , early-18.1336.8 Fixed , late-14.2350.2 Early + late687.0 Linear model in  (-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 (95 th %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 99 th 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 SvegMalungKarlstad Sundsvall191213 Sveg2511 Malung10 Common coldest day in 48 years 5 common to all 4 northern stations

24. Max stable processes Independent processes Y i (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 kind of 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 (y i ), b -i (y i ) so that where G -i has non-degenerate margins. Pick a ji (y i ) so that and where h ji is the conditional hazard function of Y j given Y i =y i.

34. Asymptotic properties Let. Then given Y i >u, Y i - 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