Chris Ferro Climate Analysis Group Department of Meteorology University of Reading

1. Extremes in a Varied Climate Chris Ferro Climate Analysis Group Department of Meteorology University of Reading • Significance of distributional changes • Extreme-value analysis of gridded data

2. Significance of Changes Compare daily data at single grid point in Assume X1, …, Xn have same distribution Assume Y1, …, Yn have same distribution

3. Quantiles xp is the p-quantile of the control if An estimate of xp is where are the order statistics.

4. Daily Tmin at Wengen (DJF) Dots mark the 1, 5, 10, 25, 50, 75, 90, 95 and 99% quantiles

5. Confidence Intervals Quantify the uncertainty due to finite samples. A (1 – α)-confidence interval for is (Lp, Up) if

6. Resampling Replicating the experiment would reveal sampling variation of the estimate. Mimic replication by resampling from data. Must preserve any dependence • time (e.g. pre-whitening, blocking) • space (e.g. pairing)

7. Bootstrapping 1 For b = 1, …, B (large) resample {X1*, …, Xn*} and {Y1*, …, Yn*} compute Then

8. Daily Tmin at Wengen (DJF) 90% confidence intervals

9. Descriptive Hypotheses • No change: YX dp = 0 for all p reject unless • Location change: YX + m for some m dp = m for some m and all p reject unless • Location-scale change: YsX + m • Scale change (if natural origin): YsX

10. Simultaneous Intervals Simultaneous (1 – α)-confidence intervals for dp1, …, dpm are (Lp1, Up1), …, (Lpm, Upm) if Wider than pointwise intervals, e.g.

11. Bootstrapping 2 For each p bootstrap set with k chosen to give correct confidence level: estimate level by proportion of the B sets with at least one point outside the intervals.

12. Daily Tmin at Wengen (DJF) 90% pointwise intervals 90% simultaneous intervals

13. Simplest hypothesis not rejected Tmin (DJF) Tmax (JJA)

14. Other issues • Interpretation tricky when distributions change within samples: adjust for trends • Bootstrapping quantiles is difficult: more sophisticated bootstrap methods • Field significance • Computational cost

15. Conclusions • Quantiles describe entire distribution • Confidence intervals quantify uncertainty • Bootstrapping can account for dependence

16. Extreme-value Analysis • Summarise extremes at each grid point • Estimate return levels and other quantities • Framework for quantifying uncertainty • Summarise model output • Validate and compare models • Downscale model output

17. Annual Maxima Let Ys,t be the largest daily precipitation value at grid point s in year t. s = 1, …, 11766 grid points t = 1, …, 31 years Assume Ys,1, …, Ys,31 are independent and have the same distribution.

18. GEV Distribution Probability theory suggests the generalised extreme-value distribution for annual maxima: with parameters

19. Return Levels zs(m) is the m-year return level at s if i.e. exceeded once every m years on average.

20. Fitting the GEV Find to maximise likelihood: Estimate has variance Estimate is obtained from

21. Results 1 – Parameters μ σ γ

22. Results 1 – Return Levels z(100) % standard error

23. Pooling Grid Points Often if r is close to s. Assume if r is a neighbour (r ~ s). Maximise Using 9 × 31 observations increases precision.

24. Standard Errors Taylor expansion yields Independence of grid points implies H = V, leaving For dependence, estimate V using variance of

25. Results 2 – Parameters Original μ σ γ Pooled

26. Results 2 – Return Levels Original z(100) % standard error Pooled

27. Local Variations Potential bias if for r ~ s. Reduce bias by modelling, e.g. Should exploit physical knowledge.

28. Results 3 – Parameters Original μ σ γ Pooled II

29. Results 2 – Parameters Original μ σ γ Pooled

30. Results 3 – Return Levels Original z(100) % standard error Pooled II

31. Results 2 – Return Levels Original z(100) % standard error Pooled

32. Conclusions • Pooling can clarify extremal behaviour increase precision introduce bias • Careful modelling can reduce bias

33. Future Directions • Diagnostics for adequacy of GEV model • Threshold exceedances; k-largest maxima • Size and shape of neighbourhoods • Less weight on more distant grid points • More accurate standard error estimates • Model any changes through time • Clustering of extremes in space and time

34. References • Davison & Hinkley (1997) Bootstrap Methods and their Application. Cambridge University Press. • Davison & Ramesh (2000) Local likelihood smoothing of sample extremes. J. Royal Statistical Soc. B, 62, 191–208. • Smith (1990) Regional estimation from spatially dependent data. www.unc.edu/depts/statistics/faculty/rsmith.html • Wilks (1997) Resampling hypothesis tests for autocorrelated fields. J. Climate, 10, 65 – 82. c.a.t.ferro@reading.ac.uk www.met.rdg.ac.uk/~sws02caf

35. Commentary Annual maximum at grid point s is GEV. • Estimate using data at s. • Estimate using data in neighbourhood of s. Bias can occur if • Allow for local variation in parameters.