Chris Ferro Climate Analysis Group Department of Meteorology University of Reading Extremes in a Varied Climate 1.Significance of distributional changes 2.Extreme-value analysis of gridded data

Significance of Changes Compare daily data at single grid point in Assume X 1, …, X n have same distribution Assume Y 1, …, Y n have same distribution

Quantiles x p is the p-quantile of the control if An estimate of x p is whereare the order statistics.

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

Confidence Intervals Quantify the uncertainty due to finite samples. A (1 – α)-confidence interval for is (L p, U p ) if

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)

Bootstrapping 1 For b = 1, …, B (large) resample {X 1 *, …, X n * } and {Y 1 *, …, Y n * } compute Then

Daily T min at Wengen (DJF) 90% confidence intervals

Descriptive Hypotheses 1.No change: Y  X d p = 0 for all p reject unless 2.Location change: Y  X + m for some m d p = m for some m and all p reject unless 3.Location-scale change: Y  sX + m 4.Scale change (if natural origin): Y  sX

Simultaneous Intervals Simultaneous (1 – α)-confidence intervals for d p 1, …, d p m are (L p 1, U p 1 ), …, (L p m, U p m ) if Wider than pointwise intervals, e.g.

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.

Daily T min at Wengen (DJF) 90% pointwise intervals 90% simultaneous intervals

Simplest hypothesis not rejected T min (DJF)T max (JJA)

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

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

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

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

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

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

Fitting the GEV Findto maximise likelihood: Estimate has variance Estimate is obtained from

Results 1 – Parameters μσγ

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

Pooling Grid Points Oftenif r is close to s. Assumeif r is a neighbour (r ~ s). Maximise Using 9 × 31 observations increases precision.

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

Results 2 – Parameters μσγ Original Pooled

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

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

Results 3 – Parameters μ Original Pooled II σγ

Results 2 – Parameters μσγ Original Pooled

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

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

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

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

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. Wilks (1997) Resampling hypothesis tests for autocorrelated fields. J. Climate, 10, 65 – 82.

Commentary Annual maximum at grid point s is GEV. 1.Estimateusing data at s. 2.Estimate using data in neighbourhood of s. Bias can occur if 3.Allow for local variation in parameters.