# Estimation and attribution of changes in extreme weather and climate events

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1 IPCC workshop on extreme weather and climate events, June 2002, Beijing. Estimation and attribution of changes in extreme weather and climate events Dr. David B. Stephenson Department of Meteorology University of Reading cag

2 Plan of the talk 1. Introduction 2. Descriptive methods 3. Understanding the changes 4. Modelling the tail of the pdf 5. Recommendations

3

4 The definition problem: Extreme events can be defined by: Gare Montparnasse, 22 October 1895 Train crash here Maxima/ minima Magnitude Rarity Impact/ losses Man can believe the impossible, but man can never believe the improbable. - Oscar Wilde

5 IPCC 2001 definition of extreme event: An extreme weather event is an event that is rare within its statistical reference distribution at a particular place. Definitions of "rare" vary, but an extreme weather event would normally be as rare or rarer than the 10th or 90th percentile.

6 Daily CET winter temperatures (Nov-Mar) Sample 1: n= values Sample 2: n= values

7

8 Change in the p.d.f. of winter temp. Above freezing Below freezing Note shift in probability density for cold temperatures

9 Order statistics x [i] Simply rank sample values in ascending order Largest values in sample from

10 Empirical Distribution Function (e.d.f.) = =

11 Empirical return level plot x = A way of plotting the distribution function x vs. 1/(1-F)

12 Empirical quantile-quantile plots Note: quantiles differ not just a shift due to means : mean=-4.31 deg C stdev= 3.45 deg C : mean=-4.94 deg C stdev= 3.26 deg C

13 Negated Central England Temperature

14 Marked point process for exceedances Marked point process = random point process in time + random amount at each event

15 Principal attributes of exceedances Rate of exceedance R= Pr{ X> u} Mean excess M= E(X-u X> u) Volatility V= Stdev(X-u X> u)/ M Clustering in time L= 1/ Extremal Index Note: In many weather/ climate studies only rate is discussed yet the other attributes also merit attention!

16 Attributes as a function of threshold Rate R Mean excess M Volatility V Cluster time C

17 Changes in the rate of exceedance R R = X > u = F u Note reduced rate of exceedance in the later period.

18 Changes in the mean excess = = > M mean excess E( X u X u)

19 Changes in the volatility of excesses Volatility = = Coefficien t of Variation of std. deviation( excess)/mean(excess) excesses Note: values less than one imply Weibull type distribution with finite upper limit

20 Clustering in time For an unclustered point process the interarrival times T(i+1)-T(i) are exponentially distributed. However, extremes often come in clusters and it is important to take account of this dependency:

21 Changes in cluster length L = (Extremal Index) 1 = 2 T ( 2 T )2

22 Summary statistics Period R Log R M V L Change Fewer extremes Less intense extremes Shorter clusters

23

24 Changes in location, scale, and shape Fig. 5.1 from PhD thesis of Tahl Kestin: a) Shift in mean b) Increase in variance c) Change in shape

25 Hypotheses about changing extremes H0: No change (variation due to sampling only) Sampling uncertainty can be tested using tail model H1: Change due to mean effect e.g. Mearns et al. (1984), Wigley (1985), H2: Change due to variance effect e.g. Katz and Brown (1992), Katz and coworkers H3: Change due to mean and variance effects e.g. Brown and Katz (1995), H4: Structural change in shape etc e.g. Kestin 2001, Antoniadou et al. 2001

26 Response to change in the mean Change in exceedance rate for standard normal distribution Dashed lines show 95% confidence intervals for estimates based on 400 samples Note: nonlinear response in p2/p1 p2/p1 is very sensitive especially for small p1 sampling uncertainty also increases for small p1

27 Signal and noise for change in mean Change in exceedance rate for shift of 0.5 in mean of the standard normal distribution. Red lines show 95% confidence intervalsfor estimates based on 400 samples Largest uncertainty for most extreme events σ 1 np 1 p p 2 1 Note: largest change ( signal ) for the most extreme events BUT also much more sampling uncertainty ( noise ) in the tail signal/noise ratio decreases for more extreme events!

28 Response to change in variance Change in exceedance rate for standard normal distribution Dashed lines show 95% confidence intervalsfor estimates based on 400 samples Note: very nonlinear response in p2/p1 p2/p1 is very sensitive especially for small p1 sampling uncertainty also increases substantially for small p1

29 Signal diagnosis When δ log p 2 R p 1 = Pr{ X = f R x 1 F e x > x} δµ { h( δµ + zδσ )} 1 F + x σ µ Example:for CET threshold x dlog R = = = 0. 5( x σ µ δσ 0 : mean and variance change both contribute = )

30 The hazard function for CET h x = f F x x

31 Structural change Antoniadou et al Tail shape parameter has become more sensitive to westerly flow NAO+ winters in the 20 th century

32

33 Why do we need to model the tail? Quantify sampling uncertainty Make predictions beyond what is in the data sample e.g. larger values Use asymptotic theory to get the most information from the tail behaviour Summarise tail behaviour at ALL thresholds using just a few parameters Note: Modelling whole of parent distribution can fail to capture proper tail behaviour e.g. rainfall extremes and gamma fits to parent.

34 Tail parameter estimation methods 1. GEV fits to block maxima Uses only one value per block! (wastes data!) Choice of block length? 2. GEV fits to r-largest order statistics Choice of r? Choice of block length? Uses r values per block (better than maxima method) 3. GPD fits to peaks over threshold Uses ALL data above threshold Results should be insensitive to choice of threshold Choice of threshold? 4. Point process fits to peaks over threshold Uses ALL data above threshold Stochastic process that can be generalised Choice of threshold?

35 The Generalised Extreme Value distribution G Maxima/minima and the extreme tail of distributions can be modelled asymptotically by the 3-parameter Generalised Extreme Value (GEV) distribution: z = x µ ξ = + ξ σ z 1 ξ

36 Parametric description of attributes = R(u) ξ ξ µ ξ σ σ µ ξ ξ ) ( 1 ) ( ) ( 1 1 exp ) ( 1 = + = + = u V u u M u n u R Note: These expressions can be used to calculate the attributes for ANY threshold once the 3 tail parameters are known

37 The Generalised Pareto Distribution Excesses X-u above a high threshold u can be modelled using the 2-parameter Generalised Pareto Distribution: X σ u u > z = + ξ z 1 ξ σ u = σ + ξ u µ

38 Poisson process above high threshold Probability of finding m points in region above x = m! e λλ m ξ σ ξ λ 1/ ) ( ) ( where rate + = Ω u u x t t Ω

39 Point process estimates (with std. errors) µ σ ξ Period µ σ ξ ± ± ± ± ± ± ± ± ± Change -0.73(-2.03) 0.01(0.06) (-1.10) -4.3 (*) are t values change divided by standard error: t >1.96 is significant at 5% level statistically significant decrease in tail mean (p<0.05) hardly any change in tail variance large decrease in shape parameter (more convex) large reduction in upper limit of distribution

40

41 Main conclusions Extremes are fascinating but require careful statistical analysis ALL attributes of extremes should be investigated NOT just the rate of exceedence Tail parameters should be estimated using as much data as possible (e.g. not block maxima, use of pooling, etc.) Changes in attributes should be diagnosed in terms of changes in mean and variance of parent distribution (trends in variance=?) More complicated structural change might occur and can be diagnosed from regression on large-scale factors (covariates)

42

43 This document was created with Win2PDF available at The unregistered version of Win2PDF is for evaluation or non-commercial use only.

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