1. 1 Extreme Value Modelling in Climate Science: Why do it and how it can fail! Aims: What the heck do we mean by “extreme”? Summary of statistical methods used in climate science Statistics for modelling the process rather than for just making indices Some examples of extreme value modelling: Problem 1: Properties of drought indices Problem 2: Trends in extreme gridded temperatures Problem 2: Trends in largest annual skew tides; Professor David B. Stephenson U. of Exeter NCAR summer colloquium, 8 June 2011 © 2011 D.B.Stephenson@exeter.ac.uk

2. 2 Some wet and windy extremes Extra-tropical cyclone Hurricane Polar low Extra-tropical cyclone Convective severe storm

3. 3 Some dry and hot extremes Drought Wild fire Dust storm

4. 4 All are complex multivariate spatio-temporal events! So to massively simplify, it is helpful to focus in on the time evolution of single variable related to the event e.g. wind speeds of major extratropical cyclones passing by London, losses to an insurers, etc. MARKED POINT PROCESS: random times, random marks

5. 5 What do we mean by “extreme”? Large meteorological values Maximum value (i.e. a local extremum) Exceedance above a high threshold Record breaker (time-varying threshold equal to max of previously observed values) Rare event in the tail of distribution (e.g. less than 1 in 100 years – p=0.01) Large losses (severe or high-impact) (e.g. $200 billion if hurricane hits Miami) hazard, vulnerability, and exposure Gare Montparnasse, 22 Oct 1895 Stephenson, D.B. (2008): Chapter 1: Definition, diagnosis, and origin of extreme weather and climate events, In Climate Extremes and Society, R. Murnane and H. Diaz (Eds), Cambridge University Press, pp 348 pp. NOTE! Extremeness is not a binary property of an event but an ordering of a process

6. 6 IPCC 2001 definitions Simple extremes: “individual local weather variables exceeding critical levels on a continuous scale” Complex extremes: “severe weather associated with particular climatic phenomena, often requiring a critical combination of variables” Extreme weather 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.” Extreme climate event: “an average of a number of weather events over a certain period of time which is itself extreme (e.g.rainfall over a season)” X~N(0,1) Y~N(0.5,1.5) px=rank(x)/(n+1)

7. 7 How might extreme events change? Changes in location, scale, and shape all lead to big changes in the tail of the distribution. Some physical arguments exist for changes in location and scale. E.g. multiplicative change in precipitation due to increased humidity (change in scale) Scale change impacts high quantiles! Example: Normal variable 1% increase in standard deviation s shifts the 10-year return value (x 0.9 ) by 1.28s and the 200-year return value (x 0.995 ) by 2.58s.

8. 8 How can we relate the tails …

9. 9 to the bulk of the distribution? PDF =Probability Density Function Or …Probable Dinosaur Function?? Change in scale Change in shape

10. 10 Quantile attribution Describe the changes in quantiles in terms of changes in the location, the scale, and the shape of the parent distribution: The quantile shift is the sum of: a location effect (shift in median) a scale effect (change in IQR) a shape effect Ferro, C.A.T., D.B. Stephenson, and A. Hannachi, 2005: Simple non-parametric techniques for exploring changing probability distributions of weather, J. Climate, 18, 4344 4354. Beniston, M. and Stephenson, D.B. (2004): Extreme climatic events and their evolution under changing climatic conditions, Global and Planetary Change, 44, pp 1-9

11. 11 Example: Regional Model Simulations of daily T max  Changes in location, scale and shape all important T 90 ΔT 90 (2071-2100 minus 1971-2000) ΔT 90 -Δm ΔT 90 -Δm-(T 90 -m) Δs/s

12. 12 Statistical methods used in climate science Extreme indices – sample statistics Basic extreme value modelling GEV modelling of block maxima GPD modelling of excesses above high threshold Point process model of exceedances More complex EVT models Inclusion of explanatory factors (e.g. trend, ENSO, etc.) Spatial pooling Max stable processes Bayesian hierarchical models + many more Other stochastic process models

13. 13 Extreme indices are useful and easy but … They don’t always measure extreme values in the tail of the distribution! They often confound changes in rate and magnitude They strongly depend on threshold and so make model comparison difficult They say nothing about extreme behaviour for rarer extreme events at higher thresholds They generally don’t involve probability so fail to quantify uncertainty (no inferential model) More informative approach: model the extremal process using statistical models whose parameters are then sufficient to provide complete summaries of all other possible statistics (and can simulate!) See: Katz, R.W. (2010) “Statistics of Extremes in Climate Change”, Climatic Change, 100, 71-76

14. 14 Furthermore … indices are not METRICS! One should avoid the word “metric” unless the statistic has distance properties! Index, sample/descriptive statistic, or measure is a more sensible name! Oxford English Dictionary: Metric - A binary function of a topological space which gives, for any two points of the space, a value equal to the distance between them, or a value treated as analogous to distance for analysis. Properties of a metric: d(x, y) ≥ 0 d(x, y) = 0 if and only if x = y d(x, y) = d(y, x) d(x, z) ≤ d(x, y) + d(y, z)

15. 15 Universal Poisson process for extremes N=number of points with Z>z t=t 1 t=t 2  Miraculous limit theorem for tails of i.i.d. variables!

16. 16 Probability models for maxima and excesses Note: extremal properties are characterised by only three parameters (for ANY underlying distribution!)

17. 17 Why use these probability models? Model parameters are sufficient for providing a complete threshold-independent description of extremal properties. All other statistics of the extremal process are a function of these three parameters. The models provide a rigorous probability framework for making inference about extremal behaviour. Their mathematically justifiable parametric form allows more precise inference about tail properties. Model can be used to smoothly interpolate between empirical quantiles/probabilities. Such interpolation has made efficient use of all the large values; Model can be used to extrapolate out carefully to rarer less frequently (or never!) observed events AND provide intervals for such predictions!

18. 18 Problem 1: Do 2 drought series have similar extremal properties? Observed index n=90 Reconstructed index n=5000 Data example kindly provided by Eleanor Burke, Met Office

19. 19 Do 2 drought series have similar extremal properties? Observed index n=90 + Reconstructed index n=5000 Data example kindly provided by Eleanor Burke, Met Office

20. 20 Return level plots Outlier in the extended data set? Slight kink at 2.5 in d1

21. 21 Quantile-Quantile plot Empirical distributions similar except for the big outlier in d1

22. 22 Modelling the excesses using GPD

23. 23 Use of mean excess to find a suitable threshold u  GPD implies linear behaviour in mean excess for u from about 0 to 1  Try fits with u=0.5 as threshold Observed d0 Simulated d1

24. 24 Nested model approach

25. 25 Is there a statistically significant difference at 5% level? Difference in deviance 1050.4-1053.6+2*2=0.8  p=0.67 Parameter estimates 0.234/0.321=0.729  p=0.23 0.299/0.313=0.956  p=0.17 Maximum Likelihood Estimates ModelNo. of params Akaike Inf. Criterion Null21050.40.631 (0.023) ----0.107 (0.024) --- Contrast41053.60.629 (0.023) 0.234 (0.321) -0.105 (0.024) -0.299 (0.313) Contrast with outlier 41085.20.599 (0.021) 0.264 (0.321) -0.042 (0.019) -0.361 (0.313) No significant difference between the exceedances at 5% level

26. 26 Model checking: do the quantiles match? No! The null model underestimates the empirical quantiles

27. 27 Model checking: are estimates stable? No! Constant up to u=1.7 but then trends for larger values?!

28. 28 Model checking: uniform in time? Uniform distribution in time and exponential between events

29. 29 Problem 2: Extremes in surface temperature Coelho, C.A.S., Ferro, C.A.T., Stephenson, D.B. and Steinskog, D.J. (2008): Methods for exploring spatial and temporal variability of extreme events in climate data, Journal of Climate, 21, pp 2072-2092 Observed surface temperatures 1870-2005 Monthly mean gridded surface temperature (HadCRUT2v) 5 degree resolution Summer months only: June July August Grid points with >50% missing values and SH are omitted. Maximum monthly temperatures

30. 30 Long term trend in mean 75 th quantile (u y,m = 16.2ºC) 2003 exceedance Excess (T y,m – u y,m ) Non-stationarity due to seasonality and long term trends Example: Grid point in Central Europe (12.5ºE, 47.5ºN)

31. 31 GPD scale and shape estimates Shape parameter is mainly negative suggesting finite upper temperature. Spatial pooling has been used to get more reliable less noisey shape estimates Scale parameter is large over high- latitude land areas AND shows some dependence on x=ENSO.

32. 32 How significant is ENSO on extremes?  Null hypothesis of no effect can only be rejected with confidence over tropical Pacific and Northern Continents

33. 33 Use of covariates in models “with four parameters I can fit an elephant and with five I can make him wiggle his trunk.” - John von Neumann

34. 34 Model can be used to estimate return periods  return period of 133 years for August 2003 event over Europe Excess for August 2003 Return period for the excess for August 2003

35. 35 Spatial pooling Pool over local grid points but allow for spatial variation by including local spatial covariates to reduce bias (bias-variance tradeoff). For each grid point, estimate 5 GPD parameters by maximising the following likelihood over the 8 neighbouring grid points: No spatial pooling: 2 parameters from n data values Local pooling:5 parameters from 9n data values Coelho et al., 2008:Methods for Exploring Spatial and Temporal Variability of Extreme Events in Climate Data, J. Climate

36. 36 Teleconnections of extremes  association with extremes in subtropical Atlantic

37. 37 Problem 3: Is there a time trend in extreme skew tides? 10 largest skew tides for each of n=149 years Is there a time trend in the extremes? Dots show largest values Line is linear fit to the mean of the 10 values Data example kindly provided by Tom Howard, Met Office

38. 38 r Largest Order model

39. 39 Estimates for null model are similar for r=1,5,10 Estimates get more precise for larger r Null model has slightly better AIC than trend model Trend model has trouble estimating trend parameter  Could either constrain shape=0 and/or pool over more data Maximum Likelihood Estimates ModelNo. of params AIC Null r=10 3-6882.10.661 (0.0096) ---0.147 (0.0065) 0.033 (0.025) Null r=5 3-2469.20.659 (0.0099) ---0.146 (0.0068) 0.050 (0.034) Null r=1 3-91.50.658 (0.014) ---0.146 (0.0099) 0.031 (0.059) Trend r=10 4-6880.40.754 (0.0095) -4.57E-5 (???) 0.147 (0.0042) 0.032 (0.0025)

40. 40 Model checking: null model r=10 Model slightly underestimates largest r=1 and r=2 quantiles

41. 41 Model checking: trend model r=10  Including a time trend does not improve the r=1 and 2 fits

42. 42 Summary Sufficiently large values of an independent identically distributed variable can be described by a 3-parameter non-homogenous Poisson process; This leads to simple parametric forms for the distribution of maxima and r-largest values (GEV) and exceedances above a high threshold (GPD); MLE can be used to estimate the parameters (but estimates are often sensitive to individual values); Non-stationarity can be accounted forby making model parameters systematic functions of covariates; Spatial pooling can be used to obtain more precise estimates but covariates have to be included to avoid bias

43. 43 Some outstanding questions … 1. What do extreme indices really tell us about extremes? 2. How best to develop well-specified extreme value models that account for non-stationarity (non-identical distributions) caused by natural and climate change processes? 3. How to deal with large sampling uncertainty due to the rarity of events and shortness of available observational records? Robust estimation in the presence of outlier events? 4. What can imperfect climate models tell us about real world extremes? How to bias correct model errors in extremes? 5. How to develop and test well-specified inferential frameworks for prediction and attribution of real world extremes from multi-model ensembles?

44. 44 References Stephenson, D.B. (2008): Chapter 1: Definition, diagnosis, and origin of extreme weather and climate events, In Climate Extremes and Society, Cambridge University Press, pp 348 pp. Definitions of what we mean by extreme, rare, severe and high-impact events Ferro, C.A.T., D.B. Stephenson, and A. Hannachi, 2005: Simple non-parametric techniques for exploring changing probability distributions of weather, J. Climate, 18, 4344 4354. Attribution of changes in extremes to changes in bulk distribution Beniston, M. and Stephenson, D.B. (2004): Extreme climatic events and their evolution under changing climatic conditions, Global and Planetary Change, 44, pp 1-9 Time-varying attribution of changes in heat wave extremes to changes in bulk distribution Coelho, C.A.S., Ferro, C.A.T., Stephenson, D.B. and Steinskog, D.J. (2008): Methods for exploring spatial and temporal variability of extreme events in climate data, Journal of Climate, 21, pp 2072-2092 GPD fits to gridded data including covariates. Spatial pooling and teleconnection methods. Antoniadou, A., Besse, P., Fougeres, A.-L., Le Gall, C. and Stephenson, D.B. (2001): L Oscillation Atlantique Nord NAO: et son influence sur le climat europeen, Revue de Statistique Applique, XLIX (3), pp 39-60 One of the earliest papers to use climate covariates in EVT fits – NAO effect on CET extremes Stuart Coles, An Introduction to Statistical Modeling of Extreme Values, Springer. Excellent overview of extreme value theory.

45. 45 There are worse things than extreme climate … e.g. extreme ironing! Thanks for your attention d.b.stephenson@exeter.ac.uk

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47. 47 Proposed taxonomy of atmospheric extremes Rare weather/climate events Rare and Severe events Rare, Severe, Acute events e.g. hurricane in New England Rare, Severe Chronic events e.g. European blocking Rare and Non-Severe Events Rare, Non-severe, Acute events e.g. hurricane over the South Atlantic ocean Rare, Non-severe, Chronic events e.g. Atlantic blocking Rarity Severity Rapidity Acute: Having a rapid onset and following a short but severe course. Chronic: Lasting for a long period of time or marked by frequent recurrence Stephenson, D.B. (2008): Chapter 1: Definition, diagnosis, and origin of extreme weather and climate events, In Climate Extremes and Society, R. Murnane and H. Diaz (Eds), Cambridge University Press, pp 348 pp.