1. Extreme Value Analysis
FISH 558 Decision Analysis in Natural Resource Management
11/30/2015
Noble Hendrix
QEDA Consulting LLC
Affiliate Faculty UW SAFS

2. Lecture Overview Motivating examples of extreme events
Generalized Extreme Value
Statistical Development
Case Study: the white cliffs of Dover
Generalized Pareto Distribution
Case Study: whale strikes in SE Alaska
Additional resources

3. Why should we care about extreme events?
They are rare by definition, so why spend much time thinking about them?
Often the consequences of the event have significant impacts to the system – mortality, colonization, episodic recruitment
We tend to focus on averages, but extremes may be more important in some situations.
We may also be interested in estimating extremes beyond what has been observed

4. Distribution of outcomes

5. Distribution of outcomes

6. Distribution of outcomes

7. Distribution of outcomes

8. Distribution of outcomes

9. Motivation 100 year floodplain

10. Motivation Surpassing the 100 year floodplain
Road and home construction based on flood frequency and intensity i.e., 100 year floodplain

11. Motivation Hurricanes

12. Financial Markets

13. Statistical Foundations
Central Limit Theorem
Consider sequence of iid random variables, X1, … Xn We know that sum Sn = X1 + … + Xn, when normalized lead to the CLT:
Statistical Foundations

14. Generalized Extreme Value Fisher-Tippet Asymptotic Theorem
Define maxima of sequence of random variables Mn = max(X1, …, Xn) For normalized maxima, there is also a non-degenerate distribution H(x), which is a GEV distribution

15. Generalized Extreme Value Cumulative Density Function
u – location s – scale v - shape

16. Generalized Extreme Value Variants of the GEV
Shape parameter v defines several distributions: Gumbel: v = 0 Weibull: v < 0 Fréchet: v > 0

17. Generalized Extreme Value Shapes of GEV
Weibull
Gumbel
Fréchet

18. Generalized Extreme Value Applicability
Almost all common continuous distributions converge on H(x) for some value of v
Weibull – beta
Gumbel – normal, lognormal, hyperbolic, gamma, chi-squared
Fréchet – Pareto, inverse gamma, Student t, loggamma

19. Generalized Extreme Value Minima
What about minima? min(X1, …, Xn) = - max(-X1, … ,-Xn) If H(x) is the limiting distribution for maxima, then 1 – H(-x) is the limiting distribution for minima, so can also be handled

20. Generalized Extreme Value Estimation
Obtain data from an unknown distribution F
Let’s assume that there is an extreme value distribution Hv for some value of v
The true distribution of the n-block maximum Mn can be approximated for large enough n with a GEV distribution H(x)
Fit model to repeated observations of an n-block maximum, thus m blocks of size n

21. Generalized Extreme Value Example - Data
Annual sea level height at Dover, Britain between 1912 and 1992

22. Generalized Extreme Value Example - Data
Annual sea level height at Dover, Britain between 1912 and 1992

23. Generalized Extreme Value R package evd
> require(evd) > data(sealevel) > sl.no<-na.omit(sealevel[,1]) > fgev(sl.no) Call: fgev(x = sl.no) Deviance: Estimates loc scale shape Standard Errors loc scale shape

24. Generalized Extreme Value Diagnostics

25. Generalized Extreme Value Return Level Plot
Return level – “how long to wait on average until see another event equal to or more extreme” If H is the distribution of the n-block maximum, the k return level is the 1 – 1/k quantile of H

26. Generalized Extreme Value Profile likelihood of parameters

27. Generalized Extreme Value Limitations
Limitations of the GEV:
Used for block maxima, e.g., annual precipitation, annual flow,
Only 1 exceedance per block
May ignore some important observations,
Some go so far as to say it is a wasteful method! (McNeil et al Quantitative Risk Management, Princeton)

28. Generalized Pareto Distribution
GEV has largely been surpassed by another method for extremes over a threshold Pickands (1975) developed a model for excesses y over threshold a
Pickands 1975 Annals of Stats 3:119

29. Generalized Pareto Distribution
a – threshold b – scale v - shape

30. Generalized Pareto Distribution Shapes of GPD
Positive shape =
limitless loss

31. Generalized Pareto Distribution Applicability
For any continuous distributions that converge on H(x) for some value of v, which was most of the continuous distributions of interest
The same distributions will converge on G(x) as an excess distribution as the threshold a is raised

32. Generalized Pareto Distribution Estimation
Obtain data from an unknown distribution F Calculate Yj = Xj – a for Na that exceed threshold a maximize log-likelihood:

33. Generalized Pareto Distribution Threshold Estimation
Have an interesting problem:
Need a value of threshold a that must be high enough to satisfy the theoretical assumptions
Need enough data above the threshold a so that the parameters are well estimated
Use a sample mean residual life plot to help identify a reasonable threshold value a

34. Generalized Pareto Distribution Sample Mean Residual Life Plot
Let Y = X – a0. At threshold a0, if Y is GPD with parameters b and v then E(Y) = b/(1 – v), v < 1 This is true for all thresholds ai > a0, but the scale parameter bi must be appropriate to the threshold ai E(X-ai| X > ai) = (bi + v*ai)/(1-v), Thus E(X - a| X > a) is a linear function of a where GPD appropriate, so can plot E(x-ai) (where x are our observed data) versus ai. This is the sample mean residual life plot, and confidence intervals added by assuming E(x-a) are approximately normally distributed

35. Generalized Pareto Distribution Example - Data
Quantifying strike rates of whales in southeast Alaska

36. Generalized Pareto Distribution Distances to Whales
Minimum distances (i.e., D < 0) are where losses occur, so transform distance D into a positive loss metric, where value of 100 equates to D = 0

37. Generalized Pareto Distribution Whale Distance Metric

38. Generalized Pareto Distribution Threshold determination
Looking for discontinuities in the mean excess, E(x-ai), at different threshold values ai
Identified value of 70 as the threshold (equates to a distance of 300m between whales and ships)

39. Generalized Pareto Distribution Threshold determination
library(POT)
mrlplot(w.metric, xlim = c(50,90) )
tcplot(w.metric, u.range = c(50, 90) )
Mean residual life plot (previous slide) indicates a = 70
Discontinuity in scale and shape estimates when threshold a > 70

40. Generalized Pareto Distribution Estimation
> fitgpd(w.metric, thresh = 70, est = "mle") Estimator: MLE Deviance: AIC: Varying Threshold: FALSE Threshold Call: 70 Number Above: 151 Proportion Above: Estimates scale shape
Standard Error Type: observed Standard Errors scale shape Asymptotic Variance Covariance scale shape scale shape Optimization Information Convergence: successful

41. Generalized Pareto Distribution Diagnostics

42. Generalized Pareto Distribution Likelihood profiles

43. Generalized Pareto Distribution Likelihood profiles with different thresholds
relative log likelihood - likelihood relative to maximum for that threshold value

44. Generalized Pareto Distribution Empirical and Estimated
Comparison of empirical (no observed strikes) and GPD model estimates for a = 70
Since 2000, 2 confirmed strikes
GPD provides better characterization of risk
Empirical
GPD

45. Generalized Pareto Distribution Return Level
Return level – how many encounters where whales are less than 300m until a strike? Conditional return level of approx. 500 Absolute return level of approx (1 in 5 encounters has an encounter < 300m)

46. Summary: GEV and EVT Generalized Extreme Value (GEV) distribution
Used for block maxima, e.g., maximum sea-level per year
Data loss due to only block maxima
Generalized Pareto Distribution (GPD)
Used for points over a threshold
All exceedances above some limit are used
Question about how to deal with selecting a threshold value

47. Additional Resources Books and Papers
Coles, S An Introduction to Statistical Modelling of Extreme Values. Springer Series in Statistics. London.
McNeil, A. J., Frey, R., & Embrechts, P Quantitative risk management: concepts, techniques, and tools. Princeton University Press.
Embrechts, P Modelling extremal events: for insurance and finance (Vol. 33). Springer.
Bayesian GPD Modeling
Coles, S. and L. Pericchi Anticipating catastrophes through extreme value modeling. Applied Statistics 52(4): 405–416.
Jagger. T. H. and J. B. Elsne Climatology models for extreme hurricane winds near the United States. Journal of Climate 19:

48. Additional Resources Fitting models in R and BUGS
A few R packages
Points over Threshold (POT)
Extreme Value Distributions (evd)
extRemes
Quantitative Risk Management (QRM)
evdbayes
BUGS
OpenBUGS – GEV and GPD
WinBUGS/JAGS – GPD with 1’s trick