**Extreme Value Analysis** FISH 558 Decision Analysis in Natural Resource Management 12/4/2013 Noble Hendrix QEDA Consulting LLC Affiliate Faculty UW SAFS

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

**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

**Distribution of outcomes**

**Distribution of outcomes**

**Distribution of outcomes**

**Distribution of outcomes**

**Distribution of outcomes**

**Motivation100 year floodplain**

**MotivationSurpassing the 100 year floodplain** • Road and home construction based on flood frequency and intensity i.e., 100 year floodplain

**MotivationHurricanes**

**Financial Markets**

**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

**Generalized Extreme ValueFisher-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

**Generalized Extreme ValueCumulative Density Function** u – location s – scale v - shape

**Generalized Extreme ValueVariants of the GEV** Shape parameter v defines several distributions: Gumbel: v = 0 Weibull: v < 0 Fréchet: v > 0

**Generalized Extreme ValueShapes of GEV** Weibull Gumbel Fréchet

**Generalized Extreme ValueApplicability** 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

**Generalized Extreme ValueMinima** 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

**Generalized Extreme ValueEstimation** Obtain data from an unknown distribution F • Let’s assume that there is an extreme value distribution Hvfor 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

**Generalized Extreme ValueExample - Data** Annual sea level height at Dover, Britain between 1912 and 1992

**Generalized Extreme ValueExample - Data** Annual sea level height at Dover, Britain between 1912 and 1992

**Generalized Extreme ValueR package evd** > require(evd) > data(sealevel) >sl.no<-na.omit(sealevel[,1]) > fgev(sl.no) Call: fgev(x = sl.no) Deviance: -5.022368 Estimates loc scale shape 3.59252 0.20195 -0.02107 Standard Errors loc scale shape 0.02642 0.01874 0.07730

**Generalized Extreme ValueDiagnostics**

**Generalized Extreme ValueReturn 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/kquantile of H

**Generalized Extreme ValueProfile likelihood of parameters**

**Generalized Extreme ValueLimitations** • 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. 2005 Quantitative Risk Management, Princeton)

**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

**Generalized Pareto Distribution** a – threshold b – scale v - shape

**Generalized Pareto DistributionShapes of GPD** Positive shape = limitless loss

**Generalized Pareto DistributionApplicability** 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

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

**Generalized Pareto DistributionThreshold 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

**Generalized Pareto DistributionSample Mean Residual Life** Plot Let Y = X– a0. At threshold a0, if Y is GPD with parameters b and vthen 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

**Generalized Pareto DistributionExample - Data** Quantifying strike rates of whales in southeast Alaska

**Generalized Pareto DistributionDistances 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

**Generalized Pareto DistributionWhale Distance Metric**

**Generalized Pareto DistributionThreshold 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)

**Generalized Pareto DistributionThreshold determination** library(POT) mrlplot(w.metric, xlim = c(50,90) ) tcplot(w.metric, u.range = c(50, 90) ) Meanresidual life plot (previousslide) indicatesa = 70 Discontinuity in scale and shapeestimateswhenthresholda > 70

**Generalized Pareto DistributionEstimation** > fitgpd(w.metric, thresh = 70, est = "mle") Estimator: MLE Deviance: 974.4418 AIC: 978.4418 VaryingThreshold: FALSE Threshold Call: 70 Number Above: 151 Proportion Above: 0.1946 Estimates scale shape 14.8380 -0.4706 Standard Error Type: observed Standard Errors scale shape 1.53542 0.07452 Asymptotic Variance Covariance scale shape scale 2.357530 -0.106864 shape -0.106864 0.005553 Optimization Information Convergence: successful

**Generalized Pareto DistributionDiagnostics**

**Generalized Pareto DistributionLikelihood profiles**

**Generalized Pareto DistributionLikelihood profiles with** different thresholds relative log likelihood - likelihood relative to maximum for that threshold value

**Generalized Pareto DistributionEmpirical 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

**Generalized Pareto DistributionReturn 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. 2500 (1 in 5 encounters has an encounter < 300m)

**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

**Additional ResourcesBooks and Papers** Coles, S. 2001. An Introduction to Statistical Modelling of Extreme Values. Springer Series in Statistics. London. McNeil, A. J., Frey, R., & Embrechts, P. 2005. Quantitative risk management: concepts, techniques, and tools. Princeton University Press. Embrechts, P. 1997. Modellingextremal events: for insurance and finance (Vol. 33). Springer. Bayesian GPD Modeling Coles, S. and L. Pericchi. 2003. Anticipating catastrophes through extreme value modeling. Applied Statistics 52(4): 405–416. Jagger. T. H. and J. B. Elsne 2004. Climatology models for extreme hurricane winds near the United States. Journal of Climate 19: 3220-3236.

**Additional ResourcesFitting 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