ExtremeValueAnalysisericg/Extremes.pdf · ExtremeValueAnalysis SystemsThinking, Spring2011...
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Extreme Value Analysis
Systems Thinking,Spring 2011Boulder, Colorado
Eric GillelandE-mail: EricG @ ucar.eduResearch Applications Laboratory,National Center for Atmospheric Research (NCAR)Weather and Climate Impacts Assessment Science (WCIAS) Program
Photo by Everett Nychka
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Motivation
Colorado Lottery
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Motivation
Colorado LotteryPr{Winning ≥ $10, 000 in one drawing} ≈
0.000001306024
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Motivation
Colorado LotteryPr{Winning ≥ $10, 000 in one drawing} ≈
0.000001306024
In ten years, playing one ticketeveryday, Pr{Winning ≥ $10, 000} ≈
0.004793062
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Motivation
Colorado LotteryPr{Winning ≥ $10, 000 in one drawing} ≈
0.000001306024
In ten years, playing one ticketeveryday, Pr{Winning ≥ $10, 000} ≈
0.004793062
In 100 years ≈ 0.05003321
In 1000 years ≈ 0.7686185
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Motivation
Colorado LotteryPr{Winning ≥ $10, 000 in one drawing} ≈
0.000001306024
In ten years, playing one ticketeveryday, Pr{Winning ≥ $10, 000} ≈
0.004793062
In 100 years ≈ 0.05003321
In 1000 years ≈ 0.7686185
Law of small numbers: events with small probably rarely happen, buthave many opportunities to happen. These follow aPoisson distribution.
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Motivation
Colorado Lottery
Can also talk about waiting time probability. The exponentialdistribution models this.
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Motivation
Colorado Lottery
Can also talk about waiting time probability. The exponentialdistribution models this. For example, the probability that it willtake longer than a year to win the lottery (at one ticket per day) is≈ 0.999523, longer than ten years ≈ 0.9952411, longer than 500 years≈ 0.7877987, and so on (decays exponentially, but with a very slowrate).
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Motivation
Colorado Lottery
Another way to put it is that the expected number of years that itwill take to win more than $10,000 in the lottery (buying one ticketper day) is about 2,096 years. If a ticket costs $1, then we can expectto spend $765,682.70 before winning at least $10,000.
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Motivation
Taleb, N.N. 2010: The Black Swan: The impact of the highly improb-able, Random House, New York, NY, 444 pp.
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Outline
• Further motivation for why extremes are of interest, and why theyrequire careful attention to analyze them.
• Introduce the basics of statistical Extreme Value Analysis (EVA).
• Discuss some limitations for practical applications (climate heavy).
• Introduce the idea of correlation, and why this topic has caused alot of controversy regarding the current economic crisis.
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Motivation
On the eve of the events in 1914 leading to WWI, would you haveguessed what would happen next?
Archduke Franz Ferdinand of Austria
How about the rise ofHitler and WWII?
Adolf Hitler
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Motivation
The impact of computers?Spread and impact of the internet?The stock market crash of 1987, and its surprising recovery?
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Motivation
Retrospective PredictabilityDifferent from Prospective Predictability. Once something has happened,it is easier to trace the steps to find the cause and effect.
PerspectiveInsider Trading can lead to an extreme event that is well prospectivelypredicted by those on the inside, but if done right, is a surprise toeveryone else (ethics).
RiskHave you considered extreme events in your risk analysis for yourfinancial portfolio?
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Motivation
Taleb defines a Black Swan event as
• being rare
• having an extreme impact
• being predictable retrospectively, not prospectively.
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Motivation
Randomness and Large Deviations
Focus is typically on central tendencies,
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Motivation
Randomness and Large Deviations
Focus is typically on central tendencies,
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Motivation
Randomness and Large Deviations
Focus is typically on central tendencies,
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Motivation
Law of Large Numbers, Sum Stability, Central Limit TheoremAnd other results give theoretical support for use of theNormal distribution for analyzing most data.
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Motivation
Law of Large Numbers, Sum Stability, Central Limit TheoremAnd other results give theoretical support for use of theNormal distribution for analyzing most data.
But, it is the possibleextreme (or rare) events thatare the most influencial.
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Background
Extremal Types Theorem
Theoretical support for using the Extreme Value Distributions(EVD’s) for extrema .
• Valid for maxima over very large blocks, or
• Excesses over a very high threshold.
It is possible that there is no appropriate distribution for extremes,but if there is one, it must be from the Generalized Extreme Value(GEV) family (block maxima) or the Generalized Pareto (GP) family(excesses over a high threshold). The two families are related.
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Background
Extremal Types Theorem
Theoretical support for using the Extreme Value Distributions(EVD’s) for extrema .
• Valid for maxima over very large blocks, or
• Excesses over a very high threshold.
It is possible that there is no appropriate distribution for extremes,but if there is one, it must be from the Generalized Extreme Value(GEV) family (block maxima) or the Generalized Pareto (GP) family(excesses over a high threshold). The two families are related.Poisson process allows for a nice characterization of the thresholdexcess model that neatly ties it back to the GEV distribution.
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Background
Simulated Maxima
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Background
GEV
Three parameters: location, scale and shape.
Pr{X ≤ z} = exp
{−[1 + ξ
(z − µ
σ
)]−1/ξ}
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Background
GEV
Three parameters: location, scale and shape.
Pr{X ≤ z} = exp
{−[1 + ξ
(z − µ
σ
)]−1/ξ}
Three types of tail behavior:
1. Bounded upper tail (ξ < 0, Weibull),
2. light tail (ξ = 0, Gumbel), and
3. heavy tail (ξ > 0, Fréchet).
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Background
Weibull Type
Bounded upper tail is a function of parameters. Namely, µ− σ/ξ.
ξ < 0Temperature, Wind Speed,Sea Level
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Background
Weibull TypePredicted Speed Limits for:Thoroughbreds (Kentucky Derby) ≈ 38 mphGreyhounds (English Derby) ≈ 38 mphMen (100 m distance) ≈ 24 mphWomen (100 m distance) ≈ 22 mphMen (10 km distance) ≈ 15 mphWomen (marathon distance) ≈ 12 mphWomen (marathon distance, population model)
≈ 11.45 mph
Paula Radcliffe, 11.6 mph world marathon record
Denny, M.W., 2008: Limits to running speed in dogs, horses andhumans. J. Experim. Biol., 211:3836–3849.
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Background
Gumbel Type
Light-tailed distribution (i.e., decays exponentially)
ξ = 0Domain of attraction for manycommon distributions(e.g., the normal distribution)
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Background
Gumbel Type
Light-tailed distribution (i.e., decays exponentially)
ξ = 0Domain of attraction for manycommon distributions(e.g., the normal distribution)
Single point in a continuous parameter space!
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Background
Fréchet Type
Heavy-tailed distribution (i.e., decays polynomially)Bounded lower tail at µ− σ/ξ.
ξ > 0Precipitation, Stream Flow,Economic Impacts
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Background
Fréchet Type
Heavy-tailed distribution (i.e., decays polynomially)Bounded lower tail at µ− σ/ξ.
ξ > 0Precipitation, Stream Flow,Economic Impacts
Infinite mean if ξ ≥ 1!Infinite variance if ξ ≥ 1/2!
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Background
All three types together
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Background
Analogous for Peaks Over a Threshold (POT) approach
Generalized Pareto Distribution (GPD), which has two parameters:scale and shape. Threshold replaces the location parameter.Three Types:
1. Beta (ξ < 0), bounded above at threshold−σ/ξ2. Exponential (ξ = 0), light tail
3. Pareto (ξ > 0), heavy tail
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Background
MinimaSame as maxima using the relation:
min{X1, . . . , Xn} = −max{−X1, . . . ,−Xn}
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Background
MinimaSame as maxima using the relation:
min{X1, . . . , Xn} = −max{−X1, . . . ,−Xn}
Analogous for POT approach: Look at negatives of deficits under athreshold instead of excesses over a threshold.
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Background
Block Maxima vs. POT
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Background
Block Maxima vs. POT
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Background
Block Maxima vs. POT
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Examples
Fort Collins, Coloradodaily precipitation amount
• Time series of daily precipitationamount (inches), 1900–1999.
• Semi-arid region.
• Marked annual cycle in precipitation(wettest in late spring/early summer,driest in winter).
• No obvious long-term trend.
• Recent flood, 28 July 1997.(substantial damage toColorado State University)
http://ccc.atmos.colostate.edu/~odie/rain.html
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Examples
Fort Collins, Colorado precipitation
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Examples
Fort Collins, Colorado Annual Maximum Precipitation
How often is such anextreme expected?
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Examples
Fort Collins, Colorado precipitationGumbel hypothesis rejected at 5% level.
ξ ≈ 0.17, 95% CI ≈ (0.01, 0.37)
Fréchet (heavy tail)
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ExamplesFort Collins, Colorado precipitation
Risk Communication
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ExamplesFort Collins, Colorado precipitation
Risk CommunicationEasy to invert GEV distribution to get quantiles, which forblock maxima are Return Levels .
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ExamplesFort Collins, Colorado precipitation
Risk CommunicationEasy to invert GEV distribution to get quantiles, which forblock maxima are Return Levels .
The return level is the value expected to beexceeded on average once every 1/p years.
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ExamplesFort Collins, Colorado precipitation
Risk Communication
For 1/p = 10 years, the return level is ≈ 2.8 inches with95% CI ≈ (2.4, 3.2) inches.
For 1/p = 100 years, the return level is ≈ 5.1 incheswith 95% CI ≈ (3.4, 6.8) inches.
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ExamplesFort Collins, Colorado precipitation
Risk Communication
Pr{annual max. precip. ≥ 3 inches} ≈ 0.08That is, the return period for 3 inchesof accumulated rainfall at this gaugein Fort Collins is estimated to beabout 12.5 years.
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Examples
Fort Collins, Colorado precipitation
Can also obtain other information, such as
• Mean annual maximum daily precipitation accumulation ≈ 1.76inches (6= µ̂ ≈ 1.35).
• Variance is ≈ 0.84 inches2.
• Standard deviation is ≈ 0.92 inches (6= σ̂ ≈ 0.53).
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Background
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Examples
Hurricane damage
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1930 1940 1950 1960 1970 1980 1990
020
4060
Economic Damage from Hurricanes (1925−1995)
Year
billi
on U
S$
Economic damage causedby hurricanes from 1926to 1995.
Trends in societalvulnerability removed.
Excess over threshold ofu = 6 billion US$.
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Examples
Hurricane damage
10 20 30 40 50 60 70
0.00
0.05
0.10
0.15
0.20
GPD
σ̂ ≈ 4.589
ξ̂ ≈ 0.512
95% CI for shapeparameter usingprofile likelihood.≈ (0.05, 1.56)Heavy tail!
51
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Examples
Hurricane Dennis (2005)Caused at least 89 deaths and2.23 billion USD in damage.
Impactful despite being under the 6 billion USD threshold!
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Examples
Phoenix (airport) minimum temperature (oF)
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−85
−80
−75
−70
−65
−60
Phoenix airport
(neg
ativ
e) M
in. T
empe
ratu
re (
deg.
F)
July and August 1948–1990.
Urban heat island (warmingtrend as cities grow).
Model lower tail as upper tailafter negation.
Dependence over the threshold.
Temporal trend!
53
![Page 54: ExtremeValueAnalysisericg/Extremes.pdf · ExtremeValueAnalysis SystemsThinking, Spring2011 Boulder,Colorado EricGilleland ... It is possible that there is no appropriate distribution](https://reader031.fdocuments.net/reader031/viewer/2022030416/5aa2aa717f8b9aa0108d820e/html5/thumbnails/54.jpg)
Examples
Phoenix minimum temperature
1950 1960 1970 1980 1990
6570
75
Phoenix summer minimum temperature
Year
Min
imum
tem
pera
ture
(de
g. F
)
Regression-like approach.Covariate information inGEV parameters.
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Examples
http://pubs.usgs.gov/fs/FS-229-96
Rapid urbandevelopmentstarted about 1970.
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Extreme Value Problems in Climatology
2006 European Heat Wave F5 Tornado in Elie Manitoba(Fig. from KNMI) on Friday, June 22nd, 2007
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Extreme Value Problems in Climatology
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Banff Calgary
~40−km CFDDA reanalysis (1985−2005)~200−km NCAR/NCEP reanalysis (1980−1999)~150−km CCSM3 regional climate model
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Extremes vs Extreme Impacts
ExtremesMay or may not have an extreme impact depending on various factors(e.g., location, duration).
Combinations of ordinary conditionsFrozen ground and rain (e.g., 1959 Ohio statewide flood).
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Weather Spells
Many different ways to define themtechnically.
Do extremes of lengths of spells follow EV df’s?
The same type of weather spell may or may not be importantdepending on where it occurs.
Photo from NCAR Digital Image Library, DI00325.
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Defining an Extreme Event
What is a Drought?
"a period of abnormally dry weather sufficiently prolonged for the lackof water to cause serious hydrologic imbalance in the affected area."-Glossary of Meteorology (1959)
Photo from NCAR’s digital image library, DIO1492.
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Defining an Extreme Event
What is a Drought?
Meteorological–a measure of departure of precipitation from normal.Due to climatic differences, what might be considered a droughtin one location of the country may not be a drought in anotherlocation.
Agricultural–refers to a situation where the amount of moisture inthe soil no longer meets the needs of a particular crop.
Hydrological–occurs when surface and subsurface water supplies arebelow normal.
Socioeconomic–refers to the situation that occurs when physicalwater shortages begin to affect people.
http://www.wrh.noaa.gov/fgz/science/drought.php?wfo=fgz
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Defining an Extreme Event
What is a Heat Wave?(e.g., Meehl and Tebaldi, 2004, Science, 305, 994–997):
• Three-day worst heat event: mean annual 3-day warmestnighttime minima event.
• Threshold excess: The longest period of consecutive dayssatisfying:
1. daily maximum temperature above T1 for at least three days,2. average daily maximum temperature above T1 for entire period,
and3. daily maximum temperature above T2 for every day of entire
period.
T1 = 97.5th percentile of the df of maximum temperatures in observedand present-day climate simulations. T2 = 81st percentile.
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Weather Spells
Some things to consider• How should a spell be defined?
– In terms of impacts? (Varies greatly by region)– In terms of perceived impact (e.g., perceived temperature)?(Varies by person)
– By combinations of variables? (not necessarily extreme)– Duration of some persistent event?– Can/Should EVD’s be used for these types of phenomena?
• Often only seasons are examined (e.g., summer for heat waves),but times of seasons may be changing, and spells may also shift intime.
• Large-scale phenomena important, as well as local conditions andcharacteristics.
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Severe Weather
As climate models become better in resolution, they may resolve somesevere weather phenomena, such as hurricanes. However, other typesof severe weather may still require higher resolution.
• Use large-scale indicators to analyze conditions ripe for severeweather,
• Use climate models as drivers for finer scale weather models,
• Statistical approach to current trends in observations,
• Model EVD with means and variances as covariates,
• Other?
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Extreme Value Problems in Climatology: Discussion
• How should extreme events be defined? Deadliness? Perception-based? Statistically? Economically? Other?
•What is the relationship between changes in the mean and changesin extremes? What about variability? Higher order moments?
• If climate models project the distribution of atmospheric variables,then do they accurately portray them? Enough so that extremaare correctly characterized?
• If climate models only project the mean, then can anything besaid about extremes?
• How can it be determined if small changes in high values of large-scale indicators lead to a shift in the distribution of severe weatherconditional on the indicators?
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Extreme Value Problems in Climatology: Discussion
• How do we verify climate models, especially for inferring aboutextremes?
• Extremes are often largely dependent on local conditions (e.g.,topography, surface conditions, atmospheric phenomena, etc.), aswell as larger scale processes.
• Can a metric for climate change pertaining to extremes bedeveloped that makes sense, and provides reasonably accurateinformation?
• How can uncertainty be characterized? Is there too muchuncertainty to make inferences about extremes?
• How can spatial structure be taken into account for extremes?
• Many extreme events, and especially extreme impact events, resultfrom multivariate processes. How can this be addressed?
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Economic Crisis
Salmon, F., 2009: Recipe for disaster: The formula that killed WallStreet, Wired magazine, 23 February, 2009, 7 pp., Available at:http://www.wired.com/print/techbiz/it/magazine/17-03/wp_quant
Embrechts, P., Lectures on, “Did a Mathematical Formula ReallyBlow Up Wall Street?"
Gaussian copula (aka, David X. Li’s formula)“His method was adopted by everybody from bond investors and WallStreet banks to ratings agencies and regulators. And it became sodeeply entrenched–and was making people so much money–thatwarnings about its limitations were largely ignored." –Salmon, 2009.
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Economic Crisis
Correlation
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Slide from Paul EmbrechtsProfessor, ETH Zurich, Dept. of Mathematics
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Software
http://www.assessment.ucar.edu/toolkit/
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Thanks! Questions?
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