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Extreme Hurricane Winds in the United States
Thomas H. Jagger & James B. ElsnerDepartment of Geography
Florida State University
http://garnet.fsu.edu/~jelsner/www
University of Florida’s Winter Workshop on Environmental Statistics
January 12, 2007Gainesville, FL
• Maximum Likelihood Inference– Extreme hurricane winds– Calibrate geological records
• Bayesian Inference– US hurricanes and global temperature– US hurricanes and other covariates– Insured losses
Research QuestionsResearch Questions
1. What are the return levels of hurricane winds in the U.S. over 5, 10, 50, and 100 years?
2. Are return levels different for different regions? 3. What is the maximum possible hurricane wind
speed level? 4. How do these levels change under different climate
conditions? Although fewer hurricanes affect the United States when El Niño conditions are present, are they stronger?
5. Can we apply a similar analysis to insured hurricane losses?
6. What are the primary predictors of extreme insured losses?
7. How does the analysis of extreme insured losses differ from those of extreme winds?
We answer these questions by modeling the maximum wind speeds near the coast a using peaks-over-threshold model.
1. We estimate parameters using ML on data only over the reliable period from 1899-2004.
2. We then demonstrate the use of a Bayesian approach that allows us to incorporate an additional set of Atlantic hurricane data extending back to 1851.
3. We show that the same Bayesian extreme value model applied can be used to estimate extreme insured losses.
– We use log(loss).– Alternate truncated normal model used for yearly
losses.
• Coastal hurricanes are a serious social and economic concern for the United States. – Strong winds, heavy rainfall, and storm surge kill people and destroy
property. – Hurricane destruction rivals that from earthquakes.– Historically, 80% of all U.S. hurricane damage is caused by 20% of the most
intense hurricanes.
• The rarity of severe hurricanes implies that empirical estimates of return periods likely will be unreliable.
• Extreme value theory provides models for rare wind events and a justification for extrapolating to levels that are much greater than have already been observed. – Stuart Coles “An Introduction to the Statistical Modeling of Extreme Values”
• Definitive answers to questions about whether hurricanes will be more intense or more frequent in a future of global warming require long records.– The longest records available are near the coast.
The maximum possible hurricane wind speed is estimated to be 208 kt (183 kt) using the Bayesian (ML) model.
On average we can expect 132 (157) kt hurricane winds near the U.S. coast once every 10 (100) years.
Along the Florida coastline we can expect 108 kt (137) kt winds once every 5 (50) years on average.
Along the East coast we can expect 103 kt (120) kt hurricane winds once every 10 (300) years.
Return periods change with climate factors.
The return period for Hurricane Katrina is 21 years.
Global temperature may have some effect.
The expected worse case indicates a 94% probability of at least some loss during a year.
The expected worse case amount is $23.7 billion.
The expected best case indicates a 53% probability of at least some loss.
The expected best case amount is $1.1 billion.
The maximum 50-year single event loss under the worse case is $630 billion.
The maximum 50-year single event loss under the best case is $10 billion.
PoissonRegression
1993
RegressionTree
1996
DiscriminantModel
1997
TimeSeriesModel
1998
SingleChangePointModel
2000
WeibullModel
2001
SpaceTime
Model
2002
ExtremeValueModel
2006
ClusterModel
2003
TimeSeries
Regression
2008
MultipleChangePointModel
2009
SpaceTime
Regression
2010
Hurricane Type(Paths, Origin)
Hurricane Rate(Counts)
Hurricane Strength(Intensity)
Regional HurricaneActivity
Large Scale Predictors(AMO, NAO, ENSO)
Regional Scale Predictors(SST, SLP, etc)
● Darling (1991): empirical model to estimate local probabilities of hurricane wind speeds exceeding specified thresholds.
● Rupp and Lander (1996): method of moments on annual peak winds
over Guam to determine the parameters of an extreme value model leading to estimates of recurrence intervals for extreme typhoon winds.
● Heckert et al. (1998): peaks-over-threshold method and a reverse Weibull distribution to obtain mean recurrence intervals for extreme wind speeds at consecutive mileposts along the U.S. coastline.
● Chu and Wang (1998): various parametric distributions to model return periods for tropical cyclone wind speeds in the vicinity of Hawaii.
● Jagger et al. (2001): maximum likelihood (ML) estimation to determine a linear regression for the scale and shape parameters of the Weibull distribution for hurricane wind speeds in coastal counties.
● Pang et al. (2001): Bayesian approach to estimating parameters from a Weibull distribution using wind speed data.
We enhance previous research efforts by
1. Interpolating 6-hourly positions and intensities hourly. • For each tropical storm in each region, find maximum wind
speed using interpolated values.• Interpolation remove some boundary bias.
2. Examining the effect of climate variables on the distribution of extreme winds. ● The model employed by Jagger et al. (2001) captures the
variation of hurricane frequency as a function of climate variables using the Weibull distribution, which is appropriate for wind speeds above some threshold but not necessarily for the most extreme winds.
● Here we attempt to put extreme hurricane winds in the context of climate variability and climate-change.
1. Demonstrating the feasibility of a Bayesian approach for adding older, less reliable, data into the analysis.
Generalized Pareto Distribution
Limit family for extremes
P(x
> v
| x
> u
)
60 100 140 180
0.0
0.2
0.4
0.6
0.8
1.0shape = 0shape = +0.5shape = -0.5
v [kt]
GPD threshold estimation
• Which observations do we keep?
• Graphical Tools:– Mean residual life plot
Smallest value.
– , 0 versus threshold plot (not shown).
• Thresholds: – 83 kt for Florida and Gulf.– 64 kt for East.– 96 kt for Entire Coast.
• Best Track HURDAT center fixes• Time period:1851 through 2004.
– 10 kt rounding error prior to 1900
– 5 kt 1900 and later
– 1900-2004 data used in initial analysis, 1851-1899 used in Bayesian analysis.
• Interpolated to hourly fixes using natural splines.– Initial smoothing: Use polynomial
local fit: Savitsky-Golay
– Try smoothing splines with uniform Wmax ± 2.5 kt distribution on true Wmax.
Data Sets:Regions
Anatomy of a Tropical Storm
Atlantic Sea Surface Temperature (SST)
ºC
Hurricane Season Global Warming
1860 1880 1900 1920 1940 1960 1980 2000
Year
-0.4
-0.2
0.0
0.2
0.4
0.6
Degrees C
Global Temperature Anomoly July through October
Results:Results: Models for ExtremesModels for Extremes
Curves are based on an extreme value model and asymptote to finite levels as a consequence of the shape parameter having a negative value. Parameter estimates are made using the ML approach. The thin lines are the 95% confidence limits. The return level is the expected maximum hurricane intensity over p-years. Points are empirical estimates and fall close to the curves.
Ret
urn
leve
l plo
ts b
y re
gio
n
Return level plots for the entire U.S. coast (Region 4) by climate factors.
Curves are based on an extreme value model using a ML estimation procedure. Data is partitioned separately by predictor. Red (blue) lines and points indicate above (below) normal climate conditions for each predictor.
15
More hurricanes with
La Ni ña
More extreme hurricanes with
El Ni ñoENSO NAO
SST
More hurricanes with
-NAO
More strong hurricanes with
high SST
More strong hurricanes with global warming
15
More hurricanes with
La Ni ña
More extreme hurricanes with
El Ni ñoENSO NAO
SST
More hurricanes with
-NAO
More s trong hurricanes with
high SST
More s trong hurricanes with global warming
15
Theoretical predictions of the influence of global warming on hurricane activity suggest:– Increased maximum intensity.– Increased level of storminess is uncertain.
Impact of Global Warming?
Return Level (kt)
1 10 100 1,0002 3 4 5 678 2 3 4 5 67 2 3 4 5 6
Return Period (yr)
60
80
100
120
140
160
180
200
Warm years
Cold years
For a given return period (> 5 yr), warm
years result in higher return levels.
Entire coast
137 kt
Magnitude of the difference in return level is consistent with climate
models
Warm Years
ColdYears
Saffir-Simpson Category
14 yr
No change in frequency of
weaker hurricanes
11
Hurricane Katrina
Bayesian Inference
Bayesian Inference
Bayesian Inference of Extremes
Bayesian Inference of Extremes:
Hurricane Intensity Component
Hurricane Frequency Component
Covariate Component
Observed maximum
wind speed.
True maximum
wind speed.
Bayesian Bayesian Extreme Value ModelsExtreme Value Models
Bayesian model for coastal hurricane wind speeds
Intercept: j=1 X[,1]=1 for all
WinBUGS Code
How Gibbs Sampling Works
Gibbs sampling algorithm in two
parameter dimensions starting from an initial point
and completing three iterations.
(0)
(1)
(2)
(3)
The contours in the plot represent the joint distribution of and the labels (0),
(1) etc., denote the simulated values. One iteration of the algorithm is
complete after both parameters are revised. Each parameter is revised
along the direction of the coordinate axes---problematic if the two
parameters are correlated (contours compressed) as movement along the
axes tend to produce small changes in parameter values.
POT WinBUGS code Part IModel{
for(j in 1:M) {
#For each year calculate log sigma and xi: lsigma2[j] <- inprod(ls.x[],X[j,]) xi2[j] <- inprod(xi.x[],X[j,]) #Threshold Crossing Rate for each year: H[j] ~ dpois(lambda[j]) log(lambda[j]) <- inprod(tc[],X[j,])
} #Extreme value for each exceedance, note censoring:
for( i in 1:N){
yy[i]~dGPD(u,sigma[i],xi[i])I(ys[i],yl[i])
offset[i] <- Yr[i] - Yr0 lsigma[i] <- lsigma2[offset[i]] sigma[i] <- exp(lsigma[i]) xi[i] <- xi2[offset[i]] yl[i] <- y[i] + e[i] ys[i] <- y[i] - e[i]
}
POT WinBUGS code Part II#Initializations and Missing Data model #In our case Np=2, intercept and global temperature
for(k in 1:Np) { #Parameter initializations for model parameters #tau.xx[], data used for model selection:
tc[k] ~ dnorm(0,tau.tc[k]) ls.x[k] ~ dnorm(0,tau.ls[k])
xi.x[k] ~ dnorm(0,tau.xi[k])
#Missing data model, normal no trend, for(j in 1:M) { X[j,k] ~ dnorm(xmu[k],xtau[k]) }
#Missing data model initializations: xmu[k] ~ dnorm(0,.001) xtau[k] ~ dgamma(.01,.01)
} } #Close Model (Demo: GPD1)
Posterior Densities
Model Output Comparison Data
Probability Parameter is less than zero
tc[1] tc[2] ls.x[1] ls.x[2] xi.x[1] xi.x[2] 1.000000 0.277300 0.000000 0.128425 0.961925 0.150350
GT Scale GT Shape
Maximum Likelihood values (MLE) from Coles ismev gpd.fit():Bayesian samples: burn in 5000, sample 40000:
MLE se mean sd MC_error 2.5% median 97.5% tc[1] -0.381 0.113 -0.395 0.108 0.00078 -0.613 -0.394 -0.188 tc[2] 0.287 0.451 0.255 0.430 0.00291 -0.600 0.263 1.086ls.x[1] 3.167 0.138 3.134 0.143 0.00462 2.853 3.136 3.407ls.x[2] 0.699 0.601 0.724 0.624 0.01945 -0.472 0.726 1.965xi.x[1] -0.264 0.088 -0.205 0.103 0.00337 -0.372 -0.215 0.028xi.x[2] 0.543 0.425 0.532 0.538 0.01691 -0.572 0.541 1.572
Regions Parameters Covariates+ Intercept
-9 5 -9 0 -8 5 -8 0 -7 5 -7 0
lo n g itu d e
25
30
35
40
45
latitude
H ourly In terpo lated Positions in R egions of S tudy
R e g io n 1 : G u lf C o a stR e g io n 2 : Flo rid aR e g io n 3 : Ea st C o a stR e g io n 4 : N e w En g la n d
Region 4: Northeast Coast
Region 3: Southeast Coast
Region 2: FloridaRegion 1: Gulf Coast
Hourly interpolated
hurricanepositions
(1851-2004)
Bayesian Model for Coastal Hurricane Winds 2
Hurricane Intensity Component
Hurricane Frequency Component
Covariate Component
Observed maximum
wind speed.
True maximum
wind speed.
Results: Raw Climatology
P(>0) = 0.22 P(>0) < 0.01
P(>0) < 0.01 P(>0) < 0.01
Assuming the model is correct, the data support a super-intense hurricane threat only in the Gulf of Mexico.
Region 1: Gulf Coast: Shape
Region 2: Florida: Shape
Region 3: Southeast: Shape Region 4: Northeast: Shape
Probability that hurricane intensity is
unbounded
Fre
qu
ency
Dis
trib
uti
on
Freq
uen
cy Distrib
utio
n
Important Results: Conditional Climatology
log()
log()
log()
log()
log()
log ()
Fre
qu
ency
Dis
trib
uti
on
Freq
uen
cy Distrib
utio
n
Region 1: AMO: Scale Region 3: AMO: Threshold
Region 1: NAO: Threshold Region 2: NAO: Threshold
Region 1: SOI: Threshold Region 3: SOI: Threshold
Stronger Hurricanes
More Hurricanes
More Hurricanes
More Hurricanes
More Hurricanes
More Hurricanes
Simulated Hurricane Seasons
• A Monte Carlo procedure is employed on the posterior samples to generate a large number (50K) simulated hurricane seasons based on sampled covariate and parameter values.
Results from the Gulf Coast show that the simulated data match the empirical data through Category 4 wind speeds but for winds in excess of 135 kt (68 m/s) the simulated data indicate a higher frequency (by a factor of 2 to 3).
Category (SS) > I II III IV V V+ V++
Threshold (kt) 80 83 96 114 135 150 170
Empirical data* 0.463 0.418 0.313 0.142 0.037 0.007 0.007
Simulation* 0.469 0.423 0.279 0.143 0.054 0.028 0.015
*mean annual exceedence rate
Region 1: Gulf Coast
• Modeled 100-year return levels do not appear to match empirical evidence.
– Might be a problem with the regression structure of the shape parameter. Using simpler models where we replace the covariates with discrete factors (above/below normal) we produce a better match.
• Convergence is not guaranteed when modeling the log() and scale as a linear combination of covariates.
– Problem in region 4 (NE) where there are fewer hurricanes.
• Threshold is not estimated in the current model.
– Behrens, Lopes and Gamerman(2004) "Bayesian analysis of extreme events with threshold estimation“ Statistical Modelling, 4, 227-244.
• The nature of the shape parameter is such that its value is a function of the support of the underlying distribution.
– Attempts to model this using a Reversible Jump Markov Chain MC approach reduces this problem but introduces significant autocorrelation into the sampler.
– Model assigns positive probability for discrete values of xi, uses RJMCMC
Outstanding Issues
• To be useful to risk models, the relationship between climate and hurricane activity needs to be forecast in advance of the season.
Fortunately, three important climate variables related to hurricanes can be used in a prediction model; but each variable enters the prediction model in a unique way.
NAO: Natural precursor signal to hurricane activity.
SST: Slowly varying (persistent).
ENSO: Can be predicted with some skill by dynamical models.
16
Predicting Insured LossesPredicting Insured Losses
21
22
Peaks Over Threshold
Small LossEvents(36.5%)
Large LossEvents(63.5%)
99.4%Losses
0.6%Losses
Reference line indicates 80/17 splitWe split losses (red line: $100 Million)Allows us to examine significant events
Splitting Insured Losses
24
Large Loss Potential Evenly Distributed
SST
SS
T
25
Preseason Predictors Insured Loss Model
Model Distributions:log(loss): Truncated Normal. dnorm(,1/2)Rate: Poisson dpois()
May June averaged values of predictors
+ NAO, -SST
- NAO, +SST
26
Yearly Insured Losses Depends on Climate
SST
27
Extreme Loss Model and Results
Predictors set at maximum values of covariates with least
favorable climate:
+NAO, +SOI
-NAO, -SOI
Model Distributions:log(loss): GPD distribution. dGPD(u,,)
u=9 (log(1 Billion))Rate: Poisson dpois()
● The yearly model and extreme loss model use the same Peaks over Threshold approach.
● Yearly model used SST and NAO predictors with truncated Normal distributions:– Smallest DIC for truncated Normal, with SST and NAO predictors– Better estimates of single year insured losses.
● Extreme Loss Model uses SST, NAO and SOI with GPD distribution:– SST and NAO used in regression for logarithm of threshold
crossing rate– Log(loss) assumed to have GPD distribution for loss > $1 Billion.
● Mean Residual Life Plot used to estimate threshold.– NAO used in log() regression– SOI used in regression
● More Information: “Forecasting U.S. Insured Hurricane Loses”, available at our website, in “Forecasting Insured Losses” in press.
Comments on POT BUGS Models:
● Model mixes well.● Model must be initialized carefully.
– Once model compiles it runs smoothly.– Posterior mean and MLE sometimes very different.– MLE may be approximated by posterior sample values at
minimum deviance…, but not well in this model.● Return level sampling not recommended. ● Data and R code using BRugs available.● POT model using U.S. Hurricane loss data
– Demo and R/OpenBUGS code available. ● Examples of truncated normal distributions in POT type model
as well as truncated normal/GPD.
Evidence of Prehistoric HurricanesEvidence of Prehistoric Hurricanes
peat layer
sand layer1954 H
peat layer
peat layer
peat layer
sand layer1938 H
sand layer1635 H
Courtesy: Jeff Donnelly
A sediment core from a back barrier marsh in New England.– These data are currently not used in risk models.– Need calibration.
Increasing Radius beginning at 45 km
• Combining extreme value theory with a Bayesian specification provides a practical way to assess return periods of extreme hurricane winds.– Results are expressed in terms of posterior distributions of the
parameters of interest.– Less reliable, but still useful information is incorporated into the model
in a natural way.– A large number of simulated hurricane seasons can be generated by
repeated sampling from the posterior distributions. – Analysis of hurricane winds suggests the possibility that the highest
winds estimated for early storms may be under reported.– Results suggests the possibility that the highest winds estimated for
early storms may be under reported. • The approach can be used to better understand the
projected impact of global warming on extreme hurricane winds.
• The POT approach can be extended to insured losses and the logarithm of extreme losses can be modeled using a GPD distribution.
Summary
32
Taken as a random event along the Gulf coast, the return period of Hurricane Katrina is 21 years.
Katrina might be a sign of things to come as the observed and modelled effect of global warming appears to start at Katrina’s intensity.
Preseason climate signals provide information about the nature of the upcoming hurricane season.
The hurricane risk index quantifies this information geographically.
Insured losses (both expected and maximum) show a statistical link to preseason climate signals.
More InformationMore Information
Google hurricane climate
http://garnet.fsu.edu/~jelsner/www
[email protected]; [email protected]
35
Future Work I• Use more data and new covariates
– Incorporate historical climate data with geological evidence and historical data sets. (last slide)
– Use proxy data for covariates– Use Quasi-Biennial Oscillation (QBO) Zonal Wind
Index .• Only from 1953, very predictable, especially west
phase• 50 mb, East phase, more vertical wind shear.• El-Nino may interact with West descent phase.
– New wind index: • Saunders, M. A. and A. S. Lea, 2005: Seasonal
prediction of US landfalling hurricane activity from 1 August, Nature 434, 1005–1008.
Future Work II• Use alternative prior formulation:
– Choose three return periods, rp[1:3]= [2,10, 100] years
– Assign reasonable multivariate prior to the return levels for rp[1:3] with rl[1] < rl[2] < rl[3] (using rl[1], rl[2]-rl[1],rl[3]-rl[2])
– Use rl[i]=u+ ¢((z[i]¢u)-1 ) with z[i]=1/log(rp[i]/(rp[i]-1)) ¼ rp[i]-1/2
– Solve for u, , as a function of rl[1:3], rp[1:3]
– Coles, S. G. and J. A. Tawn, (1996): A Bayesian analysis of extreme rainfall data. Appl. Statist., 45, 463-478
Future Work III• Additional analyses using extreme value theory and
Bayesian methods:– Use Bayesian model averaging and median models
selection– Use RJMCMC to examine spatial and temporal parameter
changes, e.g. smoothing splines (WinBUGS 1.4 RJMCMC)– Apply to pressure, rainfall and sea surge.– Apply to hurricane rapid intensification (RI) process.
• Bivariate GPD, maximum intensity, intensification per storm• Western Pacific RI density, conditional probability of intensification.• How does global temperature affect RI.
– Use spatial model for distribution of GPD parameters.
• Spatial and Temporal Models:– Clustering models on a lattice. (Jagger Dissertation)– Point process models (Cox Processes). (R/Splus)– Apply to hurricane activity and intensity full models.