Statistical Software for Weather and Climate
Transcript of Statistical Software for Weather and Climate
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Statistical Software for Weather and Climate
Eric Gilleland,National Center forAtmospheric Research,Boulder, Colorado, U.S.A.http://www.ral.ucar.edu/staff/ericg
Effects of climate change: coastal systems, policy implications, andthe role of statistics. Interdisciplinary Workshop 18-20 March 2009,Preluna Hotel and Spa, Sliema, Malta
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The R programming language
R Development Core Team (2008). R: A language and environmentfor statistical computing. R Foundation for Statistical Computing,Vienna, Austria. ISBN 3-900051-07-0,http://www.r-project.org
Vance A, 2009. Data analysts captivated by Rβs power. New YorkTimes, 6 January 2009. Available at:http://www.nytimes.com/2009/01/07/technology/business-computing/07program.html?_r=2
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Already familiar with R?
Advanced (potentially useful) topics:
β’ Reading and Writing NetCDF file formats:http://www.image.ucar.edu/Software/Netcdf/
β’ A Climate Related Precipitation Example for Colorado:http://www.image.ucar.edu/~nychka/FrontrangePrecip/
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R preliminaries
Assuming R is installed on your computer...
In linux, unix, and Mac (terminal/xterm) the directory in which Ris opened is (by default) the current working directory. In Windows(Mac GUI?), the working directory is usually in one spot, but can bechanged (tricky).
Open an R workspace:Type R at the command prompt (linux/unix, Mac terminal/xterm)or double click on Rβs icon (Windows, Mac GUI).
getwd() # Find out which directory is the current working directory.
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R preliminaries
Assigning vectors and matrices to objects:
# Assign a vector containing the numbers -1, 4 and 0 to anobject called βxβx < β c( -1, 4, 0)
# Assign a 3Γ 2 matrix with column vectors: 2, 1, 5 and# 3, 7, 9 to an object called βyβ.y < β cbind( c( 2, 1, 5), c(3, 7, 9))
# Write βxβ and βyβ out to the screen.xy
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R preliminaries
Saving a workspace and exiting
# To save a workspace without exiting R.save.image()
# To exit R while also saving the workspace.q("yes")
# Exit R without saving the workspace.q("no")
# Or, interactively...q()
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R preliminaries
Subsetting vectors:
# Look at only the 3-rd element of βxβ.x[3]
# Look at the first two elements of βxβ.x[1:2]
# The first and third.x[c(1,3)]
# Everything but the second element.x[-2]
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R preliminaries
Subsetting matrices:
# Look at the first row of βyβ.y[1,]
# Assign the first column of βyβ to a vector called βy1β.# Similarly for the 2nd column.y1 < β y[,1]y2 < β y[,2]
# Assign a "missing value" to the second row, first column# element of βyβ.y[2,1] < β NA
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R preliminaries
Logicals and Missing Values:
# Do βxβ and/or βyβ have any missing values?any( is.na( x))any( is.na( y))
# Replace any missing values in βyβ with -999.0.y[ is.na( y)] < β -999.0
# Which elements of βxβ are equal to 0?x == 0
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R preliminaries
Contributed packages
# Install some useful packages. Need only do once.install.packages( c("fields", # A spatial stats package.
"evd", # An EVA package."evdbayes", # Bayesian EVA package.
"ismev", # Another EVA package."maps", # For adding maps to plots.
"SpatialExtremes")
# Now load them into R. Must do for each new session.library( fields)library( evd)library( evdbayes)library( ismev)library( SpatialExtremes)
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R preliminaries
See hierarchy of loaded packages:
search()
# Detach the βSpatialExtremesβ package.detach(pos=2)
See how to reference a contributed package:
citation("fields")
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R preliminaries
Simulate some random fieldsFrom the help file for the fields function sim.rfhelp( sim.rf)
#Simulate a Gaussian random field with an exponential# covariance function, range parameter = 2.0 and the# domain is [0, 5]Γ [0, 5] evaluating the field at a 100Γ100# grid.grid < β list( x= seq( 0,5β100), y= seq(0,5β100))obj < β Exp.image.cov( grid=grid, theta=.5, setup=TRUE)look < β sim.rf( obj)
# Now simulate another ...look2 < β sim.rf( obj)
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R preliminaries
Plotting the simulated fields
# setup the plotting device to have two plots side-by-sideset.panel(2,1)
# Image plot with a color scale.image.plot( grid$x, grid$y, look)title("simulated gaussian field")image.plot( grid$x, grid$y, look2)title("another (independent) realization ...")
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R preliminaries
Basics of plotting in R:
β’ First must open a device on which to plot.
β Most plotting commands (e.g., plot) open a device (that youcan see) if one is not already open. If a device is open, it willwrite over the current plot.
β X11() will also open a device that you can see.β To create a file with the plot(s), use postscript, jpeg, png,
or pdf (before calling the plotting routines. Use dev.off() toclose the device and create the file.
β’ plot and many other plotting functions use the par values to de-fine various characteristics (e.g., margins, plotting symbols, char-acter sizes, etc.). Type help( plot) and help( par) for moreinformation.
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R preliminaries
Simple plot example.
plot( 1:10, z <- rnorm(10), type="l", xlab="", ylab="z",main="Std Normal Random Sample")
points( 1:10, z, col="red", pch="s", cex=2)lines( 1:10, rnorm(10), col="blue", lwd=2, lty=2)
# Make a standard normal qq-plot of βzβ.qqnorm( z)
# Shut off the device.dev.off()
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Background on Extreme Value Analysis (EVA)
Motivation
Sums, averages and proportions (Normality)
β’ Central Limit Theorem (CLT)β’ Limiting distribution of binomial distribution
Extremes
β’ Normal distribution inappropriateβ’ Bulk of data may be misleadingβ’ Extremes are often rare, so often not enough data
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Background on Extreme Value Analysis (EVA)
Simulations# Simulate a sample of 1000 from a Unif(0,1) distribution.U < β runif( 1000)hist( U)
# Simulate a sample of 1000 from a N(0,1) distribution.Z < β rnorm( 1000)hist( Z)
# Simulate a sample of 1000 from a Gumbel distribution.M < β rgev( 1000)hist( M)
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Background on Extreme Value Analysis (EVA)
Simulations# Simulate 1000 maxima from samples of size 30 from# the normal distribution.Zmax < β matrix( NA, 30, 1000)dim( Zmax)for( i in 1:1000) Zmax[,i] < β rnorm( 30)Zmax < β apply( Zmax, 2, max)dim( Zmax)class( Zmax)length( Zmax)class( Zmax)hist( Zmax, breaks="FD", col="blue")
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Background on Extreme Value Analysis (EVA)
Simulations# Simulate maxima from samples of size 30 from# the Unif(0,1) distribution.Umax <- matrix( NA, 30, 1000)for( i in 1:1000) Umax[,i] < β runif( 30)Umax < β apply( Umax, 2, max)hist( Umax, breaks="FD", col="blue")
# Simulate 1000 maxima from samples of size 30 from# the FrΓ©chet distribution.Fmax < β matrix( NA, 30, 1000)for( i in 1:1000) Fmax[,i] < β rgev( 30,
loc=2, scale=2.5, shape=0.5)Fmax < β apply( Fmax, 2, max, finite=TRUE, na.rm=TRUE)hist( Fmax, breaks="FD", col="blue")
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Background on Extreme Value Analysis (EVA)
Simulations Last one# Simulate 1000 maxima from samples of size 30 from# the exponential distribution.Emax <- matrix( NA, 30, 1000)for( i in 1:1000) Emax[,i] <- rexp( 30)Emax <- apply( Emax, 2, max)
# This time, compare with samples from the# exponential distribution.par( mfrow=c(1,2))hist( Emax, breaks="FD", col="blue")hist( rexp(1000), breaks="FD", col="blue")
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Background on Extreme Value Analysis (EVA)
Extremal Types Theorem
Let X1, . . . , Xn be a sequence of independent and identically dis-tributed (iid) random variables with common distribution function,F . Want to know the distribution of
Mn = max{X1, . . . , Xn}.
Example: X1, . . . , Xn could represent hourly precipitation, daily ozoneconcentrations, daily average temperature, etc. Interest for now is inmaxima of these processes over particular blocks of time.
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Background on Extreme Value Analysis (EVA)
Extremal Types Theorem
If interest is in the minimum over blocks of data (e.g., monthly mini-mum temperature), then note that
min{X1, . . . , Xn} = βmax{βX1, . . . ,βXn}
Therefore, we can focus on the maxima.
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Background on Extreme Value Analysis (EVA)
Extremal Types Theorem
Could try to derive the distribution for Mn exactly for all n as follows.
Pr{Mn β€ z} = Pr{X1 β€ z, . . . , Xn β€ z}
indep.= Pr{X1 β€ z} Γ Β· Β· Β· Γ Pr{Xn β€ z}
ident. dist.= {F (z)}n.
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Background on Extreme Value Analysis (EVA)
Extremal Types Theorem
Could try to derive the distribution for Mn exactly for all n as follows.
Pr{Mn β€ z} = Pr{X1 β€ z, . . . , Xn β€ z}
indep.= Pr{X1 β€ z} Γ Β· Β· Β· Γ Pr{Xn β€ z}
ident. dist.= {F (z)}n.
But! If F is not known, this is not very helpful because small dis-crepancies in the estimate of F can lead to large discrepancies forF n.
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Background on Extreme Value Analysis (EVA)
Extremal Types Theorem
Could try to derive the distribution for Mn exactly for all n as follows.
Pr{Mn β€ z} = Pr{X1 β€ z, . . . , Xn β€ z}
indep.= Pr{X1 β€ z} Γ Β· Β· Β· Γ Pr{Xn β€ z}
ident. dist.= {F (z)}n.
But! If F is not known, this is not very helpful because small dis-crepancies in the estimate of F can lead to large discrepancies forF n.Need another strategy!
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Background on Extreme Value Analysis (EVA)
Extremal Types Theorem
If there exist sequences of constants {an > 0} and {bn} such that
Pr{
Mn β bn
anβ€ z
}ββ G(z) as n βββ,
where G is a non-degenerate distribution function, then G belongs toone of the following three types.
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Background on Extreme Value Analysis (EVA)
Extremal Types Theorem
I. Gumbel
G(z) = exp
{β exp
[β
(z β b
a
)]}, ββ < z < β
II. FrΓ©chet
G(z) =
{0, z β€ b,
exp{β
(zβba
)βΞ±}
, z > b;
III. Weibull
G(z) =
{exp
{β
[β
(zβba
)Ξ±]}, z < b,
1, z β₯ b
with parameters a, b and Ξ± > 0.
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Background on Extreme Value Analysis (EVA)
Extremal Types Theorem
The three types can be written as a single family of distributions,known as the generalized extreme value (GEV) distribution.
G(z) = exp
{β
[1 + ΞΎ
(z β Β΅
Ο
)]β1/ΞΎ
+
},
where y+ = max{y, 0}, ββ < Β΅, ΞΎ < β and Ο > 0.
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Background on Extreme Value Analysis (EVA)
GEV distribution
Three parameters: location (Β΅), scale (Ο) and shape (ΞΎ).
1. ΞΎ = 0 (Gumbel type, limit as ΞΎ ββ 0)
2. ΞΎ > 0 (FrΓ©chet type)
3. ΞΎ < 0 (Weibull type)
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Background on Extreme Value Analysis (EVA)
Gumbel type
β’ Light tail
β’ Domain of attraction for many common distributions(e.g., normal, lognormal, exponential, gamma)
0 1 2 3 4 5
0.0
0.1
0.2
0.3
0.4
0.5
Gumbel
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Background on Extreme Value Analysis (EVA)
FrΓ©chet type
β’ Heavy tail
β’ E [Xr] = β for r β₯ 1/ΞΎ (i.e., infinite variance if ΞΎ β₯ 1/2)
β’ Of interest for precipitation, streamflow, economic impacts
0 1 2 3 4 5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Frechet
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Background on Extreme Value Analysis (EVA)
Weibull type
β’ Bounded upper tail at Β΅β ΟΞΎ
β’ Of interest for temperature, wind speed, sea level
0 1 2 3 4 5 6
0.0
0.1
0.2
0.3
0.4
0.5
Weibull
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Background on Extreme Value Analysis (EVA)
Normal vs. GEV## The probability of exceeding increasingly high values## (as they double).# Normalpnorm( c(1,2,4,8,16,32), lower.tail=FALSE)
# Gumbelpgev( c(1,2,4,8,16,32), lower.tail=FALSE)
# FrΓ©chetpgev( c(1,2,4,8,16,32), shape=0.5, lower.tail=FALSE)
# Weibull (note bounded upper tail!)pgev( c(1,2,4,8,16,32), shape=-0.5, lower.tail=FALSE)
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Background on Extreme Value Analysis (EVA)
Normal vs. GEV# Find Pr{X < x} for X = 0, . . . , 20.cdfNorm < β pnorm( 0:20)cdfGum < β pgev( 0:20)cdfFrech < β pgev( 0:20, shape=0.5)cdfWeib < β pgev( 0:20, shape=-0.5)
# Now find Pr{X = x} for X = 0, . . . , 20.pdfNorm < β dnorm( 0:20)pdfGum < β dgev( 0:20)pdfFrech < β dgev( 0:20, shape=0.5)pdfWeib < β dgev( 0:20, shape=-0.5)
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Background on Extreme Value Analysis (EVA)
Normal vs. GEV
par( mfrow=c(2,1), mar=c(5,4,0.5,0.5))plot( 0:20, cdfNorm, ylim=c(0,1), type="l", xaxt="n",
col="blue", lwd=2, xlab="", ylab="F(x)")lines( 0:20, cdfGum, col="green", lty=2, lwd=2)lines( 0:20, cdfFrech, col="red", lwd=2)lines( 0:20, cdfWeib, col="orange", lwd=2)legend( 10, 0.05,
legend=c("Normal", "Gumbel", "Frechet", "Weibull"),col=c("blue", "green", "red", "orange"),lty=c(1,2,1,1), bty="n", lwd=2)
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Background on Extreme Value Analysis (EVA)
Normal vs. GEVplot( 0:20, pdfNorm, ylim=c(0,1), typ="l", col="blue",
lwd=2, xlab="x", ylab="f(x)")lines( 0:20, pdfGum, col="green", lty=2, lwd=2)lines( 0:20, pdfFrech, col="red", lwd=2)lines( 0:20, pdfWeib, col="orange", lwd=2)
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Example
Fort Collins, Colorado daily precipitation amounthttp://ccc.atmos.colostate.edu/~odie/rain.html
β’ Time series of daily precipitation amount (in), 1900β1999.
β’ Semi-arid region.
β’ Marked annual cycle in precipitation(wettest in late spring/early summer, driest in winter).
β’ No obvious long-term trend.
β’ Recent flood, 28 July 1997.(substantial damage to Colorado State University)
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Example
Fort Collins, Colorado precipitation
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0.00
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Fort Collins daily precipitation
Pre
cipi
tatio
n (in
)
Jan Mar May Jul Sep Nov
β
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Example
Fort Collins, Colorado precipitationAnnual Maxima
1900 1920 1940 1960 1980 2000
12
34
Fort Collins annual maximum daily precipitation
Year
Pre
cipi
tatio
n (in
)
1997
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Example
Fort Collins, Colorado precipitation
How often is such an extreme expected?
β’ Assume no long-term trend emerges.
β’ Using annual maxima removes effects of annual trend in analysis.
β’ Annual Maxima fit to GEV.
# Fit Fort Collins annual maximum precipitation to GEV.fit < β gev.fit( ftcanmax$Prec/100)
# Check the quality of the fit.gev.diag( fit)
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Example
Fort Collins, Colorado precipitation
Fit looks good (from diagnostic plots).
Parameter Estimate (Std. Error)Location (Β΅) 1.347 (0.617)Scale (Ο) 0.533 (0.488)Shape (ΞΎ) 0.174 (0.092)
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Example
Fort Collins, Colorado precipitation
# Is the shape parameter really not zero?# Perform likelihood ratio test against Gumbel type.fit0 < β gum.fit( ftcanmax$Prec/100)Dev < β 2*(fit0$nllh - fit$nllh)pchisq( Dev, 1, lower.tail=FALSE)
Likelihood ratio test for ΞΎ = 0 rejects hypothesis of Gumbel type(p-value β 0.038).
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Example
Fort Collins, Colorado precipitation
95% Confidence intervals for ΞΎ, using profile likelihood, are:(0.009, 0.369).
β0.1 0.0 0.1 0.2 0.3 0.4
β11
1β
110
β10
9β
108
β10
7β
106
β10
5
Shape Parameter
Pro
file
Logβ
likel
ihoo
d
Use gev.profxi and locator(2)to find CIβs.
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Example
Fort Collins, Colorado precipitationReturn Levels
# Currently must assign the class "gev.fit" to the# fitted object, βfit,β in order to use the "extRemes"# function, βreturn.levelβ.class( fit) < β "gev.fit"fit.rl < β return.level( fit)Return Estimated Return 95% CIPeriod Level (in)10 2.81 (2.41, 3.21)100 5.10 ?(3.35, 6.84)... ... ...
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Example
Fort Collins, Colorado precipitationReturn Levels
CIβs from return.level are based on the delta method, which as-sumes normality for the return levels. For longer return periods (e.g.,beyond the range of the data), this assumption may not be valid. Cancheck by looking at the profile likelihood.gev.prof( fit, m=100, xlow=2, xup=8)
Highly skewed! Using locator(2), a better approximation for the(95%) 100-year return level CI is about (3.9, 8.0).
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Example
Fort Collins, Colorado precipitationProbability of annual maximum precipitation at least as large as thatduring the 28 July 1997 flood (i.e., Pr{max(X) β₯ 1.54 in.}).
# Using the βpgevβ function from the "evd" package.pgev( 1.54, loc=fit$mle[1],
scale=fit$mle[2],shape=fit$mle[3],lower.tail=FALSE)
pgev( 4.6, loc=fit$mle[1],scale=fit$mle[2],shape=fit$mle[3],lower.tail=FALSE)
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Peaks Over Thresholds (POT) Approach
Let X1, X2, . . . be an iid sequence of random variables, again withmarginal distribution, F . Interest is now in the conditional probabil-ity of X βs exceeding a certain value, given that X already exceeds asufficiently large threshold, u.
Pr{X > u + y|X > u} =1β F (u + y)
1β F (u), y > 0
Once again, if we know F , then the above probability can be com-puted. Generally not the case in practice, so we turn to a broadlyapplicable approximation.
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Peaks Over Thresholds (POT) Approach
If Pr{max{X1, . . . , Xn} β€ z} β G(z), where
G(z) = exp
{β
[1 + ΞΎ
(z β Β΅
Ο
)]β1/ΞΎ}
for some Β΅, ΞΎ and Ο > 0, then for sufficiently large u, the distribution[X β u|X > u], is approximately the generalized Pareto distribution(GPD). Namely,
H(y) = 1β(
1 +ΞΎy
ΟΜ
)β1/ΞΎ
+
, y > 0,
with ΟΜ = Ο + ΞΎ(uβ Β΅) (Ο, ΞΎ and Β΅ as in G(z) above).
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Peaks Over Thresholds (POT) Approach
β’ Pareto type (ΞΎ > 0)heavy tail
β’ Beta type (ΞΎ < 0)bounded above atuβ Ο/ΞΎ
β’ Exponential type (ΞΎ = 0)light tail
0 2 4 6 8
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Pareto (xi>0)Beta (xi<0)Exponential (xi=0)
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Peaks Over Thresholds (POT) Approach
Hurricane damage
ββ
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1930 1940 1950 1960 1970 1980 1990
020
4060
Economic Damage from Hurricanes (1925β1995)
Year
billi
on U
S$
Economic damage causedby hurricanes from 1926to 1995.
Trends in societalvulnerability removed.
Excess over threshold ofu = 6 billion US$.
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Peaks Over Thresholds (POT) Approach
Hurricane damage
10 20 30 40 50 60 70
0.00
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GPD
ΟΜ =β 4.589
ΞΎΜ β 0.512
Likelihood ratio test for
ΞΎ = 0 (p-valueβ 0.018)
95% CI for shapeparameter usingprofile likelihood.0.05 < ΞΎ < 1.56
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Peaks Over Thresholds (POT) Approach
Choosing a threshold
Variance/bias trade-off
Low threshold allows for more data (low variance).
Theoretical justification for GPD requires a high threshold (low bias).
β
β β β ββ
ββ β
β
2 3 4 5 6 7
β5
05
10
Threshold
Mod
ified
Sca
le
β
β β β ββ
ββ β
β
2 3 4 5 6 7
0.0
0.5
1.0
Threshold
Sha
pe
gpd.fitrange( damage$Dam, 2, 7)
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Peaks Over Thresholds (POT) Approach
Dependence above thresholdOften, threshold excesses are not independent. For example, a hotday is likely to be followed by another hot day.
Various procedures to handle dependence.
β’ Model the dependence.
β’ De-clustering (e.g., runs de-clustering).
β’ Resampling to estimate standard errors(avoid tossing out information about extremes).
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Peaks Over Thresholds (POT) Approach
Dependence above threshold
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β65
β60
Phoenix airport
(neg
ativ
e) M
in. T
empe
ratu
re (
deg.
F)
Phoenix (airport) minimumtemperature (oF).
July and August 1948β1990.
Urban heat island (warmingtrend as cities grow).
Model lower tail as upper tailafter negation.
54
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Peaks Over Thresholds (POT) Approach
Dependence above threshold
# Fit without de-clustering.phx.fit0 < β gpd.fit( -Tphap$MinT, -73)
# With runs de-clustering (r=1).phx.dc < β dclust( -Tphap$MinT, u=-73, r=1,
cluster.by=Tphap$Year)phxdc.fit0 < β gpd.fit( phx.dc$xdat.dc, -73)
55
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Peaks Over Thresholds (POT) Approach
Point Process: frequency and intensity of threshold excesses
Event is a threshold excess (i.e., X > u).
Frequency of occurrence of an event (rate parameter), Ξ» > 0.
Pr{no events in [0, T ]} = eβΞ»T
Mean number of events in [0, T ] = Ξ»T .
GPD for excess over threshold (intensity).
56
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Peaks Over Thresholds (POT) Approach
Point Process: frequency and intensity of threshold excessesRelation of parameters of GEV(Β΅,Ο,ΞΎ) to
parameters of point process (Ξ»,Οβ,ΞΎ).
β’ Shape parameter, ΞΎ, identical.
β’ log Ξ» = β1ΞΎ log
(1 + ΞΎuβΒ΅
Ο
)β’ Οβ = Ο + ΞΎ(uβ Β΅)
More detail: Time scaling constant, h. For example, for annual max-imum of daily data, h β 1/365.25. Change of time scale, h, forGEV(Β΅,Ο,ΞΎ) to hβ²
Οβ² = Ο
(h
hβ²
)ΞΎ
and Β΅β² = Β΅ +1
ΞΎ
{Οβ²
[1β
(h
hβ²
)βΞΎ]}
57
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Peaks Over Thresholds (POT) Approach
Point Process: frequency and intensity of threshold excessesTwo ways to estimate PP parameters
β’ Orthogonal approach (estimate frequency and intensity separately).
Convenient to estimate.Difficult to interpret in presence of covariates.
β’ GEV re-parameterization (estimate both simultaneously).
More difficult to estimate.Interpretable even with covariates.
58
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Peaks Over Thresholds (POT) Approach
Point Process: frequency and intensity of threshold excessesFort Collins, Colorado daily precipitation
Analyze daily data instead of just annual maxima(ignoring annual cycle for now).
Orthogonal Approach
Ξ»Μ = 365.25 Β· No. Xi > 0.395
No. Xiβ 10.6 per year
ΟΜβ β 0.323, ΞΎΜ β 0.212
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Peaks Over Thresholds (POT) Approach
Point Process: frequency and intensity of threshold excessesFort Collins, Colorado daily precipitation
Analyze daily data instead of just annual maxima(ignoring annual cycle for now).
Point Process
Β΅Μ β 1.384
ΟΜ = 0.533
ΞΎΜ β 0.213
Ξ»Μ =
[1 +
ΞΎΜ
ΟΜ(uβ Β΅Μ)
]β1/ΞΎΜ
β 10.6 per year
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Risk Communication Under Stationarity
Unchanging climate
Return level, zp, is the value associated with the return period,1/p. That is, zp is the level expected to be exceeded on averageonce every 1/p years.
That is, Return level, zp, with 1/p-year return period is
zp = Fβ1(1β p).
For example, p = 0.01 corresponds to the 100-year return period.
Easy to obtain from GEV and GP distributions (stationary case).
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Risk Communication Under Stationarity
Unchanging climate
For example, GEV return level is given by
zp = Β΅β Ο
ΞΎ[1β (β log(1β p))]βΞΎ
0.0
0.1
0.2
0.3
Return level with (1/p)βyear return period
GE
V p
df
z_p
1βp p
Similar for GPD, but must take Ξ»into account.
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Risk Communication Under Stationarity
Unchanging climate
Compare previous GPD fits (with and without de-clustering). Mustfirst assign the class name, βgpd.fit" to each so that the extRemesfunction, return.level, will know whether the objects refer to aGEV or GPD fit.
# Without de-clustering.class( phx.fit0) < β "gpd.fit"return.level( phx.fit0, make.plot=FALSE)
# With de-clustering.class( phxdc.fit0) < β "gpd.fit"return.level( phxdc.fit0, make.plot=FALSE)
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Non-Stationarity
Sources
β’ Trends:climate change: trends in frequency and intensity of extreme weatherevents.
β’ Cycles:Annual and/or diurnal cycles often present in meteorologicalvariables.
β’ Other.
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Non-Stationarity
Theory
No general theory for non-stationary case.
Only limited results under restrictive conditions.
Can introduce covariates in the distribution parameters.
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Non-Stationarity
Phoenix minimum temperature
1950 1960 1970 1980 1990
6570
75
Phoenix summer minimum temperature
Year
Min
imum
tem
pera
ture
(de
g. F
)
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Non-Stationarity
Phoenix minimum temperature
Recall: min{X1, . . . , Xn} = βmax{βX1, . . . ,βXn}.Assume summer minimum temperature in year t = 1, 2, . . . has GEV
distribution with:Β΅(t) = Β΅0 + Β΅1 Β· t
log Ο(t) = Ο0 + Ο1 Β· t
ΞΎ(t) = ΞΎ
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Non-Stationarity
Phoenix minimum temperatureNote: To convert back to min{X1, . . . , Xn},
change sign of location parameters. But note that model isPr{βX β€ x} = Pr{X β₯ βx} = 1β F (βx).
Β΅Μ(t) β 66.170 + 0.196t
log ΟΜ(t) β 1.338β 0.009t
ΞΎΜ β β0.21
Likelihood ratio test
for Β΅1 = 0 (p-value < 10β5),
for Ο1 = 0 (p-value β 0.366).
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Non-Stationarity
Phoenix minimum temperatureModel Checking. Found the best model from a range of models,
but is it a good representation of the data? Transform data to acommon distribution, and check the qq-plot.
1. Non-stationary GEV to exponential
Ξ΅t =
{1 +
ΞΎΜ(t)
ΟΜ(t)[Xt β Β΅Μ(t)]
}β1/ΞΎΜ(t)
2. Non-stationary GEV to Gumbel (used by ismev/extRemes
Ξ΅t =1
ΞΎΜ(t)log
{1 + ΞΎΜ(t)
(Xt β Β΅Μ(t)
ΟΜ(t)
)}
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Non-Stationarity
Phoenix minimum temperatureModel Checking. Found the best model from a range of models,
but is it a good representation of the data? Transform data to acommon distribution, and check the qq-plot.
β
β β
β
ββββββ
βββββββ
ββββββββ
ββ
βββββ β
β
ββ β
β
β
β
β
β
β
β1 0 1 2 3
β1
01
23
4
QβQ Plot (Gumbel Scale): Phoenix Min Temp
Model
Em
piric
al
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Non-Stationarity
Physically based covariatesWinter maximum daily temperature at Port Jervis, New York
Let X1, . . . , Xn be the winter maximum temperatures, and Z1, . . . , Zn
the associated Arctic Oscillation (AO) winter index. Given Z = z,assume conditional distribution of winter maximum temperature isGEV with parameters
Β΅(z) = Β΅0 + Β΅1 Β· z
log Ο(z) = Ο0 + Ο1 Β· z
ΞΎ(z) = ΞΎ
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Non-Stationarity
Physically based covariatesWinter maximum daily temperature at Port Jervis, New York
Β΅Μ(z) β 15.26 + 1.175 Β· z
log ΟΜ(z) = 0.984β 0.044 Β· z
ΞΎ(z) = β0.186
Likelihood ratio test for Β΅1 = 0 (p-value < 0.001)Likelihood ratio test for Ο1 = 0 (p-value β 0.635)
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Non-Stationarity
Cyclic variationFort Collins, Colorado precipitation
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ββ βββββββββββββββ
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ββββββββ βββββββββββββββββββββββββββββββ
0.00
0.01
0.02
0.03
0.04
Fort Collins daily precipitation
Pre
cipi
tatio
n (in
)
Jan Mar May Jul Sep Nov
β
73
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Non-Stationarity
Cyclic variationFort Collins, Colorado precipitation Orthogonal approach. First fit
annual cycle to Poisson rate parameter (T = 365.25):
log Ξ»(t) = Ξ»0 + Ξ»1 sin
(2Οt
T
)+ Ξ»2 cos
(2Οt
T
)Giving
log Ξ»Μ(t) β β3.72 + 0.22 sin
(2Οt
T
)β 0.85 cos
(2Οt
T
)Likelihood ratio test for Ξ»1 = Ξ»2 = 0 (p-value β 0).
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Non-Stationarity
Cyclic variationFort Collins, Colorado precipitation Orthogonal approach. Next fit
GPD with annual cycle in scale parameter.
log Οβ(t) = Οβ0 + Οβ1 sin
(2Οt
T
)+ Οβ2 cos
(2Οt
T
)Giving
log ΟΜβ(t) β β1.24 + 0.09 sin
(2Οt
T
)β 0.30 cos
(2Οt
T
)Likelihood ratio test for Οβ1 = Οβ2 = 0 (p-value < 10β5)
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Non-Stationarity
Cyclic variationFort Collins, Colorado precipitation
Annual cycle in location and scale parameters of theGEV re-parameterization approach point process model witht = 1, 2, ..., and T = 365.25.
Β΅(t) = Β΅0 + Β΅1 sin(
2ΟtT
)+ Β΅2 cos
(2ΟtT
)log Ο(t) = Ο0 + Ο1 sin
(2ΟtT
)+ Ο2 cos
(2ΟtT
)ΞΎ(t) = ΞΎ
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Non-Stationarity
Cyclic variationFort Collins, Colorado precipitation
Β΅Μ(t) β 1.281β 0.085 sin(
2ΟtT
)β 0.806 cos
(2ΟtT
)log ΟΜ(t) β β0.847β 0.123 sin
(2ΟtT
)β 0.602 cos
(2ΟtT
)ΞΎΜ β 0.182
Likelihood ratio test for Β΅1 = Β΅2 = 0 (p-value β 0).Likelihood ratio test for Ο1 = Ο2 = 0 (p-value β 0).
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Non-Stationarity
Cyclic variationFort Collins, Colorado precipitation
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββ
ββββββββββββββββ
βββββββββββββββββββββββ
βββββββ
ββββββββ
βββββ
ββββββββββ
ββββ
βββββββ
ββββ
βββββ
β ββ β
β
β
0 1 2 3 4 5 6 7
01
23
45
67
Residual quantile Plot (Exptl. Scale)
model
empi
rical
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Risk Communication (Under Non-Stationarity)
Return period/level does not make sense anymore because of chang-ing distribution (e.g., with time). Often, one uses an βeffective" re-turn period/level instead. That is, compute several return levels forvarying probabilities over time. Can also determine a single returnperiod/level assuming temporal independence.
1β 1
m= Pr {max(X1, . . . , Xn) β€ zm} β
nβi=1
pi,
where
pi =
{1β 1
nyβ1/ΞΎii , for yi > 0,
1 , otherwise
where yi = 1 + ΞΎiΟi
(zm β Β΅i), and (Β΅i, Οi, ΞΎi) are the parametrs of thepoint process model for observation i. Can be easily solved for zm
(using numerical methods). Difficulty is in calculating the uncertainty(See Coles, 2001, chapter 7).
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References
Coles S, 2001. An introduction to statistical modeling of extremevalues. Springer, London. 208 pp.
Katz RW, MB Parlange, and P Naveau, 2002. Statistics of extremesin hydrology. Adv. Water Resources, 25:1287β1304.
Stephenson A and E Gilleland, 2006. Software for the analysis ofextreme events: The current state and future directions. Extremes,8:87β109.
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