1. This document summarizes statistics on climate extremes, including time series plots and extreme value analyses of temperature and precipitation data from Houston, Texas. 2. Fitted generalized extreme value (GEV) distributions to one-day, three-day, and seven-day maximum precipitation values show increasing return levels with longer durations. 3. Bayesian and frequentist methods are demonstrated for fitting GEV distributions and estimating return levels of extreme precipitation events.

1. STATISTICS OF CLIMATE EXTREMES
Richard L Smith
Departments of STOR and Biostatistics, University
of North Carolina at Chapel Hill
and
Statistical and Applied Mathematical Sciences
Institute (SAMSI)
SAMSI Graduate Class
November 7, 2017
1
1

2. 1900 1950 2000
25.526.026.5
Annual Mean SST (July−June) in Gulf of Mexico
Year
SST
2

3. 1950 1960 1970 1980 1990 2000 2010
24681012
Time Series Plot:
Houston Hobby Annual Maxima One−Day Precipitation
Year
1−dayMaximumPRCP
3

4. −1 0 1 2 3 4 5
24681012
Gumbel Plot of One−day Maxima
Straight line fitted to first n−1 data points
Reduced Value
OrderedValue
4

5. 1950 1960 1970 1980 1990 2000 2010
51015202530
Time Series Plot:
Houston Hobby Annual Maxima Three−Day Precipitation
Year
3−dayMaximumPRCP
5

6. −1 0 1 2 3 4 5
51015202530
Gumbel Plot of Three−day Maxima
Straight line fitted to first n−1 data points
Reduced Value
OrderedValue
6

7. 1950 1960 1970 1980 1990 2000 2010
5101520253035
Time Series Plot:
Houston Hobby Annual Maxima Seven−Day Precipitation
Year
7−dayMaximumPRCP
7

8. −1 0 1 2 3 4 5
5101520253035
Gumbel Plot of Seven−day Maxima
Straight line fitted to first n−1 data points
Reduced Value
OrderedValue
8

9. EXTREME VALUE DISTRIBUTIONS
X1, X2, ..., i.i.d., F(x) = Pr{Xi ≤ x}, Mn = max(X1, ..., Xn),
Pr{Mn ≤ x} = F(x)n.
For non-trivial results must renormalize: find an > 0, bn such that
Pr
Mn − bn
an
≤ x = F(anx + bn)n → H(x).
The Three Types Theorem (Fisher-Tippett, Gnedenko) asserts
that if nondegenerate H exists, it must be one of three types:
H(x) = exp(−e−x), all x (Gumbel)
H(x) =
0 x < 0
exp(−x−α) x > 0
(Fr´echet)
H(x) =
exp(−|x|α) x < 0
1 x > 0
(Weibull)
In Fr´echet and Weibull, α > 0.
9

10. The three types may be combined into a single generalized ex-
treme value (GEV) distribution:
H(x) = exp − 1 + ξ
x − µ
σ
−1/ξ
+
,
(y+ = max(y, 0))
where µ is a location parameter, σ > 0 is a scale parameter
and ξ is a shape parameter. ξ → 0 corresponds to the Gumbel
distribution, ξ > 0 to the Fr´echet distribution with α = 1/ξ, ξ < 0
to the Weibull distribution with α = −1/ξ.
ξ > 0: “long-tailed” case, 1 − F(x) ∝ x−1/ξ,
ξ = 0: “exponential tail”
ξ < 0: “short-tailed” case, finite endpoint at µ − ξ/σ
10

11. EXCEEDANCES OVER THRESHOLDS
Consider the distribution of X conditionally on exceeding some
high threshold u:
Fu(y) =
F(u + y) − F(u)
1 − F(u)
.
As u → ωF = sup{x : F(x) < 1}, often find a limit
Fu(y) ≈ G(y; σu, ξ)
where G is generalized Pareto distribution (GPD)
G(y; σ, ξ) = 1 − 1 + ξ
y
σ
−1/ξ
+
.
Equivalence to three types theorem established by Pickands (1975).
11

12. The Generalized Pareto Distribution
G(y; σ, ξ) = 1 − 1 + ξ
y
σ
−1/ξ
+
.
ξ > 0: long-tailed (equivalent to usual Pareto distribution), tail
like x−1/ξ,
ξ = 0: take limit as ξ → 0 to get
G(y; σ, 0) = 1 − exp −
y
σ
,
i.e. exponential distribution with mean σ,
ξ < 0: finite upper endpoint at −σ/ξ.
12

13. POISSON-GPD MODEL FOR
EXCEEDANCES
1. The number, N, of exceedances of the level u in any one
year has a Poisson distribution with mean λ,
2. Conditionally on N ≥ 1, the excess values Y1, ..., YN are IID
from the GPD.
13

14. Relation to GEV for annual maxima:
Suppose x > u. The probability that the annual maximum of the
Poisson-GPD process is less than x is
Pr{ max
1≤i≤N
Yi ≤ x} = Pr{N = 0} +
∞
n=1
Pr{N = n, Y1 ≤ x, ... Yn ≤ x}
= e−λ +
∞
n=1
λne−λ
n!
1 − 1 + ξ
x − u
σ
−1/ξ n
= exp −λ 1 + ξ
x − u
σ
−1/ξ
.
This is GEV with σ = σ +ξ(u−µ), λ = 1 + ξu−µ
σ
−1/ξ
. Thus the
GEV and GPD models are entirely consistent with one another
above the GPD threshold, and moreover, shows exactly how the
Poisson–GPD parameters σ and λ vary with u.
14

15. ALTERNATIVE PROBABILITY MODELS
1. The r largest order statistics model
If Yn,1 ≥ Yn,2 ≥ ... ≥ Yn,r are r largest order statistics of IID
sample of size n, and an and bn are EVT normalizing constants,
then
Yn,1 − bn
an
, ...,
Yn,r − bn
an
converges in distribution to a limiting random vector (X1, ..., Xr),
whose density is
h(x1, ..., xr) = σ−r exp − 1 + ξ
xr − µ
σ
−1/ξ
− 1 +
1
ξ
r
j=1
log 1 + ξ
xj − µ
σ



.
15

16. 2. Point process approach (Smith 1989)
Two-dimensional plot of exceedance times and exceedance levels
forms a nonhomogeneous Poisson process with
Λ(A) = (t2 − t1)Ψ(y; µ, σ, ξ)
Ψ(y; µ, σ, ξ) = 1 + ξ
y − µ
σ
−1/ξ
(1 + ξ(y − µ)/σ > 0).
16

17. Illustration of point process model.
17

18. An extension of this approach allows for nonstationary processes
in which the parameters µ, σ and ξ are all allowed to be time-
dependent, denoted µt, σt and ξt.
This is the basis of the extreme value regression approaches
introduced later
Comment. The point process approach is almost equivalent to
the following: assume the GEV (not GPD) distribution is valid for
exceedances over the threshold, and that all observations under
the threshold are censored. Compared with the GPD approach,
the parameterization directly in terms of µ, σ, ξ is often easier
to interpret, especially when trends are involved.
18

19. ESTIMATION
GEV log likelihood:
Y (µ, σ, ξ) = −N log σ −
1
ξ
+ 1
i
log 1 + ξ
Yi − µ
σ
−
i
1 + ξ
Yi − µ
σ
−1/ξ
provided 1 + ξ(Yi − µ)/σ > 0 for each i.
Poisson-GPD model:
N,Y (λ, σ, ξ) = N log λ − λT − N log σ − 1 +
1
ξ
N
i=1
log 1 + ξ
Yi
σ
provided 1 + ξYi/σ > 0 for all i.
Usual asymptotics valid if ξ > −1
2 (Smith 1985)
19

20. Some Sample Code in R
# to define likelihood function:
lh1=function(par){
# neg log likelihood function for 3-parameter GEV with covariates
#
# The following variables must be defined in the calling program:
#
# Y0: vector of annual maxima observations
# n0: length of Y0
# npar: length of par
# X0, n, p: if npar=3+p, then X0 is nxp matrix of covariates
# if p=0, X0 can be omitted
l0=0
mu0=par[1]
sig=exp(par[2])
xi=par[3]
# reject if sig or xi are outside reasonable ranges (could be adjusted)
if(abs(par[2])>20|abs(par[3]>1){
l0=1e10
return(list=l0)}
for(i in 1:n0){
mu=mu0
p=npar-3
if(p>0){mu=mu0+sum(par[4:(3+p)]*X0[i,1:p])}
aa=1+xi*(Y0[i]-mu)/sig
if(aa<0){
l0=1e10
return(list=l0)}
20

21. Some Sample Code in R (continued)
aa=log(aa)/xi
if(abs(aa)>20){
l0=1e10
return(list=l0)}
l0=l0+par[2]+(1+xi)*aa+exp(-aa)
}
return(list=l0)
}
#
# To run the code: fit GEV to data in vector Y0 assumed given
#
# First make rough guess about mu, log sigma, xi, then optimize
par1=c(mean(Y0),0.5*log(var(Y0)),0.01)
npar=3
n0=length(Y0)
lh1(par1) # value of lh1 after optimizing
opt1=optim(par1,lh1,method=’BFGS’,control=list(ndeps=rep(1e-6,npar),maxit=10000),hessian=T)
opt1$value # value of lh1 after optimizing
# display parameters and SEs
library(’MASS’)
covm=ginv(opt1$hess)
out1=matrix(nrow=npar,ncol=4)
for(i in 1:npar){
out1[i,1]=opt1$par[i]
out1[i,2]=sqrt(covm[i,i])
out1[i,3]=out1[i,1]/out1[i,2]
out1[i,4]=2*pnorm(-abs(out1[i,3]))}
21

22. Some Sample Code in R (continued)
# calculate probability of a new value exceeding "valnew"
mu=opt1$par[1]
sig=exp(opt1$par[2])
xi=opt1$par[3]
aa=1+xi*(valnew-mu)/sig
aa=1-exp(-aa^(-1/xi))
For a well-designed package to perform lots of extreme value
analysis tasks, look up the “extRemes” package in R:
https://cran.r-project.org/web/packages/extRemes/extRemes.pdf
22

23. Bayesian approaches
An alternative approach to extreme value inference is Bayesian,
using vague priors for the GEV parameters and MCMC samples
for the computations. Bayesian methods are particularly useful
for predictive inference, e.g. if Z is some as yet unobserved ran-
dom variable whose distribution depends on µ, σ and ξ, estimate
Pr{Z > z} by
Pr{Z > z; µ, σ, ξ}π(µ, σ, ξ|Y )dµdσdξ
where π(...|Y ) denotes the posterior density given past data Y
23

24. GUMBEL PLOTS
• Wi is ith order statistic (i = 1 smallest, i = n largest)
• Vi = − log − log i−0.5
n
• Plot (Wi against Vi
Rationale:
• ξ → 0 limit of GEV is F(x) = exp − exp −x−µ
σ
• − log (− log (F(x))) = x−µ
σ
• Assume “plotting position” for i’th order statistic is pn,i =
i−0.5
n
• Plot of Wi versus Vi approx. linear, intercept −µ
σ , slope 1
σ.
• Linearity of plot indicates Gumbel or curve up (ξ > 0) or
curve down (ξ < 0).
• Also use to assess outliers
24

25. ALTERNATIVE PLOTTING POSITIONS
pn,i = i
n+1 is often recommended in elementary statistical texts
(in connection with, for example, Kolmogorov-Smirnov test of
fit) but in practice it does not work so well in probability plotting.
Another formula is Gringorten’s formula: pn,i = i−0.44
n+0.12
This was dervied by a rather complicated procedure to mini-
mize bias (Gringorten 1963, Journal of Geophysical Research).
It works perfectly well but the pn,i = i−0.5
n formula is easier to
remember and universal in procedures of this nature.
25

26. PLOTS FOR THRESHOLD EXCEEDANCES
Main idea is “mean excess over threshold” plot
(From: R.L. Smith (2003), Statistics of extremes, with applications in environment, insur-
ance and finance. Chapter 1 of Extreme Values in Finance, Telecommunications and the
Environment, edited by B. Finkenstadt and H. Rootzen, Chapman and Hall/CRC Press,
London, pp. 1–78)
26

27. GEV FITTED TO ONE-DAY MAXIMA
(based on data 1949–2016)
Parameter Estimate S.E. t-ratio p-value
µ 3.50 0.18 19.03 0.000
log σ 0.27 0.11 2.37 0.02
ξ 0.20 0.11 1.74 0.08
Estimated probability of exceeding 2017 level: 0.0152
(66-year return value)
27

28. GEV FITTED TO THREE-DAY MAXIMA
(based on data 1949–2016)
Parameter Estimate S.E. t-ratio p-value
µ 5.09 0.28 18.06 0.000
log σ 0.70 0.11 6.48 0.00
ξ 0.12 0.11 1.06 0.29
Estimated probability of exceeding 2017 level: 0.00028
(3,515-year return value)
28

29. GEV FITTED TO SEVEN-DAY MAXIMA
(based on data 1949–2016)
Parameter Estimate S.E. t-ratio p-value
µ 6.07 0.31 19.28 0.000
log σ 0.85 0.10 8.39 0.00
ξ 0.10 0.08 1.25 0.21
Estimated probability of exceeding 2017 level: 0.00024
(4,129-year return value)
29

30. From now on, use 7-day maxima, following example of M.D.
Risser and M.F. Wehner, “Attributable human-induced changes
in the likelihood and magnitude of observed extreme precipita-
tion in the Houston, Texas region during Hurricane Harvey”, to
appear in Geophysical Research Letters
30

31. BAYESIAN ANALYSIS
(based on data 1949–2016)
Bayesian analysis assuming uniform prior on (µ, log σ, ξ)
Use adaptive Metropolis algorithm (Haario et al. 2001)
Parameter Estimate S.E. t-ratio p-value
µ 5.96 0.41 14.62 0.00
log σ 0.82 0.12 7.04 0.00
ξ 0.13 0.08 1.64 0.10
MLE probability of exceeding 2017 level: 0.00024
Posterior median probability of exceeding 2017 level: 0.00039
90% credible interval for probability of exceeding 2017 level:
(9.2 × 10−6, 0.0034)
Posterior mean probability of exceeding 2017 level: 0.00083
31

32. 0.000 0.005 0.010 0.015 0.020 0.025 0.030
0.000.050.100.150.200.25
Posterior Density of 2017 Exceedance Probability
Exceedance Probability
ProbabilityDensity
32

33. ADDING A LINEAR TREND
GEV in year t has parameters µt = α + βt, σ, ξ
(t measured in centuries)
Parameter Estimate S.E. t-ratio p-value
α 5.01 0.56 8.97 0.00
log σ 0.81 0.10 8.26 0.00
ξ 0.10 0.07 1.39 0.16
β 3.25 1.40 2.33 0.02
Estimated probability of exceeding 2017 level: 0.00028
Same probability for 1950: 0.00015
Ratio: 1.89
33

34. BAYESIAN ANALYSIS WITH A LINEAR
TREND
Parameter Estimate S.E. t-ratio p-value
α 5.13 0.57 9.02 0.00
log σ 0.83 0.09 8.84 0.00
ξ 0.08 0.06 1.26 0.21
β 3.01 1.22 2.48 0.01
Mean/median probability of exceeding 2017 value: 0.00035/0.00027
Mean/median probability of same event in 1960: 0.000085/0.000039
Ratio of probabilities (Relative Risk or RR) has mean 1.75, median 1.60
90% credible interval 1.06–3.06
34

35. 0 1 2 3 4 5
0.00.20.40.60.81.01.2
Posterior Density of Relative Risk
(P(RR>1)=0.9986)
N = 4923 Bandwidth = 0.06728
Density
35

36. PAPER BY WEHNER AND RISSER
36

37. lfldlsa
37

38. 38

39. lfldlsa
MODELS IN RISSER-WEHNER ANALYSIS
AIC prefers M2
39

40. 40