Lecture presented during the Climate Extremes Workshop

1. Extreme Value theory and the Re-assessment in
the Caribbean: Lessons from Hurricane Maria
David Torres N´u˜nez 1
1University of Puerto Rico at Rio Piedras Campus
Inst´ıtuto de Estad´ısticas y Sistemas Computadorizados de la Informaci´on
Facultas de Administraci´on de Empresas
May 17, 2018

2. Acknowledgment
This is an ongoing work within the guidance and discussion of Dr.
Luis Pericchi and Dr. Jorge Ortiz both from University of Puerto
Rico. Any comments can be sent to
David Torres (david.torres9@upr.edu)
Dr. Luis Pericchi Guerra (luis.pericchi@upr.edu)
Dr. Jorge Ortiz (jorge.ortiz@upr.edu)

3. Contents
Introduction
Some historical remarks
Estimators
Implementation and Results
Conclusions and Further Work
Questions
Cited Literature

4. Risk assessment has been the corner stone in resilient
planning.
Pericchi, Coles Sissons Conjecture state that standard Gumbel
analyses routinely assign near-zero probability to subsequently
observed disasters, and that for San Juan, Puerto Rico,
standard 100-year predicted rainfall estimates may be
routinely underestimated by a factor of two by the Maximum
likelihood estimators.
Using a extended the San Juan rain data before Hurricane
Maria event predictions will be stated using Maximum
Likelihood Estimators(MLE) and more generalized Bayesian
Models including Hierarchical Modeling to restate the
conjecture with more variate models.
It is shown that using Bayesian Analysis models a slightly
more precise results predicting extreme weather events can be
established.

5. San Juan International Airport Station
The San Juan International Airport Station(SAN JUAN INTL AP),
Identification number GHCND:RQW00011641
Latitude 18.4325◦
Longitude -66.01083◦
Elevation:2.7 ft
www.ncdc.noaa.gov

6. Precipitation San Juan data.
Time
Precipitation[mm/month]
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Monthly time series
0200400
Jan 1956 Jan 1970 Jan 1985 Jan 2000 Jan 2015
Monthly series
Time
Precipitation[mm/year]
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Annual time series
5001500
1956 1964 1972 1980 1988 1996 2004 2012

7. The hurricane Maria had sustained winds of 250km (155.34
miles)per hour when it made landfall on Puerto Rico on September
20, 2017. Most of the rain stay in those basins at the center of the
island.

8. Name Date Intensity
H. San Ciriaco Aug.8, 1899 23in/24h(Adjuntas)
San Felipe Sept13 − 14, 1928 19in/24hr
H. Donna Sept.6, 1960 19in/24hr(Luquillo)
Tropical D. Oct.5 − 10, 1970 17in/24hr(Aibonito)
T. Eloisa Sept.15 − 17, 1975 23in/24hr(Maricao)
Low preasure sys. May17 − 18, 1985 25in/5dy(Jayuya)
H. Hortense Sept.10, 1996 24.6in/24hr
H. Georges Sept.21 − 22, 1998 24.6in/48hr
Tropical D. Nov.12 − 14, 2003 21.0in/48hr
H. Maria Sept.19 − 21, 2017 36.2in/24hr(Caguas)
Table: Maximum registered rain in Puerto Rico.1
1
Col´on, E. Torres, Acevedo. 1991, Puerto Rico: Floods and droughts;
National Water Summary, US Geological Survey.

9. San Juan Station daily, month and year.
Time
stationPRCP/25.4[inches/day]
Daily time series
02468
Jan 01 1956 Jan 01 1980 Jan 01 2005
Daily series
Time
stationPRCP/25.4[inches/month]
Monthly time series
02468
Jan 1956 Jan 1975 Jan 1995 Jan 2015
Monthly series
stationPRCP/25.4[inches/year]
Annual time series
2468
1956 1967 1978 1989 2000 2011
02468
Daily Boxplot
stationPRCP/25.4[inches/day]
Jan Mar May Jul Sep Nov
02468
Monthly Boxplot
stationPRCP/25.4[inches/month]
2468
Annual Boxplot
stationPRCP/25.4[inches/year]
Daily Histogram
stationPRCP/25.4 [inches/day]
Pbb
0 2 4 6 8
01234567
Monthly Histogram
stationPRCP/25.4 [inches/month]
Pbb
0 2 4 6 8 10
0.00.10.20.30.4
Annual Histogram
Pbb
0.000.100.200.30

12. USGS 50999961 LA PLAZA RAINGAGE, CAGUAS PR report a
maximum of 14.49 in per hr from 8:15am to 9:15am at September
20, 2017.

14. Fisher - Tippett, Gnedenko Theorem
In 1928 Fisher and Tippett is their seminal work describe limiting
extreme distributions and 15 years later (1943)Gedenko formalized
rigorously defining their domain of attraction.
Let X1, X2, . . . Xn are independent random variables with the
same probability distribution and Mn = max{X1, X2, . . .}. For
an > 0, bn such that
P(
mn − bn
an
≤ x) = F(anx + bn)n
=⇒ H(x)
If such function exist then it will be one of the following; Gumbel
Law, Weibull Law, and Frechet Law.

15. Type I, II and III
H(x) = exp{− exp{−x}}, for all x ∈ R. Gumbel Law
H(x) =
0, for x < 0.
exp{−xα}, x > 0.(Frechet’s Law)
H(x) =
0, for x < 0.
exp{−xα}, x > 0.(Weibull’s Law)

16. The Generalized Extreme Value Distribution(GEV)
Combining all the cases we can write the GEV as
H(x) = exp{−(1 + ξ(
x − µ
ψ
)
−1/ξ
+ )}
where y+ = max(y, 0) and is defined in the set
{x : 1 + ξ(x − µ)/ψ > 0}.

17. The conversion is to summarize an extreme value analysis, the m
return level zp satisfies for the m year
P(Z ≤ zp) = 1/p
corresponding to the level that is expected to occur every m years.
The zp are such that H(zp) = 1 − p, were H(x) as defined. Hence
zp =
µ + (ψ/ξ)[−ln(1 − p)]−ξ − 1, para ξ = 0.
µ + ψ{−ln[−ln(1 − p)]}, para ξ = 0.

18. Estimating using MLE
Based on the data x1, x2, . . . , xn the likelihood function takes the
form
L(µ, σ, ξ) =
n
i=n
g(xi; µ, σ, ξ)
where
g(xi; µ, σ, ξ) =
H(u), if x ≤ u
dH
dx (x) if x > u.
There is not an analytic solution but numerical to assets the
maximum values estimators(Coles and Pericchi 2004 ).

19. Estimating parameters using Bayesian Methods
Suppose q(x, y) is a random walk: q(x, y) = q∗(y − x) for some
distribution q∗. The Metropolis-Hastings ratio is
min(π(y)q∗
(x − y)/π(x)q∗
(y − x), 1)
(or 1 if π(x)q∗(y − x) = 0).
If q∗ is symmetric, then the Metropolis-Hastings ratio reduces to
min(π(y)/π(x), 1)
(or 1 if π(x) = 0).

20. 2
MLE 95% lower CI Estimate 95% upper CI
2-year level 2.7865 3.1555 3.5244
20-year level 5.2893 7.0945 8.8997
100-year level 5.8725 10.8318 15.7910
1000-year level 3.3764 18.7031 34.0298
2000-year level 1.3301 21.8694 42.4087
Bayesian 25% Posterior mean 97.5%
2-year level 2.7468 3.1789 3.6896
20-year level 5.8275 7.7611 11.9216
100-year level 7.7990 13.0586 27.8001
1000-year level 10.4522 28.5611 95.0695
2000-year level 11.2098 36.9192 138.3462
2
Using the Statistical Package extRemes from R, calculations of the MLE
has been done and are presented here.

21. MLE Diagnostic
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50 100 150 200 250
50100150200
Model Quantiles
EmpiricalQuantiles
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50 100 150 200
50100150200
San.Juan Empirical Quantiles
QuantilesfromModelSimulatedData
1−1 line
regression line
95% confidence bands
0 50 100 150 200 250
0.0000.0040.0080.012
N = 61 Bandwidth = 11.3
Density
Empirical
Modeled
2 5 10 50 200 1000
200400600800
Return Period (years)
ReturnLevel(mm)
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
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q q q
fevd(x = San.Juan, type = "GEV", method = "MLE", units = "mm")

22. MCMC Diagnostic
2 4 6 8 10
2468
Model Quantiles
EmpiricalQuantiles
2 4 6 8
51015
San.Juan.MaxRain Empirical Quantiles
QuantilesfromModelSimulatedData
1−1 line
regression line
95% confidence bands
0 2 4 6 8 10
0.000.100.200.30
N = 61 Bandwidth = 0.4451
Density
Data
Model
2 5 10 20 50 100 200 500 1000
020406080
Return Period (years)
ReturnLevel(inches)
fevd(x = San.Juan.MaxRain, method = "Bayesian", units = "inches",iter = 1e+05)

23. Mean Residual life plot(with 95% CI) for daily rainfall
data. Commentary on Priors.
The mean residual start in almost u = 50mm which is appears to
be constant behavior up to a 150 period. This helps for a choice of
robust priors(Coles et. al. 2003).
50 100 150 200
020406080
MeanExcess

24. Metropolis Hasting
Using a Metropolis Hasting algorithm with a robust prior a
exponential prior in the localization parameter, a normal and
uniform prior for scale and shape parameters, implemented using C
a more stable estimators where founded.
0 200 400 600 800 1000
5060708090100110
Index
sigma
Histogram of Sigma
sigma
Frequency
50 60 70 80 90 100 110
050100150200
0 200 400 600 800 1000
707580859095100
Index
mu
Histogram of Mu
mu
Frequency
70 80 90 100
0100200300400
0 200 400 600 800 1000
0.81.01.21.41.6
Index
epsilon
Histogram of Epsilon
epsilon
Frequency
0.8 1.0 1.2 1.4 1.6 1.8
050100150200250

25. Parameters statistics
Bayesian MH 2.5% Posterior Mean Estimate 95%
Location 79.7698 82.9296 90.6705
Scale 66.2329 72.9550 89.5822
Family 1.1170 1.2268 1.5022

26. Remarks and Conclusions I
Information about the maximum historical behavior is crucial
for a more robust Bayesian Model.
Using a more complete maximum data a predicted rainfall
estimates may be routinely underestimated by a factor of less
than a factor of two compared with the Maximum likelihood
estimators. More study has to be done taking in account
Spatial assessment and different robust priors for scale and
shape parameters.
Missing data due to the event represent a valuable
information lost.
Most of the instruments are not prepared for Hurricanes
greater intensity. Investment in proper rain gauge on safe
zones are needed.

27. Thanks for having me here and for you patience.
Muchas gracias!

28. Cited Literature
SA Sisson, LR Pericchi, SG Coles(2006). A case for a
reassessment of the risks of extreme hydrological hazards in
the Caribbean. Stochastic Environmental Research and Risk
Assessment 20 (4), 296-306.
S Coles, LR Pericchi, S Sisson(2003). A fully probabilistic
approach to extreme rainfall modeling. Journal of Hydrology
273 (1-4), 35-50.
Coles S. G. and L.R. Pericchi(2003). Anticipating
catastrophes through extreme value modeling. Applied
Statistics 52(405 - 416).
Coles, S. (2001) An introduction to statistical modeling of
extreme values, London, U.K.: Springer-. Verlag, 208 pp.

29. R. A. Fisher and L. H. C. Tippett, Limiting forms of the
frequency distribution of the largest and smallest member of a
sample, Math. Proc. Cambridge Philos. Soc. 24 (1928),
180–190.
B. V. Gnedenko, Sur la distribution limite du terme maximum
d’une s´erie al´eatoire. Ann. Math. 44 (1943), 423–453.
English translation: On the limiting distribution of the
maximum term in a random series, in Breakthroughs in
Statistics, Volume 1: Foundations and Basic Theory. Springer,
1993, pp. 185–225.
E. J. Gumbel, Statistics of Extremes, Columbia University
Press, 1958. Dover, 2004.
Gilleland, E. Package extRemes Extreme Value Analysis.
Extremes 2016, 18, 1.