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Filling the gap between point-in-time and extreme value distributions
M´
ınguez, R., Guanche, Y., Jaime, F. F., M´
endez, F. J. & Tom´
as, A.
Environmental Hydraulics Institute “IH Cantabria”, Universidad de Cantabria, Spain
ABSTRACT: Engineering structures must satisfy different requirements during its life-cycle: construction,
useful life and dismantling. The satisfaction of those requirements is checked by defining failure probabilities
associated with the different ways in which the structure might fail, also known as limit states. Threshold or
maximum allowed probabilities are defined by expert committees according to the consequences produced in
case each limit state is trespassed. From the practical point of view, this distinction requires the characterization
of both the point-in-time and extreme probability (right tail) distribution of all random variables involved, and
the selection of the appropriate distribution depending on the limit state. However, there are still international
code regulations and design standards that do not clarify this issue and might lead to get invalid designs from
the engineering perspective. This paper provides a practical guide based on M´ınguez, Guanche, & M´endez
(2012) work in order to: i) analyze if distinction between both distributions is relevant for each failure mode,
ii) which distribution should be used, and iii) howto work with both distributions at the same time. In addition,
an example of the importance of these considerations using the IEC international standards for the definition of
design requirements for offshore wind turbines (IEC 61400-3) is given.
1 INTRODUCTION
Engineering structures must satisfy several design
conditions during its lifetime with a certain probabil-
ity of failure. Those acceptable rates are established
by codes and expert committees (Baker 1976, Lind
1976, Horne & H. 1977, ROM 0.0 2001) on the ba-
sis of the consequences of failure for each limit state,
and trying to counter-balance safety and costs (direct,
societal and environmental). These different proba-
bility thresholds encompass the consideration of dif-
ferent probability distributions for agents, while ser-
viceability or operating stop limit conditions depend
on regular or mean values (point-in-time) of those
agents, damage and ultimate limit states require ex-
treme conditions or conditions in the right tail. Tra-
ditionally, both problems are treated independently,
which makes difficult to understand the link between
point-in-time and extreme distributions and their im-
plications from the practical point of view.
There are several attempts in the literature to in-
corporate the information from both distributions in
the same probability model (mixture models), see
for instance, Frigessi, Haug, & Rue (2002) ,Vaz de
Melo Mendes & Freitas Lopes (2004), Behrens,
Lopes, & Gamerman (2004), Tancredi, Anderson, &
Ohagan (2006), Cai, Gouldby, Hawkes, & Dunning
(2008), Furrer & Katz (2001), Solari (2011). How-
ever, they are applied to specific distributions,and the
parameter estimation is fuzzy, not providing a general
framework to deal with this problem.
Recently, M´ınguez, Guanche, & M´endez (2012)
presented some findings about this issue, focusing on
three aspects: i) the development of a Monte Carlo
simulation method for reproducing both the point-in-
time (mean values) and extreme value distributions
of random variables involved, ii) presenting a graph-
ical interpretation to merge both distributions on a
compatible scale, and iii) providing new insights for
the use of the point-in-time and extreme regimes si-
multaneously within First-Order-Reliability methods
(FORM).
An example of this problem is shown using the sea
level record at San Francisco (see Figure 1), where an-
nual maxima are plotted using triangle marker spec-
ifiers. This data set consists of an hourly time series
of sea level in meters from January 1, 1901 to De-
cember 31, 2003. The particular record used in this
paper only contains information on storm surge lev-
els. Astronomical tidal and mean sea levels have been
removed from the initial data record for illustrative
purposes.
Fitting the storm surge sea level, the correspond-
ing annual maxima, and exceedances over threshold
u=0.446 mto three different parametric distribu-
tions: i) a Gaussian Mixture with 4 components for
the point-in-time distribution, ii) a generalized ex-
treme value (GEV) model for the annual maxima, and
1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
0.5
0
0.5
1
Time
x(m)
Storm surge at San Francisco
Sea level
xmax
Figure 1: Hourly sea level (storm surge) record at San Francisco from January 1, 1901 to December 31, 2003, and annual maxima.
0 0.5 1
0
2
4
6
8
10
12
x(m)
Density
San Francisco PDFs
0 0.5 1 1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x(m)
Cumulative probability
San Francisco CDFs
ydata
Point-in-time fit
ythr data
Pareto fit
ymax data
GEV fit
xdata
xthr data over thr.
xmax daannual max. ta
Point-in-time fit
Pareto fit
GEV fit
Figure 2: Graphical representation of the point-in-time, GEV
(annual maxima) and Pareto distributions for the sea level record
at San Francisco.
iii) Pareto distribution for exceedances, it is possi-
ble to plot their histograms, fitted densities, empirical
cumulative and fitted distributions, respectively (see
Figure 2). Note that they all present very good fitting
diagnostic plots, specially for the point-in-time distri-
bution, where the model is almost undistinguishable
from the data all over its range. However, it is difficult
to establish whether the fitted point-in-time distribu-
tion is capable of reproducing the tail of interest.
Plotting data and fits from San Francisco using
the graphical representation proposed by M´ınguez,
Guanche, & M´endez (2012), results shown in Figure 3
are obtained. Note that this representation allows es-
tablishing the range of validity of the fitted point-in-
time regime, which starts distorting results for ex-
ceedances above the selected threshold u=0.446 m
of sea level. The probability of not exceeding this
value within the year is 0.9987. Above these quantile
and probability thresholds the point-in-time regime is
no longer valid. It can be observed that the GEV and
Pareto fits allow reproducing appropriately the tail of
the distribution for their corresponding data sets, re-
spectively, but providing inconsistent results between
them.
This paper tries to go deeper into the relationship
between both distributions; establishingthe appropri-
ate methodology to work with point-in-time and the
right-tail distribution simultaneously, and understand-
ing the link between them from the practical perspec-
tive. In addition, an example of the importance of
these considerations when using the IEC international
standards regarding design requirements for offshore
1 2 5 10 20 50 100 250 1000
0
0.5
1
1.5
2
2.5
Return Period T(years)
x(m)
Point-in-time
GEV fit
fit
95% conf.b.
Pareto fit
95% conf.b.
xmax data
xdata
Threshold
Figure 3: Graphical representation of the point-in-time, GEV and Pareto distributions for the sea level record at San Francisco using
the method proposed by M´ınguez, Guanche, & M´endez (2012).
wind turbines (IEC 61400-3) is given. Results demon-
strate that the use of the IFORM (Winterstein, Ude,
Cornel, Bjerager, & Haver 1993) to determine 50-year
environmental contours of significantwave height and
wind speed for hourly sea states, without distinction
between point-in-time and the right-tail distributions,
may lead to invalid results from the engineering per-
spective.
2 RELATIONSHIP BETWEEN POINT-IN-TIME
AND THE RIGHT TAIL OF THE
DISTRIBUTION
Consider the stochastic process Xt, whose point-
in-time and right-tail probability distributions are
FPT(x)and FEV(x), respectively.Let consider the ex-
ample given for the storm surge at San Francisco,
where the FEV(x)distribution corresponds to the
Pareto fit. Alternatively, the GEV fit could also be
selected but there would be a non smooth transition
between the point-in-time and right-tail distribution.
Note that the method proposed in this paper is inde-
pendent of the right-tail distribution selection, there-
fore the issue about which model is more convenient
is out of the scope of this paper. This question is
treated in detail in Davidson & Smith (1990) and the
discussions associated with that paper. Besides, the
selection of exceedances over a threshold is consistent
with Pickands–Balkema–de Haan theorem (Pickands
1975), also known as the second theorem in extreme
value theory.
In this particular case the threshold xlim, which
corresponds to the maximum value governed by the
point-in-time distribution, is equal to the threshold
u=0.446 m. This value has been selected in order
to ensure an smooth transition between both distribu-
tions in terms of equivalent return periods. According
to M´ınguez, Guanche, & M´endez (2012) the proba-
bility distribution function of the stochastic process
considering both distributions is equal to:
FPT (x)if x≤xlim
pPT
lim +FEV (x)−pEV
lim
1−pEV
lim (1 −pPT
lim)if x>x
lim,(1)
where pPT
lim is the probability of not exceeding the
maximum value xlim within the point-in-time distri-
bution and is equal to pPT
lim =FPT(xlim), and pEV
lim =
FEV(xlim)is the probability of not exceeding the xlim-
value within the extreme distribution. Note that the
second term in (1) re-scales the probability given by
the right tail distribution to be consistent with that re-
lated to the point-in-time because fitting results pro-
ceed from different samples.
The quantile associated with the probability pis
equal to:
FPT−1(p)if p≤pPT
lim
FEV−1pEV
lim +p−pPT
lim
1−pPT
lim (1 −pEV
lim)if p>p
PT
lim.(2)
It is relevant to point out that the probabilities as-
sociated with equations (1) and (2) are related to the
sample frequency of the stochastic process, and there-
fore, the equivalent return periods obtained using the
expression T=1
1−pis given in the same temporal
units, i.e. hours for the San Francisco example. This
equivalent return period is transformed into years di-
viding by 8766. Note that we use the terminology
“equivalent” because it truly corresponds to a return
period only if data is independent. We assume that the
extremal index tends to 1, which is always the case
for the largest data values. Otherwise, it corresponds
to an approximation which is still valuable to check
the relationship of data in the right tail.
From expressions (1) and (2), it comes clear that if
both the point-in-time and the right tail distributions
are intended to be used within first order reliability
methods (FORM), the associated Rosenblatt transfor-
mation (Rosenblatt 1952) becomes:
Φ(z)=FPT (x)if x≤xlim
Φ(z)=pPT
lim +FEV(x)−pEV
lim
1−pEV
lim (1 −pPT
lim)if x>x
lim,(3)
where zis an standard independent normal random
variable and Φ(·)is its corresponding cumulative dis-
tribution function.
3 IEC61400-3 STANDARDS FOR OFF-SHORE
WIND TURBINE DESIGN
To show the importance of considering both the
point-in-time and right tail distributions in engineer-
ing design, herein we present an example from the
IEC61400-3 Standards, which establishes the set of
design requirements made to ensure that off-shore
wind turbines are appropriately engineered against
damage from hazards. This code divide external ma-
rine conditions related to agents into normal and ex-
treme categories.
In particular, and for the case of waves, it pro-
poses the consideration of severe sea states, which
shall be considered in combination with normal wind
conditions for calculation of the ultimate loading of
an offshore wind turbine during power production.
The model should associate a severe sea state with
each wind speed in the range corresponding to power
production. The significant wave height for each se-
vere sea state shall be determined by extrapolation of
appropriate site-specific metocean data such that the
combination of the significant wave height and the
wind speed has a recurrence period of 50 years. For
all wind speeds, the unconditional extreme significant
wave height with a recurrence period of 50 years may
be used as a conservative value for Hs.
It is recommended by this guide to extrapolate
metocean data using the so-called Inverse First Or-
der Reliability Method (IFORM). This method pro-
duces an environmental contour defining, in a cer-
tain sense, 50-year recurrence period combinations of
1 2 5 10 50 250
0
10
20
30
40
50
60
70
80
Return Period T(years)
V(m/s)
0 5 10 15 20 25 30 35 40
0
0.02
0.04
0.06
0.08
0.1
0.12
Dens ity
Point-in-time data
Point-in-time fit: gev
Pareto fit re-scaled
V(m/s)
Point-in-time fit: gev
Point-in-time data
Figure 4: Histogram and fitted probability density function of
wind speed V, and graphical representation in terms of equiv-
alent return periods using the method proposed by M´ınguez,
Guanche, & M´endez (2012).
mean wind speeds, v, and significant wave heights,
Hs. A common way to construct this transformation
is to apply the so called Rosenblatt transformation as
follows:
Φ(z1)=FV(v)
Φ(z2)=FHs|V(Hs),(4)
where FV(v)is the marginal distribution of mean
wind speed, and FHs|V(Hs)is the conditional distri-
bution of significant wave heights for given values of
the mean wind speed. Using First Order Reliability
Methods (Hasofer & Lind 1974, Ditlevsen & Madsen
1996) the points satisfying the equation z2
1+z2
2=β2,
is transformed into a curve in the v−Hsplane, which
constitutes the environmental contour. βis the relia-
bility index, whose required value is obtained from
the following equation:
Φ(β)=1−1
N,(5)
where Nis the number of independent sea states in
50 years.
In our particular case, and according to the IEC
61400-3, for a sea state of 1 hour N=50×365 ×24
and the required reliability index from expression (5)
is equal to β≈4.6.
H(m)
s
0
5
10
0
5
10
0
5
10
0
5
10
0
5
10
0
5
10
0
5
10
0
5
10
1 10 50 500
0
5
10
Return Period T
(
years
)
1 1050 500
Return Period T
(
years
)
1 10 50 500
Return Period T
(
years
)
Point-in-time data
Point-in-time fit: gev
Pareto fit re-scaled
<V<
01 <V<
12 <V<
23
<V<
34 <V<
45 <V<
56
<V<
67 <V<
78 <V<
89
<V<
910 <V<
10 11 <V<
11 12
<V<
12 13 <V<
13 14 <V<
14 15
<V<
15 16 <V<
16 17 <V<
17 18
<V<
18 19 <V<
29 20 <V<
20 21
<V<
21 22 <V<
22 23 <V<
23 24
<V<
24 25 <V<
25 26 <V
26
Density
02468
0
0.2
0.4
0.6
0.8
Density
0
0.2
0.4
0.6
Density
0
0.2
0.4
0.6
0246802468 02468 02468
H(m)
s
02468 02468 02468 02468
<V<
01<V<
12 <V<
23<V<
34 <V<
45<V<
56<V<
67<V<
78<V<
89
<V<
910
<V<
10 11 <V<
11 12 <V<
12 13 <V<
13 14 <V<
14 15 <V<
15 16 <V<
16 17 <V<
17 18
<V<
18 19 <V<
29 20 <V<
20 21 <V<
21 22 <V<
22 23 <V<
23 24 <V<
24 25 <V<
25 26 <V
26
Figure 5: Histograms and fitted probability density functions of significant wave height Hsfor given values of the wind speed V.
4 APPLICATION OF THE EVALUATION OF
50-YEAR RECURRENCE PERIOD
ENVIRONMENTAL CONTOURS
Let consider a specific location in the Northern Span-
ish coast as possible candidate for an off-shore wind
farm. We have at our disposal two times series of
hourly significant wave heights and hourly mean wind
speeds at 10 meters height. Both data sets come from
reanalysis data bases GOW (Reguero et al. (2012)),
DOW (Camus et al. (2013)) and SeaWind (Men´endez,
Garc´ıa-D´ıez, Fita, Fern´andez, M´endez, & M. 2013),
respectively, calibrated using instrumental data (see
Espejo et al. (2011), M´ınguez et al. (2011), M´ınguez
et al. (2012)).
First of all, marginal and conditional distributions
of Hsand Vgiven in expression (4) must be defined.
The best diagnostic fit for the wind speed data corre-
sponds to the Generalized Extreme Value (GEV) dis-
tribution. The maximum likelihood estimates are ˆμ=
6.0019,ˆσ=3.5812 and ˆ
ψ=0.0236 for the location,
scale and shape parameters, respectively. The his-
togram and fitted probability density functions shown
in Figure 4 (upper panel) apparently present good fit-
ting diagnostics, however, if we analyze in detail the
right tail in terms of equivalent return periods (lower
panel of Figure 4), the GEV distribution does not
appropriately reproduce extreme winds (gray dashed
line) with respect to data. This problem is solved by
fitting the Pareto distribution above threshold Vlim =
20 m/s. The maximum likelihood estimates for the
Pareto distribution are ˆ
θ=3.1004 and ˆ=−0.1467
for the scale and shape parameters, respectively. Note
that the tail behavior is completely different for the
GEV and Pareto fits, while the one related to GEV
defines a heavy tail (Frechet), the one associated with
Pareto exhibits a bounded tail (Weibull).
The conditional distribution of significant wave
height for given valuesof the wind speed has been fit-
ted using 27 different GEV fits. Each data set Ωi;i=
1,...,27 is conformed choosing Hsi-values so that
their corresponding wind speeds Vihold the follow-
ing conditions: j−1≤Vi<j;∀j=1,...,26 and
Vi>j−1; j= 27. Figure 5 (upper panel) shows the
histograms and fitted probability density functions for
each significant wave height data set. Note that they
present good fitting diagnostics for the bulkof data.
However, if we take a closer look at the right tail of
the distributions (Figure 5, lower panel), it is clear that
the GEV distribution does not appropriately repro-
duce extreme significant wave heights (gray dashed
line) with respect to data for medium-low values of
wind speed. Analogously to the previous case, this
problem is solved by fitting the Pareto distribution
above the threshold associated with the 95% per-
centile. Lower panel of Figure 5 shows how the Pareto
fit reproduces the right tail of the distribution. Note
that the GEV distribution is not valid for significant
wave height values associated with wind speeds be-
ˆp1(×10−5)ˆp2ˆp3ˆp4
μ(V)−26.150 0.01603 −0.11791.191
σ(V)−7.240 0.003438 −0.0044290.4573
ψ(V) 6.429−0.003046 0.020690.05389
θ(V) 4.362 −0.002161 0.03037 0.4693
(V) 3.545−0.001666 0.01482 −0.1192
Hlim
s(V)−22.120 0.01277 −0.01653 2.568
Table 1: Optimal parameter estimates of the regression models
given in (6) and (7).
Figure 6: 50-year environmental contour plot according to IEC
61400-3 standards, without specific tail fitting.
low 16m/s. Above this threshold value, the GEV may
be considered appropriate although the use Pareto fit
on the tail is more convenient.
To get an smooth transition for the GEV parameters
of the probability density function Hs|V, the location,
scale and shape parameters are fitted to a third order
polynomial:
μ(V)=pμ
1V3+pμ
2V2+pμ
3V+pμ
4
σ(V)=pσ
1V3+pσ
2V2+pσ
3V+pσ
4
ψ(V)=pψ
1V3+pψ
2V2+pψ
3V+pψ
4.
(6)
Analogously, Pareto distribution parameters (scale
and shape) and threshold Hlim
sare also smoothed as
follows:
θ(V)=pθ
1V3+pθ
2V2+pθ
3V+pθ
4
(V)=p
1V3+p
2V2+p
3V+p
4
Hlim
s(V)=pHs
1V3+pHs
2V2+pHs
3V+pHs
4.
(7)
Maximum likelihood estimates of these parameters
are given in Table 1.
The 50-year environmental contour for a 1-hour
sea state duration, using expressions (4) and (6), and
only considering the GEV fittings related to both the
wind speed and significant wave height, are shown
in Figure 6. Contours are associated with equivalent
return periods of 100 hours, 1, 5, 50, 100 and 500
years, respectively. 50-year environmental contour is
in black. This result confirms that the 50-year envi-
ronmental contour overestimate in excess significant
wave heights for wind speeds lower than ≈20 m/s,
which is precisely the interval where the GEV fit does
not appropriately reproduce the tail of the distribution.
In contrast, for wind speeds above ≈20 m/s results
are in accordance with data.
Figure 7: 50-year environmental contour plot according to IEC
61400-3 standards, but including specific tail fitting.
Alternatively, if we could calculate the 50-year en-
vironmental contour using the following Rosenblatt
transformation:
Φ(z1)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
FPT
V(V)if V≤Vlim,
pPT
V,lim +FEV
V(V)−pEV
V,lim
1−pEV
V,lim (1 −pPT
V,lim)
if V>V
lim,
Φ(z2)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
FPT
Hs|V(Hs)if Hs≤Hs,lim,
pPT
Hs,lim +FEV
Hs|V(Hs)−pEV
Hs,lim
1−pEV
Hs,lim (1 −pPT
Hs,lim)
if Hs>H
s,lim.
(8)
In this particular case, we made distinction be-
tween the point-in-time distribution, that represents
the probabilistic behavior of all data range but the
right tail, and the extreme-value distribution, that
characterizes the right tail above the selected thresh-
old. Note that the point-in-time distributions FPT
V
and FPT
Hs|Vcorrespond, respectively, to FV(v)and
FHs|V(Hs)from expression (4). The new 50-year en-
vironmental contour for a 1-hour sea state duration
improving the probability density functions at the
right tails is shown in Figure 7. The improvement in
the region of interest, i.e. the one associated with high
values of the significant wave height and wind speeds
in the range corresponding to power production, is
significant. Note that with previous approach signif-
icant wave heightsrelated to the 50-year environmen-
tal contour for low wind speed values are above 10
meters, even higher than those related to high wind
speeds, which is physically unlikely.
The IEC 61400-3 standards also recommend to use
for all wind speeds of the unconditional extreme sig-
nificant wave height with a recurrence period of 50
years as a conservativevalue for Hs. Note that accord-
ing to the fitting shown in Figure 8, this conservative
value is equal to 9.6483 m (white circle marker spec-
ifier). Since this value is above the 50-year environ-
mental contour (see Figure 7), it is the one designer
must take if the second analysis is performed. In con-
trast, in case using the first analysis, designer would
use significant wave heights considerable above this
threshold for low wind speeds, leading to an exces-
sive conservationist.
0
5
10
15
20
25
30
data
fit logn
Pareto fit re-scaled
1 10 50 250
Return Period T
(
years
)
012345678910
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Hs(m)
Density
Hsdata
Point-in-time fit: logn
Hs(m)
Figure 8: Histogram and fitted probability density function of
unconditional significant wave height Hs, and graphical repre-
sentation in terms of equivalent return periods using the method
proposed by M´ınguez, Guanche, & M´endez (2012).
These results clearly demonstrated the importance
of considering both the point-in-time and right-tail
distributions.
It is important to clarify that in both cases, the
left tail of the distributions is reproduced inappro-
priately. Contour plot intercepts negative wind speed
and significant wave height values, which is impossi-
ble. However, this does not distort the relevant results
from the engineering perspective, i.e. those in the up-
per tail. Alternatively, an specific distribution for the
left tail could be used instead.
5 CONCLUSIONS
This paper reveals the importance of considering
both the point-in-time and right-tail distributions from
the engineering perspective, to appropriately repro-
duce the probability density function of environmen-
tal variables all over their range. In addition, it pro-
poses a practical guide to check whether this fact may
be relevant or not for specific cases, and how to work
with both distributions at the same time. To further
reinforce our arguments, a practical example about
how to construct environmental contours for the def-
inition of design requirements for offshore wind tur-
bines (IEC 61400-3) is given, emphasizing possible
problems which may lead to unsafe or excessively
conservatism designs.
Nevertheless, further work is required to improve
the methodology in terms of:
1. Which is the appropriate threshold value for the
definition of the right-tail distribution?
2. How to transform equivalent return periods into
real return periods? This issue is related to the
dependence assumption of extremes, and could
be solved introducing the extremal index con-
cept.
ACKNOWLEDGEMENTS
This work was partially funded by projects “AM-
VAR” (CTM2010-15009), “GRACCIE” (CSD2007-
00067, CONSOLIDER-INGENIO 2010), “IMAR21”
(BIA2011-2890) and “PLVMA” (TRA2011-28900)
from the Spanish Ministry MICINN, “MARUCA”
(E17/08) from the Spanish Ministry MF and “C3E”
(200800050084091) from the Spanish Ministry
MAMRM. The support of the EU FP7 Theseus “In-
novative technologies for safer European coasts in a
changing climate”, contract ENV.2009-1, n. 244104,
is also gratefully acknowledged. Y. Guanche is in-
debted to the Spanish Ministry of Science and Inno-
vation for the funding provided in the FPI Program
(BES-2009-027228). R. M´ınguez is also indebted to
the Spanish Ministry MICINN for the funding pro-
vided within the “Ramon y Cajal” program.
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