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BRABSON AND PALUTIKOF
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Tests of the Generalized Pareto Distribution for Predicting Extreme Wind Speeds
B. B. BRABSON
Department of Physics, Indiana University, Bloomington, Indiana
J. P. PALUTIKOF
Climatic Research Unit, University of East Anglia, Norwich, United Kingdom
(Manuscript received 27 June 1998, in final form 21 October 1999)
ABSTRACT
Extreme wind speed predictions are often based on statistical analysis of site measurements of annual maxima,
using one of the Generalized Extreme Value (GEV) distributions. An alternative method applies one of the
Generalized Pareto Distributions (GPD) to all measurements over a chosen threshold (peaks over threshold).
This method increases the number of measurements included in the analysis, and correspondingly reduces the
statistical uncertainty of quantile variances, but raises other important questions about, for example, event
independence and the choice of threshold. Here an empirical study of the influence of event independence and
threshold choice is carried out by performing a GPD analysis of gust speed maxima from five island sites in
the north of Scotland. The expected invariance of the GPD shape parameter with choice of threshold is utilized
to look for changes of characteristic wind speed behavior with threshold. The impact of decadal variability in
wind on GEV and GPD extreme wind speed predictions is also examined, and these predictions are compared
with those from the simpler Gumbel and exponential forms.
1. Introduction
a. The need
Safe design of structures at exposed locations (bridges, radio masts, wind turbines, etc.) requires knowledge
of maximum expected wind gust speeds. The design
criteria for such structures are often stated in terms of
quantile values. For example, a structure only able to
withstand the 50-yr quantile x 50 would have an unacceptably high predicted failure rate of 2% yr21 .
A time series of gust speed measurements can be used
to extract both the parameters of the statistical distribution of the extreme gust speeds and the corresponding
quantiles. For example, the Generalized Extreme Value
(GEV) distribution has been used to quantify the behavior of measured annual maximum (AM) gust speeds
(Davison 1984; Smith 1989; Stedinger et al. 1993). As
the standard errors for both distribution parameters and
quantiles are proportional to 1/Ï n, where n is the number of measurements used, a longer time series gives
smaller standard errors, the desired result. Unfortunately, long gust speed time series are not always available
at the site of interest. Ten years of data are regarded as
Corresponding author address: B. B. Brabson, Department of
Physics, Indiana University, Bloomington, IN 47405.
E-mail: [email protected]
q 2000 American Meteorological Society
the absolute minimum for a GEV analysis based on
annual maxima (Cook 1985).
To overcome the problem of a time series of inadequate length, several alternatives to a GEV analysis of
annual maxima have been developed. Among these are
analyses of the parent distribution directly (Gomes and
Vickery 1977; Milford 1987), the use of statistical simulation techniques (Kirchoff et al. 1989; Kaminsky et
al. 1991; Dukes and Palutikof 1995), r-largest events
analysis (Weissman 1978; Smith 1986), the Method of
Independent Storms (Cook 1982; Harris, 1999), and the
Peaks over Threshold (POT) technique (Hosking and
Wallis 1987; Abild et al. 1992; Simiu and Heckert 1996)
where all measurements above a chosen threshold, j,
are included. The Generalized Pareto Distribution
(GPD), introduced by Pickands (1975) and explored by
several authors including Hosking and Wallis (1987), is
often the distribution of choice because of its special
relationship to GEV, as explained in part b of this section. Using a POT analysis with GPD, a number of
authors have carried out extreme value analyses based
on as few as 10 yr of data (e.g., Coles and Walshaw
1994; Abild et al. 1992).
In this paper using 13 yr of hourly gust speed data
from Sumburgh, Shetland, and some 40 yr of daily maximum gust speed data from each of four additional island
sites in northern Scotland, we examine the following
questions.
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JOURNAL OF APPLIED METEOROLOGY
R With the greater number of gust speed measurements
above threshold in a POT selection, more time-correlated events (events from a single storm, for example) are potentially included in the analysis. In section 2 we describe an empirical method, based on the
Poisson distribution of independent events above a
sufficient threshold, to examine and quantify this clustering. Using this method, we measure the number of
such correlated events as a function of choice of
threshold and event separation, and examine the influence of these events on the extreme speed quantile
estimates x T . An additional measure of this event clustering is provided by an autocorrelation analysis.
R When performing a GPD analysis on POT selected
data, the choice of threshold is important. Only at
sufficiently high thresholds does the GPD asymptotic
behavior approach GEV. Using the long time series
data with clustering removed, we are able to explore
the GPD threshold dependence across a range from
values well below GPD to GEV convergence to gust
speeds of 40 m s21 . This long range allows us to
examine the convergence process itself.
R A characteristic of the GPD distribution that makes
it suitable for extreme value analysis is termed the
‘‘threshold stability property’’ (Hosking and Wallis
1987; Dargahi-Nougari 1989). If the random variable
X is GPD distributed, then the conditional distribution
of (X 2 j), where X $ j, is also a GPD with the same
shape parameter k. That is, for a single underlying
mechanism following a GPD distribution, k should
not change with choice of threshold. However, a mixture of underlying climatic components with significantly different gust speed distribution shapes and intensities could result in a change in shape parameter
with threshold. In section 3 we investigate the evidence for the presence of multiple storm mechanisms
by examining the shape parameter as a function of j.
R For short time series the preferred use of GPD over
GEV is based on the assumption that the quantile
estimates will be improved by increasing the sample
size. In section 4 we investigate the validity of this
assumption.
The study of event independence and the GEV/GPD
comparisons were done using an hourly time series of
maximum 3-s gust speeds from 1972–84, and annual
maxima for the overlapping period from 1972–96, both
from the Sumburgh Airport in the Shetland Islands.
These data are likely to be simpler than at other onshore sites. The site has relatively few obstacles to wind,
and is surrounded by sea on three sides. The wind speed
diurnal cycle is weak. We ignore the effects of obstacles,
roughness, and terrain. To give a flavor for the Sumburgh data set, Fig. 1 shows a histogram of the hourly
gust speed maxima, and Fig. 2 the time series during
two high wind speed episodes.
An investigation of GPD threshold dependence and
the corresponding evidence for a mixed climate requires
VOLUME 39
FIG. 1. The logarithm of the histogram of 1972–84 Sumburgh
gusts.
longer time series with greater numbers of events at
large gust speeds. Thus, the analysis was performed
using gust speed maxima from Sumburgh and four additional 37- to 39-yr daily maximum time series from
the island sites of Benbecula and Stornoway Airport
(Western Islands), Kirkwall (Orkney), and Lerwick
(Shetland).
b. GPD extreme gust speed calculations
The Sumburgh distribution of 3-s maximum hourly
gust speeds, shown in Fig. 1, illustrates the Weibull-like
distribution typical of wind speeds. A GPD extreme gust
analysis of these data focuses on the high-end tail, by
selecting only those independent events above a threshold value. A GPD distribution of Type I has an exponential tail and a shape parameter k of zero. For positive
values of k, Type III, the tail falls faster than exponential
and unlike the exponential is finite in extent. Conversely, a negative shape parameter gives the Type II GPD
a faster falling slope near threshold and an infinitely
long, thicker tail. As explained by Hosking and Wallis
(1987), for Z 5 max(0, X1 , . . . , X N ) where X i are iid
according to GPD, Z is essentially GEV distributed with
corresponding types (I, Gumbel, k 5 0; II, Frechet, k
, 0; III, reversed Weibull, k . 0). We investigate this
correspondence further in section 4 of this paper.
Traditionally, the distribution of extreme gust and
wind speed data for temperate latitudes is assumed to
be Type I (e.g., Ross 1987; Gusella 1991; Abild et al.
1992), although Walshaw (1994) disputes this assumption. Lechner et al. (1992) found that of 100 wind records in the United States, 61 showed a Type III limiting
form, 36 were of Type I, and 3 were of Type II. The
Type II distribution is seldom found, and might be indicative of a mixed wind series (Gomes and Vickery
1978). No assumptions about distribution type are made
here.
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BRABSON AND PALUTIKOF
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FIG. 2. The gust speed (m s21 ) time line during two high-wind episodes at Sumburgh.
The GPD has a cumulative distribution function (cdf )
F(x; a, k) 5 1 2 [1 2 (k/ a)(x 2 j)]1/k ,
(1)
where j # x # j 1 a/ k for k . 0, and j # x , ` for
k # 0. For the special case when k 5 0,
F(x; a) 5 1 2 exp[2(x 2 j)/a],
(2)
where j # x # `. The derivative of the cdf in Eq. (1)
gives the probability density function (pdf ),
f (x; a, k) 5
[
]
1
k(x 2 j )
12
a
a
(12k)/k
.
(3)
To the extent that gust speed exceedances follow a
GPD distribution, we can estimate the GPD quantiles.
With the definition of crossing rate, l, as the expected
number of peaks above threshold per year, the expression for the number of exceedances in t yr is
l x 5 lt(1 2 F).
(4)
Taking (1 2 F) from Eq. (1), setting l x to unity, and
t to T, that is, one exceedance in T yr, we can solve for
the quantile, x T ,
xGPD
5 j 1 a/ k[1 2 (lT)2k ].
T
(5)
The problem of estimating quantiles then devolves to
estimating the GPD parameters a and k for a given
threshold, j. In this study we use two techniques for
parameter estimation, the Probability Weighted Moments (PWM) method, (Hosking and Wallis 1987; Abild
et al. 1992; Stedinger et al. 1993), and the Maximum
Likelihood (ML) method (Davison 1984; Smith 1984;
Hosking and Wallis 1987; Davison and Smith 1990).
The PWM estimators have been shown to perform
well when k , 0, particularly when k ø 20.2, while
ML estimators have higher efficiency when k is near 0
and outperform PWM for k . 0 (Hosking and Wallis
1987). Therefore, both are presented in this analysis. As
will be shown, the two methods can produce quite different results.
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2. The question of statistical independence
It is a condition of extreme value analysis that the
selected extremes must be independent (Poisson distributed). Related to this condition, it is important to
understand the quantitative impact of nonindependent
extremes on the calculations of extreme quantiles. Annual maximum wind speeds chosen from each wind year
(July–June) are virtually always statistically independent. However, when several data points are taken from
each year, as in a POT analysis, there may well be
several clustered hourly maximum gust speeds above
threshold from a single storm (see Fig. 2). Such events
are unlikely to be statistically independent. Here, we
pose the question of the effect of using nonindependent,
correlated extremes in a POT analysis.
Various strategies are invoked to remove dependent
events before proceeding with a statistical analysis. A
simple strategy is to require a minimum time separation
or ‘‘deadtime’’ between selected events. Working with
European wind climates, Cook (1985) and Gusella
(1991) use a deadtime of 48 h, whereas Coles and Walshaw (1994) extend this to 60 h.
In a second strategy, Walshaw (1994) uses a modification of the mean residual life (MRL) plot to establish
that a dataset of extremes consists of truly independent
events. The MRL plot is simply x 2 j, the sample mean
of the events above threshold minus the threshold, plotted against the threshold (Davison and Smith 1990). His
modification of MRL he calls the reclustered excess plot
and he uses the linear behavior of this plot to identify
acceptable values of the threshold where extreme events
can be deemed independent.
Alternatively, Smith and Weissman (1994) discuss the
concept of an extremal index, a measure of the degree
of clustering of extreme events in a time series. The
cluster size is taken to be the number of observations
of extreme events that occur within a chosen storm
length or ‘‘block size.’’ The extremal index is the reciprocal of the mean cluster size, lies between 0 and 1,
and can be thought of as the fraction of independent
events in a selected sample. They give two methods for
determining this extremal index, and describe how it
relates the parent distribution to the asymptotic GEV
distribution.
In this section we take an empirical and complementary approach of examining the different timescales of
clustering of extremes in our sample data, deciding
which to regard as dependent, and measuring the fraction of such dependent events in the sample. We then
calculate the desired extreme quantile wind speeds x T
for that sample, and adjust both extreme event threshold
and separation in time until stable x T are achieved. In
this manner, an acceptable level of dependent event contamination is determined.
We begin our analysis by choosing as our parent distribution the hourly maximum gust speeds (of 3 s duration) recorded at Sumburgh, Shetland, from 1972–84.
FIG. 3. Number of events above threshold for the Sumburgh maximum gust speed data, 1972–84.
The event separation is then varied in nine steps from
short (3-h) to longer (96-h) intervals. A sample of extreme wind speed events is chosen by selecting an observation above a threshold j only when this observation
is not followed by a higher wind speed in a time period
of duration d. Figure 3 shows the number of events
above threshold as a function of these two parameters.
The number of events included in the data sample rises
rapidly as the deadtime between events is allowed to
decrease below about 18 h.
a. The distribution of time intervals between storms
Associated with the rapid rise in number of exceedances for low thresholds and short deadtimes, we expect
an increase in the number of statistically dependent
(clustered) events. A simple consistency check for statistical independence can be made. Because such events
should be Poisson distributed in time, the frequency
distribution of the time intervals Dt between events
should demonstrate exponential behavior
f (Dt) 5 le2lDt ,
(6)
where l is the crossing rate defined as the average number of selected events per unit time.
A time interval plot provides a simple way of identifying event correlations at different timescales. For
example, Fig. 4 shows the Dt plot for all events above
a threshold of 24 m s21 and event separation of 3 h.
By examining the Dt plot at different scales, bin
widths of 2 h in Fig. 4a and of 10 h in Fig. 4b, we
discover several interesting time structures in the Sumburgh wind data. This plot is well represented by exponentials with different slopes, each corresponding to
a different Poisson event crossing rate l, and each characteristic of an underlying correlating mechanism (i.e.,
average storm duration). The two exponential lines in
Fig. 4a correspond to crossing rates of one event every
SEPTEMBER 2000
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BRABSON AND PALUTIKOF
4 h (1/l 5 4 h) and one event every 27 h (1/l 5 27
h), and represent the short time interval part of the spectrum quite well. When these same events are shown on
a longer timescale in Fig. 4b, an additional exponential
is needed, corresponding to a crossing rate of an event
every 160 h.
At this point we wish to develop an index giving a
measure of event independence of a set of POT data
and to use this index in identifying appropriate deadtimes and thresholds in our subsequent GPD analysis.
This amounts to counting the fraction of independent
events in a given dataset. Thus, we must decide which
components of our data to identify as independent. A
conservative choice to assure event independence at the
individual storm level is to exclude both of the short
time interval 4-h and 27-h components when counting
independent events. This choice, we will find is consistent with the autocorrelation calculations in part d of
this section and also with the literature on storm lengths
in the United Kingdom, a summary of which is given
in Palutikof et al. (1999). Thus, the area under the 160-h
exponential in Fig. 4b represents independent events,
while the events in the excess at the shorter Dt above
the 160-h exponential are not considered to be independent. This corresponds to our understanding of the
behavior of the atmospheric circulation over the eastern
Atlantic. Families of depressions track from west to east
across the Atlantic, arriving over the United Kingdom
at intervals of around 6 days (Cook 1985).
We define the independent event index « as the ratio
of the area under the 160-h exponential (events without
short-term correlation) to the total area.
«5
independent events
.
total events
(7)
b. Event independence, thresholds, and dead times
We are now in a position to quantitatively identify
proper combinations of dead time d and threshold j,
where the selected events have « near 1.0, free of shortterm correlations. The quantity « and its errors are tabulated in Table 1. At relatively low thresholds of 24 to
26 m s21 , « exceeds 0.85, for example, only for d 5
48 h or longer. At higher thresholds of 34–36 m s21 ,
on the other hand, a dead time of 12 h is sufficient to
ensure « ; 0.9. Dead times less than 12 h do not work.
c. Quantile variation with dead time
We first illustrate the nature of the 100-yr quantile
dependence on d and on «. Figure 5 shows the 100-yr
←
FIG. 4. Semilogarithmic plots of the time interval frequency histogram for a 3-h dead time and 24 m s21 threshold, using (a) 2-h
bins, and (b) 10-h bins. The superimposed exponential lines are explained in the text.
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TABLE 1. Independent event index at Sumburgh for different thresholds (m s 21) and dead times in hours. Standard errors are given in
parentheses.
Threshold
dead time
3
6
12
18
24
36
48
72
96
h
h
h
h
h
h
h
h
h
24 m s21
0.31
0.45
0.61
0.68
0.73
0.79
0.88
1.05
1.10
(0.02)
(0.03)
(0.03)
(0.04)
(0.04)
(0.04)
(0.04)
(0.05)
(0.04)
26 m s21
0.37
0.54
0.73
0.69
0.83
0.81
0.86
0.98
1.07
(0.02)
(0.03)
(0.04)
(0.05)
(0.04)
(0.04)
(0.04)
(0.05)
(0.04)
28 m s21
0.39
0.55
0.68
0.71
0.77
0.81
0.83
0.94
1.01
(0.03)
(0.04)
(0.04)
(0.04)
(0.04)
(0.04)
(0.04)
(0.04)
(0.04)
30 m s21
0.49
0.71
0.88
0.92
0.95
0.90
0.90
1.01
1.03
32 m s21
(0.04)
(0.04)
(0.04)
(0.04)
(0.04)
(0.04)
(0.04)
(0.04)
(0.04)
quantile x100 as a function of d for a threshold of 24 m
s21 on the left and the same data as a function of « on
the right. Note that quantiles from the ML method have
less variability with dead time than the PWM method.
As the dead time increases, the 100-yr quantile x100
0.49
0.71
0.86
0.90
0.91
0.89
0.85
0.98
0.98
(0.04)
(0.05)
(0.05)
(0.05)
(0.04)
(0.05)
(0.05)
(0.04)
(0.04)
34 m s21
0.54
0.72
0.91
0.88
0.92
0.91
0.96
0.82
1.08
(0.05)
(0.06)
(0.05)
(0.05)
(0.05)
(0.05)
(0.05)
(0.14)
(0.04)
36 m s21
0.68
0.76
0.91
0.85
0.90
0.88
0.91
0.75
1.02
(0.08)
(0.07)
(0.06)
(0.06)
(0.06)
(0.06)
(0.06)
(0.16)
(0.03)
38 m s21
0.59
0.81
0.95
0.95
0.95
0.66
0.66
0.68
0.95
(0.09)
(0.09)
(0.05)
(0.05)
(0.05)
(0.20)
(0.20)
(0.21)
(0.05)
first drops and then stabilizes for dead times above 36
h. As a function of «, the quantiles stabilize above « of
0.8. Thus, « is a useful parameter for identifying quantile
stability and indicates that a clustered event contamination of as much as 20% has relatively little impact
on x T . Table 1 also indicates that at thresholds as high
as 30 m s21 , a dead time of 24 h is sufficient to remove
clustered events. This interval is short in comparison
with the usual choice of 48–60 h.
To understand the higher x T at small «, we compared
the probability density distributions for event samples
with high and low «. The clustered events occur predominantly at relatively low gust speeds, just above
threshold. When included in a GPD analysis, they force
a less positive shape parameter, corresponding to higher
gust speed quantiles, thus explaining the artificial rise
in x100 observed in Fig. 5 at low dead time.
A second negative consequence of an event sample
with low « is an underestimation of the quantile standard
errors. These errors are proportional to 1/Ïnindependent
(Hosking and Wallis 1987; Abild et al. 1992), and will
be underestimated if nindependent is inflated by the addition
of clustered events.
In summary, we have identified two reasons to avoid
heavy contamination from nonindependent events in
this kind of statistical analysis. First, these events distort
the shape of the gust speed distribution producing anomalously high gust speed quantiles, and second, the quantile standard error, proportional to 1/Ï n, will be underestimated by an inflated n. However, provided a sufficiently high threshold is selected, shorter dead times
than are normally found in the literature can be utilized.
d. Event independence, autocorrelation, and longer
time series
FIG. 5. The 100-yr return quantiles x100 for gust speeds above 24
m s21 as (a) a function of dead time and (b) independent event index
«. Example PWM and ML quantile errors are included.
Up to this point the exponential behavior of the time
interval frequency and the corresponding stabilization
of the gust speed quantile x T with increasing event separation d have been used to identify a d sufficient for
Poisson event independence. As an additional check,
we show in Fig. 6 the autoregressive parameter r k (Wilks
1995) for the 24 m s21 threshold data of Fig. 5. The
value of r1 is a measure of correlation of neighboring
SEPTEMBER 2000
BRABSON AND PALUTIKOF
FIG. 6. Autoregressive parameter as a function of lag for 24 m s 21
sample data.
wind events and is seen to move to values less than
0.05 for d of 12 h or greater.
The hourly gust speed maxima from Sumburgh provided the detail necessary for this event independence
study. To confirm the usefulness of the technique, we
also calculated « for the four 37- to 39-yr time series
from the neighboring island sites. For each site and for
thresholds $24 m s21 , separating events by at least one
intervening observation is sufficient to bring « into the
range of 0.9, enough to remove the undesirable effects
of event correlation discussed for Sumburgh above.
Therefore, in discussion of quantile variation with
threshold, and the related discussions of mixed climate
in section 3, we utilize these longer time series with a
minimum of one intervening observation and shall refer
to this minimum separation as a deadtime of one day.
3. Threshold behavior GPD and the question of
mixed climate
Simiu and Heckert (1996) have examined the question of whether the finite-tailed (Reverse Weibull, k .
0) GPD distribution is an appropriate extreme wind
speed model in a POT analysis. Analyzing daily data
from 44 U.S. stations, they answer the question affirmatively by examining the threshold dependence of
deHaan’s tail-length parameter c, the negative of our
shape parameter k. We extend their study by examining
GPD k(j) for the four northern Scottish island sites
where prevailing westerlies are expected to provide the
dominant component of this high wind speed climate.
The high average wind speeds at these sites provide
an extended threshold range over which to examine the
question of GPD to GEV convergence. Upon convergence, the threshold stability property of GPD would
predict a constant k for a simple wind climate described
by a single GPD distribution. At a higher level of com-
1633
FIG. 7. (a) A histogram of Benbecula gust speed data, j $ 16 m
s21 , d $ 1 day, (b) k(j), (c) a(j), (d) x100 (j).
plexity with a mixed climate of two GPD distributions
with different values of k, k(j) should shift from one
value of k to another. As we will discuss, maximum
wind speeds at these four island sites exhibit a more
complex k behavior, where k changes continuously with
threshold. We also examine a related variable, the mean
residual life (MRL), another quantity sensitive to a
mixed climate (Davison and Smith 1990) when plotted
against threshold.
a. The threshold behavior of GPD
When GPD fits are done as a function of threshold,
they provide additional information about a time series
not available to a single GEV fit. The longer time series
from Benbecula, Kirkwall, Lerwick, and Stornoway
Airport give us a greater gust speed threshold range than
does the 13-yr time series from Sumburgh alone. Because these four time series have similar characteristics,
we begin by looking at one (Benbecula) in some detail.
Figure 7a is simply a histogram of maximum daily
gust speeds separated by a deadtime of one day. It shows
clearly the roll-over below 20 m s21 and the approximately exponential high gust speed tail. Superimposed
on this histogram are the properly normalized pdfs from
GPD maximum likelihood fits for thresholds of 20 m
s21 , 32 m s21 , and 40 m s21 . The 20 m s21 threshold
has a fit k of 10.18 and is seen to have a faster-falling
tail than the more exponential 32 m s21 fit with k of
0.03. For still lower thresholds, 16–20 m s21, the GPD
fit is forced by the turn-over at the left-hand (low gust
speed) end to still more positive values of k, and, it
turns out, to poorer fits. The 40 m s21 fit is spurious,
illustrating the behavior of a GPD when fit to a very
small dataset. At these very high thresholds there is little
constraint from low gust speeds and GPD creates a highly curved (1k) fit consistent with essentially no events
beyond those actually observed.
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Figure 7d shows the 100-yr return quantile x100 as a
function of j. In the region from 20 to 36 m s21 , x100
is roughly constant even as the individual GPD parameters are rapidly varying. In this same threshold interval
the maximum quantile speed for positive k, y limit 5 j
1 a/ k, is relatively constant because it depends only
on the ratio, a/ k. More variability is seen in the probability weighted moments estimators than in the maximum likelihood estimators, but both show the same
trend.
Effectively, we have a number of different estimates
of the extreme gust speed quantiles. Though these estimates are not independent, they do derive from different portions of the original dataset and tell us something about those portions. A conservative estimate of
the quantile x100 is given by the maximum observed
quantile over the selected threshold range. For Benbecula, such an estimate of the 100-yr quantile speed
would come from the 36 m s21 threshold point and give
a quantile estimate of x100 5 52.7 m s21 6 4.6 m s21 .
Thus, a GPD parameterization as a function of threshold
provides an estimation procedure for extreme quantiles
that includes the possibility of a number of contributing
mechanisms.
If the gust speed extremes are well described by a
single GPD distribution, then k(j) should either be constant and equal to kGEV , the GEV shape parameter, or
it should approach this value asymptotically. Alternatively, the five sites in northern Scotland might show
evidence for a second component of wind climate. We
pursue this mixed climate hypothesis in sections 3b and
3c.
b. Search for mixed climate using GPD
The question of uncovering underlying dynamical
mechanisms by observing the character of maximum
gust speed distributions has been addressed by several
authors (e.g., Gomes and Vickery 1978; Coles and Walshaw 1994; Hannah et al. 1995). For example, Gomes
and Vickery merged statistical models from different
climates (hurricanes, prevailing westerlies, and convective storms) to reproduce observed gust speed maximum
distributions, and corresponding quantiles. In this analysis, we choose to use the properties of the GPD distribution itself to search for underlying mechanisms in
these Scottish gust speed data. Although k should be
constant with threshold for simple one-GPD samples,
two-GPD wind climate data might well show shifts in
k(j).
We now examine the actual GPD maximum likelihood fits for the four long time series gust data. Figure
8 shows k(j) and a(j) and for each of these datasets,
both k and a demonstrate a decrease from 16 m s21 to
36 m s21 , with an approximately linear decrease above
20 m s21 . These data are neither constant nor does the
shape parameter k demonstrate an asymptotic approach
to the GEV shape parameter kGEV .
FIG. 8. GPD maximum likelihood parameters, (a) k(j) and (b) a(j)
for POT data with one-day separation from Benbecula, Kirkwall,
Lerwick, and Stornoway Airport. The sites are displaced vertically
by even units to facilitate their comparison.
For these four datasets the observed extreme POT
gust speed distributions do not seem to be consistent
with either a simple GPD or combinations of two GPD
distributions. The POT tail is more complex. For each
of the four datasets the highly positive shape parameters
at low threshold evolve to near zero (or even negative)
k at large thresholds. Although extreme value theory
tells us that the distribution of annual maxima taken
from a simple GPD POT distribution converges to a
GEV distribution with the same k, it does not dictate
that real wind distributions have GPD tails. We have
found examples where they do not.
Summarizing this section, we find evidence to support
the view that POT data taken from areas with high average wind speeds in northern Scotland are not well
represented by GPD with the threshold stability property
expected at sufficiently high thresholds. Nevertheless,
we find that a GPD POT analysis provides useful ex-
SEPTEMBER 2000
1635
BRABSON AND PALUTIKOF
FIG. 9. MRL plots for the four 37- to 39-yr series with one-day
event separation. Here the sites are displaced vertically by single
units to facilitate their comparison.
treme quantile predictions x T as a function of threshold,
information not available from an annual maximum
GEV analysis. Conservatively, the extreme quantiles
can be taken as the highest of the GPD determinations
with threshold.
c. Search for mixed climate using mean residual life
In an MRL plot, x 2 j, the sample mean of the events
above threshold minus the threshold, is plotted against
the threshold. For an exponential distribution, for example, the MRL has zero slope and intercept a. More
generally, Davison and Smith (1990) point out that for
a pure GPD distribution, the MRL will have a slope of
2k/(1 1 k), and an intercept of a/(1 1 k).
MRL plots for Benbecula, Kirkwall, Lerwick, and
Stornoway are shown in Fig. 9 for an event separation
of 1 day. At each site the MRL slope is negative at a
threshold of 20 m s21 and approaches zero near 36 m
s21 . Though the variability increases as the number of
data points above threshold decreases, there is evidence
near 36 m s21 for a slope change to near zero. Above
approximately 43 m s21 the MRL drops rapidly toward
zero as we reach the last events in the distribution tail.
From this information alone, we would predict a positive
and decreasing k in the region 16–36 m s21 followed
by a short interval of exponential tail (k 5 0) above 36
m s21 . Thus, we find additional evidence for the long
gradual decrease in k seen in Fig. 8 above. We do not
find an MRL with constant slope, expected if k were
constant with threshold.
4. The question of quantile estimation in a
variable climate
Extreme value analyses of wind data usually assume
the stationarity of the parent distribution, though gen-
uine long-term linear trends have been identified in such
analyses (Tawn 1988; Palutikof et al. 1999). Palutikof
et al. (1987), in a study of temporal and spatial variability of wind, found evidence for decadal variability
of average wind speeds in the United Kingdom. This
immediately raises the question of the impact of such
variation on extreme quantile predictions.
First, we begin with a comparison of GEV with GPD
for the 13 yr (1972–84) of hourly 3-s gust speed maxima
originally available at Sumburgh. Second, we include
25 yr (1972–96) of annual maximum hourly 3-s gust
speeds from Sumburgh in an attempt to validate the 13yr results from GEV and GPD. Both the generalized
distributions (GEV and GPD) and the simpler distributions (Gumbel and exponential) are included in these
comparisons. Last, we carry out a similar comparison
for the four longer time series sites of Benbecula, Kirkwall, Lerwick, and Stornoway Airport, here again investigating the influence of decadal variation.
a. The GEV distribution
Beginning with a parent distribution whose cumulative distribution function (cdf ) is F(x), we sample this
distribution n times, and take the maximum value of the
n samples. This maximum value has a cumulative distribution function of its own of simply F n (x). This relationship leads to the simplicity in extreme value theory, pointed out by Fisher and Tippett (1928), that for
sufficiently long sequences of independent and identically distributed (iid) random variables, the maxima of
these sequences can be fit to one of three limiting distributions. The cdfs of these three limiting distributions
are summarized in the GEV distribution,
5
[
F(x) 5 exp 2 1 2
]6
k(x 2 j ) 1/k
,
a
(8)
where ` , x # j 1 a/ k for k . 0, j 1 a/ k # x ,
` for k , 0, and 2` , x , ` for k 5 0. The three
classes of GEV are designated by the value of the shape
parameter, k. Gumbel or Fisher–Tippett Type I has k of
0, while Frechet or Fisher–Tippett Type II has k , 0,
and Reverse Weibull or Fisher–Tippett Type III has k
. 0. Similar to the GPD distribution, the scale parameter
a determines the width of the distribution. The GEV
distribution mode is designated here by j. For Type I,
Gumbel, the above equation takes a simpler form:
[
1
F(x) 5 exp 2exp 2
]
2
x2j
.
a
(9)
As an example we consider a sample from the Sumburgh
set of hourly gust speed maxima. The annual maxima
of the hourly gust speeds will almost certainly be independent events and should then be GEV distributed.
1636
JOURNAL OF APPLIED METEOROLOGY
FIG. 10. Sumburgh 1972–84 gust speed quantiles as a function of
return period for GEV, Gumbel, and BLUE-corrected Gumbel.
VOLUME 39
FIG. 11. Sumburgh 1972–84 gust speed quantiles (j 5 32 m s21
and d 5 24 h) as a function of return period for GPD and exponential.
Dotted lines reproduce Fig. 10.
b. The application of GEV to Sumburgh
Like the GPD distributions discussed in sections 2
and 3, the GEV shape parameter strongly influences the
nature of the upper tail of the distribution. The tail is
exponential for Gumbel (k 5 0), thicker than exponential for Frechet (k , 0), and finite in extent for Reverse
Weibull (k . 0). Thus, the GEV extreme gust speed
quantiles depend sensitively on k (Simiu and Heckert
1996).
As we did for GPD, we calculate unbiased estimates
of the GEV parameters, k, a, and j both from the probability weighted moments (b 0 , b1 , and b 2 in Stedinger
et al. 1993) and separately using maximum likelihood.
Here, these strategies agree quite well. We calculate the
quantiles of the GEV distribution from
xT 5 j 1
a
{1 2 [2ln(F )] k},
k
(10)
where F is the cumulative probability, given in terms
of the return period by F 5 (T 2 1)/T.
Figure 10 shows the PWM GEV quantile gust speed,
x T , as a function of the return period, T (dashed curve).
On the same plot we place the 13 annual gust speed
maxima for 1972–84 (plus signs). As sample estimates
of the cumulative probability for GPD given by Eq. (1),
the plotting positions are taken as F i 5 1 2 (i 2 0.44)/
(n 1 0.12), where x1 $ · · · $ x n are the ordered annual
maxima (Hosking et al. 1985; Linacre 1992; Hosking
and Wallis 1995). The PWM GEV quantile plot does
indeed follow the annual maxima from which it is derived. However, the shape parameter has the peculiar
value of 10.61 corresponding to a strongly truncated
tail, with a predicted maximum gust speed of 45.8 m
s21 , an unlikely physical result, given that wind speeds
as high as 44.0 m s21 are already observed during the
13-yr recording period.
For purposes of comparison we superimpose the
PWM Gumbel quantile plot for these same data. The
Gumbel distribution (Stedinger et al. 1993) with its
shape parameter k 5 0, has an exponential high-end tail
and is shown as the solid curve in Fig. 10. The Gumbel
quantile function is simply
x T 5 j 2 a ln[2ln(F)].
(11)
The Gumbel distribution follows these data less well.
Part of the difficulty, as explained in Cook (1985, p.
111), lies in the asymmetry of the Gumbel distribution
and the consequent unequal confidence among the estimates of the ranked annual maxima. The largest of 13
annual maxima is also Gumbel distributed, and on average, will have a value higher than the mode of this
(skewed) distribution. A method devised by Lieblein
(1974) defines a corrected set of weights called ‘‘best
linear unbiased estimators’’ (BLUE) to remove this bias.
As Cook explains, the higher ranked data points are
weighted less. We plot the BLUE corrected Gumbel plot
(dash-dotted curve in Fig. 10), as well. Though this
BLUE Gumbel fit gives higher quantile values for large
return periods, as expected, it differs relatively little
from the uncorrected Gumbel. By eye, neither fits the
13 yr of annual maxima well.
c. A comparison of GEV with GPD
Figure 11 shows the quantiles of Fig. 10 (dotted
curves) with the added quantile predictions from the
peaks-over-threshold GPD ML fit (solid line), described
in section 1b. The GPD parameters for this fit are k 5
0.262, and a 5 4.69 m s21 . With these parameters, we
calculate the GPD quantiles as a function of return period using Eq. (5). For this comparison we select the
GPD POT analysis with a threshold of 32 m s21 and
dead time of 24 h, comfortably located in the region of
independent event index « near one, where the quantile
SEPTEMBER 2000
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BRABSON AND PALUTIKOF
TABLE 2. Sumburgh: a comparison of 100-yr quantiles from GEV, Gumbel, GPD, and exponential analyses. GPD values in the table are
done with a threshold of 32 m s21 and dead time of 24 h.
Parameters
Sumburgh analysis type and period
k
GEV, PWM Param, 13 yrs of AM
Gumbel, PWM Param, 13 yr of AM
GEV, PWM Param, 25 yr of AM
GEV, ML Param, 25 yr of AM
Gumbel, PWM Param, 25 yr of AM
10.61
GPD, ML Param, 13 yr of POT
EXP, PWM Param, 13 yr of POT
GPD, PWM Param, 13 yr of POT
10.36
10.005
10.008
10.26
speeds are stable. These choices yield 86 events above
threshold.
We plot these 86 data points using the relationship
lT 5 1/(1 2 F) from Eq. (4), and the plotting position
F i 5 1 2 (i 2 0.44)/(n 1 0.12) as above. Note that
GEV
xGPD
100 (solid line) lies close to x100 (lower dotted line).
To complete these comparisons from the 13-yr dataset, we add the exponential GPD quantile prediction
(dashed line in Fig. 11). This lies well above both the
GEV and GPD predictions, in the region of the Gumbel
predictions (upper dotted lines).
Critical to the usefulness of these maximum gust
speed predictions are their associated standard errors
[see Hosking et al. (1985) and Abild et al. (1992) for
calculation procedures]. Table 2 contains both the GEV
and GPD 100-yr quantiles and standard errors, based
on the 13 yr from 1972 to 1984. We first draw attention
to the 1.8 standard deviation separation between xGEV
100 5
45.3 6 2.0 m s21 and xGumbel
5 51.7 6 3.3 m s21 . The
100
GEV analysis fits the annual maxima significantly better
21
than Gumbel. The ML xGPD
agrees
100 5 45.4 6 2.4 m s
GEV
well with the x100 but is approximately 1.5 standard
deviations away from the xGumbel
and 2.1 standard de100
viations away from xexp.
5
53.8
6
3.3 m s21 . Based on
100
FIG. 12. Sumburgh annual maximum 3-s gust speeds for 1972–96.
a
j
x100
(m s21 )
SE
(m s21 )
4.06
2.96
4.00
3.99
3.97
39.1
38.1
38.7
38.7
38.6
45.3
51.7
56.8
56.4
56.9
2.0
3.3
5.6
7.6
3.2
5.39
3.35
4.92
32.0
32.0
32.0
45.4
53.8
47.3
2.4
3.3
3.1
the 13-yr dataset alone, one would conclude that the
generalized distributions (GEV and GPD) give a better
fit, and disagree with the more highly constrained (Gumbel and exponential) distributions with shape parameter,
k 5 0.
d. A longer dataset at Sumburgh
In an attempt to validate this analysis of the 1972–
84 Sumburgh data, we now examine the annual maxima
beyond 1984, taken from the Monthly Weather Report
produced by the U.K. Meteorological Office, and shown
in Fig. 12. Using 25 yr of annual maxima from 1972
to 1996, we look again at both GEV and Gumbel predictions of quantiles for Sumburgh. Figure 13 shows
these quantile functions, superimposed on the 25 annual
maxima. During this extended period four very large
annual maxima occurred in the period, 1991–94 (see
Sumburgh in Fig. 12). These dramatically affect the
GEV quantile predictions, raising xGEV
100 from 45.3 6 2.0
m s21 to 56.8 6 5.6 m s21 , a change well outside the
standard errors calculated on the basis of the 13-yr sample.
However, the Gumbel predictions are less affected by
these new data. The xGumbel
5 51.7 6 3.3 m s21 from
100
the 13-yr sample changes to 56.9 6 3.2 m s21 . The
Gumbel BLUE analysis de-weights the extreme data
points and changes the 100-yr quantile even less, from
xBLUE
5 52.7 m s21 for 13 yr to xBLUE
5 53.0 m s21 for
100
100
25 yr. As expected, we find that the Gumbel and Gumbel
BLUE quantile analyses are less sensitive than the more
flexible GEV quantile analysis.
A key comparison can now be made between a POT
analysis of 13 yr of data (with 86 points) and an AM
analysis of 25 yr (Fig. 14). Though more data points
are used in the GPD analysis, with concomitant smaller
standard errors, the 25-yr GEV and Gumbel analyses
and the 13-yr exponential analysis are in better agreement with the 1972–96 annual maxima. We conclude
that the additional data points from a POT analysis are
not necessarily a replacement for a longer time series.
The 25-yr dataset shows that both of the generalized
models, GEV and GPD, when based on 13 yr of data,
though very good at following the detailed shape of the
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JOURNAL OF APPLIED METEOROLOGY
VOLUME 39
FIG. 14. Annual maxima for the five northern Scotland island sites.
FIG. 13. (a) Quantile functions from the 25-yr 1972–96 period.
Note that Gumbel and GEV are superimposed. (b) A comparison of
the 13-yr GPD and exponential quantiles with the 25-yr GEV and
Gumbel quantiles from (a) (dotted lines).
gust speed tails, fail to predict maximum gust speeds
accurately. The k 5 0 versions of these models have
less flexibility, and in this case, make a more accurate
prediction of high speed winds, even when based on
only 13 yr of data. This, of course, does not detract from
the utilization of the generalized models as a useful
parameterization for mixed climates analysis as described in section 3 of this paper (Abild et al. 1992;
Gomes and Vickery 1978), but does caution against their
use in quantile prediction with datasets as short as 13
yr.
e. Decadal variability and longer time series
In Fig. 14 we plot the annual maxima for the longer
daily records for Benbecula, Kirkwall, Lerwick, and
Stornoway Airport, provided by the British Atmospheric
Data Centre (BADC). At the Sumburgh site we found
that the average annual maximum gust speed during the
13-yr period from 1972 to 1984 was exceeded by each
of the years in the period from 1989 to 1992. This
apparent variability on a decadal timescale is also observed at the other four sites in this study. For example,
the annual maxima at Lerwick, 25 km farther north on
Shetland have enhanced maximum gust speeds in the
1989–92 period and an additional period of high speed
annual maxima in the period from 1962 to 1967. This
decadal variability in wind in the United Kingdom has
been studied by Palutikof et al. (1987), where they state,
‘‘In recent years we find that the years 1962–67 were
dominated by westerly circulation patterns and display
generally higher windspeeds over Britain, whereas from
1968 onward an increased frequency of anticyclonic
days has been linked to lower windspeeds.’’ A number
of authors have suggested that this low-frequency variability at the decadal rate may be linked to the North
Atlantic oscillation (Hurrell 1995).
Using the 13- and 25-yr time series from Sumburgh
we have demonstrated the difficulty in making quantile
exceedance predictions using either a GEV or a GPD
analysis in a wind climate with decadal variations. With
an even longer time series at the neighboring site of
Lerwick, we can now look further into the question of
length of time series needed in GEV and GPD analyses
to provide valid quantile predictions in a wind climate
with this level of variability.
Lerwick has an average annual maximum gust speed
close to that of Sumburgh (39.8 m s21 at Lerwick compared to 40.9 m s21 at Sumburgh). We have carried out
GEV and GPD at 13, 25, and 39 yr in an attempt to
identify the point of quantile stability. The Lerwick 100yr quantiles with errors are given in Table 3 and show
that the large difference between 13 and 25 yr is not
repeated when the analysis is extended to 39 yr, either
for GEV or GPD. For example, the x100 quantiles from
a GPD analysis of 13, 25, and 39 yr are 47.3 6 3.4,
SEPTEMBER 2000
1639
BRABSON AND PALUTIKOF
TABLE 3. Lerwick: a comparison of 100-yr quantiles from GEV, Gumbel, and GPD analyses. The GPD analyses were done with a
minimum separation of 1 day and threshold of 30 m s21 .
Lerwick analysis type and period
GEV, PWM with 13 yr of AM
GEV, PWM with 25 yr of AM
GEV, PWM with 39 yr of AM
GPD, ML with 13 yr of POT
GPD, ML with 25 yr of POT
GPD, ML with 39 yr of POT
Gumbel, PWM with 13 yr of AM
Gumbel, PWM with 25 yr of AM
Gumbel, PWM with 39 yr of AM
k
10.305
10.119
10.058
10.083
10.016
10.057
52.8 6 4.1, and 52.5 6 3.3 yr, respectively. A similar
pattern is seen in Table 3 for the corresponding three
GEV analyses, with similar errors. In both cases we find
evidence that a 25-yr analysis yields stable quantile predictions at return periods of 100 yr.
Summarizing this section, we find that although 13
yr of data are not sufficient to get stable quantile estimates, 25 yr or more do give stable quantiles within
errors for both GEV and GPD analyses.
5. Conclusions
This study examines extreme gust speeds at five island sites in northern Scotland. It focuses on hourly data
from Sumburgh, in Shetland, but uses daily data from
four similarly exposed Scottish sites to substantiate its
conclusions. It has centered on three issues in extreme
value analysis: the importance of event independence,
the existence of mixed climate indicators, and the use
of generalized extreme value distributions. As part of
this study we have examined three methods for the determination of GPD parameters and have examined the
sensitivities of each to choice of threshold, dead time,
and length of the time series. The main conclusions can
be summarized as follows.
R Event independence: Using a simple method to extract
the fraction of independent events in a time series, we
find that correlated or nonindependent events in a gust
speed time series have two specific effects on extreme
gust speed quantile estimates. First, these events distort the shape of the gust speed distribution producing
anomalously high gust speed quantiles, and second,
the quantile standard errors, proportional to 1/Ï n, are
underestimated when n, the total number of events,
is inflated by the addition of nonindependent events.
Also, for high thresholds, a 24-h dead time between
events is sufficient to remove the effects of correlated
events for these data.
R Mixed climate: The analysis of the four relatively long
(37 to 39 yr) daily time series at high average wind
speed sites provides evidence that a single GPD model
or simple combination of GPD components is not able
to describe the actual wind climate at these sites. Categorizing extreme wind data from this region by GEV/
a
(m s21 )
j
(m s21 )
x100
(m s21 )
SE
(m s21 )
4.16
4.39
3.93
3.77
3.79
3.98
3.36
3.97
3.73
38.0
37.7
37.5
30.4
30.4
30.4
37.4
37.5
37.4
48.2
53.2
53.4
47.3
52.8
52.5
52.9
55.7
54.6
2.3
4.0
3.1
3.4
4.1
3.3
3.8
3.2
2.4
GPD type is therefore not easily done because the
result depends on the threshold chosen. This result is
seen in both a GPD analysis and an MRL analysis at
these four sites.
R Generalized distribution: The generalized models,
GEV and GPD, have been brought to bear on 13 yr
of Sumburgh gust speed maxima. Though very good
at following the detailed shape of the 13-yr gust speed
tail, when extrapolated, both fail to predict the actual
maximum gust speeds observed over a longer 25-yr
period. We attribute this failure largely to the apparent
nonstationarity in the wind climate in this region. This
does not detract from the utilization of the generalized
models as a useful parameterization for mixed climate
analysis as described in section 3 of this paper but
does caution against their use as long-term quantile
estimators with datasets as short as 13 yr. Using the
longer time series from neighboring Lerwick, Shetland, we find that in this wind regime with apparent
decadal variation, extreme quantiles have already stabilized for both GEV and GPD analyses of 25 yr.
Acknowledgments. The daily wind speed data for
Benbecula, Kirkwall, Lerwick, and Stornoway Airport
were provided by the BADC. Also, B. B. Brabson wishes to acknowledge the generous hospitality and stimulating scientific environment of both the Climatic Research Unit and the School of Environmental Science
at the University of East Anglia.
REFERENCES
Abild, J., E. Y. Andersen, and D. Rosbjerg, 1992: The climate of
extreme winds at the Great Belt of Denmark. J. Wind Eng. Ind.
Aerodyn., 41–44, 521–532.
Coles, S. G., and D. Walshaw, 1994: Directional modelling of extreme
wind speeds. Appl. Stat., 43, 139–157.
Cook, N. J., 1982: Towards better estimation of extreme winds. J.
Wind Eng. Ind. Aerodyn., 9, 295–323.
, 1985: The Designer’s Guide to Wind Loading of Building Structures. Butterworths, 371 pp.
Dargahi-Nougari, G. R., 1989: New methods for predicting extreme
wind speeds. J. Eng. Mech., 115, 859–866.
Davison, A. C., 1984: Modelling excesses over high thresholds, with
an application. Statistical Extremes and Applications, J. Tiago
de Oliviera, Ed., Reidel, 461–482.
1640
JOURNAL OF APPLIED METEOROLOGY
, and R. L. Smith, 1990: Models for exceedances over high
thresholds. J. Roy. Stat. Soc., B52, 393–442.
Dukes, M. D. G., and J. P. Palutikof, 1995: Estimation of extreme
wind speeds with very long return periods. J. Appl. Meteor., 34,
1950–1961.
Fisher, R. A., and L. H. C. Tippett, 1928: Limiting forms of the
frequency distributions of the largest or smallest members of a
sample. Proc. Cambridge Philos. Soc., 24, 180–190.
Gomes, L., and B. J. Vickery, 1977: On the prediction of extreme
wind speeds from the parent distribution. J. Ind. Aerodyn., 2,
21–36.
, and
, 1978: Extreme wind speeds in mixed wind climates.
J. Ind. Aerodyn., 2, 331–344.
Gumbel, E. J., 1958: Statistics of Extremes. Columbia University
Press, 375 pp.
Gusella, V., 1991: Estimation of extreme winds from short-term records. J. Struct. Eng., 117, 375–390.
Hannah, P., P. Rainbird, J. Palutikof, and K. Shein, 1995: Prediction
of extreme wind speeds at wind energy sites. ETSU Rep. W/11/
00427/REP, 250 pp.
Harris, R. I., 1999: Improvements to the ‘‘Method of Independent
Storms.’’ J. Wind Eng. Ind. Aerodyn., 80, 1–30.
Hosking, J. R., and J. R. Wallis, 1987: Parameter and quantile estimation for the generalized Pareto distribution. Technometrics,
29, 339–349.
, and
, 1995: A comparison of unbiased and plotting-position
estimators of L moments. Water Resour. Res., 31, 2019–2025.
,
, and E. F. Wood, 1985: Estimation of the generalized
extreme-value distribution by the method of probability weighted
moments. Technometrics, 27, 251–261.
Hurrell, J., 1995; Decadal trends in the North Atlantic oscillation:
Regional temperatures and precipitation. Science, 269, 676–679.
Kaminsky, F. C., R. H. Kirchoff, C. Y. Syu, and J. F. Manwell, 1991:
A comparison of alternative approaches for the synthetic generation of wind speed time series. J. Sol. Energy Eng., 113, 280–
289.
Kirchoff, R. H., F. C. Kaminsky, and G. Y. Syu, 1989: Synthesis of
wind speed using a Markov process. Proc. Eighth ASME Wind
Energy Symp., Houston, TX, American Society of Mechanical
Engineers, 17–22.
Lechner, J. A., S. D. Leigh, and E. Simiu, 1992: Recent approaches
VOLUME 39
to extreme value estimation with application to wind speeds.
Part I: The Pickands method. J. Wind Eng. Ind. Aerodyn., 41–
44, 509–519.
Lieblein, J., 1974: Efficient methods of extreme-value methodology.
National Bureau of Standards Report, NBSIR Rep. 74–602.
Linacre, E., 1992: Climate Data and Resources. Routledge Press, 366
pp.
Milford, R. V., 1987: Annual maximum wind speeds from parent
distribution functions. J. Wind Eng. Ind. Aerodyn., 25, 163–178.
Palutikof, J. P., P. M. Kelly, T. D. Davies, and J. A. Halliday, 1987:
Impacts of spatial and temporal windspeed variability on wind
energy output. J. Climate Appl. Meteor., 26, 1124–1133.
, B. B. Brabson, D. H. Lister, and S. J Adcock, 1999: A review
of methods to calculate extreme wind speeds. Meteor. Appl., 6,
119–132.
Pickands, J., 1975: Statistical inference using extreme order statistics
Ann. Stat., 3, 119–131.
Ross, W. H., 1987: A peaks-over-threshold analysis of extreme wind
speeds. Can. J. Stat., 15, 328–335.
Simiu, E., and N. A. Heckert, 1996: Extreme wind distribution tails:
A ‘‘peaks-over-threshold’’ approach. J. Struct. Eng., 122, 539–
547.
Smith, R. L., 1984: Threshold methods for sample extremes. Statistical Extremes and Applications, J. Tiago de Oliviera, Ed., Reidel, 621–638.
, 1986: Extreme value theory based on the r largest annual
events. J. Hydrol., 86, 27–43.
, 1989: Extreme value theory. Handbook of Applicable Mathematics, W. Ledermann, Ed., John Wiley and Sons, 437–472.
, and I. Weissman, 1994: Estimating the extremal index. J. Roy.
Stat. Soc., B56, 515–528.
Stedinger, J. R., R. M. Vogel, and E. Foufoula-Georgiou, 1993: Frequency analysis of extreme events. Handbook of Hydrology, D.
R. Maidment, Ed., McGraw Hill, 18.1–18.66.
Walshaw, D., 1994: Getting the most from your extreme wind data:
A step by step guide. J. Res. Nat. Inst. Stand. Technol., 99, 399–
411.
Weissman, I., 1978: Estimation of parameters and large quantities
based on the k largest observations. J. Amer. Stat. Assoc., 73,
812–815.
Wilks, D. S., 1995: Statistical Methods in the Atmospheric Sciences:
An Introduction. Academic Press, 467 pp.