PEAKS OVER THRESHOLD PLOT
Name:
PEAKS OVER THRESHOLD PLOT
Type:
Purpose:
Generates a peaks over threshold plot.
Description:
In univariate extreme value analysis, there are two basic
approaches for extracting the extreme data.
 We find the maximum (or minimum) value in equal
length intervals. For example, we could extract the
maximum wind speed for each year and then develop a
distributional model for these yearly maximums.
This is commonly referred to as the "epochal" method.
 An alternative is to define an overall threshold.
We then extract all points above that threshold and
develop a distributional model for these points. Note
that in this case, the number of points extracted in
each interval is not necessarily equal.
The generalized Pareto distribution provides a useful
distributional model for univariate extreme value data since
it indicates what type of extreme value model is appropriate
(\( \gamma \) denotes the shape parameter):
 \( \gamma \) = 0
This is equivalent to an extreme value type I (Gumbel)
distribution.
 \( \gamma \) > 0
This is equivalent to an extreme value type II (Frechet)
distribution.
 \( \gamma \) < 0
This is equivalent to a reverse Weibull distribution (in
Dataplot, this is a Weibull with SET MINMAX MINIMUM).
The shape parameter for the reverse Weibull is
1/\( \gamma \).
The purpose of the PEAKS OVER THRESHOLD PLOT is to see
how the estimated value of gamma changes as the threshold
is changed. Specifically, the plot is generated as follows:
 Define an initial threshold. In Dataplot, you can
specify either the starting number of points above the
threshold or a particular value for the threshold.
To specify the intial number of points above the
threshold, enter the command
SET PEAKS OVER THRESHOLD INITIAL POINTS
To specify a starting value for the threshold, enter
the command
SET PEAKS OVER THRESHOLD INITIAL THRESHOLD
If neither command is given, Dataplot will start
with a threshold that gives 2.5% of the data set.
If both are specified, the INITIAL THRESHOLD takes
precedence over the INITIAL POINTS.
 For the points above the threshold, estimate the
parameters for the generalized Pareto distribution
(see the Note section below for details on how this
is done).
In addition, calculate a confidence interval for
the shape parameter. For the de Haan and CME methods
(see the Note section below), we compute the standard
deviation of the estimate of gamma. We then use
\( \gamma \pm 2s \gamma \)
as an estimate of the confidence interval. For the PPCC
plot method, the confidence interval is computed using
bootstrapping.
 Decrement the threshold. To specify how much to
decrement the threshold at each interval, enter the
command
SET PEAKS OVER THRESHOLD INCREMENT
The default increment is 1.
 This is repeated for a prespecified number of
iterations. The default number of iterations is 30.
To change the number of iterations, enter the command
SET PEAKS OVER THRESHOLDS ITERATIONS
The plot then consists of three curves:
 The point estimates of gamma.
 The lower confidence limit for gamma.
 The upper confidence limit for gamma.
Each of these is plotted against the number of points above
the threshold. To have the actual threshold plotted on the
horizontal axis, enter the command
SET PEAKS OVER THRESHOLD X AXIS THRESHOLD
To restore the default of the number of points above the
threshold, enter the command
SET PEAKS OVER THRESHOLD X AXIS POINTS
The basic interpretation of this plot is:
When the threshold is high, few points are included so
the variance of gamma is also high (and so the resulting
confidence intervals are wide). As the threshold
decreases (and more points are included), the variance
of gamma decreases with resulting narrower confidence
intervals. However, as the number of points increases,
the bias of the estimate of gamma increases. This will
often be indicated by a downward slope of the graph.
Over intervals where the bias error is small, the
graph will be nearly horizontal. When choosing a
reasonable value of gamma from the graph, it should
be noted that larger estimates of gamma imply a
longer tail and are therefore conservative from a
structural engineering point of view.
In addition to the plot, this command will also generate
the following tables at each iteration.
 The first table contains the threshold, the number
of points above the threshold, and the parameter
estimates.
 In engineering applications, the mean recurrence
interval (this is also referred to as the return
interval or recurrence interval) is of interest.
The return interval of a given wind speed, in years, is
defined as the inverse of the probability that the wind
speed will be exceeded in any one year. It is defined as
with F(x) denoting the cumulative
distribution function. Mean recurrence intervals are
discussed in more detail in Simiu and Scanlon (see
References section below).
More often, we would like to compute the wind speed
that corresponds to a given return interval. The
solution to this is given by solving the above
equation for x.
with G and R denoting the percent point
function and the desired mean recurrence interval,
respectively.
The above formula is for the case of a single yearly maximum.
If \( \lambda \) is the mean number of threshold crossings per
year, the formula is
\( X(R) = G(1  (\frac{1}{\lambda R})) \)
In the Dataplot PEAKS OVER THRESHOLD PLOT command, you
can optionally give a second variable that specifies the
desired mean recurrence intervals. If you specify mean
return intervals, Dataplot will print a table showing
the mean return interval along with the corresponding
wind speed (called XR).
 In wind engineering applications, the load factor is
also of interest. This is commonly computed as
with XMAX denoting the maximum value of the fitted
generalized Pareto distribution (the generalized Pareto
is bounded above if the shape parameter is negative) and
XR_{50} denoting the wind speed
corresponding to a mean recurrence interval of 50 years.
To print the value of the maximum wind speed and the
load factor, enter the command
SET PEAKS OVER THRESHOLD LOAD FACTOR OFF
Since this is specific to extreme wind applications,
it is OFF by default.
Syntax 1:
PEAKS OVER THRESHOLD PLOT <y>
<SUBSET/EXCEPT/FOR qualification>
where <y> is a response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Syntax 2:
PEAKS OVER THRESHOLD PLOT <y> <r>
<SUBSET/EXCEPT/FOR qualification>
where <y> is a response variable;
<r> is a variable containing mean recurrence
intervals;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
PEAKS OVER THRESHOLD PLOT Y
PEAKS OVER THRESHOLD PLOT Y R
PEAKS OVER THRESHOLD PLOT Y SUBSET TAG > 1
Note:
There are a number of methods for estimating the parameters
for the generalized Pareto distribution. To specify the
method, enter the command
SET PEAKS OVER THRESHOLD METHOD <value>
where <value> is one of the following:
DEHAAN

 use the de Haan method (enter HELP DEHAAN
for the details of this method)

CME

 use the conditional mean exceedance method
(enter HELP CME for the details of this method)

PPCC

 use the PPCC (probability plot correlation
coefficient plot) to estimate gamma and a
probability plot to estimate the location and
scale parameters. A 95% confidence interval
for the gamma parameter is obtained via the
bootstrap. For details of these methods, enter
HELP PPCC PLOT, HELP PROBABILITY PLOT, and
HELP DISTRIBUTIONAL BOOTSTRAP).
You can obtain more accurate estimates for
gamma by restricting the range for the PPCC
plot. Enter the commands
LET GAMMA1 = <lower limit>
LET GAMMA2 = <upper limit>
One recommendation is to run the plot with the
default limits and then rerun it with tighter
limits based on the first plot (be sure to keep
them wide enough to accomodate the bootstrap
estimates).

Additional methods will be added in future releases of
Dataplot. The default is DEHAAN.
Note:
Since this command generates a large number of tables, it
is typically desired to save them to file. To do this,
enter the commands
CAPTURE POT.OUT
PEAKS OVER THRESHOLD PLOT Y R
END OF CAPTURE
You can alternatively choose to save the output in HTML,
Latek, or RTF (Rich Text Format) formats. For example,
CAPTURE HTML POT.HTM
PEAKS OVER THRESHOLD PLOT Y R
END OF CAPTURE
CAPTURE LATEX POT.TEK
PEAKS OVER THRESHOLD PLOT Y R
END OF CAPTURE
CAPTURE RTF POT.RTF
PEAKS OVER THRESHOLD PLOT Y R
END OF CAPTURE
The HTML and Latex formats can incorporate the plot as
well. For details, enter
HELP CAPTURE HTML
HELP CAPTURE LATEX
Note:
Dataplot automatically writes the following values to the
file dpst1f.dat (there is one row for each distinct value
of the threshold, the following correspond to the columns
in dpst1f.dat):
 Number of points above the threshold
 The threshold
 The estimate of the shape parameter, gamma
 The estimate of the location parameter
 The estimate of the scale parameter
If mean recurrence intervals have been specified (see
Syntax 2), Dataplot additionally writes the following values
to the
file dpst2f.dat:
 Iteration
 Number of points above the threshold
 The threshold
 The requested mean recurrence interval
 The XR corresponding to the requested mean
recurrence interval
 The load factor corresponding to the mean recurrence
interval
Note that the last column, the load factor, is only printed
if the SET LOAD FACTOR ON command is entered. This load
factor is
A 99.0 is printed for mean recurrence intervals less than
or equal to 50.
The purpose of writing these values to files is to allow
you to perform additional analyses. For example, you may
want to plot the XR corresponding to the various return
intervals. This is demonstrated in the program example
below.
Note:
Some sources reverse the sign in the definition of the
generalized Pareto distribution. For details, enter
the command
If you use the reversed sign definition, then adjust the
role of positive and negative values of gamma in the
discussion above.
Default:
Synonyms:
POT is a synonym for PEAKS OVER THRESHOLD
Related Commands:
DEHAAN

= Compute the estimates for the parameters of the
generalized Pareto distribution using the de Haan
method.

CME

= Compute the estimates for the parameters of the
generalized Pareto distribution using the CME method.

GEPPPF

= Computes the percent point function of the
generalized Pareto distribution.

PPCC PLOT

= Generates a ppcc plot.

PROBABILITY PLOT

= Generates a probability plot.

DISTRIBUTIONAL BOOTSTRAP

= Perform a bootstrap analysis for a univariate
distribution.

CME PLOT

= Generates a conditional mean exceedance plot.

Reference:
E. Simiu, N. A. Heckert, and T. Whalen (April, 1996). "Estimates
of Hurricane Wind Speeds by the 'Peaks Over Threshold' Method",
NIST Technical Note 1416.
E. Simiu and N. A. Heckert (March 1995). "Extreme Wind
Distribution Tails: A 'Peak Over Threshold' Approach",
NIST Building Science Series 174.
Alan Heckert, Emil Simiu, and Tim Whalen (April,1998).
"Estimates of Hurricane Wind Speeds by the 'Peaks Over
Threshold' Approach", Journal of Structural Engineering,
pp. 445449.
J. A. Lechner, E. Simiu, N. A. Heckert (1993). "Assessment of
'peak over threshold' Methods for Estimating Extreme Value
Distribution Tails", Structural Safety, 12, pp. 305314.
E. Simiu and N. A. Heckert (1996). "Extreme Wind Distribution
Tails: A 'Peaks Over Threshold' Approach", Journal of Structural
Engineering, Vol. 122, No. 5, 1996.
Applications:
Implementation Date:
Program:
DIMENSION 40 COLUMNS
SKIP 2
READ PARAMETER MPOST550.DAT URATE
SKIP 3
READ MPOST550.DAT Y1 TO Y17
.
TITLE Peaks Over Threshold Plot (Milepost 550)CR()de Haan Method
TITLE DISPLACEMENT 5
Y1LABEL Gamma
X1LABEL Number of Points Above Threshold
TITLE CASE ASIS
LABEL CASE ASIS
LINE SOLID DOT DOT
LINE THICKNESS 0.2 0.1 0.1
.
SET PEAKS OVER THRESHOLD ITERATIONS 50
SET PEAKS OVER THRESHOLD PERIOD URATE
SET PEAKS OVER THRESHOLD METHOD DEHAAN
.
LET R = DATA 25 50 100 200 500 1000
.
CAPTURE POT.OUT
PEAKS OVER THRESHOLD PLOT Y17 R
END OF CAPTURE
.
SKIP 0
READ DPST2F.DAT ITER NPOINTS THRESH R2 XR
.
TITLE Mean Recurrence Intervals (Milepost 550)
TITLE DISPLACEMENT 2
Y1LABEL XR
LINE SOLID ALL
LINE THICKNESS 0.1 ALL
.
PLOT XR NPOINTS R2
.
CRLF ON
MARGIN 87
MOVE 87 88
TEXT 1,000  yr
TEXT 500
TEXT 200
TEXT 100
TEXT 50
TEXT 25
