Once the analysis name has been entered, the user can begin defining the analysis. The Peaks Over Threshold Analysis editor has three tabs. The tabs are labeled General, Filtering, and Tabular Results. The first tab contains general settings for performing the Peaks Over Threshold Analysis. These settings include:

• Annualization Method
• Distribution Fitting Method
• Year Specification
• Log Transform
• Output Frequency Ordinates
• Time Window Modification
• Output Labeling
• Confidence Limits and Expected Probability Options

Annualization Method

This option lets the user choose which annualization method will be used. Annualization refers to the conversion of partial duration probabilities to annual exceedance probabilities (AEP). Design quantiles of flow, precipitation, and other hydrologic variables are expressed in terms of AEP. However, the probabilities obtained from the Generalized Pareto distribution are conditional exceedances. The non-exceedance probability for the Generalized Pareto distribution describe the probabilities that a value is exceeded, given that the value already exceeds the threshold:

$//<![CDATA[ \begin{array}{l}F(x) = Pr(X \le x | x > u) = 1 - [1 - \frac{\kappa}{\alpha}(x - u)] ^ \frac{1}{\kappa}\end{array} //]]>$

where $//<![CDATA[ \begin{array}{l}u\end{array} //]]>$ is the location parameter (threshold), $//<![CDATA[ \begin{array}{l}\alpha\end{array} //]]>$ is the scale parameter, and $//<![CDATA[ \begin{array}{l}\kappa\end{array} //]]>$ is the shape parameter.

• 1% of floods that exceed 1,250 cfs are greater than 13,400 cfs
• Not 100-year flow
  

Two annualization methods are available: 

• Empirical: Langbein (1949)
• Analytical: Madsen et al. (1997)

The Empirical: Langbein (1949) and Analytical: Madsen et al. (1997) annualization methods produce the same computed curve for quantiles greater than the threshold value. The Empirical: Langbein (1949) method cannot extrapolate to quantiles less than the threshold value while the Analytical: Madsen et al. (1997) method can. Additional information on selection of an annualization method can be found here.

Empirical: Langbein (1949)

The relationship between empirical flood expectancies in the partial duration series and the probability of the corresponding flood as an annual series is given by (Langbein, 1949):

$//<![CDATA[ \begin{array}{l}p = exp[-\lambda * (1 - p_P_D_S)]\end{array} //]]>$

where p is annual non-exceedance probability, p_PDS is the partial duration non-exceendace probability, and $//<![CDATA[ \begin{array}{l}\lambda\end{array} //]]>$ is the rate of flood events ($//<![CDATA[ \begin{array}{l}\lambda\end{array} //]]>$ = m/N where m is the number of events in the partial duration series and N is the number of years).

The Langbein method cannot extrapolate to quantiles less than the Generalized Pareto threshold value.

Analytical: Madsen et al. (1997)

The relationships between the Generalized Pareto distribution parameters location $//<![CDATA[ \begin{array}{l}\xi\end{array} //]]>$, scale $//<![CDATA[ \begin{array}{l}\alpha\end{array} //]]>$, and shape $//<![CDATA[ \begin{array}{l}\kappa\end{array} //]]>$ and Generalized Extreme Value distribution parameters location $//<![CDATA[ \begin{array}{l}\xi *\end{array} //]]>$, scale $//<![CDATA[ \begin{array}{l}\alpha *\end{array} //]]>$, and shape $//<![CDATA[ \begin{array}{l}\kappa *\end{array} //]]>$ are (Madsen et al., 1997):

$//<![CDATA[ \begin{array}{l}\xi* = \xi + \frac {\alpha}{\kappa} (1 - \lambda ^ - ^ \kappa)\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}\alpha * = \alpha \lambda ^ - ^ \kappa\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}\kappa * = \kappa\end{array} //]]>$

A tutorial illustrating the process of converting partial duration probabilities to annual exceedance probabilities (AEP) using both empirical and analytical methods is available here: Peaks Over Threshold Analysis: Annualization and Partial Duration Series using HEC-SSP and Excel.

Distribution Fitting Method

Three distribution fitting method choices are available for use: Product Moments (PM), L-Moments (LM), and Maximum Likelihood Estimation (MLE). The default distribution fitting method is L-Moments. 

As noted by Vogel et al. (2024), heavy-tailed distributions can have infinite sample moments (and moment generating functions) when using the method of product moments. For example, the variance (or coefficient of variation), coefficient of skewness, and coefficient of kurtosis of Generalized Pareto and Generalized Extreme Value distributions are infinite for shape parameters larger than 1/2, 1/3, and 1/4, respectively. As a result, if one uses the method of product moments when fitting a Generalized Pareto distribution, the PM-estimated shape parameter will always be smaller than 1/3 even though real-world phenomena have heavier tails (shape parameter > 1/3).

On the other hand, L-moments are finite for all distributions that have finite means and are therefore preferred over product moments for heavy-tailed distributions.

“Many of the features of heavy tailed phenomena would render our traditional statistical tools useless at best, dangerous at worst.”
(Cooke and Nieboer, 2011)

Additional information on the Generalized Pareto and Generalized Extreme Value distributions can be found in the Distribution Fitting and Parameter Estimation section of the User's Manual.

Year Specification

The Year Specification panel allows the user to define the beginning and ending date for the analysis year. The year specification is only used to determine the counts for the Event Count Histogram. There are three options contained in the Year Specification panel. If Water Year is selected, the program uses a starting date of October 1 and an ending date of September 30. If Calendar Year is selected, the program uses a starting date of January 1 and an ending date of December 31. The Other option lets the user define the starting date. The default is Water Year.

Log Transform

There are two options contained within the Log Transform panel: Use Log Transform and Do not use Log Transform. If the user selects Use Log Transform then the logarithms (base 10) of the data will be taken before any computations. If the user selects Do not use Log Transform, then the frequency analysis will be performed on the untransformed data values. The default setting is Use Log Transform.

Time Window Modification

This option allows the user to define a time window used for the analysis. The default is to use all of the data contained in the selected data set. The user can enter either a start date for the analysis, and end date, or both a start and end date. If a start and/or end date are used, they must be dates that are encompassed within the data stored in the selected data set. The date range for the selected data set is shown in the Editor just above the Start Date field.

Output Frequency Ordinates

This option allows the user to change or add to the frequency ordinates for which the resulting frequency curve and confidence limits are computed, as shown below. The default values listed in percent chance exceedance are 0.2, 0.5, 1, 2, 5, 10, 20, 50, 80, 90, 95, and 99. Check the box next to Use Values from Table below to change or add additional values. Once this box is checked, the user can add/remove rows and edit the frequency values. To add or remove a row from the table, select the row(s), place the mouse over the highlighted row(s), and click the right mouse button. The shortcut menu contains options to Insert Row(s), Append a Row, and Delete Row(s). The program will use the default values, even if they are not contained in the table, when the Use Values from Table below option is not checked. Finally, all values in the table must be between 0 and 100. Note that these values have no impact on the computed frequency curve, but rather only the values of the curve that are reported.

Output Labeling

This option allows the user to change the default labels for data contained in the output tables and plots. The user can change both the name of the data as well as the units of the data. The output labeling does not result in the conversion of data from one unit system to another; it only affects what is displayed on table headings and the axes of plots.

Confidence Limits and Expected Probability Options

Clicking the Confidence Limits and Expected Probability Options button will launch an editor that contains several options pertinent to the parametric bootstrap routine used to estimate confidence limits and expected probability curves for each distribution | fitting method combination.  Three bootstrap confidence limit (and expected probability) computation types are available for use: Standard, Bias Corrected, and Bias Corrected and Accelerated.  The Standard method (also known and the Percentile Bootstrap method) computes confidence limits and an expected probability curve using the following algorithm (Efron, 1982): 

1. Using the initially fitted model produced by the chosen fitting method | distribution, extract a sample of size equal to the Equivalent Record Length.
2. Using the same fitting method, fit the same distribution to the sample to produce a new fitted model.
3. Randomly sample a single value from the new fitted model.
4. Save values corresponding to flow/stage/etc at each of the Output Frequency Ordinates specified within the Plot Options window.
5. Repeat 10000 times.
6. Sort the values and build a representative histogram to store the values.
   1. This histogram can be defined using linear, log, or user defined values.
   2. By default, linear values will be used.

Confidence limits and expected probability curves produced using the Standard method are considered 1st order accurate.  The Bias Corrected method uses a bias-correction factor to correct for bias in the skewness of the estimates obtained through the Standard method (Efron and Tibshirani, 1993).  The Bias Corrected and Accelerated bootstrapping method improves upon the Bias Corrected method by introducing an acceleration constant that uses jackknife sampling, or sampling without replacement, to estimate parameters.  Confidence limits and expected probability curves produced using either the Bias Corrected or Bias Corrected and Accelerated methods are considered 2nd order accurate (McCullough and Vinod, 1998).  The Confidence Limits and Expected Probability Options window is shown in the figure below.