Extreme Wind Speeds: Overview

There are two primary models used in practice for obtaining extreme wind data from a series of wind measurements (these models are commonly used for other types of extreme data as well).

These two methods are referred to as the epochal method and the peaks over thresholds method. We will give a brief discussion of each of these.

Unlike the epochal method, the number of extreme values will not be fixed. In our example, we could have months with no extreme values and we could have other months with multiple extreme values. The number of extreme values will depend on the specific threshold chosen.

This section focuses primarily on extreme wind speeds data. Data sets are provided for both non-hurricane and hurricane prone regions.

1. Hurricane Wind Speeds

2. Standardized Non-Hurricane Wind Speeds in the Conterminous United States

3. An Automated Surface Observing System (ASOS) site list is available for about 180 primary sites listed on http://www.ncdc.noaa.gov/oa/ncdc.html. To access the ASOS site list, once you are at the NCDC web site (courtesy of William Brown of the National Climatic Data Center):
   1. Click on "Start here" along the left margin.
   2. Click on "Inventories" at the bottom right.
   3. Click on the fourth link, "Inventories/Surface lists".
   4. Click on the second link, "Surface Inventories/Lists".
   5. The ASOS site list is the first link.

   Further information on anemometer elevation history is available.

   Note 2024/04: The above web site no longer seems to be available. See anemometer elevation history for an alternative site.

Several additional archival data sets are available.

• Software for extracting data from ASOS data sets
• Extreme wind speed maps
• General purpose software for analyzing univariate extreme value data
• potMax for estimating the peak value of a stationary process based on a peaks over threshold approach
• Special purpose archival Fortran and Matlab programs

When analyzing univariate sets of data consisting of extreme winds, the following tasks typically need to be performed.

• Graph the data
• Determine an appropriate distributional model for the data
• Estimate the parameters of the distribution
• Assess the goodness of fit of the fitted distributional model
• Use the model to estimate quantities of interest

• A run sequence plot of the data is useful for showing time dependent patterns in the data. Examples of time dependent patterns would include strong autocorrelation in the data, non-constant mean, non-constant variance, or notable outliers in the data. If time dependent patterns are present, these issues should be addressed before attempting to fit a distributional model. For extreme value analysis, it can be helpful to draw reference lines at certain threshold values.

• Graphs showing the distributional shape can be useful. The most common types of distributional graphs are histograms and kernel density plots.

• Extreme Value Type I (Gumbel)
• Extreme Value Type II (Frechet)
• Weibull and reverse Weibull
• Generalized Pareto
• Generalized Extreme Value

• The Lieblein BLUE method for Extreme Value Type I (Gumbel) Estimation

• The method of moments. There are refinements to the method of moments such as the method of modified moments (see Cohen and Whitten (1988), "Parameter Estimation in Reliability and Life Span Models," Marcel Dekker, p. 31 and pp. 341-344) and L moments (see Hosking and Wallis (1997), "Regional Frequency Analysis," Cambridge University Press), that can improve the performance of standard moment estimates. Of the distributions listed above, modified moments estimates are most applicable to the Weibull distribution while L moments are applicable to Weibull, Frechet, generalized Pareto, and generalized extreme value distributions.

• The method of maximum likelihood. Maximum likelihood estimates typically have excellent statistical properties. However, they are known to be problematic for several of the extreme value distributions (in particular, the generalized Pareto and generalized extreme value distributions).

• The probability plot correlation coefficient (PPCC)/probability plot method.

  The PPCC/probability plot method works well for the five extreme value distributions considered here. In some cases, using the Kolmogorov-Smirnov plot or Anderson-Darling plot may improve upon the PPCC plot.

• The method of elemental percentiles (see Castillo, Hadi, Balakrishnan, and Sarabia (2005), "Extreme Value and Related Models with Applications in Engineering and Science", Wiley) has been successfully used for generalized Pareto and generalized extreme value distributions. The estimates based on elemental percentiles for these distributions tend to be close to the estimates from the PPCC method.

• For a given distribution, there may be other specialized approaches for estimating the parameters. For the generalized Pareto distribution, the DeHaan and conditional mean exceedance (CME) methods sometimes work well in practice.

One issue in developing distributional models for extreme winds data is that we typically want a distributional model for the extreme points (i.e., the points above a given threshold) of the data rather than the full data set.

• The Kolgmogorov-Smirnov (KS) goodness of fit test can be applied to ungrouped data. The KS test is based on comparing the maximum distance between the empirical cumulative distribution of the data set to the expected cumulative distribution from a theoretical distribution.

• The Anderson-Darling goodness of fit test is a refinement of the Kolmogorov-Smirnov test. The Anderson-Darling test is generally considered a more powerful test than the KS test and is in particular more sensitive to differences in the tails of the distribution.

  There are several other variants of the KS test in addition to the Anderson-Darling test. The Cramer von Mises test is one example.

• The chi-square goodness of fit test can be used for grouped data. If the ungrouped data is available, the KS and Anderson-Darling tests are generally preferred to the chi-square test.

• The probability plot provides a graphical assessment of goodness of fit.

• You can generate a histogram or a kernel density plot with the fitted distribution overlaid.

• Estimate specific quantiles of the distribution

  Quantiles are estimated from the percent point function (also known as the inverse cumulative distribution function).

• Estimate return intervals and wind speeds corresponding to a given return interval

  The return interval (or mean recurrence 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

  \( \frac{1} {1 - F(x)} \)

  with \( F(x) \) denoting the cumulative distribution function.

  More often, we would like to compute the wind speed that corresponds to a given mean return interval. The solution to this is given by solving the above equation for x

  \( X_{R} = G(1 - \frac{1}{R}) \)

  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 set consisting of single yearly maxima. If \( \lambda \) is the mean number of threshold crossings of the extreme speed record per year, the formula is

  \( X_{R} = G(1 - \frac{1}{\lambda R}) \)

  See Simiu and Scanlan for a more complete discussion of mean recurrence intervals.

Software is available for performing most of the graphing and distributional modeling tasks described above. For example,

• Dataplot is a freely available general purpose statistical and data analysis program that supports most of the graphics and distributional modeling capabilities described above.

• R is a general purpose software environment for statistical computing and graphics. R is widely used in the statistical community and can perform most of the graphics and distributional modeling capabilities described above. Specifically, the Pintar method for the Peaks over Threshold Poisson Process analysis was implemented in R.

  In particular, the following site documents R packages useful for extreme value analysis.

• We provide an example of modeling data with the Gumbel distribution using Excel.

Note that the above list is not exhaustive. Many commercial statistical/mathematical/spreadsheet programs can be used to analyze extreme value data. The software described above is intended to provide an example of how these analyses can be performed and is not meant to imply that these software programs are the only ones or the best ones for extreme value analysis.

• Fortran-based program for analyzing the hurricane wind speed data sets using the De Haan method.

• Fortran-based program for analyzing the daily maximum wind speed data sets using the De Haan method.

• Fortran-based programs for estimation of along-wind response.

• Matlab programs for the estimation of peaks from time series.