Frequency of Extremes

Frequency of Extremes

 Often it is of interest to look at the frequency of extreme event occurences. As the event becomes more rare, the occurence of events approaches a Poisson process, so that the relative frequency of event occurence approaches a Poisson distribution. See appendix for more details.

Fitting data to a Poisson distribution

 The Extremes Toolkit also provides for fitting data to the Poisson distribution, although not in the detail available for the GEV distribution. The Poisson distribution is also useful for data that involves random sums of rare events. For example, a dataset containing the numbers of hurricanes per year and total monetary damage is included with this toolkit named Rsum.R.

Analyze Poisson Distribution.

A window appears for specifying the details of the model, just as in the GEV fit. Without a trend in the mean, only the rate parameter, , is currently estimated; in this case, the MLE for is simply the mean of the data. If a covariate is given, the generalized linear model fit is used from the R function glm (see the help file for glm for more information). Currently, extRemes provides only for fitting data to Poissons with the log link function.
Example: Hurricane Count Data
Load the Extremes Toolkit dataset Rsum.R as per Load Data section and save it (in R) as Rsum. That is, This dataset gives the number of hurricanes per year (from 1925 to 1995) as well as the ENSO state and total monetary damage. More information on these data can be found in Pielke and Landsea (1998) or Katz (2002). A simple fit without a trend in the data is performed in the following way. For these data 1.817, indicating that on average there were nearly two hurricanes per year from 1925 to 1995. A property of the Poisson distribution is that the mean and variance are the same and are equal to the rate parameter, . As per Katz (2002), the estimated variance is shown to be 1.752, which is only slightly less than that of the mean (1.817). The statistic is shown to be 67.49 with associated p-value of 0.563 indicating that there is no significant difference in the mean and variance. Similar to the GEV distribution, it is often of interest to incorporate a covariate into the Poisson distribution. For example, it is of interest with these data to incorporate ENSO state as a covariate.

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Fitting data to a Poisson distribution with a covariate

 The procedure for fitting data to a Poisson with a trend (using the Rsum dataset from above with ENSO state as a covariate) is as follows. EN for this dataset represents the ENSO state (i.e., EN is -1 for La Niña events, 1 for for El Niño events, and 0 otherwise). A plot of the residuals is created if the plot diagnostics checkbutton is engaged. The fitted model is found to be:

log() = 0.575 - 0.25(EN)

For fitting a Poisson regression model to data, a likelihood-ratio statistic is given in the main toolkit dialog, where the ratio is the null model (of no trend in the data) to the model with a trend (in this case, ENSO). Here the addition of ENSO as a covariate is significant at the 5% level (p-value 0.03) indicating that the inclusion of the ENSO term as a covariate is reasonable.



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