Peaks Over Threshold (POT)/Point Process (PP) Approach

Peaks Over Threshold (POT)/Point Process (PP) Approach

 The GPD model from the previous chapter looks at exceedances over a threshold and those values are fit to a generalized Pareto distribution. A more theoretically appealing way to analyze extreme values is to use a point process characterization. This approach is consistent with a Poisson process for the occurrence of exceedances of a high threshold and the GPD for excesses over this threshold. Inferences made from such a characterization can be obtained using other appropriate models from above (see Coles (2001) (b)). However, there are good reasons to consider this approach. Namely, it provides a nice interpretation of extremes that unifies all of the previously discussed models. For example, the parameters associated with the point process model can be converted to those of the GEV parameterization. In fact, the point process approach can be viewed as an indirect way of fitting data to the GEV distribution that makes use of more information about the upper tail of the distribution than does the block maxima approach (Coles (2001) (b)).

Fitting data to a Point Process Model

 Figure 6.1 is not quite as easy to interpret as Figure 5.10 for the GPD because of the fewer thresholds, but it seems that a threshold anywhere in the range of 0.30 to 0.40 inches would be appropriate.


Figure 6.1: Point process model fits for a range of 15 thresholds from 0.2 inches to 0.80 inches for the Fort Collins, C.O. precipitation dataset.


To create the plot in Figure 6.1 do the following. Once a threshold is selected, a point process model can be fitted. Figure 6.2 shows diagnostic plots (probability and quantile plots) for such a fit.


Figure 6.2: Diagnostic plots for Fort Collins, C.O. precipitation (inches) data fit to a point process model.




To fit the Fort Collins precipitation data to a point process model, do the following. MLE's found for this fit are: 1.38 inches (0.043), 0.53 inches (0.037 inches) and 0.21 (0.038) parameterized in terms of the GEV distribution for annual maxima, with negative log-likelihood of about -1359.82.

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Relating the Point Process Model to the Poisson-GP

 The parameters of the point process model can be expressed in terms of the parameters of the GEV distribution or, equivalently through transformations specified in appendix, in terms of the parameters of a Poisson process and of the GPD (i.e., a Poisson-GP model).
Example 1: Fort Collins Precipitation (no covariates)
When fitting the Fort Collins precipitation data to the point process model (using the BFGS optimization method) with a threshold of 0.395 and 365.25 observations per year, the following parameter estimates are obtained.

1.38343
0.53198
0.21199

Parameters from fitting data to the GPD (using the BFGS optimization method) with a threshold of 0.395 and 365.25 observations per year are * 0.3225 and 0.21191--denoting the scale parameter of the GPD by * to distinguish it from the scale parameter of the GEV distribution. Immediately, it can be seen that the value of is very nearly identical to the estimate found for the point process approach. Indeed, the small difference can be attributed to differences in the numerical approximations. The other two parameters require a little more work to see that they correspond. Specifically, because there are 1,061 observations exceeding the threshold of 0.395 inches out of a total of 36,524 observations, the (log) MLE for the Poisson rate parameter is log = log(365.25(1061/36524)) 2.3618 per year. Plugging into Eqs. (B.3) and (B.4) gives

log = log(0.3225)+0.2119(2.3618) -0.63118 exp(-0.6311) 0.53196

= 0.395 - 0.53196/0.2119(10.61-0.2119 - 1) 1.3835


both of which are very close to the respective MLEs of the point process model.
Example 2: Phoenix summer minimum daily temperature
 The Phoenix minimum temperature data included with this toolkit represents a time series of minimum and maximum temperatures (degrees Fahrenheit) for July through August 1948 to 1990 from the U.S. National Weather Service Forecast Office at the Phoenix Sky Harbor Airport. For more information on these data, please see Tarleton and Katz (1995) or Balling et al. (1990). Temperature is a good example of data that may have dependency issues because of the tendency of hot (or cold) days to follow other hot (or cold) days. However, we do not deal with this issue here (see Extremes of Dependent and/or Nonstationary Sequences). For this example, load the Tphap.R dataset and save it (in R) as Tphap. The minimum temperatures (degrees Fahrenheit) are shown in Figure 6.3. Note the increasing trend evident from the superimposed regression fit. Again, we will not consider this trend here, instead we defer this topic to the chapter on Extremes of Dependent and/or Nonstationary Sequences.


Figure 6.3: Scatter plot of minimum temperature (degrees Fahrenheit), with regression line, for the summer months of July through August at Sky Harbor airport in Phoenix, A.Z.


It is of interest with this dataset to look at the minimum temperatures. To do this, we must first transform the data by taking the negative of the MinT variable so that the extreme value distribution theory for maxima can be applied to minima. That is,

-max(-X1,...,-Xn) = min(X1,...,Xn)

This transformation can be easily made using extRemes. For the Phoenix minimum temperature series, the Poisson log-rate parameter for a threshold of -73 degrees (using the negative of minimum temperature, MinT.neg) is log = log(62.(262/2666)) 1.807144 per year, where there are 62 days in each "year" or summer season (covers two months of 31 days each; see appendix) and 262 exceedances out of 2,666 total data points. MLEs (using the BFGS method) from fitting data to the GPD are * 3.91 degrees (0.303 degrees) and -0.25 (0.049), and from fitting data to the point process model: -67.29 degrees (0.323 degrees), 2.51 degrees (0.133 degrees) and -0.25 (0.049). Clearly, the shape parameters of the two models match up. Using Eq. (B.3) the derived scale parameter for the point process model is log 0.92, or 2.51 degrees (the same as that of the point process estimate fitted directly). Using Eq. (B.4) gives -67.29 degrees (also equivalent to the point process estimate fitted directly).


Figure 6.4: Diagnostic plots of GPD fit for Phoenix Sky Harbor airport summer minimum temperature (degrees Fahrenheit) data (Tphap).



Figure 6.5: Diagnostic plots of point process fit for Phoenix Sky Harbor airport summer minimum temperature (degrees Fahrenheit) data (Tphap).


Clearly, the probability and quantile plots (Figure 6.4 and Figure 6.5) are identical, but the curvature in the plots indicates that the assumptions for the point process model may not be strictly valid--although, the plots are not too far from being straight.



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