chapter - Atmospheric and Oceanic Science

Statistical analysis of extreme events in a non-stationary context
shape parameter ξ of the GEV distribution. In the case of the dispersion parameter,
a trend would be explored using σ = exp(α + βt), the exponential being used to
ensure that the dispersion parameter is always non-negative. Also, instead of looking
at trends where the parameters vary in time, it is also possible to use the same
approach to determine whether other kinds of trend exist; for example, it might be
of interest to test whether annual maximum one-hour rainfall is related in some way
to the Southern Oscillation Index (SOI). In this case, SOI would take the place of
the time variable t in the above example.
The method of Maximum Likelihood, used in the above example, underpins
a very large part of estimation and hypothesis-testing procedures in parametric statistics,
and it is not proposed here to re-present the theory that is already fully presented
in many text-books. Briefly, given independent observations x1, x2… xN of a
random variable X with probability distribution fx(x,θ) where θ is a parameter to be
estimated, the likelihood function L(θ, x1, x2… xN) is the product fx(x1,θ) fx(x2,θ)…
fx(xN,θ), and the ML estimate of θ is the value that maximizes this function. The
variance of the estimate of θ is approximately - 1/ (second derivative of logeL with
respect to θ). ML estimates have desirable statistical properties which makes ML
estimation the procedure of choice, when the distribution fx(x,θ) can be specified.
The brief account given here assumes just one parameter θ; extension to the case of
several parameters θ = {θ1, θ2,…} is straightforward.
To determine whether the selected distribution (in this case, the GEV) is
appropriate for the data being analyzed, diagnostic techniques are used which are
also provided by the available software packages. Figure 15.1, known as a Q-Q
plot, shows a plot of the order statistics of the Porto Alegre data (vertical axis)
against the corresponding quantiles (“scores”) for a GEV distribution; if the GEV
provides a good fit to data, the plotted points should lie close to a straight line. The
degree of agreement between the points and a straight line is ascertained by looking
at whether the plotted points lie within the 95% confidence band, which is also
shown in Fig. 15.1. This shows that there is no reason to doubt that the GEV is
appropriate for the data. Since, in the present example, there is no evidence of timetrend
in the data, return periods can be calculated as shown in Fig. 15.2; the return
period, in years, is shown on the horizontal axis, and the hourly rainfall for this
return period is shown on the vertical axis. The 95% confidence limits, also plotted,
show that there is considerable uncertainty in the calculated rainfalls.
15.4. Example: POT procedure applied to fitting daily rainfalls at Ceres,
Argentina
This example uses the 44-year record 1959-2002 of daily rainfall for Ceres,
Argentina, supplied by the National Meteorological Service. The example uses a
threshold of 100mm. Theory (e.g., Coles 2001) shows that the distribution of daily
204