chapter - Atmospheric and Oceanic Science

Statistical analysis of extreme events in a non-stationary context
In the above analysis, the test for trend was made after concluding that the
GPD could be simplified to the exponential distribution. Strictly, it would be better
to test for trend in the GPD itself; this is illustrated by Coles (2001, page 119) in an
analysis of daily rainfall data to which a GPD is fitted, with a linear trend in logscale
parameter σ of the form.
σ(t) = exp (α+βt)
In the examples given above, the data were considered as annual extremes, but
extension to the case of monthly data is straightforward as long as the monthly
extremes can be taken to be uncorrelated. For example, monthly maximum rainfall
intensities of one-hour duration could be modeled as a GEV distribution with timevariant
parameters. One such model is
µ(t)= β0 + β1 cos(2πt/12) + β2 sin(2πt/12)
in which the sine and cosine terms allow for the possibility that monthly maximum
one-hour rainfalls might vary seasonally throughout the year. Further harmonic
terms, such as cos(4πt/12), sin(4πt/12), might need to be included, if the
simplest model µ(t)= β0 + β1 cos(2πt/12) + β2 sin(2πt/12) does not fit well (although
the inclusion of the two additional harmonics would require the estimation of two
additional parameters β3, β4). If it is required to test whether a linear time-trend is
superimposed upon the simplest harmonic model, exploration of models incorporating
this trend would begin from the starting point
µ(t)=β0 + β1 cos(2πt/12) + β2 sin(2πt/12) + αt
Also, just as in the case of the dispersion parameter σ(t), an exponential was
introduced to ensure that the parameter had non-negative values; functions other
than the identity function can be used for the modeling of µ(t). Thus, in his analysis
of rainfall extremes at 187 stations over the United States, Smith (2001) used a GEV
model in which the parameters σ, µ, ξ were expressed in terms of time as follows:
where µ0, σ0, ξ0 are constants, and
vt = β1t +
µt = µ0 e v1 , σt = σ0 e v1 , ξt = ξ0
p
∑ {β2p cos(ω p) + β2p+1 sin(ω p)}
p=1
Σ
The general form is h(µ(t)) = any function in which parameters occur linearly,
with h(.) any known function; similar relationships can be used for the dispersion
parameter σ(t) and the shape parameter ξ(t). Perhaps fortunately, however, the
shape parameter appears to stay fairly constant (note that it was constant in Smith's
model, given above). Also, in the two examples given above which used rainfall
data from Porto Alegre in Brazil and Ceres in Argentina, it can be seen that the two
values of ξ were also similar: namely 0.2765 ± 0.3223 at Porto Alegre in Brazil, and
0.2638 ± 0.3344 at Ceres in Argentina.
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