4.2 The peaks over threshold (POT) approach
4.2.3 Multiple Threshold Method (MTM)
This method was developed by Deidda (2010) to infer the parameters of the GP distribution underlying the exceedances of daily rainfall records over a wide range of thresholds.
Given a set of rainy and non rainy values x at daily or any other fixed time scale is possible to describe the marginal distribution of the process by the following CDF:
F (x) = P r{X ≤ x|X ≥ 0} = (1 − ζ0) + ζ0F0(x) x≥ 0 (4.37)
where ζ0 = P r{X > 0|X ≥ 0} represents the probability of occurrence of
rainy days, while F0(x) = P r{X ≤ x|X > 0} is the CDF of only rainy values.
Commonly used distribution functions F0(x) of strictly positive rainfall
records include the exponential, Gamma (Pearson III), log-Gamma (log- Pearson III), skewed normal (i.e. a normal distribution fitted to the Box-Cox transformed data), and lognormal.
Define now the Fu(x) as the CDF of the records above a given threshold u:
Fu(x) = P r{X ≤ x|X > u}
In general the parameter estimates of Fu(x) differs from those of F0(x), even
if F0(x) and Fu(x) belong to the same family. This because the distribution
of very small values may not be clearly definite and may depart from the distribution of the bulk of higher records. The author derived relationships to parametrize equation (4.37) with threshold-invariant parameters by assuring a perfect overlapping with the distribution Fu(x) for any x > u, regardless
the value of the threshold u. The GP distribution was used as Fu(x). So we
called this model MTM-GP.
Before describing the MTM-GP is better to illustrate the relationships among F (x), F0(x), and Fu(x), as described in Deidda (2010), in order to
obtain a perfect overlapping among these CDFs for any x > u as sketched in Figure 4.1.
Using simple arguments of probability is possible to write:
Fu(x) = 1− P r{X > x|X > u} = 1 −
P r{X > x|X ≥ 0}
P r{X > u|X ≥ 0} = 1−
1− F (x) 1− F (u)
4.2 The peaks over threshold (POT) approach 49
Figure 4.1: The sketch depicts some relations among the cumulative distribu- tion functions (CDFs) F (x) = P r{X ≤ x|X ≥ 0}, F0(x) = P r{X ≤ x|X >
0}, and Fu(x) = P r{X ≤ x|X > u}. Cartesian axes of F (x) are drawn
with a thin line and characteristic values are reported on the left side, while the axes of F0(x) and Fu(x) are drawn with dashed and solid thick lines,
for x > u.
These equalities lead to the following relationship between F (x) and Fu(x)
for any x > u:
F (x) = (1− ζu) + ζuFu(x) x > u (4.38)
where
ζu = P r{X > u|X ≥ 0} = 1 − F (u)
represents the probability to observe excesses of u. We note that since Fu(u) = limx→u+Fu(x) = 0, equation (4.38) becomes valid for any x ≥ u
and thus includes also equation (4.37) as a special case for u = 0. Using similar arguments we can write:
Fu(x) = 1−
P r{X > x|X > 0}
P r{X > u|X > 0} = 1−
1− F0(x)
1− F0(u)
in order to obtain a relationship between F0(x) and Fu(x):
F0(x) = F0(u) + [1− F0(u)] Fu(x) x≥ u (4.39)
Finally, computing equations (4.37) and (4.38) for x = u, eliminating F (u) among the equations, and putting Fu(u) = 0 we obtain:
ζu = ζ0[1− F0(u)] (4.40)
The same equations can be derived by the following proportions in Fig- ure 4.1 1− F (x) 1− F (u) = 1− F0(x) 1− F0(u) = 1− Fu(x) 1− Fu(u) x > u (4.41)
The probability ζu to observe an exceedance of the threshold u is esti-
mated as:
ζu =
Nu
N (4.42)
where Nu is the number of records above the threshold u and N is the sample
size (including the zeros).
Now let us assume that also F0(x) is a GP distribution with threshold u =
0 and parameters α0 and ξ, and that it can be expressed by equation (4.22)
with u = 0.
Substituting F0(x) and Fu(x) from equation (4.22) into equation (4.39)
we can easily obtain:
4.2 The peaks over threshold (POT) approach 51
where the subscript u is used to label parameter estimates (including ξ) on the basis of the threshold used. Thus, if a suitable threshold has been selected, by virtue of equation (4.43) the α0 reparametrization should be invariant for
any higher threshold (even if αu changes with u).
Computing now F0(u) from equation (4.22), i.e. putting first u = 0
and then computing for x = u, substituting F0(u) in equation (4.40), and
(optionally) using equation (4.22) we obtain:
ζ0 = ζu 1 + ξu u α0 1/ξ = ζu 1− ξuαuu −1/ξ ξu 6= 0 ζuexp u α0 = ζuexpαuu ξu = 0 (4.44)
This last equation states that the ζ0 reparameterization is threshold-
invariant, although the probability ζu of exceeding u obviously decreases
as u increases.
The threshold-invariant GP parameterization is obtained by sub- stituting F0(x) from equation (4.22) into equation (4.37), and using α0 and
ζ0 values obtained from equations (4.43) and (4.44):
F (x; ζ0, α0, ξ0) = 1− ζ0 1 + ξ x α0 −1/ξ ξ 6= 0 1− ζ0exp −αx 0 ξ = 0 (4.45)
Assuming x as an i.i.d. random variable, the distribution function of annual maxima G(x) is related to F (x) and the yearly return period T by the relation
G(x) = F (x)n = 1− 1
T (4.46)
where n = 365.25 is the average number of days in a year. From the inversion of equation (4.45) and using equation (4.46) we obtain the expression for the T-year return period quantile:
xT = α0 ξ 1− 1 − 1 T 1n ζ0 −ξ − 1 ξ6= 0 −α0ln 1− 1 − 1 T 1n ζ0 ξ = 0 (4.47)
As remarked by Deidda (2010), equation (4.45) perfectly overlaps any GP distribution fitted on the exceedances over thresholds larger than the optimum one u∗
: the only minor drawback is that there can be small depar- tures from records smaller than u∗
, but this does not affect extreme quantile estimation using equation (4.47). Concerning the choice of the optimum threshold u∗
it should be selected large enough to reliably consider the dis- tribution of the exceedances closely approximated by a GP distribution, but low enough to keep small the estimation variance.
The MTM improves the fitting on irregularly discretized records, as often happens in presence manually collected rainfall measurements. In Deidda (2010) the performances of the MTM model is superior compared to those of standard single threshold fitting on regularly discretized data. This is very important because Deidda (2007) highlighted that many time series collected by the Sardinian Hydrological Survey contain anomalous quantities of daily rainfall records rounded off at unexpected resolutions of 0.5, 1 and 5 mm/d. Furthermore, the three parameters in equation (4.45) do not depend on the threshold used for GP fitting, but only on the local climatic features: this property is particularly helpful to investigate the spatial pattern of rainfall signature in regional analyses.
MTM-GPD estimates
The MTM-GP estimates are obtained by the following hierarchical procedure: 1. ξM estimate. Identify suitable values of equally spaced threshold can-
didates u∗
< u1 < . . . < un. Take the MTM estimate ξM of the shape
parameter as the median of the ξ estimates on the suggested range of thresholds.
2. αM
0 estimate. In order to filter out the variability of the αM0 estimates
driven by the fluctuations of ξ we estimate again the αu values condi-
tioned to ξM estimate obtained at step 1 and use again the reparam-
eterization in equation (4.43) with the new αu estimates and ξ = ξM
constant. Results from equation (4.43) are now denoted as αC
0 to re-
mark that they are conditioned on ξM. The MTM estimate αM
0 of the
scale parameter is the median of the new αC
0 estimates within the range
of thresholds. 3. ζM
0 estimate. In a similar way we can reduce the variability of ζ0
by introducing the ζu estimates provided by equation (4.42) together
with the MTM estimates ξM and αM
0 (obtained at step 1 and 2) into
equation (4.44). Results from equation (4.44) are now denoted as ζC 0
4.2 The peaks over threshold (POT) approach 53
to remark again that they are conditioned toξM and αM
0 . The MTM
estimate ζM
0 is the median of the new ζ0C estimates within the range of
thresholds.
Figure 4.2 shows an example of the MTM procedure on a daily rainfall time series of our database (station 008). The figure graphically shows the hierarchical procedure previously described.
0 5 10 15 20 25 30 0 25 50 75 100 N(x>u)/N(x>0) %
MTM GPD fit − SM − st 008 [Mandas F.C. ] − 86 years − ξ0M = 0.06 − α0M = 8.41 − ζ0M = 0.21
0 5 10 15 20 25 30 0 0.2 0.4 ξ (u) 0 5 10 15 20 25 30 0 5 10 α0 C(u) 0 5 10 15 20 25 30 0 0.5 ζ0 C(u) u
Figure 4.2: Station 008 : example of MTM application on a daily rainfall time series collected by a tipping-bucket rain gauge with a 0.2 mm resolution. The first plot from top displays the fraction of the values exceeding different thresholds u in a range from 0 to 30 mm. The second plot from top displays the ξ(u) estimates with increasing threshold u: the ξM
0 MTM estimate is the
median value (horizontal red line) within the range of thresholds between 2.5 and 12.5 mm suggested for practical applications. In the third plot the αM 0
MTM estimate is obtained as the median value of the reparameterized αC 0
estimates conditioned on the ξM
0 MTM estimate, while in the fourth plot the
ζM
0 MTM estimate is obtained by the ζ0C estimates conditioned on both ξ0M
and αM
Chapter 5
Regional and geostatistical
approaches
Initially, research on the statistical distribution of extreme rainfall events focused on obtaining the most accurate estimates at measurement sites, based on long series of observations (typically longer than 30 years). Now, one of the main challenges in this area is the spatial representation of rainfall extremes in order to obtain estimates at ungauged sites.
In order to achieve this two different approaches are generally used. The first approach merges information from different gauged sites, ac- cording to a selected procedure, to compensate for short records at a single gauged site, and to obtain rainfall quantiles at locations where no measure- ments are available.
The second approach infers the parameters of the selected distribution model at each station, and then the return levels, or the distribution pa- rameters, are spatially interpolated over the region. Different interpolation techniques can be used, like linear regression-based methods, inverse distance weighting, spline, kriging.
The regional approach is described in section 5.1 while the geostatistical approach is described in section 5.2.
5.1
Regional frequency analysis
Determination of the distribution of the annual maximum of daily precipita- tion from a single site is generally affected by large sample uncertainties. For this reason regionalization techniques have been proposed, with the purpose of using also the statistical information of the neighboring sites in order to obtain more robust estimates.
Regional frequency analysis consists in grouping the sites in homogeneous regions, choosing a frequency distribution, and then in estimating the rainfall quantiles at the sites of interest.
Several methods are commonly used for the regionalization of hydrological variables such as rainfall or floods. Multivariate techniques, such as a cluster analysis (CA), principal component analysis (PCA) and factorial analysis (FA), are very common methods for classification (Beaudoin and Rousselle, 1982; Karl et al., 1982; Mallants and Feyen, 1990; Van Regenmortel, 1995; Baeriswyl and Rebetez, 1997; Comrie and Glenn, 1998; Munoz-Diaz and Rodrigo, 2004; Pineda-Martinez et al., 2007).
Hosking and Wallis (1997) developed several tests for judging the degree of homogeneity of a group of sites and for choosing and estimating a regional distribution. This methodology is widely used for regional rainfall/flood frequency analysis, e.g. Alila (1999) applied L-moments for regionalization of 5 min to 24 hours annual rainfall extremes in Canada using the GEV distribution; Trefry et al. (2005) used this methodology to estimate intensity duration frequency (IDF) curves using two index-rainfall model, one for the annual maximum series and the other for the partial duration series, using a GEV and GP distribution respectively; Satyanarayana and Srinivas (2008) used large-scale atmospheric variables to the identification of homogeneous using a cluster analysis and the homogeneity tests described in Hosking and Wallis (1997) .
In this research we used a regional frequency analysis based on the index- rainfall method, described in sections 5.1.1 with a GEV growth curve de- scribed in section 5.1.2. For the identification of homogeneous regions we used the cluster analysis and the homogeneity tests proposed by Hosking and Wallis (1997), and reported in sections 5.1.3 and 5.1.4. The L-moment ratio diagram (Hosking, 1990) guided us in the identification of regional distri- butions. Section 5.1.5 briefly describes the Two-Component Extreme Value (TCEV) distribution. In the results section, we compare the outcomes from using the regional GEV model, with those from the TCEV model reported in Deidda and Piga (1998).
5.1 Regional frequency analysis 57