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The plotting of ordered data on extreme probability paper is done according to a general plotting position function: P = (m - a) / (N + 1 - 2a). Constant 'a' is an input variable and is default set to 0.3. Many different plotting functions are used, some of them can be reproduced by changing the constant 'a'.
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For the normal distribution the Blom plotting function can best be used, for the Gumbel distribution Gringorton gives the best results. A plotting function which can be used for all distribution functions is Chegadayev. More information on plotting positions can be found in many hydrological handbooks (i.e. Applied Hydrology: pages. 394-396).
The frequency distributions are briefly described in then next Section. For a detailed description of the frequency distributions reference is made to other text books. The selection procedure of data is dealt with in Section 10.2.3.
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- mixed moment-maximum likelihood method,
- modified maximum likelihood method on Y = ln (X - X0)
Extreme Type I or Gumbel
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Distribution:
with:
X0 = location parameter,
β = scale parameter.
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The parameters can also be determined by application of the second method for part of the data set leaving the lowest N1 and highest N2 values out of the analysis provided:
N - (N1 + N2) ³ 5.
Extreme Type II or Frechet
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Distribution:
with:
X0 = location parameter, (X0 < X) !!
β = scale parameter,
γ = shape parameter. (k < 0)
The location parameter may be given (2-parameter distribution) or estimated (3-parameter distribution). The parameters can be estimated by the modified maximum likelihood method.
Extreme Type III
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Distribution:
with:
X0 = location parameter, (X0 < X) !!
β = scale parameter,
γ = shape parameter. (k > 0)
For the estimation of parameters the same applies as for the Extreme Type II distribution.
Goodrich/Weibull
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Distribution:
with:
X0 = location parameter, (X0 < X) !!
β = scale parameter,
γ = shape parameter. (k > 0)
For the estimation of parameters the same applies as for the Extreme Type II distribution.
Pareto
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Distribution:
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with:
Z = (X - X0)/sσ
X0 = threshold (to be specified by the user),
s σ = scale parameter,
q Θ = shape parameter:
0 0: General Pareto Type-I distribution,
< 0: General Pareto Type-II distribution, and
> 0: General Pareto Type-III distribution.
The domain of X is:
for q £ Θ ≤ 0 : X > X0
for q for Θ > 0 : X0 < X < X0 + sσ/q Θ
The parameters are estimated either by the maximum likelihood method (q £ Θ ≤ 0 ) or by the method of moments (q > Θ > 0).
Pareto
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Distribution with Peaks over Threshold
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Method
The pot-method uses the Pareto distribution. An additional parameter lis introduced, which is the average number of exceedances per year. A return period of T years corresponds in the POT method to one occurrence of the extreme in a series of lT T exceedances above a fixed threshold X0 (the location parameter X0 is equal to the selected lower threshold). Hence the related probability of non-exceedance in a series of all exceedances above a threshold is 1 - 1/(lT).
Selection of
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Data
Data used for the distribution functions will be read from the hymosdatabase. All distribution functions, except the Pareto with POT method, require the numberof selected data to be equal to the number years for a correct interpretation of probability of non-exceedance versus return period. Only for the Pareto with POT method a correction is applied for the return period. For all other methods a manual correction have to be made when the number of selected data is not equal to the number of years.
Series
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Codes
Series can be selected by clicking the series in the 'series codes' list box. Only one series may be selected at a time.
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