Vízügyi Közlemények, 2002 (84. évfolyam)

4. füzet - Domokos Miklós: Még egyszer a simuló eloszlásfüggvények misztifikálásáról

Még egyszer a simuló eloszlásfüggvények misztifikálásáról 663 Once more about the mystification of fitting distribution functions (Comments on the paper of Professor Klemes) by Dr Miklós DOMOKOS, С.Eng., mathematician Perhaps the most important statement of the paper by Klemes (2002), to be commented here, is, that the duration curve of any hydrological variable, compiled from the elements, arranged in order of magnitude, of the statistical sample of that variable — which curve can be called also empirical distribution function or can be equivalently transformed into such a function (Fig. I) and has generally been used for a long time in everyday's practice of hydrology and hydrotechnics — can be extrapolated in the domain of very rare events (e.g., floods) by adopting various sophisticated methods (so-called "distribution models") of mathematical statistics with at least the same uncertainty as when extrapolating the upper tail of the duration curve - plotted either in a linear or a logarithmic coordinate system or a probability paper (Klemes 2002, Fig. 1) - "by eye" (with free hand or a ruler), as it had been done, at the first time, in 1914, by the American engineer Allan Hazen. What this comment on the paper of Klemes (2002) wants, is to awake the Hungarian readers to the fact, that the perception quoted above could have been read already in the period 1968-1973 , from the pen of Hungarian authors (Domokos-Szász 1968, Domokos 1972, Domokos-Szász 1973); a perception which was notl able to assert itself, either at that time or since then, against the modish, ornamental mainstream of hydrological statistics, making its way from the international literature also into the Hungarian textbooks. According to the opinion of the Hungarian authors quoted, theoretically the following three versions or version-groups of any (planar) distribution function - defining the relationship between the values of a random variate (RV) (e.g., water level, flow discharge) and the non-exceedance probabilities related to them - can be distinguished: — the (true) theoretical distribution function (ThDF) of the given RV, whose perfect cognition is forever impossible for the human mind, — the empirical distribution function (EDF) of the given RV, which itself (or its version flipped around the probabiity value of 0.5) is being called duration curve in hydrology and hydrotechnics (Fig. 1) and which can be compiled by arranging in increasing order the observed values of the RV, as elements of a statistical sample; and finally — a lot of fitting distribution functions (FDF) adjusted to the EDF, whose adjustment may be carried out either by free hand ("by eye"), or by using a purposfully selected probability paper (geometrical adjustment) or even by adopting a numerical mathematical apparatus aiming at the estimation of the parameters of the FDF from the sample elements (e.g., by using the moment, the maximum likelihood or the L-moment method). (Figs. 2, 3 and 4). According to the basic theorem of mathematical statistics, declared by Glivenko (1936), the EDF of a given RV stochastically converges - whenever certain conditions are fulfilled — to the (true) ThDF, if the number of the sample elements used for compiling the EDF converges to infinity (Rényi 1967). Thus, according to this theorem, there is a unique authentic and credible estimation of the (forever unknown) ThDF: namely the EDF In its original wording, there are two conditions for the validity of the Glivenkotheorem: — the sample elements must be homogeneous (i.e., originating from thesame distribution) (Figs. 5 and 6) — they must be independent from each other.