An experiment in subjective graphical quantité estimation applied to the generalized extreme value distribution

Fitting flood prediction curves by eye to the data of return period plots is an intuitive approach to estimating «-year flood discharge magnitudes from single-site data. However, the subjective nature of this "procedure" means that it is unlikely to be adopted as a recognized flood estimation technique. An experiment with a group of volunteers demonstrates that some individuals are capable of subjectively fitting generalized extreme value curves to yield flood quantile estimates from simulated data with an accuracy comparable to probability weighted moments. This encourages the development of alternative formal flood estimation procedures based on some form of pattern recognition. Such procedures might mimic the more talented individuals' approach to curve fitting, without the disadvantage of subjectivity. Une expérience d'estimation graphique subjective de quantités appliquée à la distribution généralisée des valeurs extremes Résumé Ajuster à l'œil des courbes de predetermination de crues au tracé de durées de retour est une manière intuitive d'estimer un débit de durée de retour donnée à partir de données stationnelles. 11 est toutefois difficile, en raison de sa nature subjective, de valider cette "procédure" en tant que technique d'estimation des crues. Une expérience, impliquant un groupe de volontaires, a montré de certains individus sont capables d'ajuster subjectivement les courbes de la loi généralisée des valeurs extrêmes à des estimations de quantiles issues de données simulées avec une précision comparable à celle de la méthode des moments pondérés. Ce résultat encourage la recherche de procédures formelles d'estimation des crues fondées sur des reconnaissances de forme. De telles procédures pourraient imiter la capacité des individus les plus talentueux à ajuster des courbes, en évitant l'obstacle de la subjectivité. INTRODUCTION There will always be large uncertainties involved in the estimation of extreme flood magnitudes using a limited data record from a single recording site, and there is a large literature on the development and applications of small-sample statistical estimation theory to such situations. A distinctly non-formal alternative approach was suggested by Bardsley (1989), whereby an interactive computer graphic was proposed as a means for subjective fitting of a generalized extreme value (GEV) prediction curve to flood data on a return period plot. Open for discussion until 1 December 1999 400 W. E. Bardsley & C. P. Pearson Subjective interaction with the data plot is more intuitive and convenient than the black-box approach of the many standard methods available (see, for example, Rosen & Weissman (1996) and references therein for GEV estimation methods). Also, the point was made by Bardsley (1994) that subjective extrapolation of extreme floods is as valid as any objective method, given that no amount of single-site analysis will ever reveal the "correct" distribution of flood maxima or identify the "best" method for estimating its parameters. However, the subjective approach is unlikely to find practical application in design work, because objective methods have the distinct advantage of avoiding the issue of personal liability in the event of extensive flood damage in the future. Despite this drawback, it is useful to evaluate subjective curve fitting because it leads into a new and very different objective flood estimation methodology based on automated pattern recognition in return period plots. Such automated procedures would seek to formalize the processes of subjective curve fitting by those individuals who are able to demonstrate estimation accuracy comparable to current standard statistical methods. A starting point toward the new methods, then, is demonstrating that some individuals can if fact estimate flood quantiles subjectively to the accuracy of standard methods. In that context, this paper outlines some encouraging results from a small group experiment in subjective flood quantile estimation using simulated flood data from the generalized extreme value distribution. THE EXPERIMENT The experiment focused on simulated at-site GEV estimation. The same approach could be taken for estimating the simulated distribution for dimensionless regional flood peaks (scaled by an at-site index flood). The "annual flood maxima" were generated as random variables from two selected extreme value distributions, one Type 2 and the other Type 3 (Table 1). Twenty random samples were generated from each distribution, with ten samples of size 25 and ten of size 50. The 40 simulated flood samples were all entered into an interactive computer graphic and the corresponding 40 subjective curve fits were carried out independently by each of 11 senior hydrology students at the University of Waikato. Each curve fit yielded a subjective estimate of the 0.98 and 0.99 quantiles of the distribution concerned. The data display on the interactive graphic is influenced to some extent by the choice of plotting position formula, so other choices of plotting position could lead to better or worse subjective curve fitting. For this exercise, a median-unbiased plotting position was used (Beard, 1943). This was prompted by the thought that the students Table 1 Parameter values of the extreme value distributions used for simulation. Parameter definitions follow Bardsley (1989). The Type 2 and Type 3 location parameters are lower and upper bounds, respectively. Location Shape Scale Type 2 -43.2 -0.394 96^ ' Type 3 301.2 0.112 212.5 An experiment in subjective graphical quantité estimation 401