Model selection techniques for the frequency analysis of hydrological extremes: the MSClaio2008 R function

The frequency analysis of hydrological extremes requires fitting a probability distribution to the observed data to suitably represent the frequency of occurrence of rare events. The choice of the model to be used for statistical inference is often based on subjective criteria, or it is considered a matter of probabilistic hypotheses testing. In contrast, specific tools for model selection, like the well known Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), are seldom used in hydrological applications. The paper of Laio et ̃al. (2008) verifies whether the AIC and BIC work correctly when they are applied for identifying the probability distribution of hydrological extremes, i.e. when the available samples are small and the parent distribution is highly asymmetric. An additional model selection criterion, based on the Anderson-Darling goodness-of-fit test statistic, is proposed, and the performances of the three methods are compared trough an extensive numerical analysis. In this brief document, an application of the R function MSClaio2008, part of the package nsRFA, is provided. Introduction The problem of model selection can be formalized as follows: a sample of n data, D = (x1, . . . , xn), arranged in ascending order is available, sampled from an unknown parent distribution f(x); Nm operating models, Mj , j = 1, . . . , Nm, are used to represent the data. The operating models are in the form of probability distributions, Mj = gj(x, θ̂), with parameters θ̂ estimated from the available data sample D. The scope of model selection is to identify the model Mopt which is better suited to represent the data, i.e. the model which is closer in some sense to the parent distribution f(x). Three different model selection criteria are considered here, namely, the Akaike Information Criterion (AIC), the Bayesian Information Criterion (BIC), and the Anderson-Darling Criterion (ADC). Of the three methods, the first two belong to the category of classical literature approaches, while the third derives from a heuristic interpretation of the results of a standard goodness-of-fit test (see Laio, 2004). The R function MSClaio2008, part of the package nsRFA, is used on a data sample from the FEH database: > data(FEH1000) The data of site number 69023 are used here: > sitedata serieplot(sitedata[,4], sitedata[,3], ylim=c(0,200), + xlab="year", ylab="Max annual peak [m3/s]")