EVA and Threshold Selection

One of the key challenges in extreme value analysis is to determine the threshold above which the asymptotic model provides a reliable approximation to the tail of the population distribution (Davison and Smith, 1990; Coles, 2001). From a theoretic perspective, the threshold needs to be chosen to provide a reasonable GPD. On a more pragmatic level, it has to be known how much the exceedances over a threshold affect the estimated return levels.

A suitable threshold will ensure that the distribution well-fits all data and retains sufficient data for modelling extreme values (Davison and Smith, 1990; Coles, 2001; MacDonald et al., 2011; Papalexiou et al., 2013; Wyncoll and Gouldby, 2013).

Traditionally, threshold selection methods rely on graphical tools rather than analytic procedures (Coles, 2001; Heffernan and Tawn, 2004; Wyncoll and Gouldby, 2003, Scarrott and MacDonald, 2012; Li et al., 2014). The graphical analysis requires an experimental knowledge to identify for example the breaks of linearity in the mean residual life (MRL) plot (i.e. mean exceedances over a sequence of thresholds) (for more information, see Section 4.4.1). The interpretation of a MRL plot is not always simple in practice; it can be difficult to interpret as a method of threshold selection (Scarrott and MacDonald, 2012). Since the plots are difficult to interpret to select a desirable threshold, the threshold selection by graphs tools shows a lack of robustness for its subjectivity (Davison and Smith, 1990; Coles, 2001; Solari and Losada, 2012a). However, a more useful interpretation had to be devised.

Recently, a number of analytical threshold selection methods have been proposed: parametric approach (e.g. Rosbjerg et al., 1992), percentile approach (e.g. Grabemann and Weisse, 2008; McMillan et al., 2011; Arns et al., 2013), or mixture models (e.g. Frigessi et al., 2002; Behrens et al., 2004; Mendes and Lopes, 2004; Tancredi et al., 2006; Carreau and Bengio, 2009).

The parametric and percentile approach are the simplest threshold selection, however, selecting a fixed percentile of data (e.g. 95th or 99th) is eventually a subjective procedure (Arns et al., 2013; Caballero-Megido et al., 2017). Pre-defined paramentric thresholds can be considered as initial thresholds when dealing with multiple datasets is time-consuming.

A plethora of recent articles has proposed various extreme value mixture models for threshold estimations. The mixture models typically consider the threshold as a parameter, so it can be objectively estimated using standard inference tools, avoiding the traditional graphical diagnostics which require expert subjective judgment. However, mixture models do not produce a decisive verdict as evidenced by the variety of mixtures proposed (Ghil et al., 2011; Scarrott and MacDonald, 2012; Solari and Losada, 2012a; Mazas et al., 2014;

Frigessi et al. (2002) design a dynamically weighted mixture model for datasets containing only positive values (Figure 4.2a), where one part is the GPD and the other is a light-tailed density distribution (e.g. Weibull). The threshold is assigned to be the point over which the weighted contribution of the GPD term is higher. Frigessi et al. (2002) approach has two limitations (Scarrott and MacDonald, 2012); a lack of robustness in the inversion and a tendency for the bulk to at least partially influence the estimated character of the tail.

The method of Behrens et al. (2004) is arguably the simplest of the extreme value mixture models with which to fit the entirety of the dataset, and the threshold aim to decouple the bulk and the tail. The threshold is explicitly estimated as a parameter with its own probability density function (pdf), which is what allows uncertainty in its value to be assessed.

In a related approach, Mendes et al. (2004) fit models to both of a distribution’s tails, even though this makes little difference for skew surge dataset where only the ‘upper’ tail is considered. Mendes et al. (2004) use a normal distribution for the bulk and GPD models to fit the two tails. Thresholds for both tails are based on estimating the best proportion of observations for each tail by maximising the log-likelihood over all possible pairs of proportions.

Carreau and Bengio (2009) further develop Mendes et al. (2004) approach by extending the GPD to a ‘hybrid Pareto’ and by placing a continuity constraint on the probability density function of the threshold’s location and on its first derivative at the threshold. However, as Carreau and Bengio (2009) note, the continuity requirements effectively create some linkage between the bulk and the tail.

Alternatively, to avoid the influence of assuming a form for the bulk distribution, Tancredi et al. (2006) propose a mixture model that combines non-parametric density estimation using an unknown number of uniform distributions for the bulk (Scarrott and MacDonald, 2012). Tancredi et al. (2006) method is computationally complex with difficulties of ensuring convergence (Thompson et al., 2009; Scarrott and MacDonald, 2012), and some subjectivity exists in the choice of Bayesian prior parameters (Tancredi et al., 2006).

Li et al. (2014) propose a Root Mean Square Error (RMSE) analysis to select a suitable threshold. The RMSE measures the difference between analytical and observed cumulative distribution functions (cdfs) of 𝑋 to select suitable thresholds. Arbitrarily, Li et al. (2014) also chose only to consider events with an RP greater than approximately 1 year.

Recently, Thompson et al. (2009) develop a less subjective, semi-automatic threshold selection procedure that uses elements of the manual selection approach, without replicating it. Various parameters need to be set (e.g. test significance level), and a method that directly mimics the manual approach may prove more intuitive.

In the literature, great efforts have been made in overcoming uncertainty associated with threshold selection (Ghil et al., 2011; Scarrott and MacDonald, 2012; Solari and Losada, 2012b), accounting for covariate dependence, both parametric and non-parametric, (e.g. Mendes and Lopes, 2004; Frigessi et al., 2002; Behrens et al., 2004; Carreau and Bengio, 2009; Tancredi et al., 2006) and non-stationary sequences (e.g. Smith, 1987; Davison and Ramesh, 2000; Hall and Tajvidi, 2000; Pauli and Coles, 2001; Chavez-Demoulin and Davison, 2005; Yee and Stephenson, 2007; Eastoe and Tawn, 2009), dependence among extremes (e.g. Davison and Smith, 1990; McNeil and Frey, 2000; Ferro and Segers, 2003) and multivariate extremes (e.g. Coles and Tawn, 1991; Coles and Tawn, 1994; Heffernan and Tawn, 2004).