The transformed-stationary approach: a generic and simplified methodology for non-stationary extreme value analysis

Abstract

Abstract. Statistical approaches to study extreme events require, by definition, long time series of data. In many scientific disciplines, these series are often subject to variations at different temporal scales that affect the frequency and intensity of their extremes. Therefore, the assumption of stationarity is violated and alternative methods to conventional stationary extreme value analysis (EVA) must be adopted. Using the example of environmental variables subject to climate change, in this study we introduce the transformedstationary (TS) methodology for non-stationary EVA. This approach consists of (i) transforming a non-stationary time series into a stationary one, to which the stationary EVA theory can be applied, and (ii) reverse transforming the result into a non-stationary extreme value distribution. As a transformation, we propose and discuss a simple timevarying normalization of the signal and show that it enables a comprehensive formulation of non-stationary generalized extreme value (GEV) and generalized Pareto distribution (GPD) models with a constant shape parameter. A validation of the methodology is carried out on time series of significant wave height, residual water level, and river discharge, which show varying degrees of long-term and seasonal variability. The results from the proposed approach are comparable with the results from (a) a stationary EVA on quasi-stationary slices of non-stationary series and (b) the established method for non-stationary EVA. However, the proposed technique comes with advantages in both cases. For example, in contrast to (a), the proposed technique uses the whole time horizon of the series for the estimation of the extremes, allowing for a more accurate estimation of large return levels. Furthermore, with respect to (b), it decouples the detection of non-stationary patterns from the fitting of the extreme value distribution. As a result, the steps of the analysis are simplified and intermediate diagnostics are possible. In particular, the transformation can be carried out by means of simple statistical techniques such as low-pass filters based on the running mean and the standard deviation, and the fitting procedure is a stationary one with a few degrees of freedom and is easy to implement and control. An open-source MAT-LAB toolbox has been developed to cover this methodology, which is available at https://github.com/menta78/tsEva/ (Mentaschi et al., 2016).

Global changes of extreme coastal wave energy fluxes triggered by intensified teleconnection patterns

“…The present projections show an overall stronger statistical significance, partially due to the nonstationary extreme value analysis, which allows for bigger samples and consequently less statistical fitting uncertainty, compared to a stationary approach applied on time slices [Mentaschi et al, 2016]. Moreover, dynamical models allow for a more accurate representation of extreme conditions compared to statistical methodologies, due to the scarce availability of extreme data to train the statistical models [Breivik et al, 2014;Laugel et al, 2014;Martínez-Asensio et al, 2014].…”

“…A nonstationary extreme value analysis (EVA) was performed on each WEF time series by using the approach described by Mentaschi et al [2016] in order to detect long-term trends in the extremes. The nonstationary EVA algorithm was set in order to filter out the variability on time scales below 30 years, to exclude from the analysis noise and sharp variations.…”

“…proxy for coastal flooding and erosion, studying the long-term changes of its extremes estimated from a nonstationary extreme value analysis technique [Mentaschi et al, 2016].…”

Extreme heat waves under 1.5 °C and 2 °C global warming

“…This technique consists in (i) transforming a nonstationary time series into a stationary one to which the stationary extreme value theory can be applied; and (ii) reverse-transforming the result into a nonstationary extreme value distribution, for instance a generalized extreme value (GEV) distribution. This technique returns estimations of the extremes comparable with those based on non-stationary Maximum Likelihood Estimators, but is generally more stable (Mentaschi et al 2016).…”

“…Return levels and return periods are calculated for every model run with a transformed-stationary methodology developed by Mentaschi et al (2016) and successfully applied to the projection of extreme coastal waves by Mentaschi et al (2017).…”