Sign Constrained Bayesian Inference for Nonstationary Models of Extreme Events Abhinav Prakash 1 ; Vijay Panchang, Ph.D., F.ASCE 2 ; Yu Ding, Ph.D. 3 ; and Lewis Ntaimo, Ph.D. 4 Abstract: Recent studies show that many of the extreme events in hydrology can be modeled more realistically by means of a nonsta- tionary generalized extreme value (GEV) distribution. However, existing approaches for estimating the parameters can mistake a positive trend in the data to be negative. This can lead to underdesigning in engineering projects. To address this issue, this work devises a sign constrained Bayesian inference method for nonstationary GEV distributions. This new approach ensures that the final GEV model em- bodies a trend consistent with the physical understanding of the underlying phenomenon and design requirements. The advantage of using the sign constrained Bayesian approach is twofold: first, it produces a probability distribution instead of a point estimate of the model parameters; and second, it affords a natural method of uncertainty quantification, thus giving greater confidence to engineers in selecting design parameter values for civil and mechanical structures to withstand extreme events. The merit of the proposed Bayesian approach is illustrated using two water level datasets pertaining to tidal rivers in New Jersey. The results show that the new method is capable of appropriately handling datasets for which traditional methods return a positive or negative slope in the location parameters, and produces the posterior distribution of the parameters based on the observed data and not point estimates. Further, the availability of a probability distribution for the return event gives engineering designers and planners additional information and perspective on the risks involved. DOI: 10.1061/(ASCE)WW.1943-5460.0000589. © 2020 American Society of Civil Engineers. Introduction Probabilistic modeling of extreme events is often performed using the generalized extreme value (GEV) distribution or other distributions. such as the Gumbel distribution, the Weibull distribution, etc. Engi- neers are often interested in understanding the behavior of extreme events to develop a basis for designing civil engineering structures such as floodwalls and bridges. The reliability of these structures is paramount to public safety. For instance, about 40% of the land area in the Netherlands is below sea level (Haan and Ferreira 2006). This portion of land is protected against flooding by construc- tion of storm-surge barriers, which should withstand extreme water levels. These barriers are undergoing renovations, which should be able to withstand oceanic conditions corresponding to a probability of occurrence of 10 -4 in a given year. Then the question that becomes relevant is the following: How should this probability be translated into a design quantile for the height of the dyke? In statistics, extreme value theory provides techniques for answering this type of question. The standard form of the GEV distribution is a stationary model, which means that the parameters of the distribution (i.e., the location, scale, and shape parameters) are time invariant. However, hydrological maxima often show time-dependent trends (Potter 1991; Olsen et al. 1999; Lins and Slack 1999; Douglas et al. 2000; Strupczewski et al. 2001b), thus creating a need for probabi- listic modeling to incorporate the trends observed in the data. This can be handled by using a nonstationary GEV distribution whose parameters are a function of time (Coles 2001), so that the probabil- ity distribution changes over the time. A nonstationary GEV distri- bution is generally based on prespecified trends in the parameters, enabling one to capture the change in the probability of occurrence of the underlying events over time. The commonly used technique to estimate the parameters for a nonstationary GEV distribution is the maximum likelihood estimation (MLE) method. Examples include the work of Caires et al. (2006) in the context of significant wave heights; of the studies of Salas and Obeysekera (2014) of peak discharges in the Aberjona River and Sugar Creek Basin; and of the work of Masina and Lamberti (2013) in the context of extreme sea levels in the northern Adriatic (see also Strupczewski et al. 2001a). Conversely, Mudersbach and Jensen (2010) have used the L-moments method (Gado and Nguyen 2016) to estimate the 100 year extreme water levels for the North Sea coast. Although the L-moments method is valid for stationary dis- tribution, it can be applied to a nonstationary distribution by first detrending the time-series data and then applying the method. The re- sults of Gado and Nguyen (2016) based on simulated datasets suggest that this method can perform as well as or better than the MLE method in most cases. In another related work, the moments method is used to estimate intensity–duration–frequency (IDF) rainfall curves in some African cities (De Paola et al. 2014). An alternative to the MLE and L-moments approaches is the Bayesian inference approach. The Bayesian approach considers the parameters of the GEV distribution as random variables. There- fore, it employs the concept of priors, which reflects the prior belief for any parameter before observing the data and computes a poste- rior distribution of the parameters based on the given data. The pri- ors can act as a layer of information if some properties of the 1 Ph.D. Student, Dept. of Industrial and Systems Engineering, Texas A&M Univ., 3131 TAMU, College Station, TX 77843. ORCID: https://orcid.org/0000-0002-1792-8011. Email: abhinavp@tamu.edu 2 Professor and Program Chair, Dept. of Mechanical Engineering, Texas A&M Univ. at Qatar, Education City, Doha, Qatar. Email: vijay .panchang@qatar.tamu.edu 3 Professor, Dept. of Industrial and Systems Engineering, Texas A&M Univ., 3131 TAMU, College Station, TX 77843 (corresponding author). Email: yuding@tamu.edu 4 Professor, Dept. of Industrial and Systems Engineering, Texas A&M Univ., 3131 TAMU, College Station, TX 77843. ORCID: https://orcid .org/0000-0002-9114-5170. Email: ntaimo@tamu.edu Note. This manuscript was submitted on January 28, 2019; approved on February 26, 2020; published online on June 9, 2020. Discussion period open until November 9, 2020; separate discussions must be submitted for individual papers. This paper is part of the Journal of Waterway, Port, Coastal, and Ocean Engineering, © ASCE, ISSN 0733-950X. © ASCE 04020029-1 J. Waterway, Port, Coastal, Ocean Eng. J. Waterway, Port, Coastal, Ocean Eng., 2020, 146(5): 04020029 Downloaded from ascelibrary.org by Texas A&M University on 06/09/20. Copyright ASCE. For personal use only; all rights reserved.