China Ocean Eng., 2019, Vol. 33, No. 2, P. 127–136 DOI: 10.1007/s13344-019-0013-9, ISSN 0890-5487 http://www.chinaoceanengin.cn/ E-mail: [email protected] Generalized Extreme Value-Pareto Distribution Function and Its Applica- tions in Ocean Engineering CHEN Bai-yua, ZHANG Kuang-yuanb, WANG Li-pingc, JIANG Songd, LIU Gui-line, * aCollege of Engineering, University of California Berkeley, Berkeley, USA bDepartment of Energy and Mining Engineering, Pennsylvania State University, University Park, PA, USA cSchool of Mathematical Sciences, Ocean University of China, Qingdao 266100, China dSchool of Management, Xi’an University of Architecture and Technology, Xi’an 710055, China eCollege of Engineering, Ocean University of China, Qingdao 266100, China Received September 14, 2018; revised October 30, 2018; accepted December 2, 2018 ©2019 Chinese Ocean Engineering Society and Springer-Verlag GmbH Germany, part of Springer Nature Abstract In this paper, we establish a generalized extreme Value-Pareto distribution model and derive an analytical expression of Weibull–Pareto distribution model. Based on a data sample of 26-year wave height, we adopt the new model to estimate the design wave height for 500, 700 and 1000-year return periods. Results show that the design wave height from Weibull–Pareto distribution is between that of the Weibull distribution and that of the Pearson-III distribution. For the 500-year return period design wave height, the results from the new model is 1.601% lower than those from the Weibull distribution and 1.319% higher than those from the Pearson-III distribution. The Weibull–Pareto distribution innovatively considers the fractal features, extreme-value statistics and the truncated data in the derivation process. Therefore, it is a more holistic and practical model for estimating the design parameters in marine and coastal environments. Key words: fractal, Pareto distribution, design wave height Citation: Chen, B. Y., Zhang, K. Y., Wang, L. P., Jiang, S., Liu, G. L., 2019. Generalized extreme value-pareto distribution function and its applications in ocean engineering. China Ocean Eng., 33(2): 127–136, doi: 10.1007/s13344-019-0013-9 1 Introduction and Background quired. The essence of the design lies in the creation of the In area of designing coastal and marine structures, the most reasonable and appropriate model to estimate marine estimation of parameters of the marine environment such as environment design parameters. wave height and water level is one of the critical factors. Traditionally, univariate extreme value models, such as Proper and accurate calculation of the wave height or water Gumbel, Weibull, Pearson-III type distributions and the level is very important for the research on safety and build- maximum entropy distribution have been adopted for mod- ing cost of engineering structures, especially under severe eling wave height and water level under extreme conditions marine conditions (Wang et al., 2017; Xian et al., 2018). (Liu et al., 2006; Wang et al., 2014). As the multivariate The marine environment can be very complicated. Offshore statistical theory develops, multivariate models like Logist- projects need to withstand the damage caused by natural ic model, Hüsler-Reiss model, Beta model and other mix- disasters such as strong waves and typhoons. When design- ture models have been adopted in an accumulation of literat- ing coastal structures and assessing the relevant risk, if the ure, and compound extreme value distribution under ex- designing security’s standard is not high enough, extreme treme conditions has been introduced recently (Chen et al., marine impacts such as typhoons can easily result in im- 2017a; Liu G.L. et al., 2015, 2018; Liu X.J. et al., 2018; measurable losses. On the other hand, if the designing se- Wang and Wang, 2013; Wang et al., 2013, 2016). Tradition- curity standard is too high, there may be issues like over-in- ally, the main idea of deriving design wave height (or wave vestment and financial waste. Therefore, the estimation of a height for a specific return period) is to select a certain type proper and accurate design standard, which can not only of probability distribution (such as, Weibull, Pearson-III, withstand the natural impacts but also guarantee the secur- Gumbel distribution) to model the distribution of the max- ity and stability of the designed marine structures, is re- imum annual wave height. Then observed data of maxim- Foundation item: This work was supported by the NSFC-Shandong Joint Fund (Grant No. U1706226), and Graduate Education Reform and Research Fund (Grant No. HDJG18007). *Corresponding author. E-mail: [email protected] 128 CHEN Bai-yu et al. China Ocean Eng., 2019, Vol. 33, No. 2, P. 127–136 um annual wave height and frequency is used to determine most distribution functions of extreme values/statistics have the model parameters. After the calculation of parameters, the form of a power function. Taking advantage of the the cumulative density function is used to finally determine fractal characteristics of power function when the large- the design wave height for a specific return period. Specific- value tail is truncated, this research discusses the changing ally, this modeling process is based on the following as- laws of marine environment elements over a shorter time sumptions: 1) The annual maximum wave height follows period (20 to 50 years). Then, using self-similarity of data the same probability distribution both in the short-term and statistics, this paper calculates the design wave height over the long-term; 2) the observed statistical characteristics in a an extended period of time (100 or 500 years). Second, relatively short period of time are strictly self-similar to when using above-threshold data to derive design wave those in the long-term (100-year return period or longer); 3) height, the tail of the distribution function approaches 0, a due to constrains in the distribution function, only the annu- poor fit of tail data. Therefore, this research introduces the al maximum wave height (only one data point per year) is Pareto distribution that exhibits fractal characteristics when used in the models of wave height and water level, instead low-value data is truncated. Third, this research combines of the majority of observed data. generalized extreme distribution function (de Haan and Fer- The data volume and quality are fundamental for the reira, 2006) and Pareto distribution function, to take advant- modeling and analysis and the quantity of the data often dir- age of their complementary strengths. Fourth, in Section 3, ectly influences the quality of the results. Due to practical we derived the generalized extreme value – Pareto distribu- constraints in many cases, only data over a limited time tion model, and gave the explicit express of Weibull–Pareto range (For example, 20 or 30 years) is available. Therefore, distribution, where the steps are: first create a function fam- it is crucial to make full use of all obtained data and extract ily; then, prove when random variable T follows the gener- as much useful information as possible. alized extreme value distribution, and X follows Pareto dis- The probabilistic analysis on ocean environmental ele- tribution, the expression for Weibull–Pareto distribution can ments such as wave height and water level with Pearson-III, be found. Weibull distribution and multivariate extreme value distri- bution is designed to explore the changing laws of marine 2 Fractal characteristics of the function environmental factors on different time scales (i.e., 100 Fractal refers to a system with irregularity and complex- years, 500 years) and locations, and the expansion and infer- ity, but with similarities between parts and the whole. There ences to and from different scales. In previous extreme is randomness in the formation of fractal with a certain reg- value models, if the random variable is larger than a certain ularity in the randomness. The most important fundamental threshold, the distribution function shows the form of a feature of fractal is self-similarity and scale-invariance. power function, which has a fractal characteristic if the Self-similarity means that an entity complies to the same large-value tail is truncated. Therefore, these distribution statistical distribution locally and globally. The scale-invari- functions can describe the power functions at large values ance feature refers to the entity’s structure and remains un- very well. However, the tail of the distribution function con- changed on different scales. verges to 0 so rapidly that the distribution function fits the Dimension is another important quantity to describe the data poorly near the tail. The Pareto distribution shows geometric entity. In the traditional Euclidean geometry, a fractal characteristics at the low-end of the tail and has su- single point is considered to be zero-dimension. A straight perior performance in fitting the tailing data. However, its line is one-dimensional, plane is two-dimensional, and performance for large values is mediocre. space is three-dimensional. The fractal dimension, in con- This research proposes a generalized extreme value – trast to the Euclidean geometry of the line, plane and space, Pareto distribution function to make full use of the marine has the dimension that may not only be an integer, but can data based on the extreme value statistical theory and fractal also be a fraction, which is different compared with the tra- theory. The new model contains the commonly used ex- ditional dimensions in Euclidean geometry. So far, fractal treme value distribution that shows the statistical character- has been abstracted into a theory. Data analysis based on istics of random variables in extreme conditions, as well as fractal has shown promising results (Barrs and Chen, 2018; the fractal characteristics that utilize data efficiently. If the Cai et al., 2011; Escalante et al., 2016; Liu G.L. et al., 2018; observed data are from a rather short period of time, the data Zhang S.F.