Application of the envelope peaks over threshold (EPOT) method for probabilistic assessment of dynamic stability

doi:10.1016/j.oceaneng.2016.03.006

Ocean Engineering

Volume 120, 1 July 2016, Pages 298-304

https://doi.org/10.1016/j.oceaneng.2016.03.006 Get rights and content

Highlights

•
Development of the Envelope Peaks over Threshold (EPOT) for large roll is reviewed.
•
Peaks of envelope is used to avoid dependence of consecutive peaks of roll motion.
•
Generalized Pareto distribution is used to approximate the tail of peak distribution.

Abstract

This paper reviews the research and development of the Envelope Peaks over Threshold (EPOT) method that has taken place in the last three years. The EPOT method is intended for the statistical extrapolation of ship motions and accelerations from time-domain numerical simulations, or possibly, from a model test. To model the relationship between probability and time, the large roll angle events must be independent, so Poisson flow can be used. The method uses the envelope of the signal to ensure the independence of large exceedances. The most significant development was application of the generalized Pareto Distribution (GPD) for approximation of the tail, replacing the previously used Weibull distribution. This paper reviews the main aspects of modeling the GPD, including its mathematical justification, fitting the parameters of the distribution, and evaluating the probability of exceedance and its confidence interval.

Introduction

The rarity of dynamic stability failures in realistic sea condition makes the problem of extrapolation inevitable. This can be illustrated in the following example. If we assume an hourly stability failure rate of 10⁻⁶ h⁻¹ (Kobylinski and Kastner, 2003), then we can expect to see (on average) one failure every 1,000,000 h. If we require 10 observations for a reliable statistical estimate; then we need to simulate 10,000,000 h. Even if an advanced hydrodynamic code could run in a real time and a cluster with 1000 processors is dedicated to the task, it would take 10,000 h per condition (combination of seaway, speed and heading) to perform the assessment. The cost of the calculations prohibits direct simulation in this manner.

Additionally stability failure is associated with large-amplitude motions and is expected to be nonlinear. Indeed, capsize is related to the ultimate nonlinearity – transition to another equilibrium. In order to have enough fidelity to model this problem, the hydrodynamic code must be quite sophisticated (see a review by Reed et al., 2014). The probability of capsizing is the topic of a multi-year ONR research project titled “A Probabilistic Procedure for Evaluating the Dynamic Stability and Capsizing of Naval Vessels” (Belenky et al., 2016).

IMO document SLF 54/3/1, Annex 1 (IMO, 2011) defines intact stability failure as a state of inability of a ship to remain within design limits of roll (heel, list) angle combined with high rigid body accelerations. This includes also partial stability failure when a ship is subjected to a large roll angle or excessive accelerations, but does not capsize. Following the same logic one could also include an excessive pitch angle. As this study focuses on partial stability failure, peak over threshold method (POT) was chosen (Pickands, 1975). Introducing a threshold allows considering the data that are more influenced by nonlinearity; this incorporates changing physics into the statistical estimates.

To satisfy the requirement of independent peaks over threshold, the peaks of envelope were used instead of the peaks of the process itself (Campbell and Belenky, 2010). The review of this research effort is available from Belenky and Campbell (2012). That work included consideration of the relationship between probability and time, the probabilistic properties of peaks, application of envelope theory and the extreme value distribution.

The relationship between time and probability is key to the proper treatment of the partial stability failures. It may be modeled with Poisson, which requires the independence of the failure events. In the case of capsizing, the enforcement of Poisson Flow is not required, since capsizing can only occur once per record (the possibility of several capsizings within one record can be safely ignored for practical cases). Belenky and Campbell (2012) also review different ways of statistical characterization of the rate of events, the only parameter of the Poisson flow.

Classical POT methods use the Generalized Pareto Distribution (GPD) to approximate the tail of the distribution above a threshold. However, under certain conditions the GPD may be right bounded, that is, there is some value above which the probability of exceedances is zero. This is not a problem for conventional statistical consideration, when we are interested in the quantiles of the GPD (i.e., the probability is given and the level needs to be found). In ship stability generally the failure level is known and related to down flooding or cargo shifting angles and probability is to be found. The physical meaning of the right bound was not clear at that time (and still is not completely clear). As a result, the Weibull distribution was used for modeling the tail.

Normally distributed wave elevation was the subject of study in Belenky and Campbell (2012). This was a logical first test for these techniques. The study concluded that the distribution of large absolute values of peaks can be approximated by Rayleigh law. The Rayleigh distribution is a particular case of Weibull distribution when the shape parameter equals two. Thus, deviation of this parameter from two may be suitable for representing nonlinearity in a dynamical system.

To investigate the performance of a POT scheme based on the Weibull distribution, a model representing ship motions with realistic stability variation was used (Weems and Wundrow, 2013, Weems and Belenky, 2015). It was found that Weibull distribution does not have enough flexibility to approximate the tail of large-amplitude ship motions and the consideration of the GPD was started again.

Application of the GPD with EPOT produced very reasonable results (Smith and Zuzick, 2015). The techniques used to fit GPD, estimate the probability of exceedance of a given level and evaluate its uncertainty are described in Campbell et al., 2016, Campbell et al., 2014b and Glotzer et al. (2016) and briefly reviewed in the rest of this paper.

Section snippets

Distribution of order statistics

In order to understand why statistical extrapolation is possible when the underlying distribution is unknown, we begin with order statistics.

Consider a set of n independent realizations of random variable z. Assume that the distribution is given in a form of a cumulative distribution function (CDF) and probability density function (PDF). Sorting the observed values from the largest to smallest we have: $y_{i} = s o r t (z_{i}) i = 1, .. ., n$

Indeed, for randomly selected values y and z: $p d f (y) = p d f (z); C D F (y) = C D F (z)$

Preparing independent data

Ship motions are characterized by strong dependence of the data points on each other, especially in following and stern quartering seas when a spectrum is narrow and autocorrelation function takes a long time to decay. However, the peaks of the piecewise linear envelope (see Fig. 1) represent independent data points. The difference between piecewise linear and theoretical envelope is considered by Belenky and Campbell (2012). However, this technique may not work for the cases of parametric

Extrapolated estimate

Using the GPD to extrapolate the probability of exceedance yields the conditional probability that the level of interest c has been exceeded if the threshold μ has been exceeded: $\hat{P} = {\begin{matrix} {(1 + \hat{ξ} \cdot \frac{c - μ}{\hat{σ}})}^{- \frac{1}{\hat{ξ}}} i f \hat{ξ} > - \frac{\hat{σ}}{c - μ} \\ 0 i f \hat{ξ} \leq - \frac{\hat{σ}}{c - μ} \end{matrix}$

c is the limit that constitutes the stability failure. The probability that μ has been exceeded can be estimated statistically, since it has been exceeded often enough so that we have enough data to build the distribution of peaks above it.

Now lets consider the problem of the

Current status and future work

The EPOT method has evolved significantly in the last three years. The main idea remains the same, however extrapolate peaks over a threshold and use the envelope to ensure independence. The idea of a threshold was originally aimed to emphasize influence of nonlinearity. Now it also has a mathematical interpretation – it ensures the applicability of the second extreme value theorem. However, the relation between statistics and nonlinearity of ship motions is still a matter of current scientific

Acknowledgments

The work described in this paper has been funded by the Office of Naval Research, under Dr. Patrick Purtell, Dr. Ki-Han Kim and Dr. Thomas Fu. The authors greatly appreciate their support. The participation of Prof. Pipiras was facilitated by the Summer Faculty Program supported by ONR and NSWCCD under Dr. Jack Price (David Taylor Model Basin, NSWCCD). The authors would like to acknowledge fruitful discussions with Prof. Pol Spanos (Rice University), Dr. Art Reed, Mr. Tim Smith (David Taylor

References (23)

B. Campbell et al.
Properties of the tail of envelope peaks and their use for the prediction of the probability of exceedance for ship motions in irregular waves
Belenky, V., Campbell, B.L., 2012. Statistical extrapolation for direct stability assessment. In: Proceedings of the...
Belenky, V., Weems, K., Lin, W.M., 2016. Split-time method for estimation of probability of capsizing caused by pure...
R.C. Bishop et al.
Parametric Investigation on the Influence of GM, Roll damping, and Above-Water Form on the Roll Response of Model 5613.
Report. NSWCCD-50-TR-2005/0
(2005)
D.D. Boos et al.
Essential Statistical Inference: Theory and Method
(2013)
Campbell, B., Belenky, V., 2010. Assessment of short-term risk with Monte-Carlo method. In: Proceedings of the 11th...
Campbell, B., Belenky, V., Pipiras V., 2014. On the application of the generalized Pareto distribution for statistical...
S. Coles
An Introduction to Statistical Modeling of Extreme Values
(2001)
H.A. David et al.
Wiley Series in Probability and Statistics
(2003)
D. Glotzer et al.
Confidence intervals for exceedance probabilities with application to extreme ship motions
REVSTAT Stat. J
(2016)

IMO SLF 54/3/1, 2011. Development of Second Generation Intact Stability Criteria. Report of the Working Group at SLF 53...

Cited by (23)

A multiparameter simulation-driven analysis of ship response when turning concerning a required number of irregular wave realizations
2024, Ocean Engineering
The growing implementation of Decision Support Systems on modern ships, digital-twin technology, and the introduction of autonomous vessels cause the marine industry to seek accurate modeling of vessel response. Despite the contemporary 6DOF models can be used to predict ship motions in irregular waves, the impact of their stochastic realization is usually neglected and remains under-investigated. Especially in the case of turning, differences arising from the stochastic representation of the waves may result in excessive ship motions or even stability failure during maneuver execution. Therefore, in this study, statistical distributions of maximum amplitudes of roll, pitch, and lateral acceleration calculated in two representative locations on board a passenger vessel were analyzed concerning stochastic wave realization and existing extremes. The research utilized 6DOF simulation data and numerous realizations of the irregular wave with random phases of its components. Furthermore, the required number of wave realizations allowing for capturing the actual ranges of ship response at an assumed confidence level has been determined and analyzed. Ultimately, the results were compared in the safety-critical cases concerning various wave and operational conditions. The outcome of this study may be found useful by all parties involved in developing maritime autonomous systems and modeling ship motions.
Estimation of probability of large roll angle with envelope peaks over threshold method
2023, Ocean Engineering
The development and application of the Envelope Peak-over-Threshold (EPOT) method is summarized for the roll motion of a ship in irregular waves. The method is based on the conventional Peak-over-Threshold (POT) method, but uses an envelope of the peaks for declustering the data. As suggested by a reduced-order model, a power-law tail should be applicable for the peaks of roll motions, which allows the introduction of physical considerations into the statistical model. A data-driven Generalized Pareto distribution (GPD) is applied to study the behavior of the shape parameter on large-volume samples in order to confirm the theoretical prediction of the tail structure. The physics-informed statistical model approximates data above a suitable threshold with a Pareto distribution, while the threshold is determined with a prediction error criterion. An example is included, where extrapolations with GPD are compared to the physics-informed model.
Extrapolation of failure rate over wave height
2023, Ocean Engineering
Evaluation of the probability of rare events, such as occurrence of maximum reactions over ship design life, from direct numerical simulations requires significant computational time. The paper reviews some ways (called statistical extrapolation) proposed to reduce the required computational time and considers in detail the extrapolation of failure rate over significant wave height. Methods to construct the 95%-confidence interval of the extrapolated failure rate are proposed and validated. For the validation, new criteria are proposed which require that the extrapolation should not introduce a non-conservative error. Application of these criteria shows that the extrapolation of failure rate over significant wave height does not satisfy the proposed criteria with the sample size 20 and satisfies with the sample size 200.
Wave episode based Gaussian process regression for extreme event statistics in ship dynamics: Between the Scylla of Karhunen–Loève convergence and the Charybdis of transient features
2022, Ocean Engineering
Citation Excerpt :
The reconstruction step draws from the theory of data-driven surrogate modeling techniques (Forrester et al., 2008; Vu et al., 2017; Rudy et al., 2021), and in particular theory of Gaussian Process Regression (GPR) (Rasmussen and Williams, 2006; Stevens, 2018). Like related techniques, such as Envelope Peaks Over Threshold (Campbell et al., 2016; Weems et al., 2019; Belenky et al., 2019), surrogate modeling must be carefully designed to accurately capture the tails of the distribution. The design step draws from literature on optimal experimental design both in the traditional Bayesian formulation (Chaloner and Verdinelli, 1995; Huan and Marzouk, 2013) and in forms that are either output aware or specifically tailored for recovering extreme events (Sapsis, 2020; Blanchard and Sapsis, 2021b; Sapsis, 2021).
For a number of structural and fatigue problems in marine science, engineers need accurate statistics for kinematic and dynamical quantities, such as displacements and bending moments. For many such problems the response statistics depend nonlinearly on input stochastic processes, such as sea surface elevation. Nonlinearity prevents the use of standard linear methods, leading to the adoption of expensive experiments and time-consuming numerical simulations. In order to avoid this cost we present a machine learning framework to minimize the training set requirements. This framework consists of two parts. First, we use the Karhunen–Loève theorem to represent stochastic sea states with finite-time wave episodes, which have low dimensionality that nonetheless captures the features important to hydrodynamics and structural mechanics. However, the choice of the wave episode duration is ‘caught’ between the Scylla of slow Karhunen–Loève series convergence for long time durations and the Charybdis of missing transient behaviors when the interval is short. To combat this dilemma, we propose a division into a region, designed for parametric interpolation and machine learning, and a stochastic prelude region, designed to probabilistically model transients. The second part of the framework consists of a Gaussian Process Regression (GPR) model designed to learn the mapping from wave episodes to structural outputs. GPR is able to take advantage of the low dimensional parametric representation of the sea state in order to converge with reasonably-sized training sets (on the order $n \approx 100$ ). At the same time, we use a low-dimensional representation in order to represent the stochastic response time series. The principal advantages of the Gaussian process surrogate are the blazing speed of evaluation – ten thousand times faster than the direct method – and the built in uncertainty quantification. Taken together, we can reconstruct the statistics of the responses by sampling sea states via the Karhunen–Loève construction, estimating the corresponding model outputs using the trained GPR, and estimating statistics of interest through Monte-Carlo calculation on the surrogate model. Finally the developed framework allows for trivial computation of the statistics for different sea spectra, i.e. without the need to retrain the Gaussian Process Regression scheme.
A data-driven model for nonlinear marine dynamics
2021, Ocean Engineering
Citation Excerpt :
In the case of using CFD or experimental model tests, expense of these methods limits the number of designs and environmental conditions that can be evaluated. To predict the worst-case scenario throughout the lifetime of a ship or platform, it is not possible to use random environment conditions in experiments or CFD due to the expense, and specialized methods must be used to extrapolate statistics (Belenky et al., 2012; Campbell et al., 2016) or predict the worst-case environment (Choi and Jensen, 2019). Recent activity in data-science has generated new algorithms and ideas towards developing data-driven models for complex systems, and in this paper, a data-driven method based on Long Short-Term Memory neural networks is presented for the representation of nonlinear marine dynamical systems.
The design and engineering of ships and platforms that operate in the ocean environment requires understanding of a nonlinear dynamical system that responds according to complex interaction with a wide range of sea and wind conditions. Time domain observation of nonlinear marine dynamics with either experiments or high-fidelity numerical simulation tools is costly due to the random nature of the ocean and the full range of environmental and loading conditions that are experienced in the lifetime of a ship or platform. In this paper, a data-driven method is presented to predict the complex nonlinear input–output relationship typical of marine systems. A Long Short-Term Memory neural net is used to learn nonlinear wave propagation and the nonlinear roll of a ship section in beam seas. Training data are generated with second-order wave theory or a volume-of-fluid computational fluid dynamics, although the method is directly applicable to data that is generated by other means such as nonlinear potential flow or experimental measurements. The cost and the amount of data to apply the method are estimated and measured. The data-driven results are compared with unseen data to demonstrate the accuracy and feasibility.
Pitfalls of data-driven peaks-over-threshold analysis: Perspectives from extreme ship motions
2020, Probabilistic Engineering Mechanics
A popular peaks-over-threshold (PoT) method of Extreme Value Theory to quantify the probabilities of rare events is examined here on data generated from a nonlinear random oscillator model, describing a qualitative behavior of rolling of a ship in irregular seas. The restoring force in the oscillator model has a softening shape associated with the ship rolling application, and the response is also made bounded, so as to eliminate the possibility of “capsizing.” As a result, the tail of the resulting probability density function of the response undergoes three regimes: the Gaussian core, the heavy tail and the short bounded tail. By considering several scenarios where data are available in one but not another regime, it is shown that the PoT method can produce unsatisfactory results. Some refined methods from Extreme Value Theory, for example, those based on mixture models, are also examined, but without much success. It is thus argued that a data-driven application of the PoT method may fail, if the physical aspects of the system under study are not taken into account.

View all citing articles on Scopus

View full text

Published by Elsevier Ltd.

Application of the envelope peaks over threshold (EPOT) method for probabilistic assessment of dynamic stability

Highlights

Abstract

Introduction

Section snippets

Distribution of order statistics

Preparing independent data

Extrapolated estimate

Current status and future work

Acknowledgments

Parametric Investigation on the Influence of GM, Roll damping, and Above-Water Form on the Roll Response of Model 5613.

Report. NSWCCD-50-TR-2005/0

Essential Statistical Inference: Theory and Method

An Introduction to Statistical Modeling of Extreme Values

Wiley Series in Probability and Statistics

Confidence intervals for exceedance probabilities with application to extreme ship motions

REVSTAT Stat. J