Rosenblatt Transformation Research Articles

Reliability analysis of an aircraft load mechanism considering multi-site damage dependency

Aircraft load mechanisms are the important actuators of the aircraft rudder control systems. By sharing the same operation circumstances and working conditions, the damage processes at multiple sites are dependent. In this paper, we firstly conduct a wear experiment and a stress relaxation experiment of an aircraft load mechanism. To consider the individual variability of the damage processes, the stochastic processes are adopted. Then, vine copula function is introduced to analyze the multivariate dependence of the damage processes. Furthermore, a time-varying dependent model is formed to model the dynamic damage interaction. The reliability of the aircraft load mechanism is assessed by using the Rosenblatt transformation and Monte Carlo method. The obtained results show that overlooking the dependency will lead to a conservative reliability result. Finally, the effect of the damage processes on the mechanism is analyzed, which provides effective guidance for the maintenance of the mechanism.

Application of the Rosenblatt transformation in First-Order System Reliability approximations

The reliability assessment of structural systems presents a significant challenge in structural engineering. A commonly employed approximation is the First-Order System Reliability Method (FOSRM), which estimates system reliability using the FORM component reliabilities and sensitivity factors. An essential step in FORM involves transforming the random vector X into the standard vector U, often using the Rosenblatt transformation (RT). Several studies demonstrated that different conditioning orders in the RT yield different FORM component results. This study investigates how these differences on component level propagate into the FOSRM system level. We conducted several typical engineering case studies with various failure probabilities, system sizes, and dependency structures (Gaussian and Frank Copula). For the Frank Copula, different Rosenblatt conditioning orders systematically yielded different FOSRM results, with most cases showing differences between 10% and 30% in estimated failure probability. For some systems, these differences increased with system size, suggesting that greater variations might be observed for larger systems. Notably, systems with Gaussian Copula functions also proved vulnerable to the Rosenblatt conditioning order when different components were assessed with different conditioning orders. The observed differences were larger than previously reported and should be carefully considered in uniform safety assessments.

A construction strategy of Kriging surrogate model based on Rosenblatt transformation of associated random variables and its application in groundwater remediation

When using simulation-optimization models for optimizing the design of groundwater pumping-treatment plans for pollution, building a surrogate model for the numerical simulation model has become an effective means of overcoming the computational load of such models. However, previous studies often treated pumping time as a single optimization variable, leading to unnecessary excessive pumping. This paper considers the location, pumping rate, start time, and end time of each candidate pumping well as optimization variables, and proposes a Rosenblatt-transform-based optimal Latin hypercube sampling method for the associated random variables to ensure that the start time is less than or equal to the end time. This method is coupled with an adaptive sampling method based on batch local optimal solutions to construct a dynamic adaptive Kriging surrogate model for the numerical model, ensuring that the true value of the optimal remediation scheme is not lost. The results show that, at the final stage of remediation, the pollutant concentration in the 4 scenarios achieves comprehensive compliance. However, when considering the minimization of remediation costs as the evaluation criterion, the remediation scheme in scenario 1 (the pumping start and end times are independent optimization variables for all candidate pumping wells) is optimal. In the optimization design of groundwater pumping-treatment plans, the pumping wells should be arranged in the midstream and downstream regions of the contaminant plume and parallel to the regional flow direction. This paper provides a method reference for the construction and adaptive updating of surrogate models involving associated random variables, as well as guidance for the dynamic optimization of groundwater pumping and treatment at contaminated sites.

Constrained optimization of offshore wind turbine positions under uncertain wind conditions with correlated data

Offshore wind power production is subject to substantial uncertainty due to variability in wind conditions. Planning of new wind parks should take these uncertainties into account by means of stochastic modeling and uncertainty quantification. Wind speed and wind direction exhibit dependence that needs to be properly modeled to yield reliable uncertainty estimates. In particular, the dependence is strong when data are averaged over a short time-horizon, e.g., on a monthly rather than an annual basis. In this work we introduce a stochastic model for wind speed and wind direction using Rosenblatt transformation and an empirical model. This allows efficient numerical quadrature that replaces thousands of Monte Carlos samples by a few tens of model evaluations at quadrature points in parametric space. A robust design formulation for maximizing power production honoring dependent data is proposed and demonstrated on data from the North Sea site Sørlige Nordsjø II.

The Statistical Characteristics Analysis for Overvoltage of Elevated Transmission Line under High-Altitude Electromagnetic Pulse Based on Rosenblatt Transformation and Polynomial Chaos Expansion

A High-Altitude Electromagnetic Pulse (HEMP) could induce very fast transient overvoltage (VFTO) with nanosecond level rise time and mega-volt amplitude, which severely threatens the electrical devices connected to the elevated transmission line. An elevated transmission line with different locations may suffer different levels of HEMP threat since the dip angle could influence the polarization of the HEMP wave. The combination of Rosenblatt Transformation and Polynomial Chaos Expansion (R-PCE) is introduced in this paper. With this method, the efficiency of calculating the overvoltage of an elevated transmission line under HEMP is improved, speeding up 24.75 times. The influence of different factors (dip angle, elevated height, and earth conductivity) on the overvoltage of elevated transmission lines applied in power systems is calculated and analyzed. The numerical result shows with 99.9% confidence that the overvoltage would be over 3.7 MV of amplitude and 6.7 × 1014 V/s of voltage derivative, which is much more rigorous than a lighting pulse. We also find that elevated transmission lines may have a larger HEMP threat in a small dip angle area. The corresponding data are shown at the end of the paper, which could be useful for relative researchers in pulse injection experiments and reliable evaluation.

Environmental contours based on imprecise probability distributions

Environmental contours are commonly applied in the design of marine structures to identify extreme environmental conditions that can give rise to extreme loads and responses. There are different methods to establish these contours, mainly the one based on the Rosenblatt transformation and the one based on Monte Carlo simulations, and in some cases significant differences can be obtained.In this paper, a method to determine environmental contours using imprecise probability models is proposed. The aim is incorporating the uncertainties that exist in the joint cumulative distribution function of the environmental variables. This is accomplishment through probability boxes defined by confidence intervals of the parameters of the joint cumulative distribution function of the environmental variables, which are proposed to be obtained with the maximum likelihood method. Furthermore, the dependence structure of the environmental variables is modeled using copulas.The proposed method is applied to obtain imprecise environmental contours for a site in the Gulf of Mexico, using extreme sea states defined by significant wave height and peak period. Then, such environmental contours are used to determine the imprecise most probable extreme response of a tension leg platform.

A convenient infinite dimensional framework for generative adversarial learning

In recent years, generative adversarial networks (GANs) have demonstrated impressive experimental results while there are only a few works that foster statistical learning theory for GANs. In this work, we propose an infinite dimensional theoretical framework for generative adversarial learning. We assume that the probability density functions of the underlying measure are uniformly bounded, k-times α-Hölder differentiable (Ck,α) and uniformly bounded away from zero. Under these assumptions, we show that the Rosenblatt transformation induces an optimal generator, which is realizable in the hypothesis space of Ck,α-generators. With a consistent definition of the hypothesis space of discriminators, we further show that the Jensen-Shannon divergence between the distribution induced by the generator from the adversarial learning procedure and the data generating distribution converges to zero. Under certain regularity assumptions on the density of the data generating process, we also provide rates of convergence based on chaining and concentration.

Integrated Approach for Estimating Extreme Hydrodynamic Loads on Elevated Pile Cap Foundation Using Environmental Contour of Simulated Typhoon Wave, Current, and Surge Conditions

Abstract Typhoon is a disastrous weather system, which usually induces strong waves, currents, and surges along the coastal area, and causes severe hydrodynamic loads on the elevated pile cap foundation, which is widely used to support the sea-crossing bridge. Estimating the hydrodynamic loads under typhoons is essential to ensure the bridge's safety. This paper develops an environmental contour-based framework that can estimate the extreme hydrodynamic loads induced by typhoons while considering the correlation among environmental conditions. The elevated pile cap foundation of the Xihoumen Rail-cum-road Bridge was used to illustrate the framework. The SWAN + ADCIRC model was employed to simulate the environmental conditions under typhoons. The pair-copulas were adopted to construct joint probability distributions, and the environmental contours with a given return period were then established by the inverse first-order reliability method. Given the hydrodynamic model and short-term peak value of the structural response, the AK-LHS method was then used to find the maximum hydrodynamic loads based on the environmental contours. The environmental contour constructing methods and selection methods of short-term peak values were compared and discussed. The main findings include: (1) ignoring correlations of the environmental conditions overestimates the extreme hydrodynamic loads and results in a conservative design; (2) the estimation of extreme hydrodynamic loads is affected by the selection and fitting of short-term peak values significantly; and (3) the extreme hydrodynamic loads estimated by either Rosenblatt transformation or Nataf transformation shows similar results.

The Inverse Transformation of L-Hermite Model and Its Application in Structural Reliability Analysis

In probabilistic analysis, random variables with unknown distributions are often appeared when dealing with practical engineering problem. A Hermite normal transformation model has been proposed to conduct structural reliability assessment without the exclusion of random variables with unknown probability distributions. Recently, linear moments (L-moments) are widely used due to the advantages of stability and insensitivity. In this paper, the complete expressions of the inverse transformation of L-moments Hermite (L-Hermite) model have been proposed. The criteria are proposed to derive the complete inverse transformation of performance function and the complete expressions of the inverse transformation of L-Hermite model are formulated. Moreover, a first-order reliability method for structural reliability analysis based on the proposed inverse transformation of L-Hermite model is then developed using the first four L-moments of random variables. Through the numerical examples, the proposed method is found to be efficient for normal transformations since the results of the proposed L-Hermite are in close agreement with the results of Rosenblatt transformation. Additionally, the reliability index obtained by the proposed method using the first four L-moments of random variables provides a close result to the reliability index obtained by first-order reliability method with known probability density functions in structural reliability assessment.

Stochastic analysis of structures under limited observations using kernel density estimation and arbitrary polynomial chaos expansion

Over the past decades there has been considerable interest among the scientific community in modeling random system inputs from limited observations and propagating the system responses. In this paper, we develop a novel method for the reasonable modeling of system inputs and efficient propagation of response. Our method firstly constructs the random model of system inputs from observations by developing a novel kernel density estimator (KDE) for Karhunen–Loeve (KL) variables. By further implementing the arbitrary polynomial chaos (aPC) formulation on dependent non-Gaussian KL variables, the associated aPC-based response propagation is then developed. In our method, the developed random model can accurately represent the input parameters from limited observations as the developed KDE of KL variables can incorporate the inherent relation between marginals of input parameters and KL variables, while empowering the input model to preserve the second-order correlations. Furthermore, since the statistical moments of KL variables can be analytically evaluated, the aPC formulation for achieving optimal convergence of system responses can be accurately determined. In addition, the system response can be accurately propagated in an efficient way by developing an aPC-based regression method using space-filling design, in which the conditional distributions in Rosenblatt transformation of aPC variables can be efficiently determined without evaluations of multi-dimensional integrations. In this way, the current work provides an effective framework for the reasonable stochastic modeling​ and efficient response propagation of real-life engineering systems with limited observations. Two numerical examples are presented to highlight the effectiveness of the developed method.

An advanced mixed-degree cubature formula for reliability analysis

Efficient assessment of mechanical system reliability subject to arbitrary probability distributions and dependent input parameters signifies an important yet challenging task. To tackle this problem, this study proposes a new moment-based method for reliability analysis of complex mechanical systems which incorporates the advanced mixed-degree cubature formula and vine copula function. To start with, a method integrating the vine copula function and Rosenblatt transformation is developed for transferring the uncertainty with dependent random variables, in which the vine copula is applied for depicting the correlation between random variables. An advanced mixed-degree cubature formula is then established for calculating statistical moments of performance function, which enables to capture sufficient uncertainty information of variables after rotating integral node. The Hermite polynomial model is adopted here to reconstruct the probability distribution of performance function with the statistical moments. To demonstrate the effectiveness of the proposed method, four illustrative examples are implemented in this study, in which Monte Carlo Simulation (MCS) and dimension-reduction techniques, are performed for comparisons. The results show that the proposed method can handle multi-dimensional correlation effectively and well balance computational accuracy and efficiency for both statistical moments and probability distribution evaluations.

Gegenbauer reconstruction method with edge detection for multi-dimensional uncertainty propagation

This paper proposes an edge-detection-based method for discontinuous functions in multi-dimensional uncertainty propagation problems. We develop the Gegenbauer reconstruction method for multivariate functions to resolve the Gibbs phenomenon. To this end, we extend the concentration edge detector to approximate discontinuity hypersurfaces and use the Rosenblatt transformation to treat irregular space decomposition. Numerical experiments for an algebraic test function and an aerodynamic design problem of the supersonic biplane airfoil flow show that the proposed method can reconstruct spectral expansions that are consistently accurate and free from the Gibbs phenomenon from given polynomial chaos coefficients and without additional model computation.

Spatiotemporal ETAS Model with a Renewal Main-Shock Arrival Process

AbstractWe propose a spatiotemporal point process model that enhances the classical Epidemic-Type Aftershock Sequence (ETAS) model. This is achieved with the introduction of a renewal main-shock arrival process and we call this extension the renewal ETAS (RETAS) model. This modification is similar in spirit to the renewal Hawkes (RHawkes) process but the conditional intensity process supports a spatial component. It empowers the main-shock intensity to reset upon the arrival of main-shocks. This allows for heavier clustering of main-shocks than the classical spatiotemporal ETAS model. We introduce a likelihood evaluation algorithm for parameter estimation and provide a novel procedure to evaluate the fitted model's goodness-of-fit (GOF) based on a sequential application of the Rosenblatt transformation. A simulation algorithm for the RETAS model is outlined and used to validate the numerical performance of the likelihood evaluation algorithm and GOF test procedure. We illustrate the proposed model and methods on various earthquake catalogues around the world each with distinctly different seismic activity. These catalogues demonstrate the RETAS model's additional flexibility in comparison to the classical spatiotemporal ETAS model and emphasizes the potential for superior modelling and forecasting of seismicity.

HUT‐based method for structural reliability considering the non‐normal and unknown distributions

AbstractStructural reliability analysis involving the non‐normal random variables are commonly encountered in practical engineering. Furthermore, some random variables are often unknown probability distributions, and the probabilistic characteristic of these variables may be expressed using only statistical moments. In this contribution, an efficient algorithm for structural reliability analysis considering the random variables with non‐normal and unknown probability distributions is proposed, in which high‐order unscented transformation (HUT) and fourth‐moment methods are adopted to efficiently evaluate the structural reliability index. By introducing the Rosenblatt transformation and third‐order polynomial transformation (TPT) techniques, random variables with non‐normal or unknown distributions are transformed into standard normal distributions. Accordingly, the performance functions can be reconstructed using standard normal random variables, and HUT is then employed to efficiently estimate the first four moments (i.e., mean, standard deviation, skewness, and kurtosis) of the performance function. Based on the Hermite polynomial model, the complete expression of fourth‐moment reliability index is derived for estimating the reliability index using the first four moments of the performance function. The efficiency and accuracy of the proposed algorithm are demonstrated through six numerical examples. The results show that the proposed algorithm is applicable to estimating reliability index of both static and dynamic reliability problems involving explicit and implicit performance functions.

Bivariate kernel density estimation for environmental contours at two offshore sites

ABSTRACT This paper proposes a novel environmental contour line approach based on measured ocean wave data at two offshore sites. For implementing the novel environmental contour line approach, we propose the use of bivariate kernel density estimation with rigorous bandwidth selection based on the Scott-Tapia-Thompson estimation method. The environmental contours obtained by using the proposed novel approach have been compared with those obtained by using the traditional Inverse-First-Order Reliability Method (IFORM) with Rosenblatt transformation or principal component analysis, and the effectiveness of our proposed novel approach have been clearly substantiated. The research results in this paper demonstrate that our proposed novel approach can be utilised as an effective tool for generating environmental contours if one has sufficient data extending beyond the return period of interest.

A generalized computationally efficient copula-polynomial chaos framework for probabilistic power flow considering nonlinear correlations of PV injections

This paper develops a general computationally efficient copula-polynomial chaos expansion (copula-PCE) framework for power system probabilistic power flow that can handle both linear and nonlinear correlations of uncertain power injections, such as wind and PVs. A data-driven copula statistical model is used to capture both nonlinear and linear correlations of uncertain power injections. This allows us to resort to the Rosenblatt transformation to transform correlated variables into independent ones while preserving the dependence structure. It paves the way for leveraging PCE for surrogate modeling and uncertainty quantification of power flow results. To further improve the computational efficiency of the proposed method without sacrificing accuracy, two strategies are developed and unified together. Specifically, the branch flow sensitivity analysis (BFSA) is used to select an appropriate number of outputs that need PCE surrogate modeling while the analysis of variance (ANOVA) scheme enables us to reduce the number of basis functions and coefficients evaluations for each PCE surrogate. Simulations carried out on the IEEE 57-bus and 118-bus and PEGASE 1354-bus systems show that the proposed framework can obtain better accuracy and computational efficiency than the existing PCE-based methods with linear correlation assumption and other Monte Carlo-based methods. The sensitivities to different copula types and parameters are also investigated.

A new multivariate quadrature rule for calculating statistical moments of stochastic response

As a derivative-free algorithm, the multivariate quadrature rule has been widely used to calculate statistical moments of the output of a system with uncertain inputs. In this paper, by using Rosenblatt transformation and copula method, the system inputs are related to an independent standard normal random vector U; then, a multivariate polynomial model is introduced to relate the system outputs to U, whereby a set of moment matching equations are derived to define quadrature weights and points. After reviewing the tensor product (TP) based quadrature rule and sparse grid method (SGM), the univariate dimension reduction (UDR) method is reformulated by Kronecker product. Following this routine, a new multivariate quadrature rule is derived by combining a Hadamard matrix based quadrature rule and discrete sine transformation matrix (DSTM). Compared with TP and SGM, the proposed quadrature rule can significantly alleviate the curse of dimensionality, its computational burden increases linearly with respect to the number of uncertain inputs. Besides, the proposed algorithms can match moment matching equations neglected by the UDR method, and thus perform more robustly in the uncertainty quantification problem. Finally, numerical examples are performed to check the proposed quadrature rule.

A new IFORM-Rosenblatt framework for calculation of environmental contours

Environmental contours are useful for designing and assessing risk to offshore structures. The Inverse First-Order Reliability Method (IFORM) and the Rosenblatt transformation are widely used to calculate environmental contours but have several challenges. In this approach, the conditional distribution is often expressed in a parametric form with parameters defined as functions of the independent variable. The distribution should be accurate for both extreme and frequently occurring values, for both the dependent and independent variables, and this is a challenging requirement for complex datasets. A new IFORM-Rosenblatt framework is proposed here to overcome this challenge. The framework includes a method to approximate a distribution with a complex form, including parametric, empirical, and mixed parametric/empirical forms, using a reduced set of parameters that are designed to prioritize accuracy for the quantiles of the distribution that contribute to the contour and a set of metrics to evaluate the goodness-of-fit of the resulting contour. A hindcast dataset is used as an example to calculate the 50-year contour using the proposed framework and a traditional approach. The proposed framework is demonstrated to generate accurate contours, although it requires more parameters and is more complex than conventional approaches.

Convex environmental contours

Environmental contours are widely used as a basis for e.g., ship design, especially in early design phases. The traditional approach to such contours is based on the well-known Rosenblatt transformation. Here we focus on convex contours estimated using Monte Carlo methods and establish a rigorous mathematical foundation for such contours. In the present paper we also present an improved simulation procedure based on importance sampling. In particular, we show how this procedure can be extended to cases with omission factors and where the joint distribution of the environmental variables is a discrete mixture. It is well-known that contours constructed using Monte Carlo simulation typically have certain irregularities. In particular, the sets bounded by the estimated contours appear to be convex. However, when the curves are investigated more closely, they include a large number of small loops. In the present paper we provide a precise condition for convexity, and propose a smoothing method which can be used to eliminate the loops. The methods are illustrated by a numerical example.

Environmental wave contours by inverse FORM and Monte Carlo Simulation with variance reduction techniques

Environmental contours of significant wave height and wave period are established for 10, 25 and 100 year-return periods by different approaches. The environmental contours are first derived by Direct sampling and by the Inverse First Order Reliability method (IFORM) using the Rosenblatt transformation. Then, the importance sampling technique is applied at different locations along the environmental contours to reduce the variance of the estimates and, consequently, to increase the accuracy and efficiency of the prediction method. The different environmental contours are derived from a large collection of hindcast wave data from offshore of Canary Islands, which is used as a case study to compare the different approaches.