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Copula based Damage Detection for Structural Health Monitoring
This project utilizes copulas for damage detection in a Structural Health Monitoring (SHM) application. A copula-based system was chosen for the benefit of multivariate joint distribution with a goal to detect damage based on how the system as a whole reacts rather than one or two sensors by themselves. Copulas are commonly used in the field of finance for risk modeling based on two or more random inputs. A few applications in the field of SHM and Non-Destructive Evaluation (NDE) have been researched mostly on risk or reliability of the structure. The goal of this project is to determine if a copula-based approach can be used for damage detection. An unsupervised learning method was desired to reduce the dimensionality, minimal training, and be a faster evaluation method than other unsupervised methods. If a copula method can be used to detect damage what additional information on the damage can be interpreted. The remainder of this report will go over the background needed, SHM methodology, SHM applications, conclusions, and future developments.
Stochastic Simulation of Daily Suspended Sediment Concentration Using Multivariate Copulas
Estimation of daily suspended sediment concentration (SSC) is required for water resources and environment management. In this paper, a copula-based stochastic method was proposed for daily SSC simulation. Here, the multivariate copula function, constructed based on a bivariate copula and two bivariate conditional probability distributions, was used to model the temporal and cross dependence structures in daily SSCs. Then, the daily SSCs were generated by sampling from the multivariate conditional distribution. As a result, synthetic long-term SSCs data beyond the limited observation period can be provided for water resources managers, which plays a critical role in accurately estimating frequency and magnitude of extreme SSCs events. The proposed method was under rigorous examination by applying to a case study at Pingshan station in the Jinsha River Basin, China. Results showed that the generated daily SSC sequences not only had a high degree of accuracy in preserving the statistical characteristics of the daily SSC observations, but also captured both the temporal correlation and the cross-correlation between the daily streamflow and daily SSC. Specifically, the average daily relative error values corresponding to mean, standard deviation, skewness, lag-1 temporal correlation, and cross correlation were 0.87%, 4.24%, 7.52%, 0.51% and 2.02%, respectively. The multivariate copula framework proposed here can accurately and efficiently generate long-term daily SSC data for water resources management such as frequency analysis and risk assessment of extreme SSC events.
On the quantification and efficient propagation of imprecise probabilities with copula dependence
This paper addresses the problem of quantification and propagation of uncertainties associated with dependence modeling when data for characterizing probability models are limited. Practically, the system inputs are often assumed to be mutually independent or correlated by a multivariate Gaussian distribution. However, this subjective assumption may introduce bias in the response estimate if the real dependence structure deviates from this assumption. In this work, we overcome this limitation by introducing a flexible copula dependence model to capture complex dependencies. Here, a hierarchical Bayesian multimodel approach is proposed to quantify uncertainty in dependence model-form and model parameters that result from small data sets. This approach begins by identifying, through Bayesian multimodel inference, a set of candidate marginal models and their corresponding model probabilities, and then estimating the uncertainty in the copula-based dependence structure, which is conditional on the marginals and their parameters. The overall uncertainties integrating marginals and copulas are probabilistically represented by an ensemble of multivariate candidate densities. A novel importance sampling reweighting approach is proposed to efficiently propagate the overall uncertainties through a computational model. Through an example studying the influence of constituent properties on the out-of-plane properties of transversely isotropic E-glass fiber composites, we show that the composite property with copula-based dependence model converges to the true estimate as data set size increases, while an independence or arbitrary Gaussian correlation assumption leads to a biased estimate.
Decoding the age–chemical structure of the Milky Way disc: an application of copulas and elicitable maps
In the Milky Way, the distribution of stars in the [α/Fe] versus [Fe/H] and [Fe/H] versus age planes holds essential information about the history of star formation, accretion, and dynamical evolution of the Galactic disc. We investigate these planes by applying novel statistical methods called copulas and elicitable maps to the ages and abundances of red giants in the Apache Point Observatory Galactic Evolution Experiment survey. We find that the high- and low-α disc stars have a clean separation in copula space and use this to provide an automated separation of the α sequences using a purely statistical approach. This separation reveals that the high-α disc ends at the same [α/Fe] and age at high [Fe/H] as the low-[Fe/H] start of the low-α disc, thus supporting a sequential formation scenario for the high- and low-α discs. We then combine copulas with elicitable maps to precisely obtain the correlation between stellar age τ and metallicity [Fe/H] conditional on Galactocentric radius R and height z in the range 0 < R < 20 kpc and |z| < 2 kpc. The resulting trends in the age–metallicity correlation with radius, height, and [α/Fe] demonstrate a ≈0 correlation wherever kinematically cold orbits dominate, while the naively expected negative correlation is present where kinematically hot orbits dominate. This is consistent with the effects of spiral-driven radial migration, which must be strong enough to completely flatten the age–metallicity structure of the low-α disc.
A copula-based rank histogram ensemble filter
Serial ensemble filters implement triangular probability transport maps to reduce high-dimensional inference problems to sequences of state-by-state univariate inference problems. The univariate inference problems are solved by sampling posterior probability densities obtained by combining constructed prior densities with observational likelihoods according to Bayes' rule. Many serial filters in the literature focus on representing the marginal posterior densities of each state. However, rigorously capturing the conditional dependencies between the different univariate inferences is crucial to correctly sampling multidimensional posteriors. This work proposes a new serial ensemble filter, called the copula rank histogram filter (CoRHF), that seeks to capture the conditional dependency structure between variables via empirical copula estimates; these estimates are used to rigorously implement the triangular (state-by-state univariate) Bayesian inference. The success of the CoRHF is demonstrated on two-dimensional examples and the Lorenz'63 problem. A practical extension to the high-dimensional setting is developed by localizing the empirical copula estimation, and is demonstrated on the Lorenz'96 problem.
A Generalized Copula-Polynomial Chaos Expansion for Probabilistic Power Flow Considering Nonlinear Correlations of PV Injections
This paper develops a generalized Copula-polynomial chaos expansion (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 the 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. This paves the way of leveraging the PCE for surrogate modeling and uncertainty quantification of power flow results, i.e., achieving the probabilistic distributions of power flows. Simulations carried out on the IEEE 57-bus system show that the proposed framework can get much more accurate results than other alternatives with different linear and nonlinear power injection correlations.
The co-determination of home and workplace relocation durations using survival copula analysis
Not Available
Closure to “Coincidence Risk Analysis of Floods Using Multivariate Copulas: Case Study of Jinsha River and Min River, China” by Yang Peng, Yulong Shi, Hongxiang Yan, Kai Chen, and Jipeng Zhang
We would like to thank the discusser for his comments on the coincidence flood risk modeling in Jinsha River and Min River, China. The discusser’s comment brings focus to the discussion of extreme-value selection methodology and historical flood information. In flood risk analysis, we acknowledge that there is always an alternative modeling approach that could be used to generate desired prediction results. In this comment, the discusser made two valuable points regarding the original paper, and the following two paragraphs provide our responses.
XPCA: Copula-based Decompositions for Ordinal Data.
Abstract not provided.
Multivariate Frequency Analysis of Annual Maxima Suspended Sediment Concentrations and Floods in the Jinsha River, China
Streamflow plays a critical role in sediment transport. Traditional frequency analysis of annual maximal (AM) suspended sediment concentrations (SSCs) is usually based on univariate distribution of SSCs and may provide a limited assessment of SSC state, leading to an underestimate or overestimate of the corresponding probability. To address this issue, we propose a multivariate frequency analysis framework is proposed for extreme sediment events using copula functions that account for the joint behavior of SSC and flood flow (i.e., peak discharge and volume). The uncertainties associated with copula modeling are also discussed. The Jinsha River in China is used as a case study. The results show that the copula-based multivariate method provides a more comprehensive assessment of extreme SSCs and their recurrence intervals than univariate frequency analysis. The uncertainty of copula modeling decreases with longer observation time and increases with larger return periods. The uncertainty ranges of the most likely design quantiles vary with different copulas, and the uncertainty range for the best-fit copula is not the smallest. This study is useful for the risk assessment of extreme SSC events under flood conditions.
A Data-Driven Nonparametric Approach for Probabilistic Load-Margin Assessment Considering Wind Power Penetration
A modern power system is characterized by an increasing penetration of wind power, which results in large uncertainties in its states. These uncertainties must be quantified properly; otherwise, the system security may be threatened. Facing this challenge, here we propose a cost-effective, data-driven approach to assessing a power system's load margin probabilistically. Using actual wind data, a kernel density estimator is applied to infer the nonparametric wind speed distributions, which are further merged into the framework of a vine copula. The latter enables us to simulate complex multivariate and highly dependent model inputs with a variety of bivariate copulae that precisely represent the tail dependence in the correlated samples. Furthermore, to reduce the prohibitive computational time of traditional Monte-Carlo simulations that process a large amount of samples, we propose to use a nonparametric, Gaussian-process-emulator-based reduced-order model to replace the original complicated continuation power-flow model through a Bayesian-learning framework. To accelerate the convergence rate of this Bayesian algorithm, a truncated polynomial chaos surrogate, which serves as a highly efficient, parametric Bayesian prior, is developed. This emulator allows us to execute the time-consuming continuation power-flow solver at the sampled values with a negligible computational cost. Results of simulations that are performed on several test systems reveal the impressive performance of the proposed method in the probabilistic load-margin assessment.
The hidden structure of hydrodynamic transport in random fracture networks
We study the large-scale dynamics and prediction of hydrodynamic transport in random fracture networks. The flow and transport behaviour is characterized by first passage times and displacement statistics, which show heavy tails and anomalous dispersion with a strong dependence on the injection condition. The origin of these behaviours is investigated in terms of Lagrangian velocities sampled equidistantly along particle trajectories, unlike classical sampling strategies at a constant rate. The velocity series are analysed by their copula density, the joint distribution of the velocity unit scores, which reveals a simple, albeit hidden, correlation structure that can be described by a Gaussian copula. Based on this insight, we derive a Langevin equation for the evolution of equidistant particle speeds. In this framework, particle motion is quantified by a stochastic time-domain random walk, the joint density of particle position, and speed satisfies a Klein–Kramers equation. The upscaled theory quantifies particle motion in terms of the characteristic fracture length scale and the distribution of Eulerian flow velocities. That is, it is predictive in the sense that it does not require the a priori knowledge of transport attributes. The upscaled model captures non-Fickian transport features, and their dependence on the injection conditions in terms of the velocity point statistics and average fracture length. It shows that the first passage times and displacement moments are dominated by extremes occurring at the first step. The presented approach integrates the interaction of flow and structure into a predictive model for large-scale transport in random fracture networks.
Disentangling direct from indirect relationships in association networks
Networks are vital tools for understanding and modeling interactions in complex systems in science and engineering, and direct and indirect interactions are pervasive in all types of networks. However, quantitatively disentangling direct and indirect relationships in networks remains a formidable task. Here, we present a framework, called iDIRECT (Inference of Direct and Indirect Relationships with Effective Copula-based Transitivity), for quantitatively inferring direct dependencies in association networks. Using copula-based transitivity, iDIRECT eliminates/ameliorates several challenging mathematical problems, including ill-conditioning, self-looping, and interaction strength overflow. With simulation data as benchmark examples, iDIRECT showed high prediction accuracies. Application of iDIRECT to reconstruct gene regulatory networks in Escherichia coli also revealed considerably higher prediction power than the best-performing approaches in the DREAM5 (Dialogue on Reverse Engineering Assessment and Methods project, #5) Network Inference Challenge. In addition, applying iDIRECT to highly diverse grassland soil microbial communities in response to climate warming showed that the iDIRECT-processed networks were significantly different from the original networks, with considerably fewer nodes, links, and connectivity, but higher relative modularity. Further analysis revealed that the iDIRECT-processed network was more complex under warming than the control and more robust to both random and target species removal ( P < 0.001). As a general approach, iDIRECT has great advantages for network inference, and it should be widely applicable to infer direct relationships in association networks across diverse disciplines in science and engineering.
Modeling Spatial Asymmetries in Teleconnected Extreme Temperatures
Abstract Combining strengths from deep learning and extreme value theory can help describe complex relationships between variables where extreme events have significant impacts (e.g., environmental or financial applications). Neural networks learn complicated nonlinear relationships from large datasets under limited parametric assumptions. By definition, the number of occurrences of extreme events is small, which limits the ability of the data-hungry, nonparametric neural network to describe rare events. Inspired by recent extreme cold winter weather events in North America caused by atmospheric blocking, we examine several probabilistic generative models for the entire multivariate probability distribution of daily boreal winter surface air temperature. We propose metrics to measure spatial asymmetries, such as long-range anticorrelated patterns that commonly appear in temperature fields during blocking events. Compared to vine copulas, the statistical standard for multivariate copula modeling, deep learning methods show improved ability to reproduce complicated asymmetries in the spatial distribution of ERA5 temperature reanalysis, including the spatial extent of in-sample extreme events.
Multi-Mechanism Flood Hazard Assessment: Critical Review of Current Practice and Approaches
This report documents the initial findings from the Nuclear Regulatory commission (NRC)-sponsored research project Methods for Estimating Joint Probabilities of Coincident and Correlated Flooding Mechanisms for Nuclear Power Plant Flood Hazard Assessments.1 This research project is a part of NRC’s Probabilistic Flood Hazard Assessment (PFHA) Research Program and will aid the development of guidance on the use of PFHA methods to evaluate infrastructure safety for existing and proposed US nuclear power plants (NPPs). More specifically, this project intends to provide technical background for the development of flood hazard curves for multi-mechanism floods (MMFs). MMFs are flood events caused by more than one flooding mechanism (e.g., flood events due to the simultaneous occurrence of precipitation-induced river flooding and storm surge). Project activities include three main tasks: Task 1—Survey of current concepts and methods in assessing MMF hazards; Task 2—Critical assessment of selected methods and approaches for quantifying probabilistic MMF hazard risk; Task 3—Development of example case studies to illustrate best practices for quantifying probabilistic MMF hazard risk The initial findings from Tasks 1 and 2 are documented in this report. Task 1 comprised a survey of approaches and methods that have been applied to understand and assess flood hazards due to MMFs. Task 2 involved a critical review of the selected approaches and methods. To that end, the scope of this report includes documentation of (1) a reconnaissance-level survey of the current state of concepts and practice for MMF hazard assessment; (2) a generalized MMF assessment framework to address the distinctions among various types of flood-forcing phenomena, flood mechanisms (grouped into three mechanism types), and flood severity metrics; (3) a wide-ranging survey of approaches and methods that have been applied to various flooding phenomena and settings; and (4) a critical assessment of MMF hazard assessment methods. Studies were identified involving MMFs related to coastal flooding mechanisms, fluvial (rivers/streams) flooding mechanisms, and associated combinations of coastal and fluvial flooding mechanisms. Studies were also identified that address MMFs involving coastal and fluvial flooding mechanisms as well as coastal flooding mechanisms along with extreme precipitation (without specific attribution to fluvial or pluvial mechanisms). The studies identified for review in this report included assessments at varying spatial scales (from local to global) with differing geographic regions of focus using both observed and synthetic data. The majority of studies identified and reviewed were site-specific assessments focusing on relatively short return periods. Studies considered a range of flood severity metrics, made differing assumptions regarding the occurrence of extrema, and used multiple statistical techniques; the use of copulas for the development of joint distributions was a particularly popular analysis technique. The literature review highlighted the differences among existing studies relative to terminology used, means of presenting results, framework and techniques employed, and level of sophistication regarding the number and types of variables considered. Despite the significant diversity in existing studies, the review identified several promising techniques that will be considered in future work under this project, including the development of joint distributions for MMFs using copula and Bayesian-motivated approaches.
The Arctic LDRD project: What happened?
This presentation overviews results from the 2019 LDRD Reserve project “Diagnosing near-future changes in Arctic sea ice and ocean conditions” and a follow-up 2020 CSES Rapid Response project “Multi- Model Risk Assessment for Arctic Navigability.” We analyzed Arctic simulations from 4 CMIP6 model ensembles (E3SM, CESM-CAM, CESM-WACCM and CESM-DPLE), comparing them with observations. We then applied a machine-learning approach using copulas to produce joint probabilities of adverse conditions along potential shipping transit routes through the Arctic. Here we compare and contrast the different models and examine the utility of the copula approach applied to these data sets.