Modeling asymmetrically dependent multivariate ocean data using truncated copulas

Modern cities heavily rely on complex transportation, making accurate traffic speed prediction crucial for traffic management authorities. Classical methods, including statistical techniques and traditional machine learning techniques, fail to capture complex relationships, while deep learning approaches may have weaknesses such as error accumulation, difficulty in handling long sequences, and overlooking spatial correlations. Graph neural networks (GNNs) have shown promise in extracting spatial features from non-Euclidean graph structures, but they usually initialize the adjacency matrix based on distance and may fail to detect hidden statistical correlations. The choice of correlation measure can have a significant impact on the resulting adjacency matrix and the effectiveness of graph-based models. This thesis proposes a novel approach for accurately forecasting traffic patterns by utilizing a multi-view spatio-temporal graph neural network that captures data from both realistic and statistical domains. Unlike traditional correlation measures such as Pearson correlation, copula models are utilized to extract hidden statistical correlations and construct multivariate distribution functions to obtain the correlation relationship among traffic nodes. A two-step approach is adopted, which involves selecting and testing different types of bivariate copulas to identify the ones that best fit the traffic data, and utilizing these copulas to create multi-weight adjacency matrices. The second step involves utilizing a graph convolutional network to extract spatial information and capturing temporal trends using dilated causal convolutions. The proposed ST-CopulaGNN model outperforms previous approaches such as DCRNN and Graph WaveNet, indicating the effectiveness of incorporating copulas in trafficforecasting. Experiments on theMETR-LA and PEMS-BAY datasets show that the proposed model outperforms previous approaches with a slight improvement.

Bayesian Approaches to Copula Modelling

Methods for estimating extreme loads are used in design as well as risk assessment. Regression using maximum likelihood or least squares estimation is widely used in a univariate analysis but these methods favour solutions that fit observations in an average sense. Here we describe a new technique for estimating extremes using a quantile function model. A quantile of a distribution is most commonly termed a ‘return level’ in flood risk analysis. The quantile function of a random variable is the inverse function of its distribution function. Quantile function models are different from the conventional regression models, because a quantile function model estimates the quantiles of a variable conditional on some other variables, while a regression model studies the conditional mean of a variable. So quantile function models allow us to study the whole conditional distribution of a variable via its quantile function, whereas conventional regression models represent the average behaviour of a variable.Little work can be found in the literature about prediction from a quantile function model. This paper proposes a prediction method for quantile function models. We also compare different types of statistical models using sea level observations from Venice. Our study shows that quantile function models can be used to estimate directly the relationships between sea condition variables, and also to predict critical quantiles of a sea condition variable conditional on others. Our results show that the proposed quantile function model and the developed prediction method have the potential to be very useful in practice.

Purpose This study aims to address a critical research gap in foundation pit collapse risk assessment by developing an innovative framework that integrates expert-guided knowledge with data-driven Bayesian Networks (BNs). Current methods often rely heavily on subjective expert judgment or limited post-accident data, leading to potential biases and incomplete risk assessments. This research seeks to enhance the accuracy and reliability of risk assessments by systematically combining expert insights with data-driven analysis, thereby improving safety management and reducing worker mortality rates in foundation pit construction. Design/methodology/approach The study processed 552 collapse cases via sentiment analysis to extract risk factors. A BN was developed using the Expectation–Maximization algorithm, and expert judgments were integrated via the Analytic Hierarchy Process, forming a data-expert hybrid framework to minimize subjectivity in risk evaluation. Findings The proposed method effectively identifies and quantifies risk factors associated with foundation pit collapses, as demonstrated through case validation. The integration of expert-guided and data-driven approaches not only reduces the cost of expert decision-making but also improves the accuracy of risk predictions. The framework successfully addresses the limitations of existing methods by providing a more comprehensive and objective risk assessment tool. Originality/value This study introduces a novel semi-empirical, semi-data-driven process for constructing BNs, combining the strengths of expert knowledge and data analysis. By addressing the challenges of data scarcity and expert bias, the proposed method offers a significant advancement in foundation pit risk assessment. Crucially, the framework's adaptive architecture – combining a universal risk model derived from global data with a flexible, expert-guided interface for local calibration – ensures its broad applicability across diverse international regulatory and geological contexts. This inherent versatility makes it a valuable and scalable safety management solution for various construction projects worldwide and offers a methodological template for risk analysis in other high-risk industries.

Chapter 14 - Ensemble-Based Multivariate Sensitivity Analysis of Satellite Rainfall Estimates Using Copula Model

Risk assessment is more and more important in offshore engineering. The assessment measure used today the Fuzzy Set has many shortcomings. How to measure the risk and set up the safety assessment model are urgent questions. After introducing the basic knowledge of a new mathematical tool the Unascertained Sets, the model based on unascertained mathematics is set up. Then, it was introduced into the fields of safety assessment of offshore jacket loadout. Engineering practice shows that the method proposed here can complete the safety assessment systematically and scientifically without any assumption. The work has significance in theory and practice for safety assessment of offshore engineering.

Multivariate claim data are common in insurance applications, for example, claims of each policyholder from different types of insurance coverages. Understanding the dependencies among such multivariate risks is critical to the solvency and profitability of insurers. Effectively modeling insurance claim data is challenging due to their special complexities. At the policyholder level, claim outcomes usually follow a two-part mixed distribution: a probability mass at zero corresponding to no claim and an otherwise positive claim from a skewed and long-tailed distribution. To simultaneously accommodate the complex features of the marginal distributions while flexibly quantifying the dependencies among multivariate claims, copula models are commonly used. Although a substantial body of literature focusing on copulas with continuous outcomes has emerged, some key steps do not carry over to mixed data. In particular, existing nonparametric copula estimators are not consistent for mixed data, and thus copula specification and diagnostics for mixed outcomes have been a problem. However, insurance is a closely regulated industry in which model validation is particularly important, and it is essential to develop a baseline nonparametric copula estimator to identify the underlying dependence structure. In this article, we fill in this gap by developing a nonparametric copula estimator for mixed data. We show the uniform convergence of the proposed nonparametric copula estimator. Through simulation studies, we demonstrate that the proportion of zeros plays a key role in the finite sample performance of the proposed estimator. Using the claim data from the Wisconsin Local Government Property Insurance Fund, we illustrate that our nonparametric copula estimator can assist analysts in identifying important features of the underlying dependence structure, revealing how different claims or risks are related to one another.

Copula models have become increasingly popular for modelling the dependence structure in multivariate survival data. The two-parameter Archimedean family of Power Variance Function (PVF) copulas includes the Clayton, Positive Stable (Gumbel) and Inverse Gaussian copulas as special or limiting cases, thus offers a unified approach to fitting these important copulas. Two-stage frequentist procedures for estimating the marginal distributions and the PVF copula have been suggested by Andersen (Lifetime Data Anal 11:333-350, 2005), Massonnet et al. (J Stat Plann Inference 139(11):3865-3877, 2009) and Prenen et al. (J R Stat Soc Ser B 79(2):483-505, 2017) which first estimate the marginal distributions and conditional on these in a second step to estimate the PVF copula parameters. Here we explore an one-stage Bayesian approach that simultaneously estimates the marginal and the PVF copula parameters. For the marginal distributions, we consider both parametric as well as semiparametric models. We propose a new method to simulate uniform pairs with PVF dependence structure based on conditional sampling for copulas and on numerical approximation to solve a target equation. In a simulation study, small sample properties of the Bayesian estimators are explored. We illustrate the usefulness of the methodology using data on times to appendectomy for adult twins in the Australian NH&MRC Twin registry. Parameters of the marginal distributions and the PVF copula are simultaneously estimated in a parametric as well as a semiparametric approach where the marginal distributions are modelled using Weibull and piecewise exponential distributions, respectively.

Copula modeling enables the analysis of multivariate count data that has previously required imposition of potentially undesirable correlation restrictions or has limited attention to models with only a few outcomes. This article presents a method for analyzing correlated counts that is appealing because it retains well-known marginal distributions for each response while simultaneously allowing for flexible correlations among the outcomes. The proposed framework extends the applicability of the method to settings with high-dimensional outcomes and provides an efficient simulation method to generate the correlation matrix in a single step. Another open problem that is tackled is that of model comparison. In particular, the article presents techniques for estimating marginal likelihoods and Bayes factors in copula models. The methodology is implemented in a study of the joint behavior of four categories of US technology patents. The results reveal that patent counts exhibit high levels of correlation among categories and that joint modeling is crucial for eliciting the interactions among these variables.

Variational Bayes Estimation of Time Series Copulas for Multivariate Ordinal and Mixed Data

The semiconductor manufacturing is one of the most sophisticated processes in modern industries. It contains hundreds of complex operations, and this makes semiconductor reliability a challenging task especially in recent years that integrated circuits (ICs) have been broadly used at much wider areas including the health cares, automotive, and aerospace applications, where safety is the major concern. To make things worse, due to the shorter product life cycles and severe competitions, time-consuming reliability tests become unacceptable. The industries are in urgent need of a fast and effective way to quickly assess reliability risks. Under all these stringent limitations and the high reliability targets, we propose a consolidated approach that consists of an efficient and effective reliability baseline (BL) management system and a test scheme developed based on sequential probability ratio test (SPRT). To ensure a trustworthy setting for the SPRT test scheme, we apply the Bayesian approach to continuously update the BL database. An integrated flow chart is introduced to elucidate our proposed methodology. It is the first time SPRT is applied on reliability risk assessments with two unknown parameters. Several actual cases from semiconductor arena are reported to demonstrate the merits and powers of our proposed methodology. From these real cases, as a side benefit, we show the wide applicability of our method, which does not limit to reliability but on yield and functionality checks as well. To facilitate the use of our method by field practitioners, we do the best to explain derivations in a plain manner and provide analysis codes in R (in Supplementary Material) to make our approach as portable as possible.

The performance of manufacturing systems or equipment is, to a great extent, dependent upon the condition of their components. Closely monitoring the condition of the critical components and carrying out timely system diagnosis whenever a fault symptom is detected would help to reduce system downtime and improve overall productivity. Fault tree analysis (FTA) is a powerful tool for reliability studies and risk assessment. However, most research on FTA focuses on the generation of minimum cut sets and how to calculate the probability of main events. As a result, the issue concerning the ordering of basic events in a fault tree has been largely neglected. In this paper, a novel approach based on rough set theory and a pairwise comparison table for fault diagnosis is proposed. The approach attempts to learn from the pattern of decision-making by domain experts from past experience and uses the knowledge acquired, which is in the form of a minimum decision rule set, to determine the ordering of basic events in a fault tree. The details of the approach, together with the basic concepts of rough set theory, are presented. A case study is used to illustrate the application of the proposed approach. Results show that a reasonable ordering of basic events in a fault tree can be generated easily. With the ordering of basic events determined, a maintenance engineer in a manufacturing plant can then carry out fault diagnosis in an efficient and orderly manner.

In remanufacturing practices, understanding and communicating the failure risk and reliability of a critical part, component or subsystem plays a crucial role as it has a significant impact on the lifecycle management of the product and thus determines the success of the remanufacturing process. In this respect, statistical time-to-failure analysis provides a very powerful and versatile analytical tool for reliability analysis and risk assessment. Among various statistical tools available, Weibull model which uses time series data on records of failure incidents of a product for the fitting of a parametric distribution, is a powerful approach to characterizing the time-to-failure probability function of the product. It is able to provide valuable information for optimized lifecycle management and remanufacturing process. This paper demonstrates successful applications of Weibull model for time-to-failure analysis using case studies in remanufacturing practices and proposes a statistical approach to assess the reliability of critical parts and components for remanufacturing in the product’s lifecycle management. It is envisaged that the research results are able to benefit remanufacturing practices in many ways such as reducing warranty loss by minimizing probability of failure of remanufactured critical parts and components.

Abstract. Non-parametric Bayesian networks (NPBNs) are graphical tools for statistical inference widely used for reliability analysis and risk assessment and present several advantages, such as the embedded uncertainty quantification and limited computational time for the inference process. However, their implementation in hydrological studies is still scarce. Hence, to increase our understanding of their applicability and extend their use in hydrology, we explore the potential of NPBNs to reproduce catchment-scale hydrological dynamics. Long-term data from 240 river catchments with contrasting climates across the United States from the Catchment Attributes and Meteorology for Large-sample Studies (CAMELS) data set will be used as actual means to test the utility of NPBNs as descriptive models and to evaluate them as predictive models for maximum daily river discharge in any given month. We analyse the performance of three networks, one unsaturated (hereafter UN-1), one saturated (hereafter SN-1), both defined only by hydro-meteorological variables and their bivariate correlations, and one saturated network (hereafter SN-C), consisting of the SN-1 network and including physical catchments' attributes. The results indicate that the UN-1 network is suitable for catchments with a positive dependence between precipitation and discharge, while the SN-1 network can also reproduce discharge in catchments with negative dependence. The latter can reproduce statistical characteristics of discharge (tested via the Kolmogorov–Smirnov statistic) and have a Nash–Sutcliffe efficiency (NSE) ≥0.5 in ∼40 % of the catchments analysed, receiving precipitation mainly in winter and located in energy-limited regions at low to moderate elevation. Further, the SN-C network, based on similarity of the catchments, can reproduce discharge statistics in ∼10 % of the catchments analysed. We show that once a NPBN is defined, it is straightforward to infer discharge and to extend the network itself with additional variables, i.e. going from the SN-1 network to the SN-C network. However, the results also suggest considerable challenges in defining a suitable NPBN, particularly for predictions in ungauged basins. These are mainly due to the discrepancies in the timescale of the different physical processes generating discharge, the presence of a “memory” in the system, and the Gaussian-copula assumption used for modelling multivariate dependence.

In general, distribution theory plays a crucial role in modeling various real-life phenomena, making it a fundamental tool in statistical analysis and decision-making. Over the years, extensive research has been conducted on different statistical distributions and estimation techniques. While the literature abounds with information regarding well-known distributions, there is always room for exploring new variants that can better capture the characteristics of complex phenomena. In this paper, we contribute to the field of distribution theory by introducing novel probability distribution called the Weibull-Weibull distribution. The Weibull-Weibull distribution is derived by compounding two Weibull distributions, and it offers a flexible framework for modeling phenomena that exhibits a complex interplay of factors. By combining the strength of the Weibull distribution with itself, we are able to capture a wider range of shapes and behaviour, providing more accurate representations of real-world occurrences. To facilitate the practical application of the Weibull-Weibull distribution, we employ the maximum likelihood estimation (MLE) approach to estimate its shape and scale parameters. The MLE method is a widely used statistical technique that allows us to determine the most likely values of the parameters based on observed data. By applying this estimation method to the Weibull-Weibull distribution, we enable researchers and practitioners to effectively utilize this new distribution in their analyses and modeling efforts. Furthermore, we delved into a comprehensive study of statistical theory and properties of the Weibull-Weibull distribution. We investigate its moments, cumulative distribution function, probability density function, and other key measures. Through rigorous analysis, we establish the theoretical foundations of the Weibull-Weibull distribution and provide insights into its behaviour and characteristics. This comprehensive examination equips researchers with a solid understanding of the distribution, enabling them make informed decisions and interpretations when working with real-life data. In conclusion, our research introduces the Weibull-Weibull distribution as a valuable addition to the existing repertoire of statistical distributions. By leveraging the power of compounding two Weibull distributions, we provide a flexible and robust framework of modeling complex phenomena. With the use of maximum likelihood estimation and the given Chernoff bound, practitioners are able to estimate the distribution's parameters as well as analyze its tails accurately. Our extensive statistical analysis further enhances the understanding of the Weibull-Weibull distribution, facilitating its practical application in a wide range of fields, including reliability analysis, survival analysis, and risk assessment.