On the simplified pair-copula construction — Simply useful or too simplistic?

A comprehensive review on the development of copulas in financial field

The Copula concept has long been used in many applications, especially in the financial field. This concept was first used in 1959 by Sklar in his mathematical work and greatly assisted in the applications of financial and insurance areas. The copula functions have been widely used in dependence modeling. In this study, we look at how the copula began to develop from a basic form to a more advanced form through studies that previous researchers have made. Throughout this study, we find various types of the copula, and each exhibits its own characteristics lying under two main families, Elliptical and Archimedean copulas. Our findings suggest that copula is vital in solving problems in statistical dependence measures and joint marginal distribution functions. This comprehensive study served as a review paper on the development of copulas from their initial existence to their latest evolution.

Discovering Low-Dimensional Descriptions of Multineuronal Dependencies

Coordinated activity in neural populations is crucial for information processing. Shedding light on the multivariate dependencies that shape multineuronal responses is important to understand neural codes. However, existing approaches based on pairwise linear correlations are inadequate at capturing complicated interaction patterns and miss features that shape aspects of the population function. Copula-based approaches address these shortcomings by extracting the dependence structures in the joint probability distribution of population responses. In this study, we aimed to dissect neural dependencies with a C-Vine copula approach coupled with normalizing flows for estimating copula densities. While this approach allows for more flexibility compared to fitting parametric copulas, drawing insights on the significance of these dependencies from large sets of copula densities is challenging. To alleviate this challenge, we used a weighted non-negative matrix factorization procedure to leverage shared latent features in neural population dependencies. We validated the method on simulated data and applied it on copulas we extracted from recordings of neurons in the mouse visual cortex as well as in the macaque motor cortex. Our findings reveal that neural dependencies occupy low-dimensional subspaces, but distinct modules are synergistically combined to give rise to diverse interaction patterns that may serve the population function.

A framework for exploring environmental risk of the longest inter-basin water diversion project under the influence of multiple factors: A case study in China

Multi-factor risk assessment is an important prerequisite for water quality protection and the safe operation of mega hydro-projects. As the largest long-distance inter-basin water diversion project in the world, the Middle Route of the South-to-North Water Diversion Project of China (MRSNWDPC) has been in operation for 8 years and has benefited 79 million people along the canal. However, concerns have been raised in recent years about the potential negative effects of abnormal algal proliferation in the MRSNWDPC. It is very important for the safety of water supply to carry out relevant risk analysis and formulate regulatory management. In order to quantitatively evaluate the risk of algal proliferation in the MRSNWDPC under the influence of multiple factors, a multivariate risk assessment method based on Vine Copula theory and Monte Carlo simulation was proposed. Five key factors (water temperature, flow velocity, flow rate, algal cell density, and dissolved oxygen) were used and multiple dependency models in each section of the MRSNWDPC from January 2016 to January 2019 were established to study the risk of algal proliferation under multiple scenarios. The results demonstrate that water temperature can be used as an appropriate early-warning indicator of algal proliferation. The early-warning interval (unit: °C) of water temperature in the upper, middle, and lower reaches are 26-29°C, 23-26°C, and 21-23°C, respectively. Unlike bivariate analysis, the multiple dependency model describes the relationship between variables more accurately and enriches the scenarios of multiple conditional probabilities. When the water temperature fluctuates in the early-warning interval, regulating the upstream, midstream, and downstream flow velocity to be higher than 0.6 m/s, 0.5 m/s, and 0.6 m/s, respectively, can effectively reduce the risk of algal proliferation. This research not only provides a reference for the ecological control of algae in the MRSNWDPC and similar mega hydro-projects but also enriches the application of the Vine Copula theory coupled with the random sampling method for multi-variable risk analysis.

Mixed vine copula flows for flexible modeling of neural dependencies

Recordings of complex neural population responses provide a unique opportunity for advancing our understanding of neural information processing at multiple scales and improving performance of brain computer interfaces. However, most existing analytical techniques fall short of capturing the complexity of interactions within the concerted population activity. Vine copula-based approaches have shown to be successful at addressing complex high-order dependencies within the population, disentangled from the single-neuron statistics. However, most applications have focused on parametric copulas which bear the risk of misspecifying dependence structures. In order to avoid this risk, we adopted a fully non-parametric approach for the single-neuron margins and copulas by using Neural Spline Flows (NSF). We validated the NSF framework on simulated data of continuous and discrete types with various forms of dependency structures and with different dimensionality. Overall, NSFs performed similarly to existing non-parametric estimators, while allowing for considerably faster and more flexible sampling which also enables faster Monte Carlo estimation of copula entropy. Moreover, our framework was able to capture low and higher order heavy tail dependencies in neuronal responses recorded in the mouse primary visual cortex during a visual learning task while the animal was navigating a virtual reality environment. These findings highlight an often ignored aspect of complexity in coordinated neuronal activity which can be important for understanding and deciphering collective neural dynamics for neurotechnological applications.

Vine Regression with Bayes Nets: A Critical Comparison with Traditional Approaches Based on a Case Study on the Effects of Breastfeeding on IQ

Regular vines (R-vines) copulas build high dimensional joint densities from arbitrary one-dimensional margins and (conditional) bivariate copula densities. Vine densities enable the computation of all conditional distributions, though the calculations can be numerically intensive. Saturated continuous nonparametric Bayes nets (CNPBN) are regular vines. Computing regression functions from the vine copula density is termed vine regression. The epicycles of regression-including/excluding covariates, interactions, higher order terms, multicollinearity, model fit, transformations, heteroscedasticity, bias-are dispelled. One simply computes the regressions from the vine copula density. Only the question of finding an adequate vine copula remains. Vine regression is applied to a data set from the National Longitudinal Study of Youth relating breastfeeding to IQ. The expected effects of breastfeeding on IQ depend on IQ, on the baseline level of breastfeeding, on the duration of additional breastfeeding and on the values of other covariates. A child given two weeks breastfeeding can expect to increase his/her IQ by 1.5-2 IQ points by adding 10 weeks of breastfeeding, depending on values of other covariates. A child given two years breastfeeding can expect to gain from 0.48-0.65 IQ points from 10 additional weeks. Adding 10 weeks breastfeeding to each of the 3,179 children in this data set has a net present value $50,700,000 according to the Bayes net, compared to $29,000,000 according to the linear regression.

A D-vine copula-based quantile regression model with spatial dependence for COVID-19 infection rate in Italy

The main determinants of COVID-19 spread in Italy are investigated, in this work, by means of a D-vine copula based quantile regression. The outcome is the COVID-19 cumulative infection rate registered on October 30th 2020, with reference to the 107 Italian provinces, and it is regressed on some covariates of interest accounting for medical, environmental and demographic factors. To deal with the issue of spatial autocorrelation, the D-vine copula based quantile regression also embeds a spatial autoregressive component that controls for the extent of spatial dependence. The use of vine copula enhances model flexibility accounting for non-linear relationships and tail dependencies. Moreover, the model selection procedure leads to parsimonious models providing a rank of covariates based on their explanatory power with respect to the outcome.

Parametric Copula-GP model for analyzing multidimensional neuronal and behavioral relationships

One of the main goals of current systems neuroscience is to understand how neuronal populations integrate sensory information to inform behavior. However, estimating stimulus or behavioral information that is encoded in high-dimensional neuronal populations is challenging. We propose a method based on parametric copulas which allows modeling joint distributions of neuronal and behavioral variables characterized by different statistics and timescales. To account for temporal or spatial changes in dependencies between variables, we model varying copula parameters by means of Gaussian Processes (GP). We validate the resulting Copula-GP framework on synthetic data and on neuronal and behavioral recordings obtained in awake mice. We show that the use of a parametric description of the high-dimensional dependence structure in our method provides better accuracy in mutual information estimation in higher dimensions compared to other non-parametric methods. Moreover, by quantifying the redundancy between neuronal and behavioral variables, our model exposed the location of the reward zone in an unsupervised manner (i.e., without using any explicit cues about the task structure). These results demonstrate that the Copula-GP framework is particularly useful for the analysis of complex multidimensional relationships between neuronal, sensory and behavioral variables.

Understanding the relationship between a set of variables is a common problem in many fields, such as weather forecast or stock market data. In neuroscience, one of the main challenges is to characterize the dependencies between neuronal activity, sensory stimuli and behavioral outputs. A method of choice for modeling such statistical dependencies is based on copulas, which disentangle dependencies from single variable statistics. To account for changes in dependencies, we model changes in copula parameters by means of Gaussian Processes, conditioned on a task-related variable. The novelty of our approach includes 1) explicit modeling of the dependencies; and 2) combining different copulas to describe experimentally observed variability. We validate the goodness-of-fit as well as information estimates on synthetic data and on recordings from the visual cortex of mice performing a behavioral task. Our parametric model demonstrates significantly better performance in describing high dimensional dependencies compared to other commonly used techniques. We demonstrate that our model can estimate information and predict behaviorally-relevant parameters of the task without providing any explicit cues to the model. Our results indicate that our model is interpretable in the context of neuroscience applications, scalable to large datasets and suitable for accurate statistical modeling and information estimation.

Conditional copula models for correlated survival endpoints: Individual patient data meta-analysis of randomized controlled trials

Correlations among survival endpoints are important for exploring surrogate endpoints of the true endpoint. With a valid surrogate endpoint tightly correlated with the true endpoint, the efficacy of a new drug/treatment can be measurable on it. However, the existing methods for measuring correlation between two endpoints impose an invalid assumption: correlation structure is constant across different treatment arms. In this article, we reconsider the definition of Kendall's concordance measure (tau) in the context of individual patient data meta-analyses of randomized controlled trials. According to our new definition of Kendall's tau, its value depends on the treatment arms. We then suggest extending the existing copula (and frailty) models so that their Kendall's tau can vary across treatment arms. Our newly proposed model, a joint frailty-conditional copula model, is the implementation of the new definition of Kendall's tau in meta-analyses. In order to facilitate our approach, we develop an original R function condCox.reg(.) and make it available in the R package joint.Cox (https://CRAN.R-project.org/package=joint.Cox). We apply the proposed method to a gastric cancer dataset (3288 patients in 14 randomized trials from the GASTRIC group). This data analysis concludes that Kendall's tau has different values between the surgical treatment arm and the adjuvant chemotherapy arm (p-value<0.001), whereas disease-free survival remains a valid surrogate at individual level for overall survival in these trials.

A Mixture of Regular Vines for Multiple Dependencies

To uncover complex hidden dependency structures among variables, researchers have used a mixture of vine copula constructions. To date, these have been limited to a subclass of regular vine models, the so-called drawable vine, fitting only one type of bivariate copula for all variable pairs. However, the variation of complex hidden correlations from one pair of variables to another is more likely to be present in many real datasets. Single-type bivariate copulas are unable to deal with such a problem. In addition, the regular vine copula model is much more capable and flexible than its subclasses. Hence, to fully uncover and describe complex hidden dependency structures among variables and provide even further flexibility to the mixture of regular vine models, a mixture of regular vine models, with a mixed choice of bivariate copulas, is proposed in this paper. The model was applied to simulated and real data to illustrate its performance. The proposed model shows significant performance over the mixture of R-vine densities with a single copula family fitted to all pairs.

A numerical strategy to evaluate performance of predictive scores via a copula-based approach

Assessing and comparing the performance of correlated predictive scores are of current interest in precision medicine. Given the limitations of available theoretical approaches for assessing and comparing the predictive accuracy, numerical methods are highly desired which, however, have not been systematically developed due to technical challenges. The main challenges include the lack of a general strategy on effectively simulating many kinds of correlated predictive scores each with some given level of predictive accuracy in either concordance index or the area under a receiver operating characteristic curve area under the curves (AUC). To fill in this important knowledge gap, this paper is to provide a general copula-based numeric framework for assessing and comparing predictive performance of correlated predictive or risk scores. The new algorithms are designed to effectively simulate correlated predictive scores with given levels of predictive accuracy as measured in terms of concordance indices or time-dependent AUC for predicting survival outcomes. The copula-based numerical strategy is convenient for numerically evaluating and comparing multiple measures of predictive accuracy of correlated risk scores and for investigating finite-sample properties of test statistics and confidence intervals as well as assessing for optimism of given performance measures using cross-validation or bootstrap.

Dependence modeling for recurrent event times subject to right-censoring with D-vine copulas

In many time-to-event studies, the event of interest is recurrent. Here, the data for each sample unit correspond to a series of gap times between the subsequent events. Given a limited follow-up period, the last gap time might be right-censored. In contrast to classical analysis, gap times and censoring times cannot be assumed independent, i.e., the sequential nature of the data induces dependent censoring. Also, the number of recurrences typically varies among sample units leading to unbalanced data. To model the association pattern between gap times, so far only parametric margins combined with the restrictive class of Archimedean copulas have been considered. Here, taking the specific data features into account, we extend existing work in several directions: we allow for nonparametric margins and consider the flexible class of D-vine copulas. A global and sequential (one- and two-stage) likelihood approach are suggested. We discuss the computational efficiency of each estimation strategy. Extensive simulations show good finite sample performance of the proposed methodology. It is used to analyze the association of recurrent asthma attacks in children. The analysis reveals that a D-vine copula detects relevant insights, on how dependence changes in strength and type over time.

Covariance Model Simulation Using Regular Vines

We propose a new and flexible simulation method for non-normal data with user-specified marginal distributions, covariance matrix and certain bivariate dependencies. The VITA (VIne To Anything) method is based on regular vines and generalizes the NORTA (NORmal To Anything) method. Fundamental theoretical properties of the VITA method are deduced. Two illustrations demonstrate the flexibility and usefulness of VITA in the context of structural equation models. R code for the implementation is provided.

Approximate Uncertainty Modeling in Risk Analysis with Vine Copulas

Many applications of risk analysis require us to jointly model multiple uncertain quantities. Bayesian networks and copulas are two common approaches to modeling joint uncertainties with probability distributions. This article focuses on new methodologies for copulas by developing work of Cooke, Bedford, Kurowica, and others on vines as a way of constructing higher dimensional distributions that do not suffer from some of the restrictions of alternatives such as the multivariate Gaussian copula. The article provides a fundamental approximation result, demonstrating that we can approximate any density as closely as we like using vines. It further operationalizes this result by showing how minimum information copulas can be used to provide parametric classes of copulas that have such good levels of approximation. We extend previous approaches using vines by considering nonconstant conditional dependencies, which are particularly relevant in financial risk modeling. We discuss how such models may be quantified, in terms of expert judgment or by fitting data, and illustrate the approach by modeling two financial data sets.