Gaussian Copula Research Articles

MSR57 Survival Analysis without Sharing of Individual Person Data: An Antidote to “Data Available upon Request”?

Cox regression and Kaplan-Meier estimates are often needed in post-approval clinical research which requires access to individual person data (IPD). Think about a long-term treatment-effect review or a comparative effectiveness oncology study involving the IPD owned by a competitor. Here, IPD cannot always be shared due to privacy or proprietary restrictions, which complicates the making of such estimates. I propose a method that generates pseudodata replacing the IPD by only using non-disclosive IPD aggregates which are shared with a central computer and are input parameters to a Gaussian copula (GC) generating the pseudodata, allowing analysts to access completely anonymised versions of possibly multiple datasets on which to run analyses.

Seismic activity analysis of five major earthquake source segments in the Sumatra megathrust zone

The Sumatra megathrust zone has five major earthquake sources, namely the Aceh-Andaman, Nias-Simeulue, Mentawai-Siberut, Mentawai-Pagai, and Enggano segments. This paper provides seismic activity analysis in these five segments via an unobserved process study of tectonic plate movements, which is conducted in two cases: each of the five segments independently (Case 1), and a pair of two adjacent segments (Case 2). To do this, two specific types of Hidden Markov Models (HMMs), i.e., Poisson-HMMs and Copula-HMMs, dealing with unobserved process issues, are applied. In practice, the data used is the annual frequency of mainshock earthquakes with a magnitude of >4.6 that occurred from 1971 to 2018. This data is obtained by working out the declustering process and estimating the magnitude of completeness from a particular earthquake catalogue. Due to the incompleteness of the data sets, the parameters of the two HMMs are estimated using the Expectation-Maximization algorithm. Results show that for Case 1, the model that fits the data for each of the five segments is the 3-state Poisson-HMM. The three states, in this instance, stand for the rates of seismic activity that correspond to the dynamic level of tectonic plate movements. Furthermore, in Case 2, the selected model for the Aceh-Andaman with Nias-Simeulue is the 2-state Gumbel Copula-HMM. Meanwhile, for the three groups remaining, namely Nias-Simeulue with Mentawai-Siberut, Mentawai-Siberut with Mentawai-Pagai, and Mentawai-Pagai with Enggano, the appropriate models are Gaussian, Gumbel, and Frank Copulas, respectively. In this case, the number of states represents the seismic activity association of two adjacent segments that corresponds to the association level of two adjacent tectonic plate dynamics.

Transformed Pair Copula Construction of Pareto Copula and Applications

The study introduced the transformed copula models and a D-vine structure for three variables. The numerical applications of the models are evaluated using two different sets of real-life data that exhibit nearly no dependence, highly dependent, over, and under dispersed characteristics. We examined only the volatilities of the first data using the exponentiated weighted moving average (EWMA). The parameter estimates of the models were obtained based on the maximum likelihood estimation method for the bivariate copula models and the Dissmann algorithm for sequential top-down estimation for the D-vine structure. The results showed that the introduced copula models outperformed some existing copula models in terms of their fit statistics for both real-life and simulated data sets. In addition, the Gaussian copula model gave a better fit to the D-vine structure than some existing copula models and could be recommended for modeling a D-vine structure comprising of variables that are positively weak correlated and highly correlated.

Adding Art to Your Portfolio: A New Risk Assessment Framework

This article presents a new framework to evaluate the merits of an art investment that differs substantially from previous studies. First, it assumes that the investor already holds a portfolio consisting of more traditional assets and is planning to add art to it. This is far more realistic than the usual academic set up in which it is assumed that the investor is planning to deploy a given amount of cash among many assets, one of which is art. Second, the approach departs from the traditional Markowitz’s mean-variance framework in two important ways: (i) the efficient frontier is constructed based on cumulative returns, rather than average returns, and risk is assessed via potential losses and not volatility; and (ii) it relies on a semi-parametric approach to generate synthetic data based on the Gaussian copula and historic returns. Finally, an alternative risk metric, based on losses and not volatility, is introduced. The usefulness of this framework is demonstrated with an example based on art sales auction data.

A Rapid Estimation Method for Post-earthquake Building Losses

Rapid estimation of post-earthquake building damage and loss is very important in urgent response efforts. The current approach leaves much room for improvement in estimating ground motion and correctly incorporating the uncertainty and spatial correlation of the loss. This study proposed a new approach for rapidly estimating post-earthquake building loss with reasonable accuracy. The proposed method interpolates ground motion based on the observed ground motion using the Ground Motion Prediction Equation (GMPE) as the weight. It samples the building seismic loss quantile considering the spatial loss correlation that is expressed by Gaussian copula, and kriging is applied to reduce the dimension of direct sampling for estimation speed. The proposed approach was validated using three historical earthquake events in Japan with actual loss reports, and was then applied to predict the building loss amount for the March 2022 Fukushima Mw7.3 earthquake. The proposed method has high potential in future emergency efforts such as search, rescue, and evacuation planning.

Quantifying evoked responses through information-theoretical measures.

Information theory is a viable candidate to advance our understanding of how the brain processes information generated in the internal or external environment. With its universal applicability, information theory enables the analysis of complex data sets, is free of requirements about the data structure, and can help infer the underlying brain mechanisms. Information-theoretical metrics such as Entropy or Mutual Information have been highly beneficial for analyzing neurophysiological recordings. However, a direct comparison of the performance of these methods with well-established metrics, such as the t-test, is rare. Here, such a comparison is carried out by evaluating the novel method of Encoded Information with Mutual Information, Gaussian Copula Mutual Information, Neural Frequency Tagging, and t-test. We do so by applying each method to event-related potentials and event-related activity in different frequency bands originating from intracranial electroencephalography recordings of humans and marmoset monkeys. Encoded Information is a novel procedure that assesses the similarity of brain responses across experimental conditions by compressing the respective signals. Such an information-based encoding is attractive whenever one is interested in detecting where in the brain condition effects are present.

Impact of SST Anomalies on Coral Reefs Damage Based on Copula Analysis

The condition of coral reefs in Indonesia is alarming. One of the influenting factors of coral reefs damage is extreme climate change. The aim of this study is to determine the relationship of climate change, that is Sea Surface Temperature (SST) anomaly index, and coral reefs damage in West, Central and East Region of Indonesia. The method used in this study is Copula analysis. Copula is one of the statistical methods used to determine the relationship of two or more variables, in which case the distribution can be normal or not. First, data is transformed into Uniform [0,1] domain. Then, Copula parameter is estimated to get significance parameter. Lastly, the best Copula that has the highest log likelihood value is selected to represent the relationship of data. The result indicates that percentage of coral reefs damage in West and Central Region has relationship with SST Nino 4, while coral reefs damage in East Region does not have relationship with any of SST Nino anomalies. In West Region, the best Copula represents the relationship is Gaussian Copula (parameter = -0.32); it concludes that the higher the value of SST Nino 4, the lower the percentage of coral reefs damage and otherwise. While in Central Indonesia, Frank Copula (parameter = -4.89) is selected; it does not have tail dependency so that the SST Nino 4 and the percentage of coral reefs in damage condition in Central Region has low correlation.

Socioeconomic and geospatial determinants of households’ food and non-food consumption dynamics within the West African Economic and Monetary Union

This article explores the cross-sectional properties of household's food and non-food consumption expenditures conceptualized as inter-dependent spatial random processes within the West African Economic and Monetary Union (WAEMU). To this end, the paper applies spatio-temporal econometric methods to data extracted from the first edition of the Harmonized Survey on Household Living Conditions (EHCVM, 2018/2019). The findings from the unconditional distribution of household economic wellbeing (measured by nominal per-capita food and non-food expenditures) using the inequality indicators of the Atkinson and Gini coefficients, suggest the presence of cross-sectional inequalities in household food and non-food wellness within WAEMU. However, food wellness inequality appeared lower than that of non-food wellness across the 8 countries. In addition, the economic security analysis, based on the indices of Watts, Sen, and Foster (aplha=1), highlight important spatial heterogeneity in household's vulnerability to poverty within-and-between WAEMU country members. Moreover, estimation of the semiparametric bivariate Gaussian copula model of food and non-food consumption expenditures, shows their significant dependence with household's demographic and geospatial characteristics, as well as the economic and health characteristics of the household head. Indeed, among other key findings, a gender based gradient in household's food and non-food wellness at respectively 3.6 and 4.4% is observed in favor of male headed households; along with an Urban/Rural divide in food and non-food wellness at 20.7% and 27.3%, respectively, in favor of urban households. Therefore, in line with goals 1, 2, 4, 5, 9 and 10 of the United Nations 2030 Agenda for sustainable development, the paper provides key insights for evidence based policy making within WAEMU.

The principle of two models: the cases of Black-Scholes formula for interest rates and of Gaussian copula for credit

In this article, I try to show that a given market segment needs at least two models: one as a pricing convention and one to run the portfolio of securities. These models should be related in such a way that the model to manage the books is an extension of the pricing convention. Financial modeling succeeded in the case of options, but it failed with the CDOs, resulting in the credit crunch.

Efficient and Unbiased Safety Test for Autonomous Driving Systems

Test the safety of Autonomous Driving Systems (ADS) with realistic traffic conditions is important to the insurance industry, legislators, and third-party technical services. However, the scarcity of risky driving events distributed in real-world driving often makes sampling inefficient. In this paper, we propose a unified and hierarchical testing framework for efficient and unbiased safety tests of ADS. We first extract the risk subspace from the Naturalistic Driving Data (NDD) with defined safety measures. Subsequently, we use the Gaussian Copula method to accurately model the subspace joint probability density function (PDF). We further formulate a Kriging model-based optimization problem for finding the appropriate Importance Sampling (IS) parameters, where the Kriging model is trained iteratively within limited computational resources. Eventually, the risky concrete scenarios library is generated using the obtained Importance Sampling parameters. With the simulation results skewered back to the original risk subspace, the ADS probability of failure (e.g., crash rate) can thus be accurately and unbiasedly estimated. Experiment results show that using 0.94% computational resources, the Kriging surrogate model captures 96.04% of ADS crashes with at least 90.25% precision. On crash rate estimation, our proposed method achieves a consistent result with that of Monte Carlo simulation, provided that the relative error does not exceed 10% while improving the testing efficiency by up to 6,539 times.

A copula-based method of risk prediction for autonomous underwater gliders in dynamic environments.

Autonomous underwater gliders (AUGs) are effective platforms for oceanic research and environmental monitoring. However, complex underwater environments with uncertainties could pose the risk of vehicle loss during their missions. It is therefore essential to conduct risk prediction to assist decision making for safer operations. The main limitation of current studies for AUGs is the lack of a tailored method for risk analysis considering both dynamic environments and potential functional failures of the vehicle. Hence, this study proposed a copula-based approach for evaluating the risk of AUG loss in dynamic underwater environments. The developed copula Bayesian network (CBN) integrated copula functions into a traditional Bayesian belief network (BBN), aiming to handle nonlinear dependencies among environmental variables and inherent technical failures. Specifically, potential risk factors with causal effects were captured using the BBN. A Gaussian copula was then employed to measure correlated dependencies among identified risk factors. Furthermore, the dependence analysis and CBN inference were performed to assess the risk level of vehicle loss given various environmental observations. The effectiveness of the proposed method was demonstrated in a case study, which considered deploying a Slocum G1 Glider in a real water region. Risk mitigation measures were provided based on key findings. This study potentially contributes a tailored tool of risk prediction for AUGs in dynamic environments, which can enhance the safety performance of AUGs and assist in risk mitigation for decision makers.

Robust and smoothing variable selection for quantile regression models with longitudinal data

In this paper, we propose a penalized weighted quantile estimating equations (PWQEEs) method to obtain sparse, robust and efficient estimators for the quantile regression with longitudinal data. The PWQEE incorporates the within correlations in the longitudinal data by Gaussian copulas and can also down-weight the high leverage points in covariates to achieve double-robustness to both the non-normal distributed errors and the contaminated covariates. To overcome the obstacles of discontinuity of the PWQEE and nonconvex optimization, a local distribution smoothing method and the minimization–maximization algorithm are proposed. The asymptotic properties of the proposed method are also proved. Furthermore, finite sample performance of the PWQEE is illustrated by simulation studies and a real-data example.

How many sites? Methods to assist design decisions when collecting multivariate data in ecology

Abstract Sample size estimation through power analysis is a fundamental tool in planning an ecological study, yet there are currently no well‐established procedures for when multivariate abundances are to be collected. A power analysis procedure would need to address three challenges: designing a parsimonious simulation model that captures key community data properties; measuring effect size in a realistic yet interpretable fashion; and ensuring computational feasibility when simulation is used both for power estimation and significance testing. Here, we propose a power analysis procedure that addresses these three challenges by: using for simulation a Gaussian copula model with factor analytical structure, fitted to pilot data; assuming a common effect size across all taxa, but applied in different directions according to expert opinion (to “increaser”, “decreaser” or “no effect” taxa); using a critical value approach to estimate power, which reduces computation time by a factor of 500 (if we would otherwise use 999 resamples to estimate each p‐value) with minor loss of accuracy. The procedure is demonstrated on pilot data from fish assemblages in a restoration study, where it was found that the planned study design would only be capable of detecting relatively large effects (change in abundance by a factor of 1.7 or more). The methods outlined in this paper are available in accompanying R software (the ecopower package), which allows researchers with pilot data to answer a wide range of design questions to assist them in planning their studies.

Costa’s concavity inequality for dependent variables based on the multivariate Gaussian copula

AbstractAn extension of Shannon’s entropy power inequality when one of the summands is Gaussian was provided by Costa in 1985, known as Costa’s concavity inequality. We consider the additive Gaussian noise channel with a more realistic assumption, i.e. the input and noise components are not independent and their dependence structure follows the well-known multivariate Gaussian copula. Two generalizations for the first- and second-order derivatives of the differential entropy of the output signal for dependent multivariate random variables are derived. It is shown that some previous results in the literature are particular versions of our results. Using these derivatives, concavity of the entropy power, under certain mild conditions, is proved. Finally, special one-dimensional versions of our general results are described which indeed reveal an extension of the one-dimensional case of Costa’s concavity inequality to the dependent case. An illustrative example is also presented.

Modeling the presumed joint probability density function of conditioning variables in stratified turbulent flames

Experimental data from the Cambridge-Sandia burner are used to calculate the conditional average of temperature and the mass fractions of H2O, CO, and CO2 in the context of the Uniform Conditional State (UCS) model. The performance of various presumed joint PDF models are examined for this burner working with different swirl and stratification levels. All of the presented PDF models are based on a few statistical moments of the conditioning variables which are extracted from the data at each measuring point. Primarily, only mixture fraction and reaction progress variable are employed as the conditioning variables. The β function is employed for modeling the marginal PDF of mixture fraction in all cases. For the progress variable, the β function and Modified Laminar Flamelet (MLF)-PDF are tested for modeling the marginal PDF. It is shown that even though the MLF-PDF results in more accurate predictions overall for most of the cases, its superiority with respect to the β-PDF is relatively small in the stratified flames. The performance of the product, Plackett, and Gaussian copulas are also investigated for employing the cross-correlation of mixture fraction and progress variable in building their joint PDF. It is found that the Gaussian copula shows a superior performance over the other two. Next, normalized total enthalpy is employed as a third conditioning variable, and different PDF models are examined for the three-condition version of UCS. It is found that even though adding the third conditioning variable can improve the predictions by considering the effect of heat transfer on the chemistry manifold, such improvements depend on having a proper 3D PDF model for the conditioning variables. In the vicinity of the flame brush, the models employed for the 3D PDFs represent an inferior performance compared to the two-condition version with the use of a Gaussian copula due to complications in the marginal PDF of normalized total enthalpy in these regions. To benefit from considering the impact of heat transfer on chemistry while also avoiding complexities in modeling the 3D joint PDF near the reaction zone, a switch between the two- and three-conditional models which is triggered by the variance of progress variable is proposed. It is shown that the predictions obtained by the use of this switch are almost always superior to all other PDF models investigated in this study.

A Multi-Parameter Fusion Method for Cuffless Continuous Blood Pressure Estimation Based on Electrocardiogram and Photoplethysmogram

Blood pressure (BP) is an essential physiological indicator to identify and determine health status. Compared with the isolated BP measurement conducted by traditional cuff approaches, cuffless BP monitoring can reflect the dynamic changes in BP values and is more helpful to evaluate the effectiveness of BP control. In this paper, we designed a wearable device for continuous physiological signal acquisition. Based on the collected electrocardiogram (ECG) and photoplethysmogram (PPG), we proposed a multi-parameter fusion method for noninvasive BP estimation. An amount of 25 features were extracted from processed waveforms and Gaussian copula mutual information (MI) was introduced to reduce feature redundancy. After feature selection, random forest (RF) was trained to realize systolic BP (SBP) and diastolic BP (DBP) estimation. Moreover, we used the records in public MIMIC-III as the training set and private data as the testing set to avoid data leakage. The mean absolute error (MAE) and standard deviation (STD) for SBP and DBP were reduced from 9.12 ± 9.83 mmHg and 8.31 ± 9.23 mmHg to 7.93 ± 9.12 mmHg and 7.63 ± 8.61 mmHg by feature selection. After calibration, the MAE was further reduced to 5.21 mmHg and 4.15 mmHg. The result showed that MI has great potential in feature selection during BP prediction and the proposed multi-parameter fusion method can be used for long-term BP monitoring.

A Copula-Based Meta-Stochastic Frontier Analysis for Comparing Traditional and HDPE Geomembranes Technology in Sea Salt Farming among Farmers in Phetchaburi, Thailand

This study analyzed sea salt production and compared the technical efficiency level and the technology gap between traditional technology and High-Density Polyethylene Geomembranes (HDPE GMB) technology in the Phetchaburi province using a copula-based meta-stochastic frontier technique. A total sample size of 250 was chosen, comprising 195 traditional farmers and 55 HDPE GMBs farmers. Several copula families were used to analyze the dependence structure of the two error components and the best-fit copula-based meta-frontier model used Gaussian copulas. Land, labor, and fuel energy are the most significant input variables in the Gaussian copula-based meta-frontier model with a translog production function. Compared to meta-frontier production, the average technological gap between traditional technology production and HDPE GMB technology production was 0.69 and 0.77, respectively, meaning HDPE GMB technology is more technically efficient than traditional technology. The study identified that land, market price, sex, and experience were the contributing technical inefficiency factors for traditional technology production. For HDPE GMB technology production, land, sex, and experience were found to be contributing factors. The performance of HDPE GMB technology in salt farming in the Phetchaburi province suggests that public and private sector agencies should promote greater access to this technology for salt farmers.

PC vs App vs Mweb: price discounts’ effect on customer purchases across digital channels in China

PurposeProduct discounts have been widely applied in digital commerce as a method to attract and retain customers to purchase in China. Given that digital channels differ in their attributes, customers may behave differently when they respond to the same discount across channels. However, little attention has been paid to explore the heterogeneity of customer responses to discounts across channels. This study aims to fill in the gap by exploring how customers’ purchase responses to price discounts differ across digital channels.Design/methodology/approachThis study applies a Poisson regression to an unbalanced data set of purchasing history from Chinese footwear brands with 3,510 customers in a two-year time window across the three digital channels (i.e. personal computer [PC], app and mobile website), with a correction for endogeneity by using the Gaussian copula method.FindingsThis paper finds that price discounts have the strongest positive effect on consumers’ purchase volumes on the PC channel, followed by the app channel, while discounts show the weakest impact in the mobile website channel. By so, this paper demonstrates that customers respond differently to online and mobile channels, and they also respond differently within mobile channels when they purchase products with price discounts.Originality/valueThis study is original in analyzing the difference in customers’ discount responses across digital channels, offering valuable contributions to existing research on multichannel marketing as well as mobile marketing and providing helpful insights for multichannel merchants to design digital discount strategies.

Controlling the false discovery rate by a Latent Gaussian Copula Knockoff procedure

Controlling the false discovery rate by a Latent Gaussian Copula Knockoff procedure

A Bayesian approach to study design and analysis with type I error rate control for response variables of mixed types.

There has been increased interest in the design and analysis of studies consisting of multiple response variables of mixed types. For example, in clinical trials, it is desirable to establish efficacy for a treatment effect in primary and secondary outcomes. In this article, we develop Bayesian approaches for hypothesis testing and study planning for data consisting of multiple response variables of mixed types with covariates. We assume that the responses are correlated via a Gaussian copula, and that the model for each response is, marginally, a generalized linear model (GLM). Taking a fully Bayesian approach, the proposed method enables inference based on the joint posterior distribution of the parameters. Under some mild conditions, we show that the joint distribution of the posterior probabilities under any Bayesian analysis converges to a Gaussian copula distribution as the sample size tends to infinity. Using this result, we develop an approach to control the type I error rate under multiple testing. Simulation results indicate that the method is more powerful than conducting marginal regression models and correcting for multiplicity using the Bonferroni-Holm Method. We also develop a Bayesian approach to sample size determination in the presence of response variables of mixed types, extending the concept of probability of success (POS) to multiple response variables of mixed types.