Copula Distributions Research Articles

The Cauchy Combination Test under Arbitrary Dependence Structures

Combining individual p-values to perform an overall test is often encountered in statistical applications. The Cauchy combination test (CCT) (Journal of the American Statistical Association, 2020, 115, 393–402) is a powerful and computationally efficient approach to integrate individual p-values under arbitrary dependence structures for sparse signals. We revisit this test to additionally show that (i) the tail probability of the CCT can be approximated just as well when more relaxed assumptions are imposed on individual p-values compared to those of the original test statistics; (ii) such assumptions are satisfied by six popular copula distributions; and (iii) the power of the CCT is no less than that of the minimum p-value test when the number of p-values goes to infinity under some regularity conditions. These findings are confirmed by both simulations and applications in two real datasets, thus, further broadening the theory and applications of the CCT.

Theoretical considerations when simulating data from the g-and-h family of distributions.

The g-and-h family of distributions is a computationally efficient, flexible option to model and simulate non-normal data. In spite of its popularity, there are several theoretical aspects of these distributions that need special consideration when they are used. In this paper some of these aspects are explored. In particular, through mathematical analysis it is shown that a popular multivariate generalization of the g-and-h distribution may result in marginal distributions which are no longer g-and-h distributed, that more than one set of (g,h) parameters can correspond to the same values of population skewness and excess kurtosis, and that multivariate generalizations of g-and-h distributions available in the literature are special cases of Gaussian copula distributions. A small-scale simulation is also used to demonstrate how simulation conclusions can change when different (g,h) parameters are used to simulate data, even if they imply the same population values of skewness and excess kurtosis.

Ordering fail-safe systems having dependent components with Archimedean copula and exponentiated location-scale distributions

In this paper, we study stochastic comparisons of two fail-safe systems with dependent components having an Archimedean copula for the joint survival function and general exponentiated location-scale lifetime distributions. We first present sufficient conditions for the usual stochastic order between fail-safe systems when the components have different exponentiated location-scale distributions but with the same Archimedean copula. Next, the hazard rate ordering of fail-safe systems is discussed when the components in the two systems are independent and follow multiple-outlier exponentiated location-scale models. Several examples are then presented to illustrate the theoretical results established here.

Predicting synchronous firing of large neural populations from sequential recordings.

A major goal in neuroscience is to understand how populations of neurons code for stimuli or actions. While the number of neurons that can be recorded simultaneously is increasing at a fast pace, in most cases these recordings cannot access a complete population: some neurons that carry relevant information remain unrecorded. In particular, it is hard to simultaneously record all the neurons of the same type in a given area. Recent progress have made possible to profile each recorded neuron in a given area thanks to genetic and physiological tools, and to pool together recordings from neurons of the same type across different experimental sessions. However, it is unclear how to infer the activity of a full population of neurons of the same type from these sequential recordings. Neural networks exhibit collective behaviour, e.g. noise correlations and synchronous activity, that are not directly captured by a conditionally-independent model that would just put together the spike trains from sequential recordings. Here we show that we can infer the activity of a full population of retina ganglion cells from sequential recordings, using a novel method based on copula distributions and maximum entropy modeling. From just the spiking response of each ganglion cell to a repeated stimulus, and a few pairwise recordings, we could predict the noise correlations using copulas, and then the full activity of a large population of ganglion cells of the same type using maximum entropy modeling. Remarkably, we could generalize to predict the population responses to different stimuli with similar light conditions and even to different experiments. We could therefore use our method to construct a very large population merging cells’ responses from different experiments. We predicted that synchronous activity in ganglion cell populations saturates only for patches larger than 1.5mm in radius, beyond what is today experimentally accessible.

Application of machine learning for filtered density function closure in MILD combustion

Abstract A machine learning algorithm, the deep neural network (DNN) 1 , is trained using a comprehensive direct numerical simulation (DNS) dataset to predict joint filtered density functions (FDFs) of mixture fraction and reaction progress variable in Moderate or Intense Low-oxygen Dilution (MILD) combustion. The important features of the DNS cases include mixture fraction variations, turbulent mixing lengths, exhaust gas recirculation (EGR) dilution levels, etc., posing a great challenge for data-driven modelling. The DNN architecture is built and optimised with extreme care to achieve high robustness and accuracy, resorting to dimensionality reduction techniques such as principal component analysis (PCA) to identify and remove the outliers in the training data. To better interpret the predictive ability of the DNN, two analytical joint FDF models respectively using two independent β and copula distributions, are also employed for a detailed comparison with the DNS data. The FDFs in MILD combustion behave differently compared to those in conventional flames because the reaction zones are more distributed. They generally exhibit non-regular (neither Gaussian nor bi-modal) distributions and strong cross correlations, which cannot be captured adequately by the analytical models. However, the DNN is well suited for this physico-chemically complex problem and its predictions are in excellent agreement with the DNS data for a broad range of mixture conditions and filter sizes. Furthermore, a priori assessment is conducted for filtered reaction rate closure. It is found that the DNN model significantly outperforms the analytical models for all cases showing very good predictions for the filtered reaction rate for a range of filter sizes. The DNN prediction improves as the filter size becomes larger than the characteristic reaction zone thickness while the analytical models works relatively better for smaller filter sizes. This is a clear advantage for the DNN to be used in practical LES applications.

Copula-based common cause failure models with Bayesian inferences

In general, common cause failures (CCFs) have been modeled with the assumption that components within the same group are symmetric. This assumption reduces the number of parameters required for the CCF probability estimation and allows us to use a parametric model, such as the alpha factor model. Although there are various asymmetric conditions in nuclear power plants (NPPs) to be addressed, the traditional CCF models are limited to symmetric conditions. Therefore, this paper proposes the copula-based CCF model to deal with asymmetric as well as symmetric CCFs. Once a joint distribution between the components is constructed using copulas, the proposed model is able to provide the probability of common cause basic events (CCBEs) by formulating a system of equations without symmetry assumptions. In addition, Bayesian inferences for the parameters of the marginal and copula distributions are introduced and Markov Chain Monte Carlo (MCMC) algorithms are employed to sample from the posterior distribution. Three example cases using simulated data, including asymmetry conditions in total failure probabilities and/or dependencies, are illustrated. Consequently, the copula-based CCF model provides appropriate estimates of CCFs for asymmetric conditions. This paper also discusses the limitations and notes on the proposed method.

Predicting synchronous firing of large neural populations from sequential recordings

A major goal in neuroscience is to understand how populations of neurons code for stimuli or actions. While the number of neurons that can be recorded simultaneously is increasing at a fast pace, in most cases these recordings cannot access a complete population. In particular, it is hard to simultaneously record all the neurons of the same type in a given area. Recent progress have made possible to profile each recorded neuron in a given area thanks to genetic and physiological tools, and to pool together recordings from neurons of the same type across different experimental sessions. However, it is unclear how to infer the activity of a full population of neurons of the same type from these sequential recordings. Neural networks exhibit collective behaviour, e.g. noise correlations and synchronous activity, that are not directly captured by a conditionally-independent model that would just put together the spike trains from sequential recordings. Here we show that we can infer the activity of a full population of retina ganglion cells from sequential recordings, using a novel method based on copula distributions and maximum entropy modeling. From just the spiking response of each ganglion cell to a repeated stimulus, and a few pairwise recordings, we could predict the noise correlations using copulas, and then the full activity of a large population of ganglion cells of the same type using maximum entropy modeling. Remarkably, we could generalize to predict the population responses to different stimuli and even to different experiments. We could therefore use our method to construct a very large population merging cells' responses from different experiments. We predicted synchronous activity accurately and showed it grew substantially with the number of neurons. This approach is a promising way to infer population activity from sequential recordings in sensory areas.

Control charts of mean and variance using copula Markov SPC and conditional distribution by copula

We propose control charts of mean and variance by using the Emura, Long, and Sun (2017) copula Markov statistical process control (SPC) and conditional distribution with diverse copula functions. To verify our new method, we generate bivariate simulated data by an asymmetric copula function and then make the conditional uniform transformed data by employing diverse copula distributions. We apply the conditional uniform transformed data to the Emura, Long, and Sun (2017) copula Markov SPC chart to investigate how copula directional dependence and copula tail dependence can affect the control charts of the mean and variance. For an illustrated example, we use Major League Baseball (MLB) batting average (BA) and earned run average (ERA) data from 1998 to 2016 seasons to detect a large abnormal variation of MLB statistics by using the proposed method. We show that the average run lengths (ARLs) of the control charts of conditional variance are affected by directional dependence by using the Gaussian copula beta regression (Kim and Hwang, 2017) and copula tail dependence.

Copula Model Selection of Stock Return Time Series Using Information Complexity

<em>In this paper we estimate the correlation between four different stock return prices. To accomplish this, we use the copula models to study the dependency structure between the variables. The original variables of interest are mapped into more manageable variables by considering joint and marginal distributions of these variables. Then a correlational structure between these variables are obtained. We fit several well-known copula models to the portfolio of the stock return price dataset using consistent information complexity (CICOMP) criterion along with other AIC-type criteria to choose the best copula functional model. CICOMP predominated the AIC-type criterion, both in the case when the fitted models are correctly specified. We expect to get more realistic results using other copula distributions contrary to the Gaussian copula used by Li (2000) that fails to capture the dependence between extreme events. </em>

Determination of order pattern frequency analysis using the copula method

The frequency and voluminosity of orders play a crucial role in determining the operation and scheduling, as well as the bird weight to be supplied to chicken processors. Due to the variability in frequency and voluminosity of each product, part, and size, orders have stochastic characteristics in nature; hence, the proper means to investigate orders is using probability theory and the stochastic process method. In this study, copula distributions were exploited to incorporate chicken order features into frequency analysis. This approach enabled the elucidation of the complex features of order frequency and voluminosity. These probabilistic properties provide useful information for order assessments and can be used for production/chicken supply planning and further scheduling of the industry.

From Weakly Chaotic Dynamics to Deterministic Subdiffusion via Copula Modeling

Copula modeling consists in finding a probabilistic distribution, called copula, whereby its coupling with the marginal distributions of a set of random variables produces their joint distribution. The present work aims to use this technique to connect the statistical distributions of weakly chaotic dynamics and deterministic subdiffusion. More precisely, we decompose the jumps distribution of Geisel-Thomae map into a bivariate one and determine the marginal and copula distributions respectively by infinite ergodic theory and statistical inference techniques. We verify therefore that the characteristic tail distribution of subdiffusion is an extreme value copula coupling Mittag-Leffler distributions. We also present a method to calculate the exact copula and joint distributions in the case where weakly chaotic dynamics and deterministic subdiffusion statistical distributions are already known. Numerical simulations and consistency with the dynamical aspects of the map support our results.

On the effect of long-range dependence on extreme value copula estimation with fixed marginals

ABSTRACTWe establish the existence of multivariate stationary processes with arbitrary marginal copula distributions and long-range dependence. The effect of long-range dependence on extreme value copula estimation is illustrated in the case of known marginals, by deriving functional limit theorems for a standard non parametric estimator of the Pickands dependence function and related parametric projection estimators. The asymptotic properties turn out to be very different from the case of iid or short-range dependent observations. Simulated and real data examples illustrate the results.

Risk forecasting in (T)GARCH models with uncorrelated dependent innovations

(G)ARCH-type models are frequently used for the dynamic modelling and forecasting of risk attached to speculative asset returns. While the symmetric and conditionally Gaussian GARCH model has been generalized in a manifold of directions, model innovations are mostly presumed to stem from an underlying IID distribution. For a cross section of 18 stock market indices, we notice that (threshold) (T)GARCH-implied model innovations are likely at odds with the commonly held IID assumption. Two complementary strategies are pursued to evaluate the conditional distributions of consecutive TGARCH innovations, a non-parametric approach and a class of standardized copula distributions. Modelling higher order dependence patterns is found to improve standard TGARCH-implied conditional value-at-risk and expected shortfall out-of-sample forecasts that rely on the notion of IID innovations.

Hedging strategy for ethanol processing with copula distributions

It has become important for ethanol producers to hedge input and output price risks. The purpose of this paper is to analyze an ethanol-producing firm's strategy to reduce price risks for inputs and outputs. Corn is the primary input, and the outputs are ethanol, corn oil, distillers' dried grains (DDGs), and renewable identification numbers (RINs). A theoretical model is developed including margins and risk is measured using value at risk (VaR). An empirical model is developed and extended to VaR using copulas to analyze the marginal distribution and dependence structure for input and output prices on margins. Efficient frontier curves analyzing VaR with and without copula are discussed. The results compare varying risk-strategy measures for long corn, short corn, and combining short and long corn. Sensitivity analyses are conducted for functional changes in the margin as a result of ethanol price changes.

Gaussian copula distributions for mixed data, with application in discrimination

The construction of a joint model for mixed discrete and continuous random variables that accounts for their associations is an important statistical problem in many practical applications. In this paper, we use copulas to construct a class of joint distributions of mixed discrete and continuous random variables. In particular, we employ the Gaussian copula to generate joint distributions for mixed variables. Examples include the robit-normal and probit-normal-exponential distributions, the first for modelling the distribution of mixed binary-continuous data and the second for a mixture of continuous, binary and trichotomous variables. The new class of joint distributions is general enough to include many mixed-data models currently available. We study properties of the distributions and outline likelihood estimation; a small simulation study is used to investigate the finite-sample properties of estimates obtained by full and pairwise likelihood methods. Finally, we present an application to discriminant analysis of multiple correlated binary and continuous data from a study involving advanced breast cancer patients.

Nonparametric meta-analysis for diagnostic accuracy studies.

Summarizing the information of many studies using a meta-analysis becomes more and more important, also in the field of diagnostic studies. The special challenge in meta-analysis of diagnostic accuracy studies is that in general sensitivity and specificity are co-primary endpoints. Across the studies both endpoints are correlated, and this correlation has to be considered in the analysis. The standard approach for such a meta-analysis is the bivariate logistic random effects model. An alternative approach is to use marginal beta-binomial distributions for the true positives and the true negatives, linked by copula distributions. In this article, we propose a new, nonparametric approach of analysis, which has greater flexibility with respect to the correlation structure, and always converges. In a simulation study, it becomes apparent that the empirical coverage of all three approaches is in general below the nominal level. Regarding bias, empirical coverage, and mean squared error the nonparametric model is often superior to the standard model, and comparable with the copula model. The three approaches are also applied to two example meta-analyses.

Copula Models for Sociology: Measures of Dependence and Probabilities for Joint Distributions

Often in sociology, researchers are confronted with nonnormal variables whose joint distribution they wish to explore. Yet, assumptions of common measures of dependence can fail or estimating such dependence is computationally intensive. This article presents the copula method for modeling the joint distribution of two random variables, including descriptions of the method, the most common copula distributions, and the nonparametric measures of association derived from the models. Copula models, which are estimated by standard maximum likelihood techniques, make no assumption about the form of the marginal distributions, allowing consideration of a variety of models and distributions in the margins and various shapes for the joint distribution. The modeling procedure is demonstrated via a simulated example of spousal mortality and empirical examples of (1) the association between unemployment and suicide rates with time series models and (2) the dependence between a count variable (days drinking alcohol) and a skewed, continuous variable (grade point average) while controlling for predictors of each using the National Longitudinal Survey of Youth 1997. Other uses for copulas in sociology are also described.

Smooth Tests of Copula Specifications

We present a family of smooth tests for the goodness of fit of semiparametric multivariate copula models. The proposed tests are distribution free and can be easily implemented. They are diagnostic and constructive in the sense that when a null distribution is rejected, the test provides useful pointers to alternative copula distributions. We then propose a method of copula density construction, which can be viewed as a multivariate extension of Efron and Tibshirani. We further generalize our methods to the semiparametric copula-based multivariate dynamic models. We report extensive Monte Carlo simulations and three empirical examples to illustrate the effectiveness and usefulness of our method.

Comparative efficiency research (COMER): meta-analysis of cost-effectiveness studies.

BackgroundThe aim of this study was to create a new meta-analysis method for cost-effectiveness studies using comparative efficiency research (COMER).MethodsWe built a new score named total incremental net benefit (TINB), with inverse variance weighting of incremental net benefits (INB). This permits determination of whether an alternative is cost-effective, given a specific threshold (TINB > 0 test). Before validation of the model, the structure of dependence between costs and quality-adjusted life years (QoL) was analysed using copula distributions. The goodness-of-fit of a Spanish prospective observational study (n = 498) was analysed using the Independent, Gaussian, T, Gumbel, Clayton, Frank and Placket copulas. Validation was carried out by simulating a copula distribution with log-normal distribution for costs and gamma distribution for disutilities. Hypothetical cohorts were created by varying the sample size (n: 15–500) and assuming three scenarios (1-cost-effective; 2-non-cost-effective; 3-dominant). The COMER result was compared to the theoretical result according to the incremental cost-effectiveness ratio (ICER) and the INB, assuming a margin of error of 2,000 and 500 monetary units, respectively.ResultsThe Frank copula with positive dependence (−0.4279) showed a goodness-of-fit sufficient to represent costs and QoL (p-values 0.524 and 0.808). The theoretical INB was within the 95% confidence interval of the TINB, based on 15 individuals with a probability > 80% for scenarios 1 and 2, and > 90% for scenario 3. The TINB > 0 test with 15 individuals showed p-values of 0.0105 (SD: 0.0411) for scenario 1, 0.613 (SD: 0.265) for scenario 2 and < 0.0001 for scenario 3.ConclusionsCOMER is a valid tool for combining cost-effectiveness studies and may be of use to health decision makers.Electronic supplementary materialThe online version of this article (doi:10.1186/1471-2288-14-139) contains supplementary material, which is available to authorized users.

Using Copula distributions to support more accurate imaging-based diagnostic classifiers for neuropsychiatric disorders

Many investigators have tried to apply machine learning techniques to magnetic resonance images (MRIs) of the brain in order to diagnose neuropsychiatric disorders. Usually the number of brain imaging measures (such as measures of cortical thickness and measures of local surface morphology) derived from the MRIs (i.e., their dimensionality) has been large (e.g. >10) relative to the number of participants who provide the MRI data (<100). Sparse data in a high dimensional space increase the variability of the classification rules that machine learning algorithms generate, thereby limiting the validity, reproducibility, and generalizability of those classifiers. The accuracy and stability of the classifiers can improve significantly if the multivariate distributions of the imaging measures can be estimated accurately. To accurately estimate the multivariate distributions using sparse data, we propose to estimate first the univariate distributions of imaging data and then combine them using a Copula to generate more accurate estimates of their multivariate distributions. We then sample the estimated Copula distributions to generate dense sets of imaging measures and use those measures to train classifiers. We hypothesize that the dense sets of brain imaging measures will generate classifiers that are stable to variations in brain imaging measures, thereby improving the reproducibility, validity, and generalizability of diagnostic classification algorithms in imaging datasets from clinical populations. In our experiments, we used both computer-generated and real-world brain imaging datasets to assess the accuracy of multivariate Copula distributions in estimating the corresponding multivariate distributions of real-world imaging data. Our experiments showed that diagnostic classifiers generated using imaging measures sampled from the Copula were significantly more accurate and more reproducible than were the classifiers generated using either the real-world imaging measures or their multivariate Gaussian distributions. Thus, our findings demonstrate that estimated multivariate Copula distributions can generate dense sets of brain imaging measures that can in turn be used to train classifiers, and those classifiers are significantly more accurate and more reproducible than are those generated using real-world imaging measures alone.