Joint Multivariate Distribution Research Articles

鄱阳湖与长江干流洪水遭遇规律及风险分析

本文引入von Mises分布及其混合形式拟合年最大洪水发生时间与间隔时间，应用Copula函数分别构造汉口、湖口两站年最大洪水发生时间二维联合分布和年最大洪水发生时间(间隔时间)与洪水量级的三维联合分布，分析了研究两站年最大洪水在时间、量级及过程三个方面的遭遇风险。结果表明：鄱阳湖和长江年最大洪水在7月16日遭遇的可能性最大，约为0.058％，鄱阳湖和长江年最大7d洪水过程发生遭遇的概率为7.31％。研究可为鄱阳湖流域防洪规划制定和生态经济区建设提供科学依据。The von Mises distribution was used to fit the series of annual maximum flood occurrence and in-terval dates. The Copula functions were used to establish the bivariate joint distribution of annual maximum flood occurrence dates and the multivariate joint distribution of interval dates and flood magnitude. The flood coincidence risk of annual maximum flood occurrence date, flood magnitude and flood process were analyzed based on the data series of Hankou and Hukou hydrological stations. The results show that the highest flood coincidence risk of annual maximum flood occurrence date and 7-day flood process between the middle Yangtze River and Poyang Lake may reach to 0.058% (on July 16th) and 7.31% respectively during the flood season. The research can provide scientific basis for Poyang Lake basin on flood prevention planning and ecological- economic zone construction.

Differentially Private Synthesization of Multi-Dimensional Data using Copula Functions.

Differential privacy has recently emerged in private statistical data release as one of the strongest privacy guarantees. Most of the existing techniques that generate differentially private histograms or synthetic data only work well for single dimensional or low-dimensional histograms. They become problematic for high dimensional and large domain data due to increased perturbation error and computation complexity. In this paper, we propose DPCopula, a differentially private data synthesization technique using Copula functions for multi-dimensional data. The core of our method is to compute a differentially private copula function from which we can sample synthetic data. Copula functions are used to describe the dependence between multivariate random vectors and allow us to build the multivariate joint distribution using one-dimensional marginal distributions. We present two methods for estimating the parameters of the copula functions with differential privacy: maximum likelihood estimation and Kendall's τ estimation. We present formal proofs for the privacy guarantee as well as the convergence property of our methods. Extensive experiments using both real datasets and synthetic datasets demonstrate that DPCopula generates highly accurate synthetic multi-dimensional data with significantly better utility than state-of-the-art techniques.

A note on the asymptotic behavior of the Bernstein estimator of the copula density

Copulas and their corresponding densities are functions of a multivariate joint distribution and the one-dimensional marginals. Bernstein estimators have been used as smooth nonparametric estimators for copulas and copula densities. The purpose of this note is to study the asymptotic distributional behavior of the Bernstein estimator of a copula density. Compared to the existing results, our general theorem does not assume known marginals. This makes our theorem applicable for real data.

The intrinsic dependence structure of peak, volume, duration, and average intensity of hyetographs and hydrographs.

[1] The information contained in hyetographs and hydrographs is often synthesized by using key properties such as the peak or maximum value Xp, volume V, duration D, and average intensity I. These variables play a fundamental role in hydrologic engineering as they are used, for instance, to define design hyetographs and hydrographs as well as to model and simulate the rainfall and streamflow processes. Given their inherent variability and the empirical evidence of the presence of a significant degree of association, such quantities have been studied as correlated random variables suitable to be modeled by multivariate joint distribution functions. The advent of copulas in geosciences simplified the inference procedures allowing for splitting the analysis of the marginal distributions and the study of the so-called dependence structure or copula. However, the attention paid to the modeling task has overlooked a more thorough study of the true nature and origin of the relationships that link , and I. In this study, we apply a set of ad hoc bootstrap algorithms to investigate these aspects by analyzing the hyetographs and hydrographs extracted from 282 daily rainfall series from central eastern Europe, three 5 min rainfall series from central Italy, 80 daily streamflow series from the continental United States, and two sets of 200 simulated universal multifractal time series. Our results show that all the pairwise dependence structures between , and I exhibit some key properties that can be reproduced by simple bootstrap algorithms that rely on a standard univariate resampling without resort to multivariate techniques. Therefore, the strong similarities between the observed dependence structures and the agreement between the observed and bootstrap samples suggest the existence of a numerical generating mechanism based on the superposition of the effects of sampling data at finite time steps and the process of summing realizations of independent random variables over random durations. We also show that the pairwise dependence structures are weakly dependent on the internal patterns of the hyetographs and hydrographs, meaning that the temporal evolution of the rainfall and runoff events marginally influences the mutual relationships of , and I. Finally, our findings point out that subtle and often overlooked deterministic relationships between the properties of the event hyetographs and hydrographs exist. Confusing these relationships with genuine stochastic relationships can lead to an incorrect application of multivariate distributions and copulas and to misleading results.

Forecasting liquidity-adjusted intraday Value-at-Risk with vine copulas

We propose to model the joint distribution of bid-ask spreads and log returns of a stock portfolio by using Autoregressive Conditional Double Poisson and GARCH processes for the marginals and vine copulas for the dependence structure. By estimating the joint multivariate distribution of both returns and bid-ask spreads from intraday data, we incorporate the measurement of commonalities in liquidity and comovements of stocks and bid-ask spreads into the forecasting of three types of liquidity-adjusted intraday Value-at-Risk (L-IVaR). In a preliminary analysis, we document strong extreme comovements in liquidity and strong tail dependence between bid-ask spreads and log returns across the firms in our sample thus motivating our use of a vine copula model. Furthermore, the backtesting results for the L-IVaR of a portfolio consisting of five stocks listed on the NASDAQ show that the proposed models perform well in forecasting liquidity-adjusted intraday portfolio profits and losses.

Copulas from the Fokker–Planck equation

We develop a theoretical framework addressing the joint distribution and provide a general equation for time-dependent copulas related to stochastic processes that arise in finance. The copula is a function that links univariate distributions to a joint multivariate distribution. The tractability and importance of a copula lie in the inference function for margins (IFM) method which is very suitable to use to achieve an understanding of many correlated statistical objects. We derive a parabolic equation for the copula governing the stochastic behavior with independent drifts and volatilities of multivariate objects. In fact, the Fokker–Planck equation for the stochastic differential equations with independent drifts and volatilities is modeled for the IFM. We also present numerical results which illustrate several sensitivity analyses of our scheme.

Evaluation of bivariate Archimedean and elliptical copulas to model wind power dependency structures

When modeling wind power from several sources, consideration of the dependency structure of the sources is of critical importance. Failure to appropriately account for the dependency structure can lead to unrealistic models, which may result in erroneous conclusions from wind integration studies and other analyses. The dependency structure is fully described by the multivariate joint distribution function of the wind power. However, fewif anyexplicit joint distribution models of wind power exist. Instead, copulas can be used to create joint distribution functions, provided that the selected copula family reasonably approximates the dependency structure. Unfortunately, there is little guidance on which copula family should be used to model wind power. The purpose of this paper is to investigate which copula families are best suited to model wind power dependency structures. Bivariate copulas are considered in particular. The paper focuses on power from wind plantscollections of wind turbines with a common interconnection pointbut the methodology can be generally extended to consider power from individual wind turbines or even aggregate wind power from entire systems. Twelve Archimedean and elliptical copulas are evaluated using hourly data from 500 wind plant pairs in the National Renewable Energy Laboratory's Eastern Dataset. The evaluation is based on (2) and Cramer-von Mises statistics. Application guidelines recommending which copula family to use are developed. It is shown that a default assumption of Gaussian dependence is not justified and that the use of Gumbel copulas can result in improved models. An illustrative example shows the application of the guidelines to model dependence of wind power sources in Monte Carlo simulations. Copyright (c) 2012 John Wiley & Sons, Ltd.

Joint modeling of multiple longitudinal cost outcomes using multivariate generalized linear mixed models

The common approach to modeling healthcare cost data is to use aggregated total cost from multiple categories or sources (e.g. inpatient, outpatient, prescriptions, etc.) as the dependent variable. However, this approach could hide the differential impact of covariates on the different cost categories. An alternative is to model each cost category separately. This could also lead to wrong conclusions due to failure to account for the interdependence among the multiple cost outcomes. Therefore, we propose a multivariate generalized linear mixed model (mGLMM) that allows for joint modeling of longitudinal cost data from multiple sources. We assessed four different approaches, (1) shared random intercept, (2) shared random intercept and slope, (3) separate random intercepts from a joint multivariate distribution, and (4) separate random intercepts and slopes from a joint multivariate distribution. Each of these approaches differs in the way they account for the correlation among the multiple cost outcomes. Comparison was made via goodness of fit measures and residual plots. Longitudinal cost data from a national cohort of 740,195 veterans with diabetes (followed from 2002-2006) was used to demonstrate joint modeling. Among examined models, the separate random intercept approach exhibited the lowest AIC/BIC in both log-normal and gamma GLMMs. However, for our data example, the shared random intercept approach seemed to be sufficient as the more complex models did not lead to qualitatively different conclusions.

Towards improved post‐processing of hydrologic forecast ensembles

AbstractForecast ensembles of hydrological and hydrometeorologial variables are prone to various uncertainties arising from climatology, model structure and parameters, and initial conditions at the forecast date. Post‐processing methods are usually applied to adjust the mean and variance of the ensemble without any knowledge about the uncertainty sources. This study initially addresses the drawbacks of a commonly used statistical technique, quantile mapping (QM), in bias correction of hydrologic forecasts. Then, an auxiliary variable, the failure index (γ), is proposed to estimate the ineffectiveness of the post‐processing method based on the agreement of adjusted forecasts with corresponding observations during an analysis period prior to the forecast date. An alternative post‐processor based on copula functions is then introduced such that marginal distributions of observations and model simulations are combined to create a multivariate joint distribution. A set of 2500 hypothetical forecast ensembles with parametric marginal distributions of simulated and observed variables are post‐processed with both QM and the proposed multivariate post‐processor. Deterministic forecast skills show that the proposed copula‐based post‐processing is more effective than the QM method in improving the forecasts. It is found that the performance of QM is highly correlated with the failure index, unlike the multivariate post‐processor. In probabilistic metrics, the proposed multivariate post‐processor generally outperforms QM. Further evaluation of techniques is conducted for river flow forecast of Sprague River basin in southern Oregon. Results show that the multivariate post‐processor performs better than the QM technique; it reduces the ensemble spread and is a more reliable approach for improving the forecast. Copyright © 2012 John Wiley & Sons, Ltd.

Multivariate drought characteristics using trivariate Gaussian and Student t copulas

AbstractThis study aims to investigate the changing properties of drought events in Weihe River basin, China, by modeling the multivariate joint distribution of drought duration, severity and peak using trivariate Gaussian and Student t copulas. Monthly precipitations of Xi'an gauge are used to illustrate the meta‐elliptical copula‐based methodology for a single‐station application. Gaussian and Student t copulas are found to produce a better fit comparing with other six symmetrical and asymmetrical Archimedean copulas, and, checked by the goodness‐of‐fit tests based on a modified bootstrap version of Rosenblatt's transformation, both of them are acceptable to model the multivariate joint distribution of drought variables. Gaussian copula, the best fitting, is employed to construct the dependence structures of positively associated drought variables so as to obtain the multivariate joint and conditional probabilities of droughts. A Kendall's return period (KRP) introduced by Salvadori and De Michele (2010) is then adopted to assess the multivariate recurrent properties of drought events, and its spatial distributions indicate that prolonged droughts are likely to break out with rather short recurrence intervals in the whole region, while drought status in the southeast seems to be severer than the northwest. The study is of some merits in terms of multivariate drought modeling using a preferable copula‐based method, the results of which could serve as a reference for regional drought defense and water resources management. Copyright © 2011 John Wiley & Sons, Ltd.

PEMODELAN REGRESI PROSES GAUSSIAN PEMODELAN REGRESI PROSES GAUSSIAN MENGGUNAKAN FUNGSI PERAGAM EKSPONENSIAL KUADRAT

Gaussian Process is a collection of random variables where any finite subset of that has a joint multivariate Gaussian distribution. A Gaussian Process is fully specified by its mean and its covariance function. One of the popular covariance functions is squared exponential that has two hyperparameters. In this paper Gaussian Process is used to made a prediction of  the number of clothes produced by PT. APAC INTI CORPORA based on the number of attending employes, the number of overtime employes, the number of brokendown machines and used materials.     Keywords : Gaussian Process, Covariace Functions, Squared Exponential

Liquidity Commonality and Risk Management

We propose to model the joint distribution of bid-ask spreads and log returns of a stock portfolio by using Autoregressive Conditional Double Poisson and GARCH processes for the marginals and vine copulas for the dependence structure. By estimating the joint multivariate distribution of both returns and bid-ask spreads from intraday data, we incorporate the measurement of commonalities in liquidity and comovements of stocks and bid-ask spreads into the forecasting of three types of liquidity-adjusted intraday Value-at-Risk (L-IVaR). In a preliminary analysis, we document strong extreme comovements in liquidity and strong tail dependence between bid-ask spreads and log returns across the firms in our sample thus motivating our use of a vine copula model. Furthermore, the backtesting results for the L-IVaR of a portfolio consisting of five stocks listed on the NASDAQ show that the proposed models perform well in forecasting liquidity-adjusted intraday portfolio profits and losses.

Hierarchical Gaussian graphical models: Beyond reversible jump

The Gaussian Graphical Model (GGM) is a popular tool for incorporating sparsity into joint multivariate distributions. The G-Wishart distribution, a conjugate prior for precision matrices satisfying general GGM constraints, has now been in existence for over a decade. However, due to the lack of a direct sampler, its use has been limited in hierarchical Bayesian contexts, relegating mixing over the class of GGMs mostly to situations involving standard Gaussian likelihoods. Recent work has developed methods that couple model and parameter moves, first through reversible jump methods and later by direct evaluation of conditional Bayes factors and subsequent resampling. Further, methods for avoiding prior normalizing constant calculations–a serious bottleneck and source of numerical instability–have been proposed. We review and clarify these developments and then propose a new methodology for GGM comparison that blends many recent themes. Theoretical developments and computational timing experiments reveal an algorithm that has limited computational demands and dramatically improves on computing times of existing methods. We conclude by developing a parsimonious multivariate stochastic volatility model that embeds GGM uncertainty in a larger hierarchical framework. The method is shown to be capable of adapting to swings in market volatility, offering improved calibration of predictive distributions.

Copula estimation of distribution algorithms based on exchangeable Archimedean copula

The two key operators in estimation of distribution algorithms (EDAs) are estimating the distribution model according to the selected population and sampling new individuals from the estimated model. Copula EDA introduces the copula theory into EDA. The copula theory provides the theoretical basis and the way to separate the multivariate joint distribution probability function into a function called copula and the univariate margins. The estimation operator and the sampling operator in copula EDA are discussed in this paper, and three exchangeable Archimedean copulas are used in copula EDA. The experimental results show that the three copula EDAs perform equivalently to some classical EDAs.

A latent variables approach for clustering mixed binary and continuous variables within a Gaussian mixture model

For clustering objects, we often collect not only continuous variables, but binary attributes as well. This paper proposes a model-based clustering approach with mixed binary and continuous variables where each binary attribute is generated by a latent continuous variable that is dichotomized with a suitable threshold value, and where the scores of the latent variables are estimated from the binary data. In economics, such variables are called utility functions and the assumption is that the binary attributes (the presence or the absence of a public service or utility) are determined by low and high values of these functions. In genetics, the latent response is interpreted as the `liability' to develop a qualitative trait or phenotype. The estimated scores of the latent variables, together with the observed continuous ones, allow to use a multivariate Gaussian mixture model for clustering, instead of using a mixture of discrete and continuous distributions. After describing the method, this paper presents the results of both simulated and real-case data and compares the performances of the multivariate Gaussian mixture model and of a mixture of joint multivariate and multinomial distributions. Results show that the former model outperforms the mixture model for variables with different scales, both in terms of classification error rate and reproduction of the clusters means.

Using Copulas to Introduce Dependence in Dose-Response Modeling of Multiple Binary Endpoints

We propose a method for introducing dependence in the dose-response modeling of multiple dichotomous endpoints. The method uses a copula to define a joint multivariate distribution that is consistent with predetermined marginal distributions representing the individual dose-response functions for each endpoint. Use of copulas allows the marginal dose-response functions for each dose-endpoint combination to be unrestricted in form. An application of particular relevance to risk assessment is the dose-response modeling of multiple types of tumors in test animals exposed to a carcinogen, allowing for tumors at different sites in the same animal to be statistically dependent. In addition, the method can be used to address the possibility that different tissues/organs are subject to different internal doses and possibly different active moieties. These applications are illustrated with rodent cancer bioassay data from two example compounds.

Elementary multivariate rearrangements and stochastic dominance on a Fréchet class

A Fréchet class collects all multivariate joint distribution functions that have the same marginals. Members of a Fréchet class only differ with respect to the interdependence between their marginals. In this paper, I study orders of interdependence on a Fréchet class using two multivariate generalizations of the bivariate rearrangement proposed by Epstein and Tanny (1980) [4] and Tchen (1980) [16]. I show how these multivariate rearrangements are underlying multivariate first order stochastic dominance in terms of the joint distribution function and the survival function. A combination of both rearrangements is shown to be equivalent to the concordance order proposed by Joe (1990) [9].

An uncertain regression model

PurposeThe purpose of this paper is to propose an uncertain regression model with an intrinsic error structure facilitated by an uncertain canonical process.Design/methodology/approachThis model is suitable for dealing with expert's knowledge ranging from small to medium size data of impreciseness. In order to have a rigorous mathematical treatment on the new regression model, this paper establishes a series of new uncertainty concepts sequentially, such as uncertainty joint multivariate distribution, the uncertainty distribution of uncertainty product variables and uncertain covariance and correlation based on the axiomatic uncertainty theoretical foundation. Two examples are given for illustrating a small data regression analysis.FindingsThe uncertain regression model is formulated and the estimation of the model coefficients is developed.Practical implicationsThe paper is devoted to a regression model to handle a small amount of data with mathematical rigor.Originality/valueThe theory and the methodology of the uncertain canonical process regression is proposed for the first time. It addresses the practical challenges of small data size modelling.

Dynamic Method for Portfolio Choice of Manufacturing Processes Based on Copulas

Multivariate volatility modeling is always a hot topic in academic research. It is difficult to consider how to construct multivariate joint distribution. Copulas, a statistic method, can be used to decompose multivariate joint distribution into marginal distribution and correlation structure. This advantage is applied into calculating the dynamic VAR of a portfolio in the paper. Furthermore, a new model for dynamic portfolio choice based on copulas is proposed, and empirical analysis is operated for the several typical discrete manufacturing processes in the end.

A modeling framework for the analysis of HPV incidence and persistence: a semi-parametric approach for clustered binary longitudinal data analysis.

Human papillomavirus (HPV) infection is a common sexually transmitted disease of growing public health importance, and over 40 genotypes have been identified in genital infections. Current HPV cohort studies often follow participants at pre-determined visits, such as every 6 months, and data generated from such epidemiology studies can be described as clustered longitudinal binary data where correlation arises in two ways: the directionless clustering due to the multiple genotypes tested within an individual, and the temporal correlation among the repeated measurements on the same genotype along time. Current analyses for identification of risk factors associated with HPV incidence and persistence often either do not fully utilize information in the data set or ignore the correlation between the multiple genotypes. Given the scientific definition of incidence and persistence, conditional probability modeling provides us a natural mathematical tool. We thus present a semi-parametric regression model for such data where full specification of the joint multivariate binary distribution is avoided by using conditioning argument to handle the temporal correlation and GEE to account for the correlation between the multiple genotypes. The model is applied to the HPV data from the Rakai male circumcision (MC) trial to evaluate the as-treated efficacy of MC and also identify modifiable risk factors for incidence and persistence of oncogenic HPV types in men. A simulation study is performed to provide empirical information on the number of individuals that is needed for satisfactory power and estimation accuracy of the association parameter estimates in future studies.