Marginal Distributions Research Articles

The Implication Problem for Functional Dependencies and Variants of Marginal Distribution Equivalences

We study functional dependencies together with two different probabilistic dependency notions: unary marginal identity and unary marginal distribution equivalence. A unary marginal identity states that two variables \(x\) and \(y\) are identically distributed. A unary marginal distribution equivalence states that the multiset consisting of the marginal probabilities of all the values for variable \(x\) is the same as the corresponding multiset for \(y\) . We present a sound and complete axiomatization for the class of these dependencies and show that it has Armstrong relations. The axiomatization is infinite, but we show that there can be no finite axiomatization. The implication problem for the subclass that contains only functional dependencies and unary marginal identities can be simulated with functional dependencies and unary inclusion atoms, and therefore the problem is in polynomial-time. This complexity bound also holds in the case of the full class, which we show by constructing a polynomial-time algorithm.

A convexity-constrained parameterization of the random effects generalized partial credit model.

An alternative closed-form expression for the marginal joint probability distribution of item scores under the random effects generalized partial credit model is presented. The closed-form expression involves a cumulant generating function and is therefore subjected to convexity constraints. As a consequence, complicated moment inequalities are taken into account in maximum likelihood estimation of the parameters of the model, so that the estimation solution is always proper. Another important favorable consequence is that the likelihood function has a single local extreme point, the global maximum. Furthermore, attention is paid to expected a posteriori person parameter estimation, generalizations of the model, and testing the goodness-of-fit of the model. Procedures proposed are demonstrated in an illustrative example.

Flexible Bayesian modeling for longitudinal binary and ordinal responses

Longitudinal studies with binary or ordinal responses are widely encountered in various disciplines, where the primary focus is on the temporal evolution of the probability of each response category. Traditional approaches build from the generalized mixed effects modeling framework. Even amplified with nonparametric priors placed on the fixed or random effects, such models are restrictive due to the implied assumptions on the marginal expectation and covariance structure of the responses. We tackle the problem from a functional data analysis perspective, treating the observations for each subject as realizations from subject-specific stochastic processes at the measured times. We develop the methodology focusing initially on binary responses, for which we assume the stochastic processes have Binomial marginal distributions. Leveraging the logits representation, we model the discrete space processes through continuous space processes. We utilize a hierarchical framework to model the mean and covariance kernel of the continuous space processes nonparametrically and simultaneously through a Gaussian process prior and an Inverse-Wishart process prior, respectively. The prior structure results in flexible inference for the evolution and correlation of binary responses, while allowing for borrowing of strength across all subjects. The modeling approach can be naturally extended to ordinal responses. Here, the continuation-ratio logits factorization of the multinomial distribution is key for efficient modeling and inference, including a practical way of dealing with unbalanced longitudinal data. The methodology is illustrated with synthetic data examples and an analysis of college students’ mental health status data.

On the Generating Mechanisms of Daily Precipitation in the Conterminous United States: Climatology, Trends, and Associated Marginal and Extreme Distributions

On the Generating Mechanisms of Daily Precipitation in the Conterminous United States: Climatology, Trends, and Associated Marginal and Extreme Distributions

Probabilistic modeling of multivariate extreme non-Gaussian wind loads and its applications in envelope engineering

Probabilistic modeling for non-Gaussian wind load fields under extreme winds is crucial to the quantitative risk and reliability analysis of envelope structures. Based on the multi-point synchronous wind load data during typhoons, a probabilistic model is proposed to characterize multivariate non-Gaussian wind loads to improve the accuracy of the risk analysis considering data dimension reduction. In the proposed model, the Pareto distribution function is utilized to fit the marginal distribution, and the copula function is employed to construct the dependent structure of wind loads from different locations. In the marginal fitted results, the extreme wind load value in the edge region is about 0.10–0.15 higher than those from inner measurement points at quantile p = 95 %. In addition, the variation gradient from the different variable directions is approximately equal, and a sharp variation occurs at the point (0.10, 0.10) for bivariate joint distributions. Engineering applications given in this article are composed of three parts: calculating regional exceeding probability, estimating wind load vector under a specific regional exceeding probability, and determining the location of joint extreme wind loads using the joint gradient value. Finally, risk analysis of existing roof coverings is carried out, the edge corner region is weaker, and two engineering recommendations are given. For determining the regional design wind load, a comprehensive calculation framework considering the temporal dependence of different locations is established, and the reduction rates are in the range (5 %, 25 %).

Tail index estimation for tail adversarial stable time series with an application to high‐dimensional tail clustering

For stationary time series with regularly varying marginal distributions, an important problem is to estimate the associated tail index which characterizes the power‐law behavior of the tail distribution. For this, various results have been developed for independent data and certain types of dependent data. In this article, we consider the problem of tail index estimation under a recently proposed notion of serial tail dependence called the tail adversarial stability. Using the technique of adversarial innovation coupling and a martingale approximation scheme, we establish the consistency and central limit theorem of the tail index estimator for a general class of tail dependent time series. Based on the asymptotic normal distribution from the obtained central limit theorem, we further consider an application to cluster a large number of regularly varying time series based on their tail indices by using a robust mixture algorithm. The results are illustrated using numerical examples including Monte Carlo simulations and a real data analysis.

Optimal configuration of dynamic VAR compensators considering uncertainty and correlation

AbstractThe transient voltage stability of modern power systems is challenged by the increasing uncertainty on source‐load dual sides. To cope with it, a robust optimal configuration method for dynamic VAR compensators (DVCs) is proposed. First, a series of power‐flow scenarios are generated with uncertainty and correlation considered, where nonparametric kernel density estimation is used to predict the marginal distribution of wind speeds and combined Copula function is employed to characterize correlations between nearby wind farms. Then, the variation‐of‐information indicator is introduced to assess the similarity among power‐flow (PF) scenarios, and representative PF (RPF) scenarios are screened with the help of Spectral clustering algorithm. Finally, locating and sizing of DVCs for RPF scenarios are achieved. By synthesizing the compensation schemes for RPF scenarios, the robust configuration scheme is suggested. Simulation studies in the modified New England 10‐machine 39‐bus system show the proposed method can ensure transient voltage security of the studied power system under source‐load uncertainty.

Few-shot intent detection with mutual information and contrastive learning

Few-shot intent detection is a challenging task. Most existing methods only focus on acquisition of generalization knowledge in known classes, or on the adaptation situation of meta-learning tasks. However, these methods lack both the learning of beneficial information for intent detection and the learning of discriminative representation of intent keywords. Solving these problems can effectively improve the model’s ability to recognize text intent and thus effectively execute user commands. Therefore, in this paper, we propose a few-shot intent detection framework based on Mutual Information and Contrastive Learning (MICL). It improves the robustness of the model learning of intent keywords through mutual information maximization, thus improving the accuracy of intent detection. Additionally, maximization of mutual information enhances the intent representation consistency on two text views,11In this paper, the “two text views” of the text represent the original text and its corresponding data augmentation text.which helps to promote the efficiency of sample contrastive learning. Specifically, we use a neural network to approximately estimate the joint probability distribution and marginal probability distribution of two text views and maximize the mutual information through back-propagation. In order to enable the model to focus on intent keywords in text, we introduce a sample-level contrastive learning. It helps the model to learn discriminative representations of intent keywords by comparing the similarity of text from two text views. In addition, to prevent overfitting of the model on known classes, we introduce a class-level contrastive learning. We regard it as regularization of known classes, aiming to improve the model generalization performance on unknown classes. Our method achieves state-of-the-art performance on four public datasets.22Our method code is available: https://github.com/YS19999/MICL.

Distributionally Robust Portfolio Optimization under Marginal and Copula Ambiguity

AbstractWe investigate a new family of distributionally robust optimization problem under marginal and copula ambiguity with applications to portfolio optimization problems. The proposed model considers the ambiguity set of portfolio returns in which the marginal distributions and their copula are close—in terms of the Wasserstein distance—to their nominal counterparts. We develop a cutting-surface method to solve the proposed problem, in which the distribution separation subproblem is nonconvex and includes bilinear terms. We propose three approaches to solve the bilinear formulation, namely (1) linear relaxation via McCormick inequalities, (2) exact mixed-integer linear program reformulation via disjunctive inequalities, and (3) inner approximation method via a novel iterative procedure that exploits the structural properties of the bilinear optimization problem. We further carry out a comprehensive set of computational experiments with distributionally robust portfolios featuring Conditional Value-at-Risk (CVaR) measures. These tests aim to compare the accuracy of the proposed algorithms, analyze the impact of the radius of the Wasserstein ambiguity ball on the portfolio, and assess portfolio performance. We use a rolling-horizon approach to conduct the out-of-sample tests, which show the superior performance of the portfolios under marginal and copula ambiguity over the equally weighted and ambiguity-free Mean-CVaR benchmark portfolios.

MODELING THE BENEFITS OF A MARRIAGE REVERSE ANNUITY CONTRACT WITH DEPENDENCY ASSUMPTIONS USING ARCHIMEDEAN COPULA

Social security benefits may not be enough for retirement. Equity release products like marriage reverse annuities can boost retirement income for older couples. Marriage reverse annuity’s contract convert all or part of the real estate value of elderly spouses while they are living (joint life status) or after one partner dies (last survivor status). Since husband and wife face the same death risk, the chance of death between spouses is believed to be dependent for realism. Thus, copula models the future dependency model of a husband and wife. Sklar's theorem states that copulas link bivariate distribution and marginal cumulative functions. One of the most common copulas is Archimedean copula. Clayton, Gumbel, and Frank are Archimedean copula that will be used in this investigation. The Indonesian Mortality Table IV data is used to obtain the marginal distribution of the male and female which will then be used to construct copulas (Clayton, Gumbel, and Frank) that combine two marginal distributions into a joint distribution. The marginal distribution of Indonesian Mortality Table IV is uncertain, hence Canonical Maximum Likelihood parameter estimation is utilized to estimate the parameter of copulas. Multiple-state models depict the marriage reverse annuity model for joint life and last survivor status. The probability structure is based on Sklar's theorem and copula survival function. The contract benefits calculation utilizing copulas (Clayton, Gumbel, and Frank) shows that joint life status benefits are higher than last survivor status. Joint life status uses the dependence assumption with Frank's copula to calculate the smallest annual benefit value of a marriage reverse annuity contract, while last survivor status uses the independence assumption (without copula).

Bivariate BMM-based hybrid domain image watermark detector

Robustness, imperceptibility, and watermark capacity are three indispensable and contradictory properties for any image watermarking systems. It is a challenging work to achieve the balance among the three important properties. In this paper, by using the bivariate Beta mixture model (Bivariate BMM), we present a statistical image watermark scheme in nonsubsampled Contourlet transform (NSCT)-fractional order Jacobi Fourier moments (FJFMs) amplitude hybrid domain. The whole watermarking algorithm includes two parts: watermark embedding and detection. NSCT is firstly performed on host image to obtain the frequency subbands, and the NSCT subbands are divided into non overlapping blocks. Then, the significant NSCT domain blocks are selected using local binary patterns (LBP). Meanwhile, for each selected NSCT coefficient block, FJFMs are calculated to obtain the NSCT-FJFMs amplitude. Finally, watermark signals are inserted into the amplitude hybrid domain of NSCT-FJFMs. In order to detect accurately watermark signal, the statistical characteristics of NSCT-FJFMs magnitudes are analyzed in detail. Then, NSCT-FJFMs magnitudes are described statistically by Bivariate BMM, which can simultaneously capture the marginal distribution and strong dependencies of NSCT-FJFMs magnitudes. Also, Bivariate BMM parameters are estimated accurately by the rough-enhanced-Bayes mixture estimation & expectation–maximization (REBMIX&EM) approach. Finally, a statistical watermark detector based on the locally optimum (LO) decision rule and Bivariate BMM is developed in NSCT-FJFMs magnitude hybrid domain. Also, the closed-form expressions on the LO statistic is derived and the receiver operating characteristic (ROC) about our detector is analyzed. Extensive experimental results show the superiority of the proposed image watermark detector over some state-of-the-art statistical watermarking methods.

Pricing Model Construction and Empirical Research on Catastrophe Bond Pricing for Three Major Crops in China

Agricultural disaster risks are significant for the sector's development, and existing risk management tools have limitations.This paper introduces a nationwide catastrophe bond for wheat, maize, and rice , offering broad coverage for China's agriculture. The paper utilizes marginal distribution and Copula function fitting to model the disaster losses of the three major crops from 1990 to 2022. It employs Monte Carlo simulation to generate a 10,000-year event set and rank correlation to align simulated data with real data. The bond price is then calculated using the Wang model, addressing the challenge of data scarcity and providing an innovative approach to catastrophe bond design. The study finds that bond prices increase with the coverage period and risk level, with interest-risk bonds offering higher returns and risks than principal-and-interest risk bonds. Catastrophe bonds present a more stable risk diversification method compared to reinsurance, both in terms of returns and potential losses.To further the development of agricultural catastrophe bonds, the paper recommends optimizing the issuance environment and establishing a robust market infrastructure.

Bivariate Pareto–Feller Distribution Based on Appell Hypergeometric Function

The Pareto–Feller distribution has been widely used across various disciplines to model “heavy-tailed” phenomena, where extreme events such as high incomes or large losses are of interest. In this paper, we present a new bivariate distribution based on the Appell hypergeometric function with marginal Pareto–Feller distributions obtained from two independent gamma random variables. The proposed distribution has the beta prime marginal distributions as special case, which were obtained using a Kibble-type bivariate gamma distribution, and the stochastic representation was obtained by the quotient of a scale mixture of two gamma random variables. This result can be viewed as a generalization of the standard bivariate beta I (or inverted bivariate beta distribution). Moreover, the obtained bivariate density is based on two confluent hypergeometric functions. Then, we derive the probability distribution function, the cumulative distribution function, the moment-generating function, the characteristic function, the approximated differential entropy, and the approximated mutual information index. Based on numerical examples, the exact and approximated expressions are shown.

Iterative Methods for Vecchia-Laplace Approximations for Latent Gaussian Process Models

Latent Gaussian process (GP) models are flexible probabilistic non-parametric function models. Vecchia approximations are accurate approximations for GPs to overcome computational bottlenecks for large data, and the Laplace approximation is a fast method with asymptotic convergence guarantees to approximate marginal likelihoods and posterior predictive distributions for non-Gaussian likelihoods. Unfortunately, the computational complexity of combined Vecchia-Laplace approximations grows faster than linearly in the sample size when used in combination with direct solver methods such as the Cholesky decomposition. Computations with Vecchia-Laplace approximations can thus become prohibitively slow precisely when the approximations are usually the most accurate, i.e., on large data sets. In this article, we present iterative methods to overcome this drawback. Among other things, we introduce and analyze several preconditioners, derive new convergence results, and propose novel methods for accurately approximating predictive variances. We analyze our proposed methods theoretically and in experiments with simulated and real-world data. In particular, we obtain a speed-up of an order of magnitude compared to Cholesky-based calculations and a threefold increase in prediction accuracy in terms of the continuous ranked probability score compared to a state-of-the-art method on a large satellite data set. All methods are implemented in a free C++ software library with high-level Python and R packages. 4

GenSynthPop: generating a spatially explicit synthetic population of individuals and households from aggregated data

Synthetic populations are representations of actual individuals living in a specific area. They play an increasingly important role in studying and modeling individuals and are often used to build agent-based social simulations. Traditional approaches for synthesizing populations use a detailed sample of the population (which may not be available) or combine data into a single joint distribution, and draw individuals or households from these. The latter group of existing sample-free methods fail to integrate (1) the best available data on spatial granular distributions, (2) multi-variable joint distributions, and (3) household level distributions. In this paper, we propose a sample-free approach where synthetic individuals and households directly represent the estimated joint distribution to which attributes are iteratively added, conditioned on previous attributes such that the relative frequencies within each joint group of attributes are maintained and fit granular spatial marginal distributions. In this paper we present our method and test it for the Zuid-West district of The Hague, the Netherlands, showing that spatial, multi-variable and household distributions are accurately reflected in the resulting synthetic population.

On optimal control of coupled mean-field forward-backward stochastic equations

Abstract We consider a control problem for a mean-field coupled forward-backward stochastic differential equations, called also McKean–Vlasov equation (MF-FBSDE). For this type of equations, the coefficients depend not only on the state of the system, but also on its marginal distributions. They arise naturally in mean-field control problems and mean-field games. We consider the relaxed control problem where admissible controls are measure-valued processes. We prove the existence of a relaxed optimal control by using a suitable form of Skorokhod representation theorem and Jakubowski’s topology, on the space of càdlàg functions. We use martingale measure to define the relaxed state process. Our results extend to MF-FBSDEs those already known for forward and backward stochastic equations of Itô type.

Inferences about robust measures of effect size for two dependent variables including situations where there is a covariate

In contrast to using means, there are two distinct approaches to developing robust measures effect size when dealing with two dependent variables. One is based on difference scores and the other focuses on the marginal distributions. The former approach is trivial based on extant results. This paper focuses on new robust methods based on the marginal distributions that take into account the strength of the association between the two dependent variables. Simulations are used to assess the extent a reasonably accurate confidence interval can be computed. Included is a proposed method for situations where there is a covariate.

A [formula omitted]-means triangular synthesis large margin classifier with unified pinball loss for imbalanced data

Large Margin Distribution Machine (LDM) improves the Support Vector Machine (SVM) by integrating the marginal distribution of samples into the objective function, which exhibits excellent classification and generalization performance. However, most of the existing LDM-based models exhibit inherent flaws: (1) The assumption of a category uniform distribution, resulting in an inability to effectively address the issue of class imbalance. (2) The inheritance of the hinge loss in SVM, leading to inferior anti-noise performance. Given the shortcomings above, this paper constructs a novel K-means Triangular Synthesis (KTS) method for imbalanced data classification and introduces Unified Pinball (UP) loss to ensure robust performance, demonstrated as a novel KTS large margin classifier with UP Loss (KTS-UPLMC) model. Specifically, the KTS method utilizes the triangular synthesis and generates samples based on the cluster density, effectively mitigating the problems of introducing noise samples and strip-shaped sample distribution in the traditional synthetic minority oversampling technique, further alleviating the LDM’s classification line offset. In addition, the UP Loss considers quantile distance, improving the anti-noise performance of the model. Comparative experiments on artificial datasets and benchmark datasets validate the effectiveness and noise resistance of the proposed KTS-UPLMC in imbalanced data classification problems. Furthermore, the stability of the KTS-UPLMC is substantiated by the parameter sensitivity analysis experiment. In conclusion, the KTS method effectively addresses category imbalance, while the integration of the UP Loss enhances noise resistance.

Asymptotic limits of transient patterns in a continuous-space interacting particle system

We study a discrete-time interacting particle system with continuous state space, which is motivated by a mathematical model for turnover through branching in actin filament networks. It gives rise to transient clusters reminiscent of actin filament assemblies in the cortex of living cells. We reformulate the process in terms of the inter-particle distances and characterize their marginal and joint distributions. We construct a recurrence relation for the associated characteristic functions and pass to the large population limit, reminiscent of the Fleming–Viot super processes. The precise characterization of all marginal distributions established in this work opens the way to a detailed analysis of cluster dynamics. We also obtain a recurrence relation that enables us to compute the moments of the asymptotic single particle distribution characterizing the transient aggregates. Our results indicate that aggregates have a fat-tailed distribution.

Enhancing Drought Risk Assessment in the Punjab, Pakistan: A Copula-Based Modeling Approach for Future Projections

Abstract Drought is one of the most complicated and challenging natural hazards, which occurs nearly in every part of the world and poses recurring challenges to agriculture, food security, livestock, human health, and water management. Pakistan has a long history of drought; however, this study focuses on drought analysis and projection in the province of Punjab, Pakistan, as it provides around 60% of the country’s food product, significantly contributing to the national food supply and economy. This study utilized the previous 56 years (1962–2017) of climate data to calculate the reconnaissance drought index (RDI) and then extracted the drought variables of durations and severity for each meteorological station. The best-fit marginal probability distribution and copula models were chosen for the stations based on numerical as well as graphical evaluation. Lognormal and exponential probability distributions, as well as Gumbel, are selected as the best-fit probability distributions for both drought characteristics and bivariate copula model, respectively, for projections. From the projections, we can infer that the smaller return periods indicate high vulnerability while longer return periods with low vulnerability. The results suggest that Faisalabad, Bahawalpur, Bahawalnagar, and Multan stations have the lowest return periods, indicating high vulnerability, and may experience drought more frequently in the future. Mianwali, Khanpur, Lahore, and Sialkot stations may have an intermediate vulnerability to drought events. The stations of Jhelum, Murree, and Sargodha have larger return periods, implying lower susceptibility to drought events in the future. The projected results provide insights for policymakers and stakeholders to optimize the risk of droughts on agriculture production, livestock, water management, human health, and food security in Punjab, Pakistan.