One-dimensional Marginal Distributions Research Articles

On the construction of stationary processes and random fields

Abstract We propose a new method to construct a stationary process and random field with a given decreasing covariance function and any one-dimensional marginal distribution. The result is a new class of stationary processes and random fields. The construction method utilizes a correlated binary sequence, and it allows a simple and practical way to model dependence structures in a stationary process and random field as its dependence structure is induced by the correlation structure of a few disjoint sets in the support set of the marginal distribution. Simulation results of the proposed models are provided, which show the empirical behavior of a sample path.

Bottom Multi-Parameter Bayesian Inversion Based on an Acoustic Backscattering Model

The geoacoustic and physical properties of the bottom, as well as spatial distribution, are crucial factors in analyzing the underwater acoustic field structure and establishing a geoacoustic model. Acoustic inversion has been widely used as an economical and effective method to obtain multi-parameters of the bottom. Compared with traditional inversion methods based on acoustic propagation models, acoustic backscattering models are more suitable for multi-parameter inversion, because they contain more bottom information. In this study, a Bayesian inversion method based on an acoustic backscattering model is proposed to obtain bottom multi-parameters, including geoacoustic parameters (the sound speed and loss parameter), partial physical parameters of the sediment, and statistical parameters of the seafloor roughness and sediment heterogeneity. The bottom was viewed as a kind of fluid medium. A high-frequency backscattering model based on fluid theory was adopted as the forward model to fit the scattering strength between the model prediction and the measured data. The Bayesian inversion method was used to obtain the posterior probability density (PPD) of the inversion parameters. Parameter estimation, uncertainty, and correlation were acquired by calculating the maximum a posterior (MAP), the mean values, the one-dimensional marginal distributions of the PPD, and the covariance matrix. Finally, the high-frequency bottom backscattering strength from the Quinault Range site was employed for inversion tests. The estimated values and uncertainties of various bottom parameters are presented and compared with the directly measured bottom parameters. The comparison results demonstrate that the method proposed herein can be used to estimate the sediment/water sound speed ratio, the sediment/water density ratio, and the spectral exponent of the roughness spectrum effectively and reliably.

Using Triples to Assess Symmetry Under Weak Dependence

ABSTRACT The problem of assessing symmetry about an unspecified center of the one-dimensional marginal distribution of a strictly stationary random process is considered. A well-known U-statistic based on data triples is used to detect deviations from symmetry, allowing the underlying process to satisfy suitable mixing or near-epoch dependence conditions. We suggest using subsampling for inference on the target parameter, establish the asymptotic validity of the method in our setting, and discuss data-driven rules for selecting the size of subsamples. The small-sample properties of the proposed inferential procedures are examined by means of Monte Carlo simulations. Applications to time series of output growth and stock returns are also presented.

Whole-Brain Imaging of Subvoxel T1-Diffusion Correlation Spectra in Human Subjects.

T1 relaxation and water mobility generate eloquent MRI tissue contrasts with great diagnostic value in many neuroradiological applications. However, conventional methods do not adequately quantify the microscopic heterogeneity of these important biophysical properties within a voxel, and therefore have limited biological specificity. We describe a new correlation spectroscopic (CS) MRI method for measuring how T1 and mean diffusivity (MD) co-vary in microscopic tissue environments. We develop a clinical pulse sequence that combines inversion recovery (IR) with single-shot isotropic diffusion encoding (IDE) to efficiently acquire whole-brain MRIs with a wide range of joint T1-MD weightings. Unlike conventional diffusion encoding, the IDE preparation ensures that all subvoxel water pools are weighted by their MDs regardless of the sizes, shapes, and orientations of their corresponding microscopic diffusion tensors. Accordingly, IR-IDE measurements are well-suited for model-free, quantitative spectroscopic analysis of microscopic water pools. Using numerical simulations, phantom experiments, and data from healthy volunteers we demonstrate how IR-IDE MRIs can be processed to reconstruct maps of two-dimensional joint probability density functions, i.e., correlation spectra, of subvoxel T1-MD values. In vivo T1-MD spectra show distinct cerebrospinal fluid and parenchymal tissue components specific to white matter, cortical gray matter, basal ganglia, and myelinated fiber pathways, suggesting the potential for improved biological specificity. The one-dimensional marginal distributions derived from the T1-MD correlation spectra agree well with results from other relaxation spectroscopic and quantitative MRI studies, validating the T1-MD contrast encoding and the spectral reconstruction. Mapping subvoxel T1-diffusion correlations in patient populations may provide a more nuanced, comprehensive, sensitive, and specific neuroradiological assessment of the non-specific changes seen on fluid-attenuated inversion recovery (FLAIR) and diffusion-weighted MRIs (DWIs) in cancer, ischemic stroke, or brain injury.

Analisis Pengendalian Kualitas Menggunakan Bagan Kendali Bivariat dengan Copula pada Industri Manufactur di Indonesia

Product quality control is an important factor for the industrial world because good quality control and carried out continuously will be able to detect abnormal production results quickly, so that anticipatory action can be taken immediately. Quality is a major factor in consumer decision making before buying goods / services. The problem that occurs at this time in manufacturing companies in Indonesia is how the statistical quality control process can be applied properly. The purpose of this statistical analysis is to find out the statistical quality control process that is applied to manufacturing companies in Indonesia using bivariate control charts with copula. Copula is a function that combines a multivariate distribution function with a uniform one-dimensional marginal distribution function, in this condition the Copula used is the Archimedean Copula group. The method used in this data collection is a simple random sampling with the sample used are three manufacturing companies in Indonesia which covers the areas of Jakarta, Bandung and Makassar. The implementation of Copula in this control chart results in Frank Copula being the best Copula, this supports that the use of Copula in the quality control process has a good role

On Modeling a Class of Weakly Stationary Processes

In this article, we show that a general class of weakly stationary time series can be modeled applying Gaussian subordinated processes. We show that, for any given weakly stationary time series $(z_t)_{z\in\N}$ with given equal one-dimensional marginal distribution, one can always construct a function $f$ and a Gaussian process $(X_t)_{t\in\N}$ such that $\left(f(X_t)\right)_{t\in\N}$ has the same marginal distributions and, asymptotically, the same autocovariance function as $(z_t)_{t\in\N}$. Consequently, we obtain asymptotic distributions for the mean and autocovariance estimators by using the rich theory on limit theorems for Gaussian subordinated processes. This highlights the role of Gaussian subordinated processes in modeling general weakly stationary time series. We compare our approach to standard linear models, and show that our model is more flexible and requires weaker assumptions.

Dependency in multisensory integration: a copula-based analysis.

The notion of copula has attracted attention from the field of contextuality and probability. A copula is a function that joins a multivariate distribution to its one-dimensional marginal distributions. Thereby, it allows characterizing the multivariate dependency separately from the specific choice of margins. Here, we demonstrate the use of copulas by investigating the structure of dependency between processing stages in a stochastic model of multisensory integration, which describes the effect of stimulation by several sensory modalities on human reaction times. We derive explicit terms for the covariance and Kendall's tau between the processing stages and point out the specific role played by two stochastic order relations, the usual stochastic order and the likelihood ratio order, in determining the sign of dependency. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.

Optimal Estimation of Shear Strength Parameters Based on Copula Theory Coupling Information Diffusion Technique

In geotechnical reliability analysis, random volatility in marginal distributions of shear strength parameters has been rarely considered. Unfortunately, conventional marginal distribution models cannot characterize real probability distribution accurately, leading to considerable dispersion with incomplete probabilistic information. In this paper, an estimation methodology is proposed based on copula theory coupling information diffusion technique. Firstly, information diffusion distribution is extended to represent one‐dimensional marginal distributions of shear strength parameters. Secondly, copula theory is employed to characterize the dependence structures among the parameters. Eventually, equivalent sample is yielded by information diffusion distribution that has been already established. A case study in Singapore is implemented to enunciate and validate the competence of the proposed method. The performances of the candidate copulas coupling different marginal distributions are further discussed. Results indicate that information diffusion distribution can efficiently capture the random volatility of real distributions of shear strength parameters and hold remarkable superiority in modeling marginal distributions. The equivalent sample, estimated by information diffusion technique in conjunction with Gaussian copula, has considerable consistency with original data. The proposed method can provide a reference to reliability analysis in geotechnical engineering.

A copula-based estimation of distribution algorithm for calibration of microscopic traffic models

The importance of calibration of microscopic traffic models as the main core of traffic simulation software results from the need for more realistic traffic behaviors. The latent essence of several parameters in such models as well as the uncertainties resulting from the noise in the data, make the process of calibration much more complex. Usually, the calibration process is formulated as an optimization problem. Selecting the appropriate solution algorithm due to nonlinear and non-convex nature of the problem is crucial. The importance of the issue is more significant when the matter of calibrating the medium or large-scale simulation model is considered. This is mainly due to the expensive cost that running the simulation models impose. Therefore, applying the current algorithms in which finding the appropriate solutions requires a large number of simulation runs is not deemed suitable. In this paper, an estimation of distribution algorithm based on copula theory has been suggested. In contrast with traditional solution algorithms, in the proposed algorithm complex interaction between parameters of a model has been considered by constructing and sampling from a copula-based probabilistic model. Copulas are functions that describe the dependence structure of a set of random variables and connect multivariate distribution functions to one-dimensional marginal distribution functions. The results indicate that applying an explicit probabilistic model based on copula helps the estimation of distribution algorithm to explore the search space more effectively and efficiently as well as provides the possibility of extracting the knowledge with regard to the structure of the calibration problem through analyzing the probabilistic models that are constructed during the evolution process. Furthermore, this new algorithm has been compared with the genetic algorithm and kernel-based cross-entropy method on synthetic and real trajectory data. The results confirm that the proposed algorithm is more efficient in terms of convergence rate, resource usage and more robust in terms of the probability of finding the global optimal solution.

Maximum entropy distribution of order statistics with given marginals

We consider distributions of ordered random vectors with given one-dimensional marginal distributions. We give an elementary necessary and sufficient condition for the existence of such a distribution with finite entropy. In this case, we give explicitly the density of the unique distribution which achieves the maximal entropy and compute the value of its entropy. This density is the unique one which has a product form on its support and the given one-dimensional marginals. The proof relies on the study of copulas with given one-dimensional marginal distributions for its order statistics.

A survey of kernel-type estimators for copula and their applications

Copulas have been widely used to model nonlinear dependence structure. Main applications of copulas include areas such as finance, insurance, hydrology, rainfall to name but a few. The flexibility of copula allows researchers to model dependence structure beyond Gaussian distribution. Basically, a copula is a function that couples multivariate distribution functions to their one-dimensional marginal distribution functions. In general, there are three methods to estimate copula. These are parametric, nonparametric, and semiparametric method. In this article we survey kernel-type estimators for copula such as mirror reflection kernel, beta kernel, transformation method and local likelihood transformation method. Then, we apply these kernel methods to three stock indexes in Asia. The results of our analysis suggest that, albeit variation in information criterion values, the local likelihood transformation method performs better than the other kernel methods.

Experimental demonstration of negative-valued polarization quasiprobability distribution

Polarization quasiprobability distribution defined in the Stokes space shares many important properties with the Wigner function for position and momentum. Most notably, they both give correct one-dimensional marginal probability distributions and therefore represent the natural choice for the probability distributions in classical hidden-variable models. In this context, negativity of the Wigner function is considered as proof of nonclassicality for a quantum state. On the contrary, the polarization quasiprobability distribution demonstrates negativity for all quantum states. This feature comes from the discrete nature of Stokes variables; however, it was not observed in previous experiments, because they were performed with photon-number averaging detectors. Here we reconstruct the polarization quasiprobability distribution of a coherent state with photon-number resolving detectors, which allows us to directly observe for the first time its negativity. Furthermore we derive a theoretical polarization quasiprobability distribution for any linearly polarized quantum state.

Asymptotic results for a multivariate version of the alternative fractional Poisson process

A multivariate fractional Poisson process was recently defined in Beghin and Macci (2016) by considering a common independent random time change for a finite dimensional vector of independent (non-fractional) Poisson processes; moreover it was proved that, for each fixed t≥0, it has a suitable multinomial conditional distribution of the components given their sum. In this paper we consider another multivariate process {M̲ν(t)=(M1ν(t),…,Mmν(t)):t≥0} with the same conditional distributions of the components given their sums, and different marginal distributions of the sums; more precisely we assume that the one-dimensional marginal distributions of the process ∑i=1mMiν(t):t≥0 coincide with the ones of the alternative fractional (univariate) Poisson process in Beghin and Macci (2013). We present large deviation results for {M̲ν(t)=(M1ν(t),…,Mmν(t)):t≥0}, and this generalizes the result in Beghin and Macci (2013) concerning the univariate case. We also study moderate deviations and we present some statistical applications concerning the estimation of the fractional parameter ν.

A distance test of normality for a wide class of stationary processes

A distance test for normality of the one-dimensional marginal distribution of stationary fractionally integrated processes is considered. The test is implemented by using an autoregressive sieve bootstrap approximation to the null sampling distribution of the test statistic. The bootstrap-based test does not require knowledge of either the dependence parameter of the data or of the appropriate norming factor for the test statistic. The small-sample properties of the test are examined by means of Monte Carlo experiments. An application to real-world data is also presented.

Using the Bootstrap to Test for Symmetry Under Unknown Dependence

This article considers tests for symmetry of the one-dimensional marginal distribution of fractionally integrated processes. The tests are implemented by using an autoregressive sieve bootstrap approximation to the null sampling distribution of the relevant test statistics. The sieve bootstrap allows inference on symmetry to be carried out without knowledge of either the memory parameter of the data or of the appropriate norming factor for the test statistic and its asymptotic distribution. The small-sample properties of the proposed method are examined by means of Monte Carlo experiments, and applications to real-world data are also presented.

A copula-based approach to accommodate the dependence among microscopic traffic variables

Developing microscopic traffic simulation models requires the knowledge of probability distributions of microscopic traffic variables. Although previous studies have proposed extensive mathematical distributions for describing traffic variables (e.g., speed, headway, vehicle length, etc.), these studies usually consider microscopic traffic observations to be independent variables and distributions for these variables are investigated separately. As a result, some traditional approaches consider microscopic traffic variables as independent inputs to the traffic simulation process and these methods may ignore the possible dependence among different traffic variables.The objectives of this paper are to investigate the dependence structure among microscopic traffic variables and to examine the applicability of the copula approach to the joint modeling of these variables. Copulas are functions that relate multivariate distribution functions of random variables to their one-dimensional marginal distribution functions. The concept of copulas has been well recognized in the statistics field and recently has been introduced in transportation studies. The proposed copula approach is applied to the 24-h traffic data collected on IH-35 in Austin, Texas. The preliminary data analysis indicates that there exists dependence among microscopic traffic variables. Moreover, the modeling and simulation results suggest that copula models can adequately accommodate and accurately reproduce the dependence structure revealed by the traffic observations. Overall, the findings in this paper provide a framework for generating multiple microscopic traffic variables simultaneously by considering their dependence.

On the martingale problem for degenerate-parabolic partial differential operators with unbounded coefficients and a mimicking theorem for Itô processes

Using results from a companion article by the authors (J. Differential Equations 254 (2013), 4401–4445) on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and mathematical finance, concerning degenerate diffusion processes. We show that the martingale problem associated with a degenerate-elliptic differential operator with unbounded, locally Hölder continuous coefficients on a half-space is well-posed in the sense of Stroock and Varadhan. Second, we prove existence, uniqueness, and the strong Markov property for weak solutions to a stochastic differential equation with degenerate diffusion and unbounded coefficients with suitable Hölder continuity properties. Third, for an Itô process with degenerate diffusion and unbounded but appropriately regular coefficients, we prove existence of a strong Markov process, unique in the sense of probability law, whose one-dimensional marginal probability distributions match those of the given Itô process.

Computable Bounds on the Spectral Gap for Unreliable Jackson Networks

The goal of this paper is to identify exponential convergence rates and to find computable bounds for them for Markov processes representing unreliable Jackson networks. First, we use the bounds of Lawler and Sokal (1988) in order to show that, for unreliable Jackson networks, the spectral gap is strictly positive if and only if the spectral gaps for the corresponding coordinate birth and death processes are positive. Next, utilizing some results on birth and death processes, we find bounds on the spectral gap for network processes in terms of the hazard and equilibrium functions of the one-dimensional marginal distributions of the stationary distribution of the network. These distributions must be in this case strongly light-tailed, in the sense that their discrete hazard functions have to be separated from 0. We relate these hazard functions with the corresponding networks' service rate functions using the equilibrium rates of the stationary one-dimensional marginal distributions. We compare the obtained bounds on the spectral gap with some other known bounds.

Computable Bounds on the Spectral Gap for Unreliable Jackson Networks

The goal of this paper is to identify exponential convergence rates and to find computable bounds for them for Markov processes representing unreliable Jackson networks. First, we use the bounds of Lawler and Sokal (1988) in order to show that, for unreliable Jackson networks, the spectral gap is strictly positive if and only if the spectral gaps for the corresponding coordinate birth and death processes are positive. Next, utilizing some results on birth and death processes, we find bounds on the spectral gap for network processes in terms of the hazard and equilibrium functions of the one-dimensional marginal distributions of the stationary distribution of the network. These distributions must be in this case strongly light-tailed, in the sense that their discrete hazard functions have to be separated from 0. We relate these hazard functions with the corresponding networks' service rate functions using the equilibrium rates of the stationary one-dimensional marginal distributions. We compare the obtained bounds on the spectral gap with some other known bounds.

DPSynthesizer: Differentially Private Data Synthesizer for Privacy Preserving Data Sharing.

Differential privacy has recently emerged in private statistical data release as one of the strongest privacy guarantees. Releasing synthetic data that mimic original data with Differential privacy provides a promising way for privacy preserving data sharing and analytics while providing a rigorous privacy guarantee. However, to this date there is no open-source tools that allow users to generate differentially private synthetic data, in particular, for high dimensional and large domain data. 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. We propose DPSynthesizer, a toolkit for differentially private data synthesization. The core of DPSynthesizer is DPCopula designed for high-dimensional and large-domain data. DPCopula computes a differentially private copula function from which synthetic data can be sampled. 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. DPSynthesizer also implements a set of state-of-the-art methods for building differentially private histograms, suitable for low-dimensional data, from which synthetic data can be generated. We will demonstrate the system using DPCopula as well as other methods with various data sets and show the feasibility, utility, and efficiency of various methods.