Arbitrary Correlation Research Articles

Polarized-radiation transfer in stochastic finite planar atmospheric media

In this work, we investigate the polarized-radiation transfer problem in a planar stochastic atmospheric medium. A solution is presented for an arbitrary extinction function, which was assumed to be continuous in position and to exhibit fluctuations about the Gaussian distributed mean. The joint probability distribution function of the integral transform of these Gaussian random functions was used to calculate the ensemble-averaged quantities, such as the reflectivity and transmissivity, for an arbitrary correlation function. A modified Gaussian probability distribution function was used to average the solution to exclude probable negative values of the optical space variable. The deterministic solutions for the total intensity and the difference function, which were used to describe the polarized radiation, were obtained using the Pomraning–Eddington technique. The numerical results were presented for the Gaussian and modified Gaussian probability density functions for different degrees of polarization.

The Empirical Distribution of a Large Number of Correlated Normal Variables

Motivated by the advent of high-dimensional, highly correlated data, this work studies the limit behavior of the empirical cumulative distribution function (ecdf) of standard normal random variables under arbitrary correlation. First, we provide a necessary and sufficient condition for convergence of the ecdf to the standard normal distribution. Next, under general correlation, we show that the ecdf limit is a random, possible infinite, mixture of normal distribution functions that depends on a number of latent variables and can serve as an asymptotic approximation to the ecdf in high dimensions. We provide conditions under which the dimension of the ecdf limit, defined as the smallest number of effective latent variables, is finite. Estimates of the latent variables are provided and their consistency proved. We demonstrate these methods in a real high-dimensional data example from brain imaging where it is shown that, while the study exhibits apparently strongly significant results, they can be entirely explained by correlation, as captured by the asymptotic approximation developed here. Supplementary materials for this article are available online.

Mössbauer forward scattering: The single-photon response of thick absorbers in an alternating field

The propagation of Mossbauer radiation through an absorber being in the acoustic oscillation mode under forward scattering conditions has been analyzed. A generalization of the existing models of the formation of the forward scattering spectra (frequency and time) to the case of arbitrary phase correlation of oscillations of nuclei in the sample has been proposed. An adequate description of the time experiment on 57Fe over the modulation of a single-photon wave packet in the acoustic field has been obtained in the representations of the Raman scattering of Mossbauer photons. The model can be used to control the degree of phase correlation of oscillations of nuclei (or other processes) induced in the sample by external fields.

The double scaling limit of the multi-orientable tensor model

In this paper we study the double scaling limit of the multi-orientable tensor model. We prove that, contrary to the case of matrix models but similarly to the case of invariant tensor models, the double scaling series are convergent. We resum the double scaling series of the two-point function and of the leading singular part of the four-point function. We discuss the behavior of the leading singular part of arbitrary correlation functions. We show that the contribution of the four-point function and of all the higher point functions are enhanced in the double scaling limit. We finally show that all the correlation functions exhibit a singularity at the same critical value of the double scaling parameter which, combined with the convergence of the double scaling series, suggest the existence of a triple scaling limit.

Stochastic transfer of polarized radiation in finite cluttered atmospheric media with dual statistical analysis

The stochastic transfer of polarized radiation in a finite cluttered atmospheric medium (e.g. clouds) is investigated, the solution being presented for arbitrary absorption and scattering cross sections. The extinction function of the medium is assumed to be a continuous random function of position, with fluctuations about the mean taken as Gaussian distributed. The deterministic analytical solution is obtained by using Pomraning–Eddington technique with weight function method. Two correlated random variables appear in the solution, namely the optical space variable and the medium optical thickness. The dual Gaussian probability density function of these two random variables is derived, from which the ensemble-averaged solution is calculated for an arbitrary correlation function. The first and the second statistical moments of some quantities of interest, such as radiant energy, net flux, reflectivity and transmissivity, are obtained. Numerical calculations are performed for specular-reflecting boundaries and an incident flux of radiation upon the medium from one side (x=0) and with no flux from other side (x=L).

Distribution Functions of Saturation for Stochastic Nonlinear Two-Phase Flow

We derive analytical expressions for the one-point cumulative distribution function and probability density function of water saturation in the one-dimensional stochastic immiscible two-phase (Buckley–Leverett) problem. The sources of uncertainty are the spatial distributions of porosity and total velocity. The derived distribution functions involve integrals of the input random parameters. Comparisons with standard Monte Carlo simulation demonstrate that the method is applicable for input parameters with large variance and arbitrary correlation lengths. We also show that the proposed method is superior to the low-order statistical moment equations approach. We also outline a streamline-based strategy to extend our distribution-based method to multiple spatial dimensions.

Location and scale mixtures of Gaussians with flexible tail behaviour: Properties, inference and application to multivariate clustering

The family of location and scale mixtures of Gaussians has the ability to generate a number of flexible distributional forms. The family nests as particular cases several important asymmetric distributions like the Generalized Hyperbolic distribution. The Generalized Hyperbolic distribution in turn nests many other well known distributions such as the Normal Inverse Gaussian. In a multivariate setting, an extension of the standard location and scale mixture concept is proposed into a so called multiple scaled framework which has the advantage of allowing different tail and skewness behaviours in each dimension with arbitrary correlation between dimensions. Estimation of the parameters is provided via an EM algorithm and extended to cover the case of mixtures of such multiple scaled distributions for application to clustering. Assessments on simulated and real data confirm the gain in degrees of freedom and flexibility in modelling data of varying tail behaviour and directional shape.

Spectral functions of strongly correlated extended systems via an exact quantum embedding

Density matrix embedding theory (DMET) [Phys. Rev. Lett., 109, 186404 (2012)], introduced a new approach to quantum cluster embedding methods, whereby the mapping of strongly correlated bulk problems to an impurity with finite set of bath states was rigorously formulated to exactly reproduce the entanglement of the ground state. The formalism provided similar physics to dynamical mean-field theory at a tiny fraction of the cost, but was inherently limited by the construction of a bath designed to reproduce ground state, static properties. Here, we generalize the concept of quantum embedding to dynamic properties and demonstrate accurate bulk spectral functions at similarly small computational cost. The proposed spectral DMET utilizes the Schmidt decomposition of a response vector, mapping the bulk dynamic correlation functions to that of a quantum impurity cluster coupled to a set of frequency dependent bath states. The resultant spectral functions are obtained on the real-frequency axis, without bath discretization error, and allows for the construction of arbitrary dynamic correlation functions. We demonstrate the method on the 1D and 2D Hubbard model, where we obtain zero temperature, thermodynamic limit spectral functions, and show the trivial extension to two-particle Green functions. This advance therefore extends the scope and applicability of DMET in condensed matter problems as a computationally tractable route to correlated spectral functions of extended systems, and provides a competitive alternative to dynamical mean-field theory for dynamic quantities.

Calculation of the molecular integrals with the range-separated correlation factor.

Explicitly correlated quantum chemical calculations require calculations of five types of two-electron integrals beyond the standard electron repulsion integrals. We present a novel scheme, which utilises general ideas of the McMurchie-Davidson technique, to compute these integrals when the so-called "range-separated" correlation factor is used. This correlation factor combines the well-known short range behaviour resulting from the electronic cusp condition, with the exact long-range asymptotics derived for the helium atom [Lesiuk, Jeziorski, and Moszynski, J. Chem. Phys. 139, 134102 (2013)]. Almost all steps of the presented procedure are formulated recursively, so that an efficient implementation and control of the precision are possible. Additionally, the present formulation is very flexible and general, and it allows for use of an arbitrary correlation factor in the electronic structure calculations with minor or no changes.

OMNI-Prop: Seamless Node Classification on Arbitrary Label Correlation

If we know most of Smith’s friends are from Boston, what can we say about the rest of Smith’s friends? In this paper, we focus on the node classification problem on networks, which is one of the most important topics in AI and Web communities. Our proposed algorithm which is referred to as OMNIProp has the following properties: (a) seamless and accurate; it works well on any label correlations (i.e., homophily, heterophily, and mixture of them) (b) fast; it is efficient and guaranteed to converge on arbitrary graphs (c) quasi-parameter free; it has just one well-interpretable parameter with heuristic default value of 1. We also prove the theoretical connections of our algorithm to the semi-supervised learning (SSL) algorithms and to random-walks. Experiments on four real, different network datasets demonstrate the benefits of the proposed algorithm, where OMNI-Prop outperforms the top competitors.

Effect of a large number of parties on the monogamy of quantum correlations

Monogamy is a nonclassical property that restricts the sharability of quantum correlation among the constituents of a multipartite quantum system. Quantum correlations may satisfy or violate monogamy for quantum states, which was tested mainly for three-qubit states. Here we establish a sufficient condition for monogamy of arbitrary quantum correlation measures of states of an arbitrary number of parties, using which and further numerical results, we obtain evidence for monogamy of measures such as distillable entanglement and relative entropy of entanglement, which are physically important but mathematically intractable, for almost all quantum states of a moderate number of parties. The result is generic and holds for a large class of quantum correlation measures. Nonetheless, we identify important zero Haar measure classes of pure states that remain nonmonogamous with respect to quantum discord and quantum work deficit, irrespective of the number of qubits.

Confinement effects on quasi 0D dimensional structure with AlxGa1-xAs/GaAs/AlxGa1-xAs quantum wells

In this work we study the effects of barrier height and position on symmetric AlxGa1-xAs/GaAs/AlxGa1-xAs quantum wells in absence of external influences. We use a variational method within the effective mass approximation to observe the effects on the structure. The donor trial function is taken as a product of the ground state wave function, with an arbitrary correlation function that depends only on ion-electron separation. It has been observed two peaks in the curves for the dependence of the ground-state binding energies versus the donor distance from the axis and it is shown that the impurity binding energy depends strongly on the impurity position, potential shape on both quantum wells.

On Joint Source-Channel Coding for a Multivariate Gaussian on a Gaussian MAC

In this paper, nonlinear distributed joint source-channel coding (JSCC) schemes for transmission of multivariate Gaussian sources over a Gaussian multiple access channel are proposed and analyzed. The main contribution is a zero-delay JSCC named Distributed Quantizer Linear Coder (DQLC), which performs relatively close the information theoretical bounds, improves when the correlation among the sources increases, and does not level off as the signal-to-noise ratio (SNR) becomes large. Therefore it outperforms any linear solution for sufficiently large SNR. Further an extension of DQLC to an arbitrary code length named Vector Quantizer Linear Coder (VQLC) is analyzed. The VQLC closes in on the performance upper bound as the code length increases and can potentially achieve the bound for any number of independent sources. The VQLC leaves a gap to the bound whenever the sources are correlated, however. JSCC achieving the bound for arbitrary correlation has been found for the bivariate case, but that solution is significantly outperformed by the DQLC/VQLC when there is a low delay constraint. This indicates that different approaches are needed to perform close to the bounds when the code length is high and low. The VQLC/DQLC also apply for bandwidth compression of a multivariate Gaussian transmitted on point-to-point links.

Modeling of Noisy EM Field Propagation Using Correlation Information

In this paper, an efficient method for the numerical simulation of near- and far-field propagation of stochastic electromagnetic (EM) fields is presented. The method is based on the transformation of field correlation dyadics using Green's functions or the field transfer functions computed for deterministic fields. The method accounts for arbitrary correlations between the noise radiation sources and allows to compute the spatial distribution of the spectral energy density of noisy electromagnetic sources. The introduced methodology can be combined with available electromagnetic modeling tools. It is shown that the method of moments can be applied to solve noisy electromagnetic field problems by network methods applying correlation matrix techniques. Examples demonstrating the strong influence of the correlation between the sources on the spatial distribution of the radiated noise field are presented.

The glassy phase of the complex branching Brownian motion energy model

We identify the fluctuations of the partition function for a class of random energy models, where the energies are given by the positions of the particles of the complex-valued branching Brownian motion (BBM). Specifically, we provide the weak limit theorems for the partition function in the so-called "glassy phase'' – the regime of parameters, where the behaviour of the partition function is governed by the extrema of BBM. We allow for arbitrary correlations between the real and imaginary parts of the energies. This extends the recent result of Madaule, Rhodes and Vargas (2013), where the uncorrelated case was treated. Inparticular, our result covers the case of the real-valued BBM energy model at complex temperatures.

Theory of high gain cavity-enhanced spontaneous parametric down-conversion

We compute the output of multimode cavity-enhanced spontaneous parametric down-conversion (SPDC) for sub-threshold, but otherwise arbitrary, gain. We find analytic Bogoliubov transformations that allow us to calculate arbitrary field correlation functions, including the second-order intensity correlation function $G^{(2)}(T)$. The results show evidence of increased coherence due to stimulated SPDC. We extend an earlier model [Lu and Ou, Phys. Rev. A, 62, 033804 (2000)] to arbitrary gain and finesse, and show the extension gives accurate results in most scenarios. The results will allow simple, analytic description of cavity-based nonclassical light sources for quantum networking, quantum-enhanced sensing of atoms and generation of highly non-classical field states

ON THE MARTINGALE PROPERTY IN STOCHASTIC VOLATILITY MODELS BASED ON TIME-HOMOGENEOUS DIFFUSIONS

Lions and Musiela give sufficient conditions to verify when a stochastic exponential of a continuous local martingale is a martingale or a uniformly integrable martingale. Blei and Engelbert and Mijatović and Urusov give necessary and sufficient conditions in the case of perfect correlation (). For financial applications, such as checking the martingale property of the stock price process in correlated stochastic volatility models, we extend their work to the arbitrary correlation case (). We give a complete classification of the convergence properties of both perpetual and capped integral functionals of time-homogeneous diffusions and generalize results in Mijatović and Urusov with direct proofs avoiding the use of separating times (concept introduced by Cherny and Urusov and extensively used in the proofs of Mijatović and Urusov).

Estimation of General Multistage Models From Cohort Data

Many systems involve progression through a series of distinct stages, such as disease or developmental stages. In ecological studies, often individuals such as small arthropods cannot be marked, so data are collected on cohort development. Multistage models for unmarked cohort data use a distribution for each stage duration and possibly stage-specific mortality rates. We generalize previous models and present computational methods for smoothed maximum likelihood estimation. The general model allows arbitrary distribution assumptions, stage-specific mortality, unobserved stages, and correlations between stage durations using Gaussian copulas. Monte Carlo integration of the stage distributions is used to approximate the probabilities needed for the likelihood. We establish a heuristic smoothing step for the simulated probabilities that yields a smooth approximate likelihood surface. For the case of classic grasshopper cohort data, we demonstrate AIC model selection to determine which among past arbitrary constraints are actually justified by the data. Finally, we demonstrate how estimates of stage distribution parameters depend on the unknown stage correlations.

The Private and Public Correlation Cost of Three Random Variables With Collaboration

In this paper we consider the problem of generating arbitrary three-party correlations from a combination of public and secret correlations. Two parties -- called Alice and Bob -- share perfectly correlated bits that are secret from a collaborating third party, Charlie. At the same time, all three parties have access to a separate source of correlated bits, and their goal is to convert these two resources into multiple copies of some given tripartite distribution $P_{XYZ}$. We obtain a single-letter characterization of the trade-off between public and private bits that are needed to achieve this task. The rate of private bits is shown to generalize Wyner's classic notion of common information held between a pair of random variables. The problem we consider is also closely related to the task of secrecy formation in which $P_{XYZ}$ is generated using public communication and local randomness but with Charlie functioning as an adversary instead of a collaborator. We describe in detail the differences between the collaborative and adversarial scenarios.

Hardware simulator for optical correlation spectroscopy with Gaussian statistics and arbitrary correlation functions.

We present a new hardware simulator (HS) for characterization, testing and benchmarking of digital correlators used in various optical correlation spectroscopy experiments where the photon statistics is Gaussian and the corresponding time correlation function can have any arbitrary shape. Starting from the HS developed in [Rev. Sci. Instrum. 74, 4273 (2003)], and using the same I/O board (PCI-6534 National Instrument) mounted on a modern PC (Intel Core i7-CPU, 3.07GHz, 12GB RAM), we have realized an instrument capable of delivering continuous streams of TTL pulses over two channels, with a time resolution of Δt = 50ns, up to a maximum count rate of 〈I〉 ∼ 5MHz. Pulse streams, typically detected in dynamic light scattering and diffuse correlation spectroscopy experiments were generated and measured with a commercial hardware correlator obtaining measured correlation functions that match accurately the expected ones.