Arbitrary Correlation Research Articles

Bayesian learning for a class of priors with prescribed marginals

We discuss three learning rules for generalized Bayesian updating of an imprecise probability: (a modified version of) the generalized Bayes’ rule, the maximum likelihood update rule (after Gilboa and Schmeidler) and a newly developed hybrid rule. We investigate the general methodology for a special class of multivariate probability measures with prescribed marginals but arbitrary correlation structure. Both the choice and analysis of this class are motivated by expert interviews that we conducted with modelers in the field of climatic change. We argue that both updating rules from the literature have strong limitations, the generalized Bayes’ rule is too conservative, i.e., too inclusive, while the maximum likelihood update rule being too exclusive, adding spurious information. As a powerful extension we introduce a new rule for Bayesian updating of an imprecise measure: a “weighted likelihood update method,” which bases Bayesian updating on the whole set of priors but weights the influence of its members. We study the different rules in the case of bivariate Gaussian priors. Our investigation shows that the new rule combines certain attractive features of the generalized Bayes’ rule and the maximum likelihood update rule. In this article we aim at highlighting the sequence of not yet fully resolved statistical issues a practitioner on complex mechanistic models would face when updating imprecise prior knowledge.

Compact formulas for the average error performance of noncoherent m-ary orthogonal signals over generalized rician, nakagami-m, and nakagami-q fading channels with diversity reception

This paper derives exact expressions for the average error performance of M-ary orthogonal signals with noncoherent equal-gain diversity combining over nonidentical generalized Rician, Nakagami-m, Nakagami-g, and implicitly Rayleigh fading channels. The assumption of generalized distributed fading envelopes implies that the received average signal-to-noise ratios (SNRs) and/or the fading parameters can have arbitrary nonidentical values. The derived expressions are precisely given in terms of either one-fold integral or rapidly convergent infinite series, which can be readily evaluated numerically. In addition, they can be usefully used to study the impact of arbitrary correlation among diversity branches on the system average error performance.

Noncoherent Communication in Multiple-Antenna Systems: Receiver Design and Codebook Construction

In this paper, we address the problem of space-time codebook design for noncoherent communications in multiple-antenna wireless systems. In contrast with other approaches, the channel matrix is modeled as an unknown deterministic parameter at both the receiver and the transmitter, and the Gaussian observation noise is allowed to have an arbitrary correlation structure, known by the transmitter and the receiver. In order to handle the unknown deterministic space-time channel, a generalized likelihood ratio test (GLRT) receiver is considered. A new methodology for space-time codebook design under this noncoherent setup is proposed. It optimizes the probability of error of the GLRT receiver's detector in the high signal-to-noise ratio (SNR) regime by solving a high-dimensional nonlinear nonsmooth optimization problem in a two-step approach. First, a convex semidefinite programming (SDP) relaxation of the codebook design problem yields a rough estimate of the optimal codebook. This is then refined through a geodesic descent optimization algorithm that exploits the Riemannian geometry imposed by the power constraints on the space-time codewords. The results obtained through computer simulations illustrate the advantages of our method. For the specific case of spatio-temporal white observation noise, our codebook constructions replicate the performance of state-of-the-art known solutions. The main point here is that our methodology permits extending the codebook construction to any given correlated noise environment. The simulation results show good performance of these new designed codes in colored noise setups.

A Closed-Form BER Expression for BPSK Using MRC in Correlated CCI and Rayleigh Fading

An exact closed-form expression for the average bit error rate (BER) of coherent BPSK using maximal ratio combining (MRC) with correlated branches in the presence of cochannel interference (CCI) and noise is derived. This expression is valid for an arbitrary correlation structure under the assumption that the branch gains of the desired user signal and interfering signals experience correlated Rayleigh fading with the same correlation eigenvector matrix.

Interference of Fock states in a single measurement

We study analytically the structure of an arbitrary order correlation function for a pair of Fock states and prove without any approximations that in a single measurement of particle positions interference effects must occur as experimentally observed with Bose-Einstein condensates. We also show that the noise level present in the statistics is slightly lower than for a respective measurement of phase states.

Correlation-space description of the percolation transition in composite microstructures

We explore the percolation threshold shift as short-range correlations are introduced and systematically varied in binary composites. Two complementary representations of the correlations are developed in terms of the distribution of phase bonds or, alternatively, using a set of appropriate short-range order parameters. In either case, systematic exploration of the correlation space reveals a boundary that separates percolating from nonpercolating structures and permits empirical equations that identify the location of the threshold for systems of arbitrary short-range correlation states. Two- and three-dimensional site lattices with two-body correlations, as well as a two-dimensional hexagonal bond network with three-body correlations, are explored. The approach presented here should be generalizable to more complex correlation states, including higher-order and longer-range correlations.

Magnetic properties of MnxTi1−xN thin films grown by plasma-assisted molecular beam epitaxy

High-quality MnxTi1−xN thin films were grown on MgO(001) substrates using plasma-assisted molecular beam epitaxy. Magnetic measurements evidence the presence of ferromagnetism with Curie temperature exceeding 380K. X-ray photoelectron spectroscopy indicates that the Mn ions are in a divalent state and uniformly substitute on Ti cation sites, consistent with the ferromagnetism that correlates with Mn substitution on Ti sites. The origin of the ferromagnetism might be attributed to itinerant-carrier mediated Rudermann-Kittel-Kasuya-Yosida (RKKY)-type long-range coupling which allows for arbitrary itinerant-carrier spin polarization and dynamic correlations.

Theory of [formula omitted] peaks in the IQHE regime with correlated disorder potential

The problem of σ xx peaks has been studied theoretically in the regime of integer quantum Hall effect (IQHE) for disorder potentials with arbitrary correlation length to magnetic length ratio λ / a H . We obtained analytical solutions for two limiting cases of long-wavelength disorder potential, λ ⪢ r c = n + 1 a H , and short-wavelength disorder potential, λ ⪡ a H . The case of intermediate λ / a H ratio has been studied numerically. Our results are in good agreement with available experimental data but not with the hypothesis of universality due to Khmelnitskii and Pruisken.

Hybrid diversity-beamforming technique for outage probability minimization in spatially correlated channels

In this paper, we present a hybrid multi-antenna technique that can minimize the outage probability by combining the diversity and beamforming techniques. The hybrid technique clusters the transmission antennas into multiple groups and exploit diversity among different groups and beamforming within each group. We analyze the performance of the resulting hybrid technique for an arbitrary correlation among the transmission antennas. Through the performance analysis, we derive a closed-form expression of the outage probability for the hybrid technique. This enables to optimize the antenna grouping for the given spatial correlation. We show through numerical results that the hybrid technique can balance the trade-offs between diversity and beamforming according to the spatial correlation and that the optimally designed hybrid technique yields a much lower outage probability than the diversity or beamforming technique does in partially correlated fading channels.

Steady flows driven by sources of random strength in heterogeneous aquifers with application to partially penetrating wells

Average steady source flow in heterogeneous porous formations is modelled by regarding the hydraulic conductivity K(x) as a stationary random space function (RSF). As a consequence, the flow variables become RSFs as well, and we are interested into calculating their moments. This problem has been intensively studied in the case of a Neumann type boundary condition at the source. However, there are many applications (such as well-type flows) for which the required boundary condition is that of Dirichlet. In order to fulfill such a requirement the strength of the source must be proportional to K(x), and therefore the source itself results a RSF. To solve flows driven by sources whose strength is spatially variable, we have used a perturbation procedure similar to that developed by Indelman and Abramovich (Water Resour Res 30:3385–3393, 1994) to analyze flows generated by sources of deterministic strength. Due to the linearity of the mathematical problem, we have focused on the explicit derivation of the mean head distribution G d (x) generated by a unit pulse. Such a distribution represents the fundamental solution to the average flow equations, and it is termed as mean Green function. The function G d (x) is derived here at the second order of approximation in the variance σ2 of the fluctuation $$\varepsilon \left({{\mathbf{x}}}\right) = 1- \frac{K{\left({{\mathbf{x}}}\right)}}{K_{A}}$$ (where K A is the mean value of K(x)), for arbitrary correlation function ρ(x), and any dimensionality d of the flow domain. We represent G d (x) as product between the homogeneous Green function G (0) (x) valid in a domain with constant K A , and a distortion term Ψ d (x) = 1 + σ2ψ d (x) which modifies G (0) (x) to account for the medium heterogeneity. In the case of isotropic formations ψ d (x) is expressed via one quadrature. This quadrature can be analytically calculated after adopting specific (e.g.. exponential and Gaussian) shape for ρ(x). These general results are subsequently used to investigate flow toward a partially-penetrating well in a semi-infinite domain. Indeed, we construct a σ2-order approximation to the mean as well as variance of the head by replacing the well with a singular segment. It is shown how the well-length combined with the medium heterogeneity affects the head distribution. We have introduced the concept of equivalent conductivity K eq(r,z). The main result is the relationship $$\frac{K^{\rm eq}\left(r,z\right)} {K_{A}} = 1-\sigma^{2}\psi^{\left(w\right)}\left(r,z\right)$$ where the characteristic function ψ(w)(r,z) adjusts the homogeneous conductivity K A to account for the impact of the heterogeneity. In this way, a procedure can be developed to identify the aquifer hydraulic properties by means of field-scale head measurements. Finally, in the case of a fully penetrating well we have expressed the equivalent conductivity in analytical form, and we have shown that $$K^{({\rm efu})}\leq K^{\rm eq}\left(r\right) \leq K_{A}$$ (being $$K^{({\rm efu})}$$ the effective conductivity for mean uniform flow), in agreement with the numerical simulations of Firmani et al. (Water Resour Res 42:W03422, 2006).

Cluster without fluster: The effect of correlated outcomes on inference in randomized clinical trials

Inference for randomized clinical trials is generally based on the assumption that outcomes are independently and identically distributed under the null hypothesis. In some trials, particularly in infectious disease, outcomes may be correlated. This may be known in advance (e.g. allowing randomization of family members) or completely unplanned (e.g. sexual sharing among randomized participants). There is particular concern when the form of the correlation is essentially unknown, in which case we cannot take advantage of the correlation to construct a more efficient test. Instead, we can only investigate the impact of potential correlation on the independent-samples test statistic. Randomization tends to balance out treatment and control assignments within clusters, so it is logical that performance of tests averaged over all possible randomization assignments would be essentially unaffected by arbitrary correlation. We confirm this intuition by showing that a permutation test controls the type 1 error rate in a certain average sense whenever the clustering is independent of treatment assignment. It is nonetheless possible to obtain a 'bad' randomization such that members of a cluster tend to be assigned to the same treatment. Conditioned on such a bad randomization, the type 1 error rate is increased.

Simulation of vector fields with arbitrary second-order correlations

Temporally-stationary electromagnetic fields with arbitrary second-order coherence functions are simulated using standard statistical tools. In cases where the coherence function takes a commonly-used separable form, a computationally-efficient variation of the approach can be applied. This work provides a generalization of previous spatio-temporal simulators which model only scalar fields and require either restrictions on the coherence function or consider only two points in space. The simulation of a partially-polarized Gaussian Schell-model beam and a partially-radially-polarized beam are demonstrated.

Information processing in generalized probabilistic theories

I introduce a framework in which a variety of probabilistic theories can be defined, including classical and quantum theories, and many others. From two simple assumptions, a tensor product rule for combining separate systems can be derived. Certain features, usually thought of as specifically quantum, turn out to be generic in this framework, meaning that they are present in all except classical theories. These include the nonunique decomposition of a mixed state into pure states, a theorem involving disturbance of a system on measurement (suggesting that the possibility of secure key distribution is generic), and a no-cloning theorem. Two particular theories are then investigated in detail, for the sake of comparison with the classical and quantum cases. One of these includes states that can give rise to arbitrary nonsignaling correlations, including the superquantum correlations that have become known in the literature as nonlocal machines or Popescu-Rohrlich boxes. By investigating these correlations in the context of a theory with well-defined dynamics, I hope to make further progress with a question raised by Popescu and Rohrlich, which is why does quantum theory not allow these strongly nonlocal correlations? The existence of such correlations forces much of the dynamics in this theory to be, in a certain sense, classical, with consequences for teleportation, cryptography, and computation. I also investigate another theory in which all states are local. Finally, I raise the question of what further axiom(s) could be added to the framework in order to identify quantum theory uniquely, and hypothesize that quantum theory is optimal for computation.

Stochastic Liouville equation treatment of the electron paramagnetic resonance line shape of an S-state ion in solution

The current approaches used for the analysis of electron paramagnetic resonance spectra of Gd3+ complexes suffer from a number of drawbacks. Even the elaborate model of [Rast et al., J. Chem. Phys. 113, 8724 (2000)] where the electron spin relaxation is explained by the modulation of the zero-field splitting (ZFS), by molecular tumbling (the so called static contribution), and deformations (transient contribution), is only readily applicable within the validity range of the Redfield theory [Advances in Magnetic Resonance, edited by J.-S. Waugh (Academic, New York, 1965), Vol. 1, p. 1], that is, when the ZFS is small compared to the Zeeman energy and the rotational and vibrational modulations are fast compared to the relaxation time. Spin labels (nitroxides and transition metal complexes) have been studied for years in systems that violate these conditions. The theoretical framework commonly used in such studies is the stochastic Liouville equation (SLE). The authors shall show how the physical model of Rast et al. can be cast into the SLE formalism, paying special attention to the specific problems introduced by the [Uhlenbeck and Ornstein, Phys. Rev. 36, 823 (1930)] process used to model the transient ZFS. The resulting equations are very general and valid for arbitrary correlation times, magnetic field strength, electron spin S, or symmetry. The authors demonstrate the equivalence of the SLE approach with the Redfield approximation for two well-known Gd3+ complexes.

Precoded DOSTBC over Rayleigh Channels

Differential orthogonal space-time block codes (DOSTBC) sent over correlated Rayleigh fading channels are considered in this paper. Approximate expressions for the symbol error rate (SER) are derived for DOSTBC withM-PSK,M-PAM, andM-QAM constellations assuming arbitrary correlation between the transmit and receive antennas. A full memoryless precoder is designed to improve the performance of the DOSTBC over correlated Rayleigh MIMO channels. The proposed precoder design differs from the previous work: (1) our precoder design considers arbitrary correlation in the channels, whereas the previously proposed precoder design considers only transmit correlations in the Kronecker correlation model; (2) the proposed precoder is based on minimizing proposed SER, whereas the previously proposed precoder is based on minimizing the Chernoff bound of approximate SER; (3) we propose precoder design for DOSTBC withM-PSK,M-PAM, andM-QAM constellations, whereas the previously proposed precoder works for DOSTBC withM-PSK only. Additionally, the proposed precoded DOSTBC outperforms the conventional eigenbeamforming-based precoded DOSTBC for the Kronecker model with only transmit correlation.

On the Multivariate Weibull Fading Model With Arbitrary Correlation Matrix

An efficient approximation for the multivariate Weibull distribution with arbitrary correlations is presented. By approximating the Gaussian correlation matrix with a Green's matrix, a useful analytical expression for the joint distribution of the fading envelopes is derived. As an application, the outage performance of multibranch receivers operating over arbitrarily correlated Weibull fading channels is studied. Numerical and computer simulation results are presented and compared to illustrate the accuracy of the proposed approximation

Bose condensed gas in strong disorder potential with arbitrary correlation length

We study the properties of a dilute Bose condensed gas at zero temperature in the presence of a strong random potential with arbitrary correlation length. Starting from the underlying Gross–Pitaevskii equation, we use the random phase approximation in order to get a closed integral equation for the averaged density distribution which allows the determination of both the condensate and the superfluid density. The obtained results generalise those of Huang and Meng (HM) to strong disorder. In particular, we find the critical value of the disorder strength, where the superfluid phase disappears by a first-order phase transition. We show how this critical value changes as a function of the correlation length.

Path-Based Statistical Timing Analysis Handling Arbitrary Delay Correlations: Theory and Implementation

An efficient path-based statistical timing analysis algorithm that can handle arbitrary causes of delay correlations is proposed in this paper. The algorithm derives bounds for the cumulative distribution function (cdf) of the circuit delay using a new mathematical formulation based on the theory of stochastic majorization. Structural and interchip correlations between path delays can be taken into account. Because the analytical computation of an exact cdf for a probabilistic timing graph is infeasible, tight upper and lower bounds on the true cumulative distribution are derived. The efficiency and accuracy of the algorithm is demonstrated on a set of ISCAS'85 benchmarks. Across the benchmarks, the error of the 95th-percentile delay is 1.1%-3.3%, and the root-mean-square error of the cumulative probability is 1.7%-4.5%. The run time of the proposed algorithm for the largest benchmark circuit takes less than 4 s

Ultrafast Solid-State 2D NMR Experiments via Orientational Encoding

Among the methods proposed in recent years toward the acceleration of multidimensional NMR acquisitions is an "ultrafast" approach, capable of delivering arbitrary 2D correlations within a single scan. This scheme operates by parallelizing the indirect-domain temporal incrementation involved in 2D acquisitions, using as aid an ancillary inhomogeneous frequency broadening acting in combination with a train of frequency-shifted RF pulses. So far, all implementations of this frequency broadening have relied on magnetic field gradients; yet the practical performance of gradient-based approaches is sometimes inadequate-for instance when applied on solid samples subject to magic-angle spinning. In order to deal with these cases, an alternative encoding protocol is here introduced and experimentally exemplified, based on exploiting the intrinsic anisotropy that spin interactions exhibit in the solid state as the ancillary broadening in charge of encoding the interactions to be measured. Principles and preliminary examples of the new orientationally encoded ultrafast 2D NMR principle thus resulting are presented and discussed.

Asymptotic properties of a robust variance matrix estimator for panel data when is large

Abstract I consider the asymptotic properties of a commonly advocated covariance matrix estimator for panel data. Under asymptotics where the cross-section dimension, n, grows large with the time dimension, T, fixed, the estimator is consistent while allowing essentially arbitrary correlation within each individual. However, many panel data sets have a non-negligible time dimension. I extend the usual analysis to cases where n and T go to infinity jointly and where T → ∞ with n fixed. I provide conditions under which t and F statistics based on the covariance matrix estimator provide valid inference and illustrate the properties of the estimator in a simulation study.