Independent Gaussian Research Articles

Simulational Tests of the Rouse Model.

An extensive review of literature simulations of quiescent polymer melts is given, considering results that test aspects of the Rouse model in the melt. We focus on Rouse model predictions for the mean-square amplitudes ⟨(Xp(0))2⟩ and time correlation functions ⟨Xp(0)Xp(t)⟩ of the Rouse mode Xp(t). The simulations conclusively demonstrate that the Rouse model is invalid in polymer melts. In particular, and contrary to the Rouse model, (i) mean-square Rouse mode amplitudes ⟨(Xp(0))2⟩ do not scale as sin-2(pπ/2N), N being the number of beads in the polymer. For small p (say, p≤3) ⟨(Xp(0))2⟩ scales with p as p-2; for larger p, it scales as p-3. (ii) Rouse mode time correlation functions ⟨Xp(t)Xp(0)⟩ do not decay with time as exponentials; they instead decay as stretched exponentials exp(-αtβ). β depends on p, typically with a minimum near N/2 or N/4. (iii) Polymer bead displacements are not described by independent Gaussian random processes. (iv) For p≠q, ⟨Xp(t)Xq(0)⟩ is sometimes non-zero. (v) The response of a polymer coil to a shear flow is a rotation, not the affine deformation predicted by Rouse. We also briefly consider the Kirkwood-Riseman polymer model.

Nonsymmetric examples for Gaussian correlation inequalities

In this paper, we compare two variances of maxima of N standard Gaussian random variables. One is a sequence of N i.i.d. standard Gaussians, and the other one is N standard Gaussians with covariances σ1,2=ρ∈(0,1) and σi,j=0, for other i≠j. It turns out that we need to discuss the covariance of two functions with respect to multivariate Gaussian distributions. Gaussian correlation inequalities hold for many symmetric (with respect to the origin) cases. However, in our case, the max function and its derivatives are not symmetric about the origin. We have two main results in this paper. First, we prove a specific case for a convex/log-concave correlation inequality for the standard multivariate Gaussian distribution. The other result is that the variance of maxima of standard Gaussians with σ1,2=ρ∈(0,1), while σi,j=0, for other i≠j, is larger than the variance of maxima of independent standard Gaussians. This implies that the variance of maxima of N i.i.d. standard Gaussians is decreasing in N.

Berry-Essén theorem for random determinants

Let Xn be a n×n random matrix with independent Gaussian entries. Then it is well-known that log(|det(Xn|)) satisfies a Central Limit Theorem. Our purpose is to evaluate the 1-Wasserstein distance for this limit theorem via the techniques of the Stein–Malliavin calculus.

Synthesis of Nonlinear Nonstationary Stochastic Systems by Wavelet Canonical Expansions

The article is devoted to Bayes optimization problems of nonlinear observable stochastic systems (NLOStSs) based on wavelet canonical expansions (WLCEs). Input stochastic processes (StPs) and output StPs of considered nonlinearly StSs depend on random parameters and additive independent Gaussian noises. For stochastic synthesis we use a Bayes approach with the given loss function and minimum risk condition. WLCEs are formed by covariance function expansion coefficients of two-dimensional orthonormal basis of wavelet with a compact carrier. New results: (i) a common Bayes’ criteria synthesis algorithm for NLOStSs by WLCE is presented; (ii) partial synthesis algorithms for three of Bayes’ criteria (minimum mean square error, damage accumulation and probability of error exit outside the limits) are given; (iii) an approximate algorithm based on statistical linearization; (iv) three test examples. Applications: wavelet optimization and parameter calibration in complex measurement and control systems. Some generalizations are formulated.

An Adaptive Richardson-Lucy Algorithm for Medical Image Restoration

Image restoration is the process of estimating the original image content from a degraded picture. In this paper, the Richardson-Lucy iterative algorithm was developed to improve the quality of degraded medical images. It has been assumed that medical images are exposed to two types of degradation. The first type is the blur function in the Gaussian form with different widths, i.e. σ = 1 , 2, and 3. The second type of degradation was assumed to be of the independent white Gaussian noise type with different signal-to-noise ratio values: SNR = 10, 50 , and 100. The results obtained from the adaptive filter are compared, quantitatively, with different conventional filters: inverse, Wiener, and constraint least square, by applying different measures, such as: power signal to noise ratio (PSNR), structural similarity index (SSID), and root mean square error (RMSE). The comparison showed that the adaptive recovery filter achieves better results.

Estimation of the selected treatment mean in two stage drop-the-losers design

ABSTRACT A common problem faced in clinical studies is that of estimating the effect of the most effective (e.g. the one having the largest mean) treatment among k ( ≥ 2 ) available treatments. The most effective treatment is adjudged based on numerical values of some statistic corresponding to the k treatments. A proper design for such problems is the so-called “Drop-the-Losers Design (DLD)”. We consider two treatments whose effects are described by independent Gaussian distributions having different unknown means and a common known variance. To select the more effective treatment, the two treatments are independently administered to n 1 subjects each and the treatment corresponding to the larger sample mean is selected. To study the effect of the adjudged more effective treatment (i.e. estimating its mean), we consider the two-stage DLD in which n 2 subjects are further administered the adjudged more effective treatment in the second stage of the design. We obtain some admissibility and minimaxity results for estimating the mean effect of the adjudged more effective treatment. The maximum likelihood estimator is shown to be minimax and admissible. We show that the uniformly minimum variance conditionally unbiased estimator (UMVCUE) of the selected treatment mean is inadmissible and obtain an improved estimator. In this process, we also derive a sufficient condition for inadmissibility of an arbitrary location and permutation equivariant estimator and provide dominating estimators in cases, where this sufficient condition is satisfied. The mean squared error and the bias performances of various competing estimators are compared via a simulation study. A real data example is also provided for illustration purpose.

Sparse Bayesian technique for load identification and full response reconstruction

Most load identification methods require that the load location is known in advance. A sparse Bayesian framework is proposed in this study to identify the force location and time history simultaneously and then reconstruct the responses with consideration of the uncertainties of the input force and response measurement. The prior distribution of the unknown forces is assumed to be the product of multiple independent Gaussian distributions of each individual potential force. Then, the most probable values of the unknown forces, measurement noise, and variances of the forces are derived and iteratively calculated by a self-adaptive posterior maximization strategy. In such a way, the estimated force vector is nonzero merely at the positions where loads are applied, and it thus possesses the sparsity in space. Consequently, the input forces are located and quantified simultaneously, and the full-field structural responses are sequentially reconstructed with suppressed uncertainties. The proposed approach is applied to numerical and experimental examples. The results demonstrate that the technique is able to identify the force and reconstruct the responses accurately.

A multivariate model to estimate environmental load on an offshore structure

Offshore structures such as oil platforms are subjected to significant environmental loads caused by wind, waves, and current. The complexity of offshore environment requires robust and reliable models to capture dependencies among environmental variables. A vine copula is a powerful tool that can be used to construct multivariate models by decomposing the complex structure into a series of simple pair copulas. Previous studies have shown that simple and symmetric copulas can be used as building blocks to construct vine copula models. In this study, symmetric and asymmetric copula functions are considered building blocks to capture all possible dependency structures. The c-vine model is then used to estimate the total environmental load on an offshore structure. Estimated loads are compared with those using the traditional independent variable approach and a multi-Gaussian distribution function-based method. The results reveal that both symmetric and asymmetric copula functions can be fitted to build c-vine copula models for the trivariate case. C-vine copulas, constructed using asymmetric copulas, provide a better estimation of the total environmental load than the independent and multivariate Gaussian methods. The result of this study is useful in probabilistic structural analysis of offshore structures for design and resilience analysis.

Entire Gaussian Functions: Probability of Zeros Absence

In this paper, we consider a random entire function of the form f(z,ω)=∑n=0+∞εn(ω1)×ξn(ω2)fnzn, where (εn) is a sequence of independent Steinhaus random variables, (ξn) is the a sequence of independent standard complex Gaussian random variables, and a sequence of numbers fn∈C is such that lim¯n→+∞|fn|n=0 and #{n:fn≠0}=+∞. We investigate asymptotic estimates of the probability P0(r)=P{ω:f(z,ω) has no zeros inside rD} as r→+∞ outside of some set E of finite logarithmic measure, i.e., ∫E∩[1,+∞)dlnr<+∞. The obtained asymptotic estimates for the probability of the absence of zeros for entire Gaussian functions are in a certain sense the best possible result. Furthermore, we give an answer to an open question of A. Nishry for such random functions.

Non-isometric quantum error correction in gravity

We construct and study an ensemble of non-isometric error correcting codes in a toy model of an evaporating black hole in two-dimensional dilaton gravity. In the preferred bases of Euclidean path integral states in the bulk and Hamiltonian eigenstates in the boundary, the encoding map is proportional to a linear transformation with independent complex Gaussian random entries of zero mean and unit variance. Using measure concentration, we show that the typical such code is very likely to preserve pairwise inner products in a set S of states that can be subexponentially large in the microcanonical Hilbert space dimension of the black hole. The size of this set also serves as an upper limit on the bulk effective field theory Hilbert space dimension. Similar techniques are used to demonstrate the existence of state-specific reconstructions of S-preserving code space unitary operators. State-specific reconstructions on subspaces exist when they are expected to by entanglement wedge reconstruction. We comment on relations to complexity theory and the breakdown of bulk effective field theory.

Achievable Rate of Near-Field Communications Based on Physically Consistent Models

This paper introduces a novel information-theoretic approach for studying the effects of mutual coupling (MC), between the transmit and receive antennas, on the overall performance of single-input-single-output (SISO) near-field communications (NFC). By incorporating the finite antenna size constraint using Chu’s theory and under the assumption of canonical-minimum scattering (CMS), we derive the MC between two radiating volumes of fixed sizes. Expressions for the self and mutual impedances are obtained by the use of the reciprocity theorem. Based on a circuit-theoretic two-port model for SISO radio communication systems, we first establish its input-output relationship where the noise depends on the self/mutual impedances of the antennas, unlike the conventional assumption of independent additive white Gaussian noise. We then characterise the achievable data rate for a given pair of transmit and receive antenna sizes, thereby providing an upper bound on the system performance under physical size constraints. Through the lens of these findings, we shed new light on the influence of MC on the information-theoretic limits of near-field communications using compact antennas.

SCMA Codebook for Uplink Rician Fading Channels

In this letter, the codebook design criteria for uplink Rician fading channels are explored. The probability distribution of the weighted sums of the Rician random variables from each frequency resource is derived based on the independent non-central circular complex Gaussian random variable. Then, considering the pairwise error probability, the key performance indicators of multidimensional constellations for uplink Rician channels are explored. A sub-optimal optimization approach is deployed to generate an enhanced codebook for various Rician scenarios. Finally, numerical results are provided to verify the key performance indicators in the uplink Rician fading channels.

Autocorrelated measurement processes and inference for ordinary differential equation models of biological systems

Ordinary differential equation models are used to describe dynamic processes across biology. To perform likelihood-based parameter inference on these models, it is necessary to specify a statistical process representing the contribution of factors not explicitly included in the mathematical model. For this, independent Gaussian noise is commonly chosen, with its use so widespread that researchers typically provide no explicit justification for this choice. This noise model assumes ‘random’ latent factors affect the system in the ephemeral fashion resulting in unsystematic deviation of observables from their modelled counterparts. However, like the deterministically modelled parts of a system, these latent factors can have persistent effects on observables. Here, we use experimental data from dynamical systems drawn from cardiac physiology and electrochemistry to demonstrate that highly persistent differences between observations and modelled quantities can occur. Considering the case when persistent noise arises owing only to measurement imperfections, we use the Fisher information matrix to quantify how uncertainty in parameter estimates is artificially reduced when erroneously assuming independent noise. We present a workflow to diagnose persistent noise from model fits and describe how to remodel accounting for correlated errors.

Trace-class Gaussian priors for Bayesian learning of neural networks with MCMC

AbstractThis paper introduces a new neural network based prior for real valued functions. Each weight and bias of the neural network has an independent Gaussian prior, with the key novelty that the variances decrease in the width of the network in such a way that the resulting function is well defined in the limit of an infinite width network. We show that the induced posterior over functions is amenable to Monte Carlo sampling using Hilbert space Markov chain Monte Carlo (MCMC) methods. This type of MCMC is stable under mesh refinement, i.e. the acceptance probability does not degenerate as more parameters of the function's prior are introduced, even ad infinitum. We demonstrate these advantages over other function space priors, for example in Bayesian Reinforcement Learning.

Generalizing the isothermal efficiency by using Gaussian distributions.

Unlike the Carnot heat engine efficiency published in 1824, an isothermal efficiency derived from thermodynamics and information theory can be applied to biological systems. The original approach by Pierce and Cutler in 1959 to derive the isothermal efficiency equation came from Shannon's channel capacity of 1949 and from Felker's 1952 determination of the minimum energy dissipation needed to gain a bit. In 1991 and 2010 Schneider showed how the isothermal efficiency equation can be applied to molecular machines and that this can be used to explain why several molecular machines are 70% efficient. Surprisingly, some macroscopic biological systems, such as whole ecosystems, are also 70% efficient but it is hard to see how this could be explained by a thermodynamic and molecular theory. The thesis of this paper is that the isothermal efficiency can be derived without using thermodynamics by starting from a set of independent Gaussian distributions. This novel derivation generalizes the isothermal efficiency equation for use at all levels of biology, from molecules to ecosystems.

Learning with Errors Public Key Cryptosystem with Its Security

The main purpose of this paper is to introduce the LWE public key cryptosystem with its security. In the first section, we introduce the LWE public key cryptosystem by Regev with its applications and some previous research results. Then we prove the security of LWE public key cryptosystem by Regev in detail. For not only independent identical Gaussian disturbances but also any general independent identical disturbances, we give a more accurate estimation probability of decryption error of general LWE cryptosystem. This guarantees high security and widespread applications of the LWE public key cryptosystem.

АНАЛІЗ ЗАСТОСОВНОСТІ МЕТОДУ ДВОЧАСТОТНОЇ ІНТЕРФЕРОМЕТРІЇ ДЛЯ ВИМІРЮВАННЯ КУТА МІСЦЯ ЦІЛЕЙ У ДВОКООРДИНАТНИХ РЛС

Subject and Purpose. The study deals with the dual-frequency radio interferometry technique, which is based on the employment of two fairly close frequencies with the aim to remove ambiguity of the radar target elevation estimation using 2D-radar and eliminate 2 pm-uncertainty of the signal phase difference measurement. Analysis of random noise action on the accuracy of the elevation angle estimation by the dual-frequency radio interferometry and assessment of practical applicability of the method make up the purpose of the paper. Methods and Methodology. The noise action on the elevation angle measurement accuracy is examined through a series of an- alytical calculations with the use of statistical analysis methods. The noise in each receiving channel is modeled in terms of additive, statistically independent stationary Gaussian processes with zero mean values and equal variances. The calculation results are checked via computer simulations with statistics estimations for 106 random noise realizations. Results. A correct condition has been developed for the sector width where the target elevation angle is unambiguously estimated depending on the space separation of the antennas (baselines) and the frequency ratio. Expressions for elevation angle estimation errors have been obtained, showing that the error is mainly contributed by the faults in the determination of the ambiguity interval number. A probability of the correct determination of the ambiguity interval number has been derived depending on the signal- to-noise ratio and the frequency difference, indicating that almost one hundred per cent probability of the correct determination of the ambiguity interval number is only achieved when the signal-to-noise ratio exceeds 30 dB. A comparative analysis has been performed between the methods of dual-frequency interferometry and conventional phase-difference direction finding in the case of close X-band frequencies and the same sectors of survey. Conclusions. The dual-frequency radio interferometry technique with close frequencies has been shown to outperform the stand- ard phase-difference direction-finding method only when the signal-to-noise ratio is sufficiently high (over 30 dB). In principle, the accuracy of the technique seems possible to improve by taking significantly different frequencies selected with regard to the scale negotiation condition. However, it should be mentioned that the implementation of the relevant algorithm in practice is much more complicated than the conventional scheme with a single frequency and several antenna baselines.

Tuning-Free Bayesian Estimation Algorithms for Faulty Sensor Signals in State-Space

Sensors provide insights into the industrial processes, while misleading sensor outputs may result in inappropriate decisions or even catastrophic accidents. In this article, the Bayesian estimation algorithms are developed to estimate unforeseen signals in sensor outputs without tuning. The optimal Bayesian estimation method is first derived by incorporating a Gaussian distribution specifying potential unmodeled dynamics into the measurement equation. Since its performance depends on tuning parameters, an iterative Bayesian estimation algorithm is developed using the variational inference technique. Specifically, an inverse Wishart distribution is introduced to describe the predicted covariance of abnormal signals. We then estimate it together with the other independent Gaussian distributions to conditionally approximate the joint posterior distribution, by which the effects of tuning parameters can be replaced adaptively. Testing the proposed algorithms through a simulated electromechanical brake model and a real experimental system shows that the proposed algorithm can satisfactorily estimate additive sensor faults online and services as a sensor monitor that simultaneously provides the locations and magnitudes of faulty signals without tuning.

Maximum Likelihood Estimation of Debye Relaxation Parameters With Open-Ended Coaxial Probes

In this paper, we present a new method to measure the complex dielectric permittivity of materials-under-test with an open-ended coaxial probe. The method, which explicitly takes measurement noise into account to enhance the stability of the measurements, is formulated based on the maximum likelihood estimation of the Debye parameters from calibrated reflection measurements performed at the aperture of the probe. In particular, the measurement noises at different frequencies are approximated with independent Gaussian distributions. To estimate the mean and the standard deviation of the mentioned probability distributions at each frequency, the differences among repetitive measurements of a reference sample are analyzed. Later, these statistical distributions are enforced into a cost functional to derive an optimization scheme based on the maximum likelihood estimation. Finally, the cost functional is iteratively minimized with a second-order Newton-based technique to retrieve the Debye relaxation parameters of a material. We further experimentally verified the method with various liquid samples and the accuracy of the obtained results is compared with the previous works.

Distribution of the Volume of Weighted Gaussian Simplex

Let X0,...,Xl be independent standard Gaussian vectors in ℝd such that l ≤ d. An explicit formula is obtained for the distribution of the volume of a weighted Gaussian simplex without origin, which is an l-dimensional simplex conv(σ0X0,..., σlXl) (σ0,..., σl > 0).