Gaussian Random Variables Research Articles

Dynamically selected steady states and criticality in non-reciprocal networks

We consider a simple neural network model, evolving via non-linear coupled stochastic differential equations, where neural couplings are random Gaussian variables with non-zero mean and arbitrary degree of reciprocity. Using a path-integral approach, we analyse the dynamics, averaged over the network ensemble, in the thermodynamic limit. Our results show that for any degree of reciprocity in the couplings, two types of criticality emerge, corresponding to ferromagnetic and spin-glass order, respectively. The critical lines separating the disordered from the ordered phases is consistent with spectral properties of the coupling matrix, as derived from random matrix theory. As the non-reciprocity (or asymmetry) in the couplings increases, both ordered phases diminish in size, ultimately resulting in the disappearance of the spin-glass phase when the couplings become anti-symmetric. We investigate non-fixed point steady-state solutions for uncorrelated interactions. For such solutions the time-lagged correlation function evolves according to a gradient-descent dynamics on a potential, which depends on the stationary variance. Our analysis shows that in the spin-glass region, the variance dynamically selected by the system leads the correlation function to evolve on the separatrix curve, limiting different realizable steady states, whereas in the ferromagnetic region, a fixed point solution is selected as the only realizable steady state. In the spin-glass region, stationary solutions are unstable against perturbations that break time-translation invariance, indicating chaotic behaviour in large single network instances. Numerical analysis of Lyapunov exponents confirms that chaotic behaviour emerges throughout the spin-glass region, for any value of the coupling correlations. While negative correlations increase the strength of chaos, positive ones reduce it, with chaos disappearing for reciprocal (i.e. symmetric) couplings, where marginal stability is attained. On the other hand, in finite size non-reciprocal networks, fixed points and limit cycles can arise in the spin-glass region, especially close to the critical line. Finally, we show that chaos is suppressed when the strength of external noise exceeds a certain threshold. Intriguing analogies between chaotic phases in non-equilibrium systems and spin-glass phases in equilibrium are put forward.

Efficient JPEG image steganography using pairwise conditional random field model

Recent research shows that the security performance of JPEG image steganography can be improved by developing an accurate statistical model to characterize the correlations between Discrete Cosine Transform (DCT) coefficients. In this paper, a novel JPEG steganographic scheme called J-CRF is proposed to characterize the correlations between DCT coefficients using a pairwise Conditional Random Field (CRF) model, in which the correlated DCT coefficient pairs are modeled as bivariate Gaussian random variables. Under the framework of pairwise CRF, the proposed scheme formulates the problem of minimizing the statistical detectability within the Gaussian image model as the minimization of the sum of a series of unary and pairwise potentials, both of which are utilized to measure the local detectability of embedding messages in the DCT coefficient pairs. All the potentials associated with statistical detectability are formalized as the KL divergence between the statistical distributions of the cover and the stego. Experimental results demonstrate that the proposed J-CRF can outperform other SOTA JPEG steganographic schemes, e.g., J-UNIWARD, J-MiPOD, and JEC-RL, in resisting the detection of various advanced steganalyzers. In addition, the performance of J-CRF could be further boosted by reformulating it as the weighted sum of unary and pairwise potentials, called J-CRFW.

Structural Properties of the Wyner-Ziv Rate Distortion Function: Applications for Multivariate Gaussian Sources.

The main focus of this paper is the derivation of the structural properties of the test channels of Wyner's operational information rate distortion function (RDF), R¯(ΔX), for arbitrary abstract sources and, subsequently, the derivation of additional properties for a tuple of multivariate correlated, jointly independent, and identically distributed Gaussian random variables, {Xt,Yt}t=1∞, Xt:Ω→Rnx, Yt:Ω→Rny, with average mean-square error at the decoder and the side information, {Yt}t=1∞, available only at the decoder. For the tuple of multivariate correlated Gaussian sources, we construct optimal test channel realizations which achieve the informational RDF, R¯(ΔX)=▵infM(ΔX)I(X;Z|Y), where M(ΔX) is the set of auxiliary RVs Z such that PZ|X,Y=PZ|X, X^=f(Y,Z), and E{||X-X^||2}≤ΔX. We show the following fundamental structural properties: (1) Optimal test channel realizations that achieve the RDF and satisfy conditional independence, PX|X^,Y,Z=PX|X^,Y=PX|X^,EX|X^,Y,Z=EX|X^=X^. (2) Similarly, for the conditional RDF, RX|Y(ΔX), when the side information is available to both the encoder and the decoder, we show the equality R¯(ΔX)=RX|Y(ΔX). (3) We derive the water-filling solution for RX|Y(ΔX).

On the effect of confounding in linear regression models: an approach based on the theory of quadratic forms

In the last two decades, significant research efforts have been dedicated to addressing the issue of spatial confounding in linear regression models. Confounding occurs when the relationship between the covariate and the response variable is influenced by an unmeasured confounder associated with both. This results in biased estimators for the regression coefficients reduced efficiency, and misleading interpretations. This article aims to understand how confounding relates to the parameters of the data generating process. The sampling properties of the regression coefficient estimator are derived as ratios of dependent quadratic forms in Gaussian random variables: this allows us to obtain exact expressions for the marginal bias and variance of the estimator, that were not obtained in previous studies. Moreover, we provide an approximate measure of the marginal bias that gives insights of the main determinants of bias. Applications in the framework of geostatistical and areal data modeling are presented. Particular attention is devoted to the difference between smoothness and variability of random vectors involved in the data generating process. Results indicate that marginal covariance between the covariate and the confounder, along with marginal variability of the covariate, play the most relevant role in determining the magnitude of confounding, as measured by the bias.

ANALYSIS FOR MULTIPLE RESPONSES IN A COMPLETELY RANDOMIZED EXPERIMENTAL DESIGN

Multiple responses are often generated in agricultural and forestry research. For example, the moisture content, fatty acids, carbohydrates, size, diameter, length, shape and hardness, among other characteristics are measured to cottonseed. Of course, multiple responses are observed in many other areas of research. Multivariate analysis of variance (MANOVA) can be useful for multiple response analysis when differences in treatment effects are to be determined. However, the performance of current post hoc tests in this context is not satisfactory due to the limitations of the available methods, or because they are difficult to use for non-statistician researchers. Furthermore, this methodology requires the assumptions of multivariate normality and homogeneity of variance and covariance matrices, assumptions that are difficult to verify if the sample size is small. This research proposes an alternative analysis to test the hypothesis of equality of effects between treatments and post hoc tests in the case of multiple responses. An asymptotic result is demonstrated for the random variable generated in the proposal for the case of uncorrelated normal variables and the case for correlated normal random variables is left open. A simulation study shows that the performance of the proposal with small samples is satisfactory in terms of power and that it has advantages compared to MANOVA. Furthermore, the methodological approach allows for post hoc testing in the case of multiple responses in the completely randomized experimental design.

Mutual information between joint angles and toe height in healthy subjects

Understanding the relationship between the position of the foot and the lower limb joint angles during normal gait is critical for the identification of the mechanisms involved in pathological gait. In this article, we introduce a novel framework that characterizes this relationship using mutual information in healthy subjects. The nonlinear connection between these variables is quantified using mutual information, and the MINE algorithm is used for precise estimation. Several simulations over Gaussian bivariate random variables reveal that overfitting is a critical factor affecting mutual information estimations by MINE, and we propose a simple methodology to address this issue when using gait signals. Our findings indicate that the statistical dependency between joint angles and toe height is symmetrical between the limbs for healthy subjects. Additionally, our results show that the ipsilateral knee angle holds the most significant amount of information regarding the toe height. In simpler terms, this knee angle serves as the most reliable predictor for toe height, and vice versa. To further enhance our understanding, we have developed a mutual information reference profile that enables the comparison of how the relationship between these biomechanical variables evolves under pathological conditions. The findings have significant implications for gait analysis and may aid in the identification and understanding of gait abnormalities in various contexts, contributing to the advancement of biomechanical research and clinical applications.

RANK-BASED INDICES FOR TESTING INDEPENDENCE BETWEEN TWO HIGH-DIMENSIONAL VECTORS.

To test independence between two high-dimensional random vectors, we propose three tests based on the rank-based indices derived from Hoeffding's , Blum-Kiefer-Rosenblatt's and Bergsma-Dassios-Yanagimoto's . Under the null hypothesis of independence, we show that the distributions of the proposed test statistics converge to normal ones if the dimensions diverge arbitrarily with the sample size. We further derive an explicit rate of convergence. Thanks to the monotone transformation-invariant property, these distribution-free tests can be readily used to generally distributed random vectors including heavily tailed ones. We further study the local power of the proposed tests and compare their relative efficiencies with two classic distance covariance/correlation based tests in high dimensional settings. We establish explicit relationships between and Pearson's correlation for bivariate normal random variables. The relationships serve as a basis for power comparison. Our theoretical results show that under a Gaussian equicorrelation alternative, (i) the proposed tests are superior to the two classic distance covariance/correlation based tests if the components of random vectors have very different scales; (ii) the asymptotic efficiency of the proposed tests based on and are sorted in a descending order.

ON THE CUMULATIVE DISTRIBUTION FUNCTION OF THE VARIANCE-GAMMA DISTRIBUTION

AbstractWe obtain exact formulas for the cumulative distribution function of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. From these formulas, we deduce exact formulas for the cumulative distribution function of the product of two correlated zero-mean normal random variables.

A Novel Tightly Coupled Solution for SINS/Polarized Navigation System/Odometer Integration Using Polarized and Installed Angle Errors Model

Precise and reliable autonomous navigation in a GPS-denied environment is critical to unmanned systems. The idea of combining SINS, the polarized navigation system (PNS), and the odometer (OD) inspired by desert ants has been proven to be effective for autonomous navigation. However, there are two major challenges for polarization navigation nowadays: inaccurate modeling and obtaining reliable heading information when some sensor channels are blocked. Aiming at these two problems, a tightly-coupled solution for SINS/PNS is proposed in this paper. To obtain a refined integrated navigation system model, the installation errors between the inertial units and PNS, and polarization angle calculation errors are analyzed and modeled for SINS/PNS. Then, to quickly gain the accurate state estimation, an improved iterative unscented filtering method is devised. In particular, the sigma-point updating step with the conditional distribution of high-dimensional Gaussian distribution random variables is developed, which employs partial states to sample in each iteration to reduce the calculation burden. Finally, a detection and elimination mechanism for the abnormal light channels is provided to enhance the reliability of the integration in the presence of a light blockage. The optical channels for navigation are chosen using this mechanism depending on the difference between the predicted and measured incident light intensities. The results in both simulations and outdoor experiments show that the proposed method provides a higher heading estimation accuracy than the traditional SINS/PNS navigation method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This article is motivated by the inaccurate modeling and obtaining reliable heading in the presence of light occlusion for the polarization navigation. Various integrated navigation algorithms based on polarized skylight have been widely developed. However, the complex environment and inaccurate modeling limit the application of the polarization navigation. This article gives a novel accurate modeling method and reliable navigation algorithm. Tightly-coupled model with installation errors and the polarization angle error aims to improve the accuracy of the polarization navigation model. The improved iterative unscented Kalman filter is utilized to quickly obtain accurate state estimation. And the light detection and elimination mechanism is used to select normal light channels to navigate in the presence of light blockage. This navigation method can also extend the application of the polarization navigation.

A note on the Gumbel convergence for the Lee and Mykland jump tests

The Lee and Mykland (2008, 2012) nonparametric jump tests have been widely used in the literature but its critical region is stated with reference to the asymptotic distribution of the maximum of a set of standard normal variates. However, such reference would imply a typo (of a non-negligible order) for the norming constants adopted. By using the asymptotic distribution of the maximum of a set of folded normal random variables instead, this paper shows that there is no typo at all, thus preserving the validity of all the empirical findings based on these tests.

A dynamical Gaussian, lognormal, and reverse lognormal Kalman filter

AbstractWe derive a generalization of the Kalman filter that allows for non‐Gaussian background and observation errors. The Gaussian assumption is replaced by considering that the errors come from a mixed distribution of Gaussian, lognormal, and reverse lognormal random variables. We detail the derivation for reverse lognormal errors and extend the results to mixed distributions, where the number of Gaussian, lognormal, and reverse lognormal state variables can change dynamically every analysis time. We test the dynamical mixed Kalman filter robustly on two different systems based on the Lorenz 1963 model, and demonstrate that non‐Gaussian techniques generally improve the analysis skill if the observations are sparse and uncertain, compared with the Gaussian Kalman filter.

Improved EWMA schemes for monitoring the ratio of two normal random variables

AbstractMonitoring the ratio of two normal random variables, which is usually denoted as RZ, is one of the most important directions in statistical process control (SPC). In production scenarios, when the product specification is related to the relative ratio of two components in a mixture, or when the ratio represents a property of the product, or the difference between a product's quality measurement before and after an operation (such as a chemical reaction), the product quality can be improved by applying monitoring schemes for RZ. To further enhance the sensitivity of traditional RZ schemes in detecting process changes, especially small and moderate ones, this paper proposes an improved exponentially weighted moving average (IEWMA) monitoring scheme. The run length (RL) distribution of the proposed IEWMA‐RZ scheme is obtained by using the Monte‐Carlo (MC) simulation. The superiority of the proposed monitoring scheme is shown by comparing with the existing EWMA‐RZ scheme.

Numerical Method for Stochastic Nonlinear Schrödinger Equation Driven by Multivariate Gaussian Measure: Algorithms and Applications

In this paper, we present a novel Galerkin spectral method for numerically solving the stochastic nonlinear Schrödinger (NLS) equation driven by multivariate Gaussian random variables, including the nonlinear term. Our approach involves deriving the tensor product of triple random orthogonal basis and random functions, which enables us to effectively handle the stochasticity and nonlinear term in the equation. We apply this newly proposed method to solve both one- and two-dimensional stochastic NLS equations, providing detailed analysis and comparing the results with Monte Carlo simulation. In addition, the proposed method is applied to the stochastic Ginzburg–Landau equation. Our method exhibits excellent performance in both spatial and random spaces, achieving spectral accuracy in the numerical solutions.

Semiparametric normal transformation joint model of multivariate longitudinal and bivariate time-to-event data.

Joint models for longitudinal and survival data (JMLSs) are widely used to investigate the relationship between longitudinal and survival data in clinical trials in recent years. But, the existing studies mainly focus on independent survival data. In many clinical trials, survival data may be bivariately correlated. To this end, this paper proposes a novel JMLS accommodating multivariate longitudinal and bivariate correlated time-to-event data. Nonparametric marginal survival hazard functions are transformed to bivariate normal random variables. Bayesian penalized splines are employed to approximate unknown baseline hazard functions. Incorporating the Metropolis-Hastings algorithm into the Gibbs sampler, we develop a Bayesian adaptive Lasso method to simultaneously estimate parameters and baseline hazard functions, and select important predictors in the considered JMLS. Simulation studies and an example taken from the International Breast Cancer Study Group are used to illustrate the proposed methodologies.

Quantifying Prediction Uncertainty in Regression Using Random Fuzzy Sets: The ENNreg Model

We introduce a neural network model for regression in which prediction uncertainty is quantified by Gaussian random fuzzy numbers (GRFNs), a newly introduced family of random fuzzy subsets of the real line that generalizes both Gaussian random variables and Gaussian possibility distributions. The output GRFN is constructed by combining GRFNs induced by prototypes using a combination operator that generalizes Dempster's rule of Evidence Theory. The three output units indicate the most plausible value of the response variable, variability around this value, and epistemic uncertainty. The network is trained by minimizing a loss function that generalizes the negative log-likelihood. Comparative experiments show that this method is competitive, both in terms of prediction accuracy and calibration error, with state-of-the-art techniques such as random forests or deep learning with Monte Carlo dropout. In addition, the model outputs a predictive belief function that can be shown to be calibrated, in the sense that it allows us to compute conservative prediction intervals with specified belief degree.

Diagnostic Method for Short Circuit Faults at the Generator End of Ship Power Systems Based on MWDN and Deep-Gated RNN-FCN

Synchronous generators with three phases are crucial components of modern integrated power systems in ships. These generators provide power for the entire operation of the vessel. Therefore, it is of paramount importance to diagnose short-circuit faults at the generator terminal in the ship’s power system to ensure the safe and stable operation of modern ships. In this study, a generator terminal short-circuit fault diagnosis method is proposed based on a hybrid model that combines the Multi-Level Wavelet Decomposition Network, Deep-Gated Recurrent Neural Network, and Fully Convolutional Network. Firstly, the Multi-Level Wavelet Decomposition Network is used to decompose and denoise the collected electrical signals, thus dividing them into sub-signals and extracting their time-domain and frequency-domain features. Secondly, synthetic oversampling based on Gaussian random variables is employed to address the problem of imbalance between normal data and fault data, resulting in a balanced dataset. Finally, the dataset is fed into the hybrid model of the Deep-Gated Recurrent Neural Network and Fully Convolutional Network for feature extraction and classification of faults, ultimately outputting the fault diagnosis results. To validate the performance of the proposed method, simulations and comparative analysis with other algorithms are conducted on the fault diagnosis method. The proposed algorithm’s accuracy reaches 96.82%, precision reaches 97.35%, and the area under curve reaches 0.85, indicating accurate feature extraction and classification for identifying short-circuit faults at the generator terminals.

Bernstein–Jackson Inequalities on Gaussian Hilbert Spaces

Estimates of best approximations by exponential type analytic functions in Gaussian random variables with respect to the Malliavin derivative in the form of Bernstein–Jackson inequalities with exact constants are established. Formulas for constants are expressed through basic parameters of approximation spaces. The relationship between approximation Gaussian Hilbert spaces and classic Besov spaces are shown.

An example showing that the sum of two normal random variables may not be normal

Two novel proofs show that the sum of a specific pair of normal random variables is not normal are established in this note. This is one of the most often misunderstood facts by first-year students in probability theory and statistics. The first proof is concise using the moment generating function. The second proof checks whether the moments of the sum have the property of normal distribution.

High-Order Spectral Method of Density Estimation for Stochastic Differential Equation Driven by Multivariate Gaussian Random Variables

There are some previous works on designing efficient and high-order numerical methods of density estimation for stochastic partial differential equation (SPDE) driven by multivariate Gaussian random variables. They mostly focus on proposing numerical methods of density estimation for SPDE with independent random variables and rarely research density estimation for SPDE is driven by multivariate Gaussian random variables. In this paper, we propose a high-order algorithm of gPC-based density estimation where SPDE driven by multivariate Gaussian random variables. Our main techniques are (1) we build a new multivariate orthogonal basis by adopting the Gauss–Schmidt orthogonalization; (2) with the newly constructed orthogonal basis in hand, we first assume the unknown function in the SPDE has the stochastic general polynomial chaos (gPC) expansion, second implement the stochastic gPC expansion for the SPDE in the multivariate Gaussian measure space, and third we obtain and numerical calculation deterministic differential equations for the coefficients of the expansion; (3) we used high-order algorithm of gPC-based for density estimation and moment estimation. We apply the newly proposed numerical method to a known random function, stochastic 1D wave equation, and stochastic 2D Schnakenberg model, respectively. All the presented stochastic equations are driven by bivariate Gaussian random variables. The efficiency is compared with the Monte-Carlo method based on the known random function.

Supremum Distribution of Weighted Sum of Random Processes from Orlicz Spaces of Exponential Type with Continuous Deviation

The paper studies distribution of sum of random processes from Orlicz spaces of exponential type weighted by continuous functions, in particular, processes from spaces Subϕ (Ω), SSubϕ (Ω) and class V (ϕ, ψ) are considered. Such spaces and classes of random variables and corresponding stochastic processes provide generalizations of Gaussian and sub-Gaussian random variables and processes and are important for various applications, for example, in queuing theory and financial mathematics. We derive the estimates for the distribution of supremum of weighted sum of such processes deviated by a continuous monotone function using the entropy method. As examples, weighted sum of Wiener and weighted sum of fractional Brownian motion processes with different Hurst indices from classes V (ϕ, ψ) are considered. Corresponding estimates of the probability of exceeding by trajectories of such weighted sums a positive level determined by a linear function are obtained. In the insurance risk theory, such aproblem arises during estimating a ruin probability of the corresponding risk process with a constant premium income, and in the communications theory, it appears for the buffer overflow probability for a single server with a constant service rate.