non-Gaussian Noise Models Research Articles

Extension of the characterization of non-Gaussianity in gravitational wave detectors with a statistical hypothesis test

Abstract In gravitational wave (GW) astronomy, non-Gaussian noise, such as scattered light noise disturbs stable interferometer operation, limiting the interferometer’s sensitivity, and reducing the reliability of the analyses. In scattered light noise, the non-Gaussian noise dominates the sensitivity in a low frequency range of less than a few hundred Hz, which is sensitive to GWs from compact binary coalescence. This non-Gaussian noise prevents reliable parameter estimation, since several analysis methods are optimized only for Gaussian noise. Therefore, identifying data contaminated by non-Gaussian noise is important. In this work, we extended the conventional method to evaluate non-Gaussian noise, the Rayleigh statistic, by using a statistical hypothesis test to determine a threshold for non-Gaussian noise. First, we estimated the distribution of the Rayleigh statistic against Gaussian noise, called the background distribution, and validated that our extension serves as the hypothetical test. The threshold on the Rayleigh statistic is estimated at 0.73 and 1.28 when the significance level is 0.05, and the sample size is 39. Moreover, we investigated the detection efficiency by assuming two non-Gaussian noise models. For example, for the model with strong scattered light noise, the true positive rate (TPR) was always above 0.7 when the significance level was 0.05. For the demonstration, we applied our extension with the estimated thresholds to the real data. We confirmed our extension provides the binarized outputs of the statistical tests and contributes to finding the non-Gaussian noise in the real data. The results showed that our extension can contribute to an initial detection of non-Gaussian noise and lead to further investigation of the origin of the non-Gaussian noise.

Modeling the measurement precision of Fringe Projection Profilometry

Three-dimensional (3D) surface geometry provides elemental information in various sciences and precision engineering. Fringe Projection Profilometry (FPP) is one of the most powerful non-contact (thus non-destructive) and non-interferometric (thus less restrictive) 3D measurement techniques, featuring at its high precision. However, the measurement precision of FPP is currently evaluated experimentally, lacking a complete theoretical model for guidance. We propose the first complete FPP precision model chain including four stage models (camera intensity, fringe intensity, phase and 3D geometry) and two transfer models (from fringe intensity to phase and from phase to 3D geometry). The most significant contributions include the adoption of a non-Gaussian camera noise model, which, for the first time, establishes the connection between camera’s electronics parameters (known in advance from the camera manufacturer) and the phase precision, and the formulation of the phase to geometry transfer, which makes the precision of the measured geometry representable in an explicit and concise form. As a result, we not only establish the full precision model of the 3D geometry to characterize the performance of an FPP system that has already been set up, but also explore the expression of the highest possible precision limit to guide the error distribution of an FPP system that is yet to build. Our theoretical models make FPP a more designable technique to meet the challenges from various measurement demands concerning different object sizes from macro to micro and requiring different measurement precisions from a few millimeters to a few micrometers.

Profile-Wise Analysis: A profile likelihood-based workflow for identifiability analysis, estimation, and prediction with mechanistic mathematical models.

Interpreting data using mechanistic mathematical models provides a foundation for discovery and decision-making in all areas of science and engineering. Developing mechanistic insight by combining mathematical models and experimental data is especially critical in mathematical biology as new data and new types of data are collected and reported. Key steps in using mechanistic mathematical models to interpret data include: (i) identifiability analysis; (ii) parameter estimation; and (iii) model prediction. Here we present a systematic, computationally-efficient workflow we call Profile-Wise Analysis (PWA) that addresses all three steps in a unified way. Recently-developed methods for constructing 'profile-wise' prediction intervals enable this workflow and provide the central linkage between different workflow components. These methods propagate profile-likelihood-based confidence sets for model parameters to predictions in a way that isolates how different parameter combinations affect model predictions. We show how to extend these profile-wise prediction intervals to two-dimensional interest parameters. We then demonstrate how to combine profile-wise prediction confidence sets to give an overall prediction confidence set that approximates the full likelihood-based prediction confidence set well. Our three case studies illustrate practical aspects of the workflow, focusing on ordinary differential equation (ODE) mechanistic models with both Gaussian and non-Gaussian noise models. While the case studies focus on ODE-based models, the workflow applies to other classes of mathematical models, including partial differential equations and simulation-based stochastic models. Open-source software on GitHub can be used to replicate the case studies.

Optimization of capacity in non-Gaussian noise models with and without fading channels for sustainable communication systems

The highest rate at which information may be reliably sent via a communication link is known as its capacity. In the case of non-Gaussian noise, the capacity of the channel depends on the specific characteristics of the noise, which can cause severe errors and reduce the reliability of communication systems over a fading channel. The Gaussian mixture impulsive noise model (GMINM), which is a more general and flexible non-Gaussian model for impulsive noise, has been compared in this paper with the Middleton Class-A impulsive noise model (MCAINM) in terms of derived channel capacity normalized by channel bandwidth (C/BW) with and without Rayleigh fading (Rf) channels. It also investigated the trade-off between complexity and accuracy in modeling the impulsive noise using two simplified Middleton Class-A impulsive noise models based on derived C/BW. The derived C/BW of these models under various conditions, such as different signal-to-noise ratios and impulsive noise parameters and models, have been performed and evaluated using two different scenarios: the exact method and the semi-analytical method. When the impulsive noise parameters and A are both near 0 in GMINM and MCAINM, respectively, the capacity of the impulsive noise channel is found to be equivalent to that of the Gaussian channel sustainable, as shown by the findings based on Monte-Carlo simulations. We have shown that when the impulsive noise decreases, the capacity increases in all models; however, the capacity of Gaussian noise is higher than the capacity of non-Gaussian noise, which in turn is higher than the capacity of non-Gaussian noise over the Rf channel overall values of SNR in dB. Moreover, multi-channel configuration introduces spatial diversity and multiplexing gains that have been proposed to sustainably optimize the ergodic capacity for the challenge case when the channel state information (CSI) is unknown at the transmitter in non-Gaussian noise over Rf channel. In today's rapidly evolving world, sustainable communication systems play a crucial role in ensuring efficient and responsible utilization of resources. As the demand for wireless communication continues to rise, it becomes imperative to optimize the capacity of communication channels, especially in scenarios involving non-Gaussian noise models and fading channels.

Blind receiver design-based multi-dimensional stable distribution model using the EM algorithm

Multi-antenna receivers are a key technology for modern communication systems. Signal attenuation in the very low frequency (VLF) channel is a serious problem. In addition to high signal attenuation, the resulting noise on the VLF channel is non-Gaussian. Hence, to analyze and address the issue of the normal operation of the multi-antenna receiver in the VLF channel, we must study the signal detection problem in a multi-dimensional non-Gaussian fading channel. Motivated by the existing blind receiver in an underwater submarine single-antenna receiver, we study the signal detection and estimation algorithm under the fading channel for a multi-dimensional non-Gaussian noise model. In this study, we propose a blind receiver based on the expectation-maximization (EM) algorithm. The proposed blind receiver can reduce non-Gaussian noise. In addition, we propose a nonlinear receiver that can accurately receive the transmitted signal over the high-attenuation VLF communication systems. Numerical and simulation results over uncorrelated and correlated non-Gaussian channels confirm that the design of the proposed blind receiver is close to optimal, with low computation complexity.

Robust DOA estimation of complex correlated signals in non-Gaussian CES distributed models

The sample covariance matrix (SCM) is commonly used in direction-of-arrival (DOA) estimation methods when the noise or observations are circular complex Gaussian (C CG) distributed. However, with a very heavy-tailed non-Gaussian noise model, the SCM-based DOA estimation methods fail to provide an accurate estimate of DOA. This paper presents a numerical analysis of the resolving capability of subspace-based circular (C) and non-circular (NC) multiple signal classification (MUSIC) DOA estimation methods of arbitrarily narrowband correlated signal sources corrupted by circular complex elliptical symmetric (C CES) distributed noise. It evaluates the robustness of these methods for correlated C and NC sources by employing the robust complex M-estimators instead of SCM. It study also the effects of correlation on robust MUSIC-based DOA estimation algorithms accuracy as a function of the magnitude and phase of the correlation coefficients. Simulations results show that the NC MUSIC algorithm which requires fewer sensor elements yields robust estimates of DOA for correlated sources than the C MUSIC algorithm using the M-estimators.

Error analysis of selection combining over α–μ fading with symmetric alpha-stable noise

In this paper, error performance of a communication link subjected to a non-Gaussian noise model known as symmetric alpha-stable (S α S) noise and α − μ fading environment is analyzed with selection combining at the receiver. To this end, using binary phase-shift keying (BPSK), closed-form expressions of exact and asymptotic bit-error-rate (BER) are obtained. Furthermore, closed-form expressions of outage probability and n -th moment are obtained. From the n -th moment, the average signal-to-noise ratio (SNR) and channel quality estimation index (CQEI) are calculated. The effect of the number of antennas ( L ) and the fading parameters on the BER is studied for different impulsive settings. Monte-Carlo simulations are performed to validate the derived results.

Parameter estimation of underwater impulsive noise with the Class B model

The statistical characteristic of ocean ambient noise plays an important role in developing underwater signal processors. Considering that the noise in shallow water shows the impulsive nature, non-Gaussian noise models are usually applied to model the ocean ambient noise. In this study, the ocean ambient noise is modelled by using the Middleton Class B model, which can be decomposed into Gaussian and non-Gaussian models. Then, the parameters of Class B model are estimated based on the least-square estimation method, which can be deduced by using the characteristic function of the Middleton Class B model. The processing results of simulated data and real data indicate that the presented method well estimates the parameters of Class B modelled noise. Besides, it further shows that the Middleton Class B model is suitable for modelling the impulsive noise in shallow water.

Two-Dimensional Robust Source Localization Under Non-Gaussian Noise

Various methods have been proposed to estimate the direction of arrival (DOA) of sources under the assumption of Gaussian noise. This assumption, based on the central limit theorem, has been mainly used because it offers an appropriate model in a homogeneous environment. Nevertheless, under certain conditions, the Gaussian hypothesis cannot fully represent the noisy environment. Consequently, these classical estimation methods are no longer suitable, and the use of non-Gaussian noise model is necessary. In this paper, we therefore treat the DOA estimation problem in a non-Gaussian framework based on the maximum likelihood approach using a compound-Gaussian (CG) noise model. We then propose the use of the expectation maximization (EM) algorithm to reduce the computational cost of the proposed algorithm. Several simulations are carried out to illustrate the interest of our approach compared to the state of the art.

Sample Complexity of Solving Non-Cooperative Games

This paper studies the complexity of solving two classes of non-cooperative games in a distributed manner, in which the players communicate with a set of system nodes over noisy communication channels. The complexity of solving each game class is defined as the minimum number of iterations required to find a Nash equilibrium (NE) of any game in that class with $\epsilon $ accuracy. First, we consider the class $\mathcal {G}$ of all $N$ -player non-cooperative games with a continuous action space that admit at least one NE. Using information-theoretic inequalities, a lower bound on the complexity of solving $\mathcal {G}$ is derived which depends on the Kolmogorov $2\epsilon $ -capacity of the constraint set and the total capacity of the communication channels. Our results indicate that the game class $\mathcal {G}$ can be solved at most exponentially fast. We next consider the class of all $N$ -player non-cooperative games with at least one NE such that the players’ utility functions satisfy a certain (differential) constraint. We derive lower bounds on the complexity of solving this game class under both Gaussian and non-Gaussian noise models. Finally, we derive upper and lower bounds on the sample complexity of a class of quadratic games. It is shown that the complexity of solving this game class scales according to $\Theta \left ({\frac {1}{\epsilon ^{2}}}\right)$ where $\epsilon $ is the accuracy parameter.

A Gaussian Uniform Mixture Model for Robust Kalman Filtering

This article presents a Kalman-type recursive estimator for discrete-time systems with a measurement noise modeled by a Gaussian-uniform mixture. The objective is to deal with data containing outliers that degrade the performance of the regular Kalman filter. The proposed non-Gaussian noise model takes into account the reliability of the measurement with respect to erroneous data. The Kalman-type estimator is based on Masreliez's formulation which copes with non-Gaussian noise models. Results in different simulated conditions are displayed to evaluate the performance of the newly-presented algorithm and to compare it to state-of-art alternatives.

Turbulence Error Modeling and Restriction for Satellite Attitude Determination System Based on Improved Maximum Correntropy Kalman Filter

In the process of satellite attitude determination, satellites or sensors themselves often encounter a variety of turbulence influences due to the complexity of space environments. Such influences can lead to the mutation and non-Gaussian noises for the attitude determination system. To solve these problems, in this paper, we construct a unified error model for the turbulence influences, which is a non-Gaussian noise model, and propose an improved attitude filter method to restrict the turbulence noises and the system mutation to enhance attitude determination accuracy and robustness. The unified error model combined with jitters and vibrations in the actual process of satellite attitude determination is firstly designed. Then an Improved Adaptive Kalman filter (IAKF) based on both the Strong Tracking Filter (STF) and the Maximum Correntropy Filter (MCKF) is put forward. By using of the optimization principle with both of fading factor and Maximum Correntropy Criterion (MCC), this proposed filter algorithm can suppress the influences of system mutations and non-Gaussian noises at the same time. It can eliminate the system mutations and the turbulence errors, and achieve excellent robustness and the attitude determination accuracy for the nonlinear system. Extensive simulations of the proposed filter are conducted under the conditions of the Gaussian noises, system mutation with large outliers, non-Gaussian noise with turbulence noises, and both the mutation and non-Gaussian turbulence error. The results demonstrate that our filter outperforms the existing attitude filter algorithms significantly.

Optimal signal processing algorithm at the background of non-Gaussian flicker noise

The problem of receiving and processing ultra-low-power signals of information transmission systems is being solved. High requirements for energy efficiency on the one hand and a low information transfer rate allows the use of signals with a small spectrum width, including flicker noise spectral regions. A non-Gaussian flicker noise model is used based on a stochastic differential equation with a nonlinear drift coefficient. An optimal signal processing algorithm is being developed against the background of the sum of flicker noise and thermal noise based on an estimated-correlation-compensation approach. The analysis of the effectiveness of optimal signal processing against a background of non-Gaussian flicker noise and thermal noise is carried out.

Parameter Estimation for Class A Modeled Ocean Ambient Noise

A Gaussian distribution is used by all traditional underwater acoustic signal processors, thus neglecting the impulsive property of ocean ambient noise in shallow waters. Undoubtedly, signal processors designed with a Gaussian model are sub-optimal in the presence of non-Gaussian noise. To solve this problem, firstly a quantile-quantile (Q-Q) plot of real data was analyzed, which further showed the necessity of investigating a non-Gaussian noise model. A Middleton Class A noise model considering impulsive noise was used to model non-Gaussian noise in shallow waters. After that, parameter estimation for the Class A model was carried out with the characteristic function. Lastly, the effectiveness of the method proposed in this paper was verified by using simulated data and real data.

Spherical Deconvolution of Multichannel Diffusion MRI Data with Non-Gaussian Noise Models and Spatial Regularization.

Spherical deconvolution (SD) methods are widely used to estimate the intra-voxel white-matter fiber orientations from diffusion MRI data. However, while some of these methods assume a zero-mean Gaussian distribution for the underlying noise, its real distribution is known to be non-Gaussian and to depend on many factors such as the number of coils and the methodology used to combine multichannel MRI signals. Indeed, the two prevailing methods for multichannel signal combination lead to noise patterns better described by Rician and noncentral Chi distributions. Here we develop a Robust and Unbiased Model-BAsed Spherical Deconvolution (RUMBA-SD) technique, intended to deal with realistic MRI noise, based on a Richardson-Lucy (RL) algorithm adapted to Rician and noncentral Chi likelihood models. To quantify the benefits of using proper noise models, RUMBA-SD was compared with dRL-SD, a well-established method based on the RL algorithm for Gaussian noise. Another aim of the study was to quantify the impact of including a total variation (TV) spatial regularization term in the estimation framework. To do this, we developed TV spatially-regularized versions of both RUMBA-SD and dRL-SD algorithms. The evaluation was performed by comparing various quality metrics on 132 three-dimensional synthetic phantoms involving different inter-fiber angles and volume fractions, which were contaminated with noise mimicking patterns generated by data processing in multichannel scanners. The results demonstrate that the inclusion of proper likelihood models leads to an increased ability to resolve fiber crossings with smaller inter-fiber angles and to better detect non-dominant fibers. The inclusion of TV regularization dramatically improved the resolution power of both techniques. The above findings were also verified in human brain data.

The Linearized Bregman Method via Split Feasibility Problems: Analysis and Generalizations

The linearized Bregman method is a method to calculate sparse solutions to systems of linear equations. We formulate this problem as a split feasibility problem, propose an algorithmic framework based on Bregman projections, and prove a general convergence result for this framework. Convergence of the linearized Bregman method will be obtained as a special case. Our approach also allows for several generalizations such as other objective functions, incremental iterations, incorporation of non-Gaussian noise models, and box constraints.

Least-squares independence regression for non-linear causal inference under non-Gaussian noise

The discovery of non-linear causal relationship under additive non-Gaussian noise models has attracted considerable attention recently because of their high flexibility. In this paper, we propose a novel causal inference algorithm called least-squares independence regression (LSIR). LSIR learns the additive noise model through the minimization of an estimator of the squared-loss mutual information between inputs and residuals. A notable advantage of LSIR over existing approaches is that tuning parameters such as the kernel width and the regularization parameter can be naturally optimized by cross-validation, allowing us to avoid overfitting in a data-dependent fashion. Through experiments with real-world datasets, we show that LSIR compares favorably with a state-of-the-art causal inference method.

Gaussian Processes for Nonlinear Signal Processing: An Overview of Recent Advances

Gaussian processes (GPs) are versatile tools that have been successfully employed to solve nonlinear estimation problems in machine learning, but that are rarely used in signal processing. In this tutorial, we present GPs for regression as a natural nonlinear extension to optimal Wiener filtering. After establishing their basic formulation, we discuss several important aspects and extensions, including recursive and adaptive algorithms for dealing with non-stationarity, low-complexity solutions, non-Gaussian noise models and classification scenarios. Furthermore, we provide a selection of relevant applications to wireless digital communications.

A Variational Framework for Region-Based Segmentation Incorporating Physical Noise Models

Image segmentation is one of the fundamental problems in computer vision and image processing. In the recent years mathematical models based on partial differential equations and variational methods have led to superior results in many applications, e.g., medical imaging. A majority of works on image segmentation implicitly assume the given image to be biased by additive Gaussian noise, for instance the popular Mumford-Shah model. Since this assumption is not suitable for a variety of problems, we propose a region-based variational segmentation framework to segment also images with non-Gaussian noise models. Motivated by applications in biomedical imaging, we discuss the cases of Poisson and multiplicative speckle noise intensively. Analytical results such as the existence of a solution are verified and we investigate the use of different regularization functionals to provide a-priori information regarding the expected solution. The performance of the proposed framework is illustrated by experimental results on synthetic and real data.

Non-Gaussian statistical inverse problems. Part I: Posterior distributions

One approach to noisy inverse problems is to use Bayesian methods. In this work, the statistical inverse problem of estimating the probability distribution of an infinite-dimensional unknown given its noisy indirect infinite-dimensional observation is studied in the Bayesian framework. The motivation for the work arises from the fact that the Bayesian computations are usually carried out in finite-dimensional cases, while the original inverse problem is often infinite-dimensional. A good understanding of an infinite-dimensional problem is, in general, helpful in finding efficient computational approaches to the problem. &nbsp The fundamental question of well-posedness of the infinite-dimensional statistical inverse problem is considered. In particular, it is shown that the continuous dependence of the posterior probabilities on the realizations of the observation provides a certain degree of uniqueness for the posterior distribution. &nbsp Special emphasis is on finding tools for working with non-Gaussian noise models. Especially, the applicability of the generalized Bayes formula is studied. Several examples of explicit posterior distributions are provided.