Case Of White Noise Research Articles

Instabilities and Pattern Formation in Epidemic Spread Induced by Nonlinear Saturation Effects and Ornstein–Uhlenbeck Noise

Abstract We analytically study the emergence of instabilities and the consequent steady-state pattern formation in a stochastic partial differential equation (PDE) based, compartmental model of spatiotemporal epidemic spread. The model is characterized by: (1) strongly nonlinear forces representing the infection transmission mechanism and (2) random environmental forces represented by the Ornstein–Uhlenbeck (O–U) stochastic process which better approximates real-world uncertainties. Employing second-order perturbation analysis and computing the local Lyapunov exponent, we find the emergence of diffusion-induced instabilities and analyze the effects of O–U noise on these instabilities. We obtain a range of values of the diffusion coefficient and correlation time in parameter space that support the onset of instabilities. Notably, the stability and pattern formation results depend critically on the correlation time of the O–U stochastic process; specifically, we obtain lower values of steady-state infection density for higher correlation times. Also, for lower correlation times the results approach those obtained in the white noise case. The analytical results are valid for lower-order correlation times. In summary, the results provide insights into the onset of noise-induced, and Turing-type instabilities in a stochastic PDE epidemic model in the presence of strongly nonlinear deterministic infection forces and stochastic environmental forces represented by Ornstein–Uhlenbeck noise.

Information Criteria for Signal Extraction Using Singular Spectrum Analysis: White and Red Noise

In singular spectrum analysis, which is applied to signal extraction, it is of critical importance to select the number of components correctly in order to accurately estimate the signal. In the case of a low-rank signal, there is a challenge in estimating the signal rank, which is equivalent to selecting the model order. Information criteria are commonly employed to address these issues. However, singular spectrum analysis is not aimed at the exact low-rank approximation of the signal. This makes it an adaptive, fast, and flexible approach. Conventional information criteria are not directly applicable in this context. The paper examines both subspace-based and information criteria, proposing modifications suited to the Hankel structure of trajectory matrices employed in singular spectrum analysis. These modifications are initially developed for white noise, and a version for red noise is also proposed. In the numerical comparisons, a number of scenarios are considered, including the case of signals that are approximated by low-rank signals. This is the most similar to the case of real-world time series. The criteria are compared with each other and with the optimal rank choice that minimizes the signal estimation error. The results of numerical experiments demonstrate that for low-rank signals and noise levels within a region of stable rank detection, the proposed modifications yield accurate estimates of the optimal rank for both white and red noise cases. The method that considers the Hankel structure of the trajectory matrices appears to be a superior approach in many instances. Reasonable model orders are obtained for real-world time series. It is recommended that a transformation be applied to stabilize the variance before estimating the rank.

Learning noise-induced transitions by multi-scaling reservoir computing

Noise is usually regarded as adversarial to extracting effective dynamics from time series, such that conventional approaches usually aim at learning dynamics by mitigating the noisy effect. However, noise can have a functional role in driving transitions between stable states underlying many stochastic dynamics. We find that leveraging a machine learning model, reservoir computing, can learn noise-induced transitions. We propose a concise training protocol with a focus on a pivotal hyperparameter controlling the time scale. The approach is widely applicable, including a bistable system with white noise or colored noise, where it generates accurate statistics of transition time for white noise and specific transition time for colored noise. Instead, the conventional approaches such as SINDy and the recurrent neural network do not faithfully capture stochastic transitions even for the case of white noise. The present approach is also aware of asymmetry of the bistable potential, rotational dynamics caused by non-detailed balance, and transitions in multi-stable systems. For the experimental data of protein folding, it learns statistics of transition time between folded states, enabling us to characterize transition dynamics from a small dataset. The results portend the exploration of extending the prevailing approaches in learning dynamics from noisy time series.

Robust laser phase noise measurement by integration heterodyne for coherent beam combining applications

Phase noise characteristics are critical for coherent beam combination engineering. Heterodyne with integration method for phase noise measurement has been studied numerically and experimentally, which reveals that the method is not only simple to implement with the least equipment but also capable of phase retrieval using under-sampling data. The integration method is compared with the traditional low-pass filter (LPF) method from both numerical and experimental perspectives. By introducing an evaluation criterion of measurement accuracy, the errors of the integration method are 0.44% and 0.08% for white noise and pink noise cases, respectively, which are smaller than that achieved by LPF one (1.52% and 0.25%). The errors of the integration method are below 1.4% when under-sampling data has been employed, which means that large consumption of data processing can be avoided, and the method is robust. Phase noise measurements in quiet laboratory and disturbed conditions are implemented, and the error between the results of the sampling rate of 250 MHz and 31.25 MHz is less than 0.08%, which is consistent with the simulation and demonstrates the excellent performance of the integration method.

Lossy Compression of Single-channel Noisy Images by Modern Coders

Lossy compression of remote-sensing images is a typical stage in their processing chain. In design or selection of methods for lossy compression, it is commonly assumed that images are noise-free. Meanwhile, there are many practical situations where an image or a set of its components are noisy. This fact needs to be taken into account since noise presence leads to specific effects in lossy compressed data. The main effect is the possible existence of the optimal operation point (OOP) shown for JPEG, JPEG2000, some coders based on the discrete cosine transform (DCT), and the better portable graphics (BPG) encoder. However, the performance of such modern coders as AVIF and HEIF with application to noisy images has not been studied yet. In this paper, analysis is carried out for the case of additive white Gaussian noise. We demonstrate that OOP can exist for AVIF and HEIF and the performance characteristics in it are quite similar to those for the BPG encoder. OOP exists with a higher probability for images of simpler structure and/or high-intensity noise, and this takes place according to different metrics including visual quality ones. The problems of providing lossy compression by AVIF or HEIF are shown and an initial solution is proposed. Examples for test and real-life remote-sensing images are presented.

PECULIARITIES OF NOISY IMAGE LOSSY COMPRESSION

This paper deals with lossy compression applied to grayscale noisy images using several coders. The case of additive white Gaussian noise is considered as the first step in studying the problem. A special attention is paid to quite new coders AVIF and HEIF for which, to the best of our knowledge, lossy compression of noisy images of different complexity has not been thoroughly considered yet. It is shown that optimal operation point is possible for all coders under certain conditions according to different quality metrics including conventional peak signal-to-noise ratio and visual quality metrics such as PSNR-HVS-M and MS-SSIM. The compression parameters (metrics) in optimal operation point significantly depend on image complexity and noise intensity where the existence of optimal operation point is more probable for simple structure images and/or intensive noise. Comparison of metric values in optimal operation point for the considered coders shows that slightly better characteristics are provided by better portable graphics (BPG) and advanced discrete cosine transform (ADCT) coders. The results for JPEG are significantly worse. The AVIF and HEIF encoders provide similar results and they outperform JPEG significantly. Optimal operation point for all studied coders is observed more rarely for visual quality metrics than for peak signal-to-noise ratio. General tendencies concerning dependence of optimal compression parameters on noise intensity are presented. It is shown that compression ratio in optimal operation point is often larger than 20 and can be as large as 60. The problem with AVIF and HEIF is that it is currently unclear how to choose quality factor for them to carry out lossy image compression in the neighborhood of the corresponding optimal operation point. Meanwhile, there is the tendency to optimal quality factor reduction if intensity of additive white Gaussian noise increases. The directions of further research are discussed.

Statistical inference for microstate distribution in recurrence plots

Recurrence is an important property to unveil patterns in dynamical systems. The recurrence plot (RP) is a matrix that summarizes recurrence information from data sequences. Recently, a new approach to the RP have been developed to generalize the standard absence or occurrence of recurrences in the RP. The microstate concept was introduced to statistically characterize samples of any size in the RP. We explore in this paper the relation between microstates of different sizes in the RP. We distinguish two approaches: the top-bottom and the down-up analysis. In the first we have information from larger microstates to make inference about smaller ones, in the second the opposite. We establish analytical results that statistically estimate the microstate probabilities between different sample sizes. The analytical results for the normalized entropy were tested using two time series: a simple white noise case and a series from a logistic equation in the chaotic scenario. For both cases, the top-bottom and the down-up analytical relations were tested with good results. To depict the actual probability distributions, we used the Lorenz oscillator for both, the top-down and the bottom-up strategies, showcasing the limitations and perspectives for future works. We believe our results can open new theoretical paths in the RP analysis.

Efficient Bayesian estimation of the generalized Langevin equation from data

Modeling non-Markovian time series is a recent topic of research in many fields such as climate modeling, biophysics, molecular dynamics, or finance. The generalized Langevin equation (GLE), given naturally by the Mori-Zwanzig projection formalism, is a frequently used model including memory effects. In applications, a specific form of the GLE is most often obtained on a data-driven basis. Here, Bayesian estimation has the advantage of providing both suitable model parameters and their credibility in a straightforward way. It can be implemented in the approximating case of white noise, which, far from thermodynamic equilibrium, is consistent with the fluctuation-dissipation theorem. However, the exploration of the posterior, which is done via Markov chain Monte Carlo sampling, is numerically expensive, which makes the analysis of large data sets unfeasible. In this work, we discuss an efficient implementation of Bayesian estimation of the GLE based on a piecewise constant approximation of the drift and diffusion functions of the model. In this case, the characteristics of the data are represented by only a few coefficients, so that the numerical cost of the procedure is significantly reduced and independent of the length of the data set. Further, we propose a modification of the memory term of the GLE, leading to an equivalent model with an emphasis on the impact of trends, which ensures that an estimate of the standard Langevin equation provides an effective initial guess for the GLE. We illustrate the capabilities of both the method and the model by an example from turbulence.

Resolution Limits of Non-Adaptive 20 Questions Search for a Moving Target

Using the 20 questions estimation framework with query-dependent noise, we study non-adaptive search strategies for a moving target over the unit cube with unknown initial location and velocities under a piecewise constant velocity model. In this search problem, there is an oracle who knows the instantaneous location of the target at any time. Our task is to query the oracle as few times as possible to accurately estimate the location of the target at any specified time. We first study the case where the oracle’s answer to each query is corrupted by discrete noise and then generalize our results to the case of additive white Gaussian noise. In our formulation, the performance criterion is the resolution, which is defined as the maximal L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> distance between the true locations and estimated locations. We characterize the minimal resolution of an optimal non-adaptive query procedure with a finite number of queries by deriving non-asymptotic and asymptotic bounds. Our bounds are tight in the first-order asymptotic sense when the number of queries satisfies a certain condition and our bounds are tight in the stronger second-order asymptotic sense when the target moves with a constant velocity. To prove our results, we relate the current problem to channel coding, borrow ideas from finite blocklength information theory and construct bounds on the number of possible quantized target trajectories.

Efficient Real-Time Whitening for Blind Eigenvalue-Based Detection in mmWave Full Duplex Cognitive Radio

Although combining Full Duplex (FD) and Cognitive Radio (CR) technologies can provide several benefits, the presence of the residual self-interference (RSI) at the FD transceiver may influence the spectrum sensing capabilities. In particular, focusing on the blind eigenvalue-based detectors (EVDs), this paper shows that operating at mmWave widebands, the RSI is a colored noise strongly affecting the detection performance even when it is characterized by low power. Consequently, we propose here a whitened detector whose novelty consists in the real-time computation of the whitening matrix, differently from previous works which employ an offline computation. The proposed whitening matrix computation uses a sample covariance matrix (SCM) regularization tailored for the FD scenario, instead of the conventional SCM estimator. Particularly, we prove that the proposed whitened detector is robust against RSI power uncertainty, attaining a constant false alarm rate under typical FD operating conditions. Finally, we show that our method provides computational load reduction and faster convergence to the theoretical false alarm probability characteristic of the white noise case. Numerical simulations validate the theoretical analysis confirming the better performance of the proposed whitening approach, particularly when applied to the sphericity test, in comparison to the offline-based approach.

On quadratic variations for the fractional-white wave equation

This paper studies the behaviour of quadratic variations of a stochastic wave equation driven by a noise that is white in space and fractional in time. Complementing the analysis of quadratic variations in the space component carried out in [Correlation structure, quadratic variations and parameter estimation for the solution to the wave equation with fractional noise, Electron. J. Stat. 12 (2018), no. 2, 3639–3672] and [Generalized k k -variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus, J. Statist. Plann. Inference 207 (2020), 155–180], it focuses on the time component of the solution process. For different values of the Hurst parameter a central and a noncentral limit theorems are proved, allowing to construct parameter estimators and compare them to the findings in the space-dependent case. Finally, rectangular quadratic variations in the white noise case are studied and a central limit theorem is demonstrated.

BPG-Based Lossy Compression of Three-Channel Noisy Images with Prediction of Optimal Operation Existence and Its Parameters

Nowadays, there is a clear trend toward increasing the number of remote-sensing images acquired and their average size. This leads to the need to compress the images for storage, dissemination, and transfer over communication lines where lossy compression techniques are more popular. The images to be compressed or some of their components are often noisy. They must therefore be compressed taking into account the properties of the noise. Due to the noise filtering effect obtained during lossy compression of noisy images, an optimal operating point (OOP) may exist. The OOP is a parameter that controls the compression for which the quality of the compressed image is closer (closest) to the corresponding noise-free image than the quality of the noisy (original, uncompressed) image according to some quantitative criterion (metric). In practice, it is important to know whether the OOP exists for a given image, because if the OOP exists, it is appropriate to perform the compression in the OOP or at least in its neighborhood. Since the real image is absent in practice, it is impossible to determine a priori whether the OOP exists or not. Here, we focus on three-channel-remote-sensing images and show that it is possible to easily predict the existence of the OOP. Furthermore, it is possible to predict the metric values or their improvements with appropriate accuracy for practical use. The BPG (better portable graphics) encoder is considered a special case of an efficient compression technique. As an initial design step, the case of additive white Gaussian noise with equal variance in the three components is considered. While previous research was mainly focused on predicting the improvement (reduction) of the PSNR and PSNR-HVS-M metrics, here we focus on the modern visual quality metrics, namely PSNR-HA and MDSI. We also discuss what to do if, according to the prediction, an OOP is absent. Examples of lossy compression of noisy three-channel remote sensing images are given. It is also shown that the use of three-dimensional compression provides a compression ratio increase by several times compared with component-wise compression in the OOP.

Level of noises and long time behavior of the solution for space-time fractional SPDE in bounded domains

In this paper we study the long time behavior of the solution to a certain class of space-time fractional stochastic equations with respect to the level $ \lambda $ of a noise and show how the choice of the order $ \beta \in (0, \,1) $ of the fractional time derivative affects the growth and decay behavior of their solution. We consider both the cases of white noise and colored noise. Our results extend the main results in Foondun [12] to fractional Laplacian as well as higher dimensional cases.

ОЦІНКА ОСОБЛИВОСТЕЙ БАГАТОВИМІРНИХ ВИПАДКОВИХ ПРОЦЕСІВ НА ОСНОВІ ЇХНІХ ВІЗУАЛЬНИХ ОБРАЗІВ

Despite the wide implementation of information technologies in diagnostic systems, visual control of the processes that characterize the object state remains relevant, especially when the number of such processes is large. An analysis of the possi-bilities of visual control for multidimensional random processes, the realizations of which are processed in measurement infor-mation systems on complex technical objects, was carried out. For this, these processes are transformed into a visual image, that represented a set of curves on a plane, which are obtained for each discrete moment of time. Any curve is the sum of products of the realizations values for each random process at a specified discrete moment of time on the corresponding or-thogonal function that depends on some generalized argument. Lagrangian functions are used in the article.The purpose of the article is to check the effectiveness of a new method of visualizing realizations of a multidimensional random process for extreme cases and to develop proposals for detecting the parameters jumps of one or more processes.For the first time, a method of visual detection of the parameters jumps of realizations for one or more random processes is proposed. The essence of the method is based on the subtraction of a visual image, that is, a set of functions in the absence of a jump of one or more process parameters from a similar image that concludes jumps of parameters. In practice, the first image should be a statistical average for the conditions of normal functioning of the technical object. After subtraction, a new visual image is created, which shows anomalies caused by jumps in process parameters. The usefulness of the method is demonstrated on the example of processing multidimensional experimental data, which were matched with generalized visual images. The functioning of this method is verified also for extreme cases of processing white noise and a deterministic process. The obtained visual images also reveal the features of the behavior of multidimensional random processes.

An Expectation-Maximization-Based Estimation Algorithm for AOA Target Tracking With Non-Gaussian Measurement Noises

This paper considers angle-of-arrival (AOA) target tracking in the two-dimensional plane with non-Gaussian noise. Practical situations arise where sensor measurement noise is non-Gaussian and the standard extended Kalman filter (EKF) based tracker may not be robust, resulting in large tracking error or even lack of convergence. This motivated development of a new tracker, designed for white non-Gaussian noise cases. The noise is decomposed into a sum of Gaussian components using expectation maximization (EM), and the target state is estimated by adapting an extended Kalman filter (EKF). Adaptations are necessary to combine the EM with the iterative EKF. From the analysis of the estimation error, a closed-form expression was derived revealing the presence of a bias in the estimate. On investigation, a sensor path optimization scheme is found that can eliminate bias. Furthermore, a compensation algorithm is presented, which reduces bias for the case of static sensors. The new tracker is suitable for real-time implementation and simulation results demonstrate significant improvements for the target tracking with non-Gaussian noise.

High-power narrow-linewidth fiber lasers using optical spectrum broadening based on high-order phase modulation of inversion probability-tuning sequence.

Continuous fiber laser with ultra-high power and narrow linewidth is one of the key devices in the field of high-precision industrial processing, beam combining, and nonlinear frequency conversion. Under the premise of ensuring the signal quality, continuously increasing the output power is the focus of high-power narrow-linewidth fiber lasers. Driven by the white noise or pseudo-random binary sequence (PRBS), using cascaded phase modulations to broaden the spectrum of the seed source to suppress the stimulated Brillouin scattering (SBS) effect in the master oscillator power amplifier (MOPA) structure is an effective solution to increase the output power. However, this type of optical spectrum needs to be optimized, and the randomness of the driving signal causes a self-pulsing effect, which limits the further increase of the output power. In this paper, the influence of the frequency interval and randomness of the driving signal on the SBS effect in the laser system is analyzed. The modulated spectral type can be simply adjusted through changing the bit rate and inversion probability. Combining with high-order phase modulation, an approximate rectangular spectral broadening of the seed source with a tunable bandwidth up to 30 GHz is achieved. Compared with the cascaded white noise case, the output power of this scheme is increased by 600 W under the extended bandwidth of 27 GHz. It is fully verified that the seed source spectrum with high in-band flatness and low randomness can effectively suppress the SBS effect in the fiber laser and increase the output power.

Learning-Based Noise Component Map Estimation for Image Denoising

A problem of image denoising, when images are corrupted by a non-stationary noise, is considered in this paper. Since, in practice, no a priori information on noise is available, noise statistics should be pre-estimated prior to image denoising. In this paper, deep convolutional neural network (CNN) based method for estimation of a map of local, patch-wise, standard deviations of noise (so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sigma-map</i> ) is proposed. It achieves the state-of-the-art performance in accuracy of estimation of sigma-map for the case of non-stationary noise, as well as estimation of a noise variance for the case of an additive white Gaussian noise. Extensive experiments on image denoising using estimated sigma-maps demonstrate that our method outperforms recent CNN-based blind image denoising methods by up to 6 dB in PSNR, as well as other state-of-the-art methods based on sigma-map estimation by up to 0.5 dB, providing, at the same time, better usage flexibility. A comparison with the ideal case, when denoising is applied using ground-truth sigma-map, shows that a difference of corresponding PSNR values for the most of noise levels is within 0.1-0.2 dB, and does not exceed 0.6 dB.

Стохастическое параметрическое возбуждение конвекции Рэлея-Бенара

The paper is devoted to the problem of the thermal convection excitation in a horizontal layer with isothermal free boundaries caused by a random modulation of the gravity acceleration. Equations for the stochastic dynamics of the amplitude of small perturbations of the temperature field and the stream function are derived for the system. For these equations, the conditions for the growth of mean-square values are derived; these conditions are used as a criterion for the excitation of convective motions in the system. The excitation of flows is considered both for the case of heating from below and for heating from above. It is verified that, for any parameter values, the obtained modes of the fastest growth of mean-square values lie within the physically meaningful domain of the phase space. In contrast to the case of high-frequency periodic vibrations, white Gaussian noise always exerts a destabilizing effect on the heat-conducting state of the system. The cases of white Gaussian noise and harmonic high-frequency vibrations are also compared in a general form, without reference to a particular form of thermal convection equations.

Persisting asymmetry in the probability distribution function for a random advection–diffusion equation in impermeable channels

In this paper, we study the effect of impermeable boundaries on the symmetry properties of a random passive scalar field advected by random flows. We focus on a broad class of nonlinear shear flows multiplied by a stationary, Ornstein–Uhlenbeck (OU) time varying process, including some of their limiting cases, such as Gaussian white noise or plug flows. For the former case with linear shear, recent studies (Camassa et al., 2019) numerically demonstrated that the decaying passive scalar’s long time limiting probability distribution function (PDF) could be negatively skewed in the presence of impermeable channel boundaries, in contrast to rigorous results in free space which established the limiting PDF is positively skewed (McLaughlin and Majda, 1996). Here, the role of boundaries in setting the long time limiting skewness of the PDF is established rigorously for the above class using the long time asymptotic expansion of the N-point correlator of the random field obtained from the ground state eigenvalue perturbation approach proposed in Bronski and McLaughlin (1997). Our analytical result verifies the conclusion for the linear shear flow obtained from numerical simulations in Camassa et al. (2019). Moreover, we demonstrate that the limiting distribution is negatively skewed for any shear flow at sufficiently low Péclet number. We demonstrate the convergence of the Ornstein–Uhlenbeck case to the white noise case in the limit γ→∞ of the OU damping parameter, which generalizes the results for free space in Resnick (1996) to the channel domain problem. We show that the long time limit of the first three moments depends explicitly on the value of γ, which is in contrast to the conclusion in Vanden Eijnden (2001) for the limiting PDF in free space. To find a benchmark for theoretical analysis, we derive the exact formula of the N-point correlator for a flow with no spatial dependence and Gaussian temporal fluctuation, generalizing the results of Bronski et al. (2007). The long time analysis of this formula is consistent with our theory for a general shear flow. All results are verified by Monte-Carlo simulations.

Effects of turbulent environment and random noise on self-organized critical behavior: Universality versus nonuniversality.

Self-organized criticality in the Hwa-Kardar model of a "running sandpile" [Phys. Rev. Lett. 62, 1813 (1989)10.1103/PhysRevLett.62.1813; Phys. Rev. A 45, 7002 (1992)10.1103/PhysRevA.45.7002] with a turbulent motion of the environment taken into account is studied with the field theoretic renormalization group (RG). The turbulent flow is modeled by the synthetic d-dimensional generalization of the anisotropic Gaussian velocity ensemble with finite correlation time, introduced by Avellaneda and Majda [Commun. Math. Phys. 131, 381 (1990)10.1007/BF02161420; Commun. Math. Phys. 146, 139 (1992)10.1007/BF02099212]. The Hwa-Kardar model with time-independent (spatially quenched) random noise is considered alongside the original model with white noise. The aim of the present paper is to explore fixed points of the RG equations which determine the possible types of universality classes (regimes of critical behavior of the system) and critical dimensions of the measurable quantities. Our calculations demonstrate that influence of the type of random noise is extremely large: in contrast to the case of white noise where the system possesses three fixed points, the case of spatially quenched noise involves four fixed points with overlapping stability regions. This means that in the latter case the critical behavior of the system depends not only on the global parameters of the system, which is the usual case, but also on the initial values of the charges (coupling constants) of the system. These initial conditions determine the specific fixed point which will be reached by the RG flow. Since now the critical properties of the system are not defined strictly by its parameters, the situation may be interpreted as a universality violation. Such systems are not forbidden, but they are rather rare. It is especially interesting that the same model without turbulent motion of the environment does not predict this nonuniversal behavior and demonstrates the usual one with prescribed universality classes instead [J. Stat. Phys. 178, 392 (2020)10.1007/s10955-019-02436-8].