Showing papers on "Probability density function published in 2021"

Ultra-slow-roll inflation with quantum diffusion

Abstract: We consider the effect of quantum diffusion on the dynamics of the inflaton during a period of ultra-slow-roll inflation. We extend the stochastic-$\delta\mathcal{N}$ formalism to the ultra-slow-roll regime and show how this system can be solved analytically in both the classical-drift and quantum-diffusion dominated limits. By deriving the characteristic function, we are able to construct the full probability distribution function for the primordial density field. In the diffusion-dominated limit, we recover an exponential tail for the probability distribution, as found previously in slow-roll inflation. To complement these analytical techniques, we present numerical results found both by very large numbers of simulations of the Langevin equations, and through a new, more efficient approach based on iterative Volterra integrals. We illustrate these techniques with two examples of potentials that exhibit an ultra-slow-roll phase leading to the possible production of primordial black holes.

Outage Probability and Capacity Scaling Law of Multiple RIS-Aided Networks

Abstract: In this letter, we consider a network assisted by multiple reconfigurable intelligent surfaces (RISs). Assuming that the RIS with the highest instantaneous end-to-end signal-to-noise ratio (SNR) is selected to aid the communication, the outage probability (OP) and average sum-rate are investigated. Specifically, an exact analysis for the OP is developed. In addition, relying on the extreme value theory, a closed-form expression for the asymptotic sum-rate is derived, based on which the capacity scaling law is established. Our results are corroborated through simulations and insightful discussions are provided. In particular, our analysis shows that the number of RISs as well as the number of reflecting elements play a crucial role in the capacity scaling law of multiple RIS-aided networks. Also, comparisons with relay-aided systems are carried out to demonstrate that the proposed system setup outperforms relaying schemes both in terms of the OP and average sum-rate.

Logarithmic Hyperbolic Cosine Adaptive Filter and Its Performance Analysis

Abstract: The hyperbolic cosine function with high-order errors can be utilized to improve the accuracy of adaptive filters. However, when initial weight errors are large, the hyperbolic cosine-based adaptive filter (HCAF) may be unstable. In this paper, a novel normalization based on the logarithmic hyperbolic cosine function is proposed to achieve the stabilization for the case of large initial weight errors, which generates a logarithmic HCAF (LHCAF). Actually, the cost function of LHCAF is the logarithmic hyperbolic cosine function that is robust to large errors and smooth to small errors. The transient and steady-state analyses of LHCAF in terms of the mean-square deviation (MSD) are performed for a stationary white input with an even probability density function in a stationary zero-mean white noise. The convergence and stability of LHCAF can be therefore guaranteed as long as the filtering parameters satisfy certain conditions. The theoretical results based on the MSD are supported by the simulations. In addition, a variable scaling factor and step-size LHCAF (VSS-LHCAF) is proposed to improve the filtering accuracy of LHCAF further. The proposed LHCAF and VSS-LHCAF are superior to HCAF and other robust adaptive filters in terms of filtering accuracy and stability.

The Kumaraswamy Generalized Half-Normal Distribution for Skewed Positive Data

Abstract: For the first time, we propose and study the Kumaraswamy generalized half-normal distribution for modeling skewed positive data. The half-normal and generalized half-normal (Cooray and Ananda, 2008) distributions are special cases of the new model. Various of its structural properties are derived, including explicit expressions for the density function, moments, generating and quantile functions, mean deviations and moments of the order statistics. We investigate maximum likelihood estimation of the parameters and derive the expected information matrix. The proposed model is modified to open the possibility that long-term survivors may be presented in the data. Its applicability is illustrated by means of four real data sets.

Analysis of stochastic process to model safety risk in construction industry

Abstract: There are many factors leading to construction safety accident. The rule presented under the influence of these factors should be a statistical random rule. To reveal those random rules and study the probability prediction method of construction safety accident, according to stochastic process theory, general stochastic process, Markov process and normal process are respectively used to simulate the risk-accident process in this paper. First, in the general-random-process-based analysis the probability of accidents in a period of time is calculated. Then, the Markov property of the construction safety risk evolution process is illustrated, and the analytical expression of probability density function of first-passage time of Markov-based risk-accident process is derived to calculate the construction safety probability. In the normal-process-based analysis, the construction safety probability formulas in cases of stationary normal risk process and non-stationary normal risk process with zero mean value are derived respectively. Finally, the number of accidents that may occur on construction site in a period is studied macroscopically based on Poisson process, and the probability distribution of time interval between adjacent accidents and the time of the nth accident are calculated respectively. The results provide useful reference for the prediction and management of construction accidents.

Statistical Load Forecasting Using Optimal Quantile Regression Random Forest and Risk Assessment Index

Abstract: To support daily operation of smart grid, the stochastic load behavior is analyzed by a day-ahead prediction interval (PI) which is built from predictor’s probability density function, computed in statistical mean-variance, and achieves a symmetrical PI. However, this approach lacks for intended risk information on the predictors’ uncertainty, e.g., weather condition and load variation. This article proposes a novel statistical load forecasting (SLF) using quantile regression random forest (QRRF), probability map, and risk assessment index (RAI) to obtain the actual pictorial of the outcome risk of load demand profile. To know the actual load condition, the proposed SLF is built considering accurate point forecasting results, and the QRRF establishes the PI from various quantiles. To correlate the uncertainty of external factors to the actual load, the probability map computes the most probable quantile happening in the training horizon. Based on the current inputs, the RAI calculates the PI’s intended risk. The proposed SLF is verified by Independent System Operator–New England data, compared to benchmark algorithms and Winkler score. The results show that the proposed method can model a more precise load PI along with the risk evaluation, as compared to results of the existing benchmark models.

Performance Analysis for Mixed κ-μ Fading and M-Distribution Dual-Hop Radio Frequency/Free Space Optical Communication Systems

Abstract: Mixed radio frequency (RF) and free space optical (FSO) communications are a promising alternative technology for backbone networks of next-generation wireless communications, but bottlenecks exist due to atmospheric turbulence. In this paper, the performance of amplified-and-forward dual-hop mixed RF/FSO systems with heterodyne detection and intensity modulation/direct detection techniques, in consideration of pointing errors, is investigated. In particular, an asymmetric fading environment is considered where the RF hop is assumed to follow $\kappa $ - $\mu $ fading, which includes Nakagami- $m$ and Rayleigh fading as special cases, while the FSO link is subjected to unified $\mathcal {M}$ -distribution fading, which has proven to be a general statistical distribution that accurately describes the fading model for the optical intensity under weak-to-strong turbulence conditions. More specifically, closed-form expressions for both the cumulative distribution function (CDF) and probability distribution function (PDF) of the end-to-end mixed RF/FSO system are derived in terms of the Meijer’s G function. Capitalizing on the derived CDF and PDF expressions, novel closed-form expressions for the outage probability, the average bit error rate (BER), and the ergodic capacity under various modulation schemes are presented. Additionally, we present tight asymptotic formulae for the outage probability and the average BER at the high signal-to-noise ratio (SNR) regime in terms of some elementary functions under various modulation schemes, which offer helpful insights into the influence of the channel parameters and system parameters on the performance of the mixed RF/FSO system. Finally, both Monte-Carlo simulation and numerical results are provided to corroborate our derived expressions.

A Novel Heavy-Tailed Mixture Distribution Based Robust Kalman Filter for Cooperative Localization

Abstract: In cooperative localization for autonomous underwater vehicles (AUVs), the practical stochastic noise may be heavy-tailed, and nonstationary distributed because of acoustic speed variation, multipath effect of acoustic channel, and changeable underwater environment. To address such noise, a novel heavy-tailed mixture (HTM) distribution is first proposed in this article, and then expressed as a hierarchical Gaussian form by employing a categorical distributed auxiliary vector. Based on that, a novel HTM distribution based robust Kalman filter is proposed, where the one-step prediction, and measurement likelihood probability density functions are, respectively, modeled as an HTM distribution, and a Normal-Gamma-inverse Wishart distribution. The proposed filter is verified by a lake experiment about cooperative localization for AUVs. Compared with the cutting-edge filter, the proposed filter has been improved by 50.27% in localization error but no more than twice computational time is required.

Performance Analysis of FSO Communications Over F Turbulence Channels With Pointing Errors

Abstract: Recently, the Fisher-Snedecor $\cal {F}$ distribution was proposed to model the turbulence in free-space optical (FSO) communications. However, the existing model does not consider pointing error impairment. To fill this gap, in this letter, we derive novel closed-form expressions for the probability density function (PDF) and cumulative distribution function (CDF) for irradiance fluctuations in the presence of pointing error impairments. Subsequently, the PDF and CDF of the received signal-to-noise ratio (SNR) are derived and employed to obtain novel closed-form expressions for the outage probability, average bit error rate, and average ergodic capacity. To gain more insight into the impact of system and turbulence channel parameters, simple and accurate asymptotic expressions are further derived. Our analytical results are supported by Monte-Carlo simulations to validate the analysis.

Ultra-slow-roll inflation with quantum diffusion

Abstract: We consider the effect of quantum diffusion on the dynamics of the inflaton during a period of ultra-slow-roll inflation. We extend the stochastic-$\delta\mathcal{N}$ formalism to the ultra-slow-roll regime and show how this system can be solved analytically in both the classical-drift and quantum-diffusion dominated limits. By deriving the characteristic function, we are able to construct the full probability distribution function for the primordial density field. In the diffusion-dominated limit, we recover an exponential tail for the probability distribution, as found previously in slow-roll inflation. To complement these analytical techniques, we present numerical results found both by very large numbers of simulations of the Langevin equations, and through a new, more efficient approach based on iterative Volterra integrals. We illustrate these techniques with two examples of potentials that exhibit an ultra-slow-roll phase leading to the possible production of primordial black holes.

Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker--Planck Equation and Physics-Informed Neural Networks

Abstract: The Fokker--Planck (FP) equation governing the evolution of the probability density function (PDF) is applicable to many disciplines, but it requires specification of the coefficients for each case...

Analysis of RIS-Based Terrestrial-FSO Link Over G-G Turbulence With Distance and Jitter Ratios

Abstract: One of the main problems faced by communication systems is the presence of skip-zones in the targeted areas. With the deployment of the fifth-generation mobile network, solutions are proposed to solve the signal loss due to obstruction by buildings, mountains, and atmospheric or weather conditions. Among these solutions, reconfigurable intelligent surfaces (RIS), which are newly proposed modules, may be exploited to reflect the incident signal in the direction of dead zones, increase communication coverage, and make the channel smarter and controllable. This paper tackles the skip-zone problem in terrestrial free-space optical (T-FSO) systems using a single-element RIS. Considering link distances and jitter ratios at the RIS position, we carry out a performance analysis of RIS-aided T-FSO links affected by turbulence and pointing errors, for both heterodyne detection and intensity modulation-direct detection techniques. Turbulence is modeled using the Gamma-Gamma distribution. We analyze the model and provide exact closed-form expressions of the probability density function, cumulative distribution function, and moment generating function of the end-to-end signal-to-noise ratio. Capitalizing on these statistics, we evaluate the system performance through the outage probability, ergodic channel capacity, and average bit error rate for selected binary modulation schemes. Numerical results, validated through simulations, obtained for different RIS positions and link distances ratio values, reveal that RIS-based T-FSO performs better when the RIS module is located near the transmitter.

Variational Adaptive Kalman Filter With Gaussian-Inverse-Wishart Mixture Distribution

Abstract: In this article, a new variational adaptive Kalman filter with Gaussian-inverse-Wishart mixture distribution is proposed for a class of linear systems with both partially unknown state and measurement noise covariance matrices. The state transition and measurement likelihood probability density functions are described by a Gaussian-inverse-Wishart mixture distribution and a Gaussian-inverse-Wishart distribution, respectively. The system state vector together with the state noise covariance matrix and the measurement noise covariance matrix are jointly estimated based on the derived hierarchical Gaussian model. Examples are provided to demonstrate the effectiveness and potential of the developed new filtering design techniques.

First passage under restart for discrete space and time: Application to one-dimensional confined lattice random walks.

Abstract: First passage under restart has recently emerged as a conceptual framework to study various stochastic processes under restart mechanism. Emanating from the canonical diffusion problem by Evans and Majumdar, restart has been shown to outperform the completion of many first-passage processes which otherwise would take longer time to finish. However, most of the studies so far assumed continuous time underlying first-passage time processes and moreover considered continuous time resetting restricting out restart processes broken up into synchronized time steps. To bridge this gap, in this paper, we study discrete space and time first-passage processes under discrete time resetting in a general setup without specifying their forms. We sketch out the steps to compute the moments and the probability density function which is often intractable in the continuous time restarted process. A criterion that dictates when restart remains beneficial is then derived. We apply our results to a symmetric and a biased random walker in one-dimensional lattice confined within two absorbing boundaries. Numerical simulations are found to be in excellent agreement with the theoretical results. Our method can be useful to understand the effect of restart on the spatiotemporal dynamics of confined lattice random walks in arbitrary dimensions.

On the Distribution of the Sum of Double-Nakagami-m Random Vectors and Application in Randomly Reconfigurable Surfaces

Abstract: Reconfigurable intelligent surfaces (RISs) intend to improve significantly the performance of future wireless networks, by controlling the wireless propagation medium through elements that can shift the phase of the reflected signals. Although ideally the signals reflected from a RIS are added coherently at the receiver, this is very challenging in practice due to the requirement for perfect channel state information (CSI) at the RIS and phase control. To facilitate the performance analysis of more practical RIS-assisted systems, first, we present novel closed-form expressions for the probability density function, the cumulative distribution function, the moments, and the characteristic function of the distribution of the sum of double-Nakagami-m random vectors, whose amplitudes follow the double-Nakagami-m distribution, i.e., the distribution of the product of two random variables following the Nakagami-m distribution, and phases are circular uniformly distributed. We also consider a special case of this distribution, namely the distribution of the sum of Rayleigh-Nakagami-m random vectors. Then, we exploit these expressions to investigate the performance of the RIS-assisted composite channel, assuming that the two links undergo Nakagami-m fading and the equivalent phase follows the uniform distribution, which corresponds to the case where CSI is not available at the RIS and leads to a lower bound of the performance of a system with CSI. Closed-form expressions for the outage probability, the average received signal-to-noise ratio, the ergodic capacity, the bit error probability, the amount of fading, and the channel quality estimation index are provided to evaluate the performance of the considered system. These metrics are also derived for the practical special case where one of the two links undergoes Rayleigh fading.

Continuous time random walk to a general fractional Fokker–Planck equation on fractal media

Abstract: A general fractional calculus is described using fractional operators with respect to another function, and some often used propositions are presented in this framework. Together with the continuous time random walk (CTRW), a general time-fractional Fokker–Planck equation is derived and the governing equation meets the general fractional derivative. Finally, various new probability density functions are proposed in this paper.

Robust Optimal Operation of Active Distribution Network Based on Minimum Confidence Interval of Distributed Energy Beta Distribution

Abstract: With the gradual increase of distributed energy penetration, the traditional optimization model of distribution network can no longer guarantee the stable and efficient operation of the distribution network. In order to deal with the inevitable uncertainty of distributed energy, a new robust optimal operation method is proposed for active distribution network (ADN) based on the minimum confidence interval of distributed energy Beta distribution in this paper. First, an ADN model is established with second-order cone to include the energy storage device, capacitor bank, static var compensator, on-load tap changer, wind turbine and photovoltaic. Then, the historical data of related distributed energy are analyzed and described by the probability density function, and the minimum confidence interval is obtained by interval searching. Furthermore, via taking this minimum confidence interval as the uncertain interval, a less conservative two-stage robust optimization model is established and solved for ADN. The simulation results for the IEEE 33-bus distribution network have verified that the proposed method can realize a more stable and efficient operation of the distribution network compared with the traditional robust optimization method.

Secrecy Performance Analysis of Mixed α − μ and Exponentiated Weibull RF-FSO Cooperative Relaying System

Abstract: Increasing concerns regarding wireless systems’ security are leading researchers to exploit the physical properties of a medium while designing any secured wireless network. The secrecy performance of a mixed radio frequency-free space optical (RF-FSO) system with a variable gain relaying scheme is investigated in this paper under the attempt of wiretapping by an eavesdropper. We assume that the eavesdropper can intrude the target data from the RF link only. Both the RF links (main and eavesdropper) undergo the $\alpha -\mu $ fading statistics and the FSO link experiences the exponentiated Weibull fading statistics. Exploiting the amplify-and-forward (AF) relaying scheme while considering two detection techniques (i.e. heterodyne detection and intensity modulation/direct detection) with pointing error impairments, the mathematical formulations of the unified probability density function and cumulative distribution function are performed for the equivalent signal-to-noise ratio of the considered dual-hop RF-FSO link. Closed-form analytical expressions for average secrecy capacity, secrecy outage probability, and the probability of non-zero secrecy capacity are derived in terms of Meijer’s $G$ and Fox’s $H$ functions to quantify the system performance. Capitalizing on these expressions, the secrecy performance is further analyzed for various channel parameters of RF links, aperture sizes of the receiver, pointing errors, and atmospheric turbulence severity. The results reveal that aperture averaging can improve the secrecy performance remarkably by suppressing the effects of turbulence. Monte Carlo simulations are provided to justify the accuracy of the proposed model.

Entanglement distribution in the quantum symmetric simple exclusion process.

Abstract: We study the probability distribution of entanglement in the quantum symmetric simple exclusion process, a model of fermions hopping with random Brownian amplitudes between neighboring sites. We consider a protocol where the system is initialized in a pure product state of $M$ particles, and we focus on the late-time distribution of R\'enyi-$q$ entropies for a subsystem of size $\ensuremath{\ell}$. By means of a Coulomb gas approach from random matrix theory, we compute analytically the large-deviation function of the entropy in the thermodynamic limit. For $qg1$, we show that, depending on the value of the ratio $\ensuremath{\ell}/M$, the entropy distribution displays either two or three distinct regimes, ranging from low to high entanglement. These are connected by points where the probability density features singularities in its third derivative, which can be understood in terms of a transition in the corresponding charge density of the Coulomb gas. Our analytic results are supported by numerical Monte Carlo simulations.

Performance Analysis of Multi-Branch Reconfigurable Intelligent Surfaces-Assisted Optical Wireless Communication System in Environment With Obstacles

Abstract: Since optical links are easily blocked by obstacles in the environment, they have been considered difficult to directly perform wireless communications. However, reconfigurable intelligent surfaces (RISs), as a new type of digital coding meta-materials, can significantly improve the coverage of optical communication by establishing new links. We propose a controllable multi-branch wireless optical communication system based on optical RISs. By setting up multiple optical RISs in the environment, multiple artificial channels are built to improve the system performance and to reduce the outage probability. In this paper, we investigate three factors affecting channel coefficients, including beam jitter, RIS jitter, and obstruction probability. We derive closed-form probability density function of channel coefficients, asymptotic average bit-error rate (BER), and the outage probability for systems with multiple branches. Our analysis reveals that the probability density function contains an impulse function, which causes irreducible error rate and outage probability floors. Based on our numerical results, the BER and outage probability floor of the multi-branch system are significantly reduced compared with the single direct path system. Therefore, the optical RIS assisted multi-branch wireless communication is a promising solution against obstacles of optical channel.

A Novel Data-based Stochastic Distribution Control for Non-Gaussian Stochastic Systems

Abstract: This note presents a novel data-based approach to investigate the non-Gaussian stochastic distribution control problem. As the motivation of this note, the existing methods have been summarised regarding to the drawbacks, for example, neural network weights training for unknown stochastic distribution and so on. To overcome these disadvantages, a new transformation for dynamic probability density function is given by kernel density estimation using interpolation. Based upon this transformation, a representative model has been developed while the stochastic distribution control problem has been transformed into an optimisation problem. Then, data-based direct optimisation and identification-based indirect optimisation have been proposed. In addition, the convergences of the presented algorithms are analysed and the effectiveness of these algorithms has been evaluated by numerical examples. In summary, the contributions of this note are as follows: 1) a new data-based probability density function transformation is given; 2) the optimisation algorithms are given based on the presented model; and 3) a new research framework is demonstrated as the potential extensions to the existing st