Probability Density Function Distribution Research Articles

Channel Modeling and Performance Analysis of Directional THz Links Under Pointing Errors and α-μ Distribution

This letter focuses on providing an analytical framework for the quantification and evaluation of high-directional terahertz communication links under pointing errors in terms of outage probability (OP), channel capacity, and symbol error rate (SER). To this end, the probability density function and cumulative distribution function of the end-to-end channel are derived in the presence of pointing errors and small-scale fading. Then, closed-form expressions are derived for the OP, average channel capacity, and SER along with the asymptotic expressions for all the aforementioned performance metrics. Using numerical integral results, the accuracy of the analytical derivations are confirmed. Finally, we use the provided analytical expressions to evaluate the effect of the channel parameters on the system performance in terms of OP, channel capacity, and SER.

Wireless Communication System Performance in M2M Nakagami-m Fading Channel

In this paper, we investigate a wireless mobile communication system with a cooperative network operating over a mobile-to-mobile (M2M) fading channel in the presence of co-channel interference. Since there is no direct line of sight between the source (S) and the destination (D) due to various obstacles, the signal transmission takes place by using a relay (R). We modeled and represented the desired signal and co-channel interference by using the Nakagami-m fading distribution from the source (S) to the relay (R), as well as from the relay (R) to the destination (D). As the popularity of M2M communication has increased in 5G cellular networks, the performance analysis of the proposed system is of great importance. We derived closed-form expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the signal-to-interference ratio (SIR) at the input of the destination station. Based on the cumulative distribution function (CDF), we also evaluated the outage probability (Pout) of the proposed communication system. In addition, we derived a closed-form approximate expression for the level crossing rate (LCR) by using the Laplace approximation formula for the three-fold integral. Validity of the derived theoretical results we approved by Monte Carlo simulations.

Stationary distribution, density function and extinction of stochastic vegetation-water systems

In this paper, two stochastic vegetation-water dynamic systems are proposed and studied. First, for the deterministic system, the possible equilibria and the related local stability are analyzed. Then for the stochastic system perturbed by Gaussian white noise, we establish sufficient conditions for the existence and uniqueness of ergodic stationary distribution, which reflects the long-term persistence of vegetation. By defining a stochastic quasi-positive equilibrium E¯∗ and solving the Kolmogorov–Fokker–Planck equation, an approximate expression of probability density function of the stationary distribution around E¯∗ is derived. Besides, we obtain some sufficient criteria for vegetation extinction. In terms of the stochastic system perturbed by both white and colored noises, the related extinction law and stationary distribution are investigated. Finally, several numerical examples are performed to substantiate our theoretical results and analyze the mean first passage time (MFPT).

The generalized EPC method for the non-stationary probabilistic response of nonlinear dynamical system

It is well known that the non-stationary response probability density function (PDF) plays an important role in the reliability and failure analysis. With current approximate or numerical methods, the non-stationary approximate PDF is generally obtained in terms of the ones at the discrete time instants. Repeated computation must be conducted for the ones at other time instants since the response PDF at only one time instant can be obtained after performing the solution procedure. Thus, the computational efficiency suffers a major setback. In this paper, the exponential polynomial closure (EPC) method is further improved and enhanced to obtain the completely non-stationary PDF solution, which is distributed continuously in the time domain. It takes the temporal base function into the PDF approximation. The unknown coefficients in the EPC solution are generalized to be explicit functions of the time parameter. With the least squares method, the explicit time functions can be determined based on the few simulated response PDF values. This implementation makes the continues PDF distribution in the time domain available, which greatly improves the computational efficiency without the repeating computation. Two typical nonlinear systems under stationary and non-stationary random excitations are taken as examples to illustrate the efficiency of the proposed method. Numerical results show that the results obtained by the proposed method agree well with the simulated results. In addition, the relationship between the explicit time function and the modulate function is discussed.

The most massive stars in very young star clusters with a limited mass: Evidence favours significant self-regulation in the star formation processes

The stellar initial mass function (IMF) is commonly interpreted to be a scale-invariant probability density distribution function (PDF) such that many small clusters yield the same IMF as one massive cluster of the same combined number of stars. Observations of the galaxy-wide IMF challenge this as dwarf galaxies do not form as many massive stars as expected. This indicates a highly self-regulated star formation process in which stellar masses are not stochastically sampled from the IMF and are instead related to the environment of star formation. Here, we study the nature of star formation using the relation between the most massive star born in a star cluster and its parental stellar cluster mass (the mmax − Mecl relation). This relation has been argued to be a statistical effect if stars are sampled randomly from the IMF. By comparing the tightness of the observed mmax − Mecl distribution with synthetic star clusters with stochastically sampled stellar masses, we find that the expected dispersion of the mock observations is much larger than the observed dispersion. Assuming that mmax and Mecl uncertainties from the literature are correct, our test rejects the hypothesis that the IMF is a PDF at a more than 4.5σ confidence level. Alternatively, we provide a deterministic stellar mass sampling tool that reproduces the observed mmax − Mecl distribution and compares well with the luminosities of star-forming molecular clumps. In addition, we find that there is a significant flattening of the mmax − Mecl relation near mmax = 13 M⊙. This may suggest strong feedback of stars more massive than about 13 M⊙, and/or that the ejections of the most massive stars from young clusters in the mass range 63 to 400 M⊙ are likely important physical processes in forming clusters.

Direct Air-to-Underwater Optical Wireless Communication: Statistical Characterization and Outage Performance

In general, a buoy relay is used to connect the underwater communication to the terrestrial network over a radio or optical wireless communication (OWC) link. The use of relay deployment may pose security and deployment issues. This paper investigates the feasibility of direct air-to-underwater (A2UW) communication from an over-the-sea OWC system to an underwater submarine without deploying a relaying node. We analyze the statistical performance of the direct transmission over the combined channel fading effect of atmospheric turbulence, random fog, air-to-water interface, oceanic turbulence, and pointing errors. We develop novel analytical expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the resultant signal-to-noise ratio (SNR) in terms of bivariate Meijer-G and Fox-H functions. We use the derived statistical results to analyze the system performance by providing exact and asymptotic results of the outage probability in terms of system parameters. We use computer simulations to demonstrate the performance of direct A2UW transmissions compared to the relay-assisted system.

The complex multiscale structure in simulated and observed emission maps of the proto-cluster cloud G0.253+0.016 (‘the Brick’)

ABSTRACT The Central Molecular Zone (the central ∼500 pc of the Milky Way) hosts molecular clouds in an extreme environment of strong shear, high gas pressure and density, and complex chemistry. G0.253+0.016, also known as ‘the Brick’, is the densest, most compact, and quiescent of these clouds. High-resolution observations with the Atacama Large Millimetre/submillimetre Array (ALMA) have revealed its complex, hierarchical structure. In this paper we compare the properties of recent hydrodynamical simulations of the Brick to those of the ALMA observations. To facilitate the comparison, we post-process the simulations and create synthetic ALMA maps of molecular line emission from eight molecules. We correlate the line emission maps to each other and to the mass column density and find that HNCO is the best mass tracer of the eight emission lines within the simulations. Additionally, we characterize the spatial structure of the observed and simulated cloud using the density probability distribution function (PDF), spatial power spectrum, fractal dimension, and moments of inertia. While we find good agreement between the observed and simulated data in terms of power spectra and fractal dimensions, there are key differences in the density PDFs and moments of inertia, which we attribute to the omission of magnetic fields in the simulations. This demonstrates that the presence of the Galactic potential can reproduce many cloud properties, but additional physical processes are needed to fully explain the gas structure.

Uncertain Dynamic Characteristic Analysis for Structures with Spatially Dependent Random System Parameters.

This work presents a robust non-deterministic free vibration analysis for engineering structures with random field parameters in the frame of stochastic finite element method. For this, considering the randomness and spatial correlation of structural physical parameters, a parameter setting model based on random field theory is proposed to represent the random uncertainty of parameters, and the stochastic dynamic characteristics of different structural systems are then analyzed by incorporating the presented parameter setting model with finite element method. First, Gauss random field theory is used to describe the uncertainty of structural material parameters, the random parameters are then characterized as the standard deviation and correlation length of the random field, and the random field parameters are then discretized with the Karhunen-Loeve expansion method. Moreover, based on the discretized random parameters and finite element method, structural dynamic characteristics analysis is addressed, and the probability distribution density function of the random natural frequency is estimated based on multi-dimensional kernel density estimation method. Finally, the random field parameters of the structures are quantified by using the maximum likelihood estimation method to verify the effectiveness of the proposed method and the applicability of the constructed model. The results indicate that (1) for the perspective of maximum likelihood estimation, the parameter setting at the maximum value point is highly similar to the input parameters; (2) the random field considering more parameters reflects a more realistic structure.

Sensor scheduling with orbital uncertainty for space target tracking

The orbit determination accuracy of space debris can be remarkably improved when it is observed by two space-based optical (SBO) sensors simultaneously. The key to observation with two SBO sensors is optimizing the pointings of sensors to maximize the length of the double coverage arc. The traditional sensor scheduling method is only based on the mean of the predicted target orbit, and its coverage performance degrades when the error of the predicted target orbit increases. In this paper, the double coverage probability (DCP) is employed as a sensor scheduling criterion with considered the uncertainty of the predicted target orbit. The DCP is obtained by calculating an integral of the probability density function of target position distribution. Furthermore, an approximate expression of the DCP is derived by simplifying the integral region to reduce the computational burden. The simulation results show that the proposed sensor scheduling method based on the DCP can provide a longer double coverage arc and higher orbit determination accuracy than the traditional method.

Mepolizumab improvements in health-related quality of life and disease symptoms in a patient population with very severe chronic rhinosinusitis with nasal polyps: psychometric and efficacy analyses from the SYNAPSE study

BackgroundAlthough the psychometric properties of patient-reported outcome measures (e.g. the 22-item Sino-nasal Outcomes Test [SNOT-22]) in chronic rhinosinusitis with nasal polyps (CRSwNP) have been defined, these definitions have not been extensively studied in patients with very severe CRSwNP, as defined by recurrent disease despite ≥ 1 previous surgery and a current need for further surgery. Therefore, the psychometric properties of the symptoms visual analogue scales (VAS) were evaluated, and meaningful within-patient change thresholds were calculated for VAS and SNOT-22.MethodsSYNAPSE (NCT03085797), a randomized, double-blind, placebo-controlled, 52-week trial, assessed the efficacy and safety of 4-weekly mepolizumab 100 mg subcutaneously added to standard of care in very severe CRSwNP. Enrolled patients (n = 407) completed symptom VAS (six items) daily and SNOT-22 every 4 weeks from baseline until Week 52. Blinded psychometric assessment of individual and composite VAS was performed post hoc, including anchor-based thresholds for meaningful within-patient changes for VAS and SNOT-22, supported by cumulative distribution function and probability density function plots. The effect of mepolizumab versus placebo for 52 weeks on VAS and SNOT-22 scores was then determined using these thresholds using unblinded data.ResultsInternal consistency was acceptable for VAS and SNOT-22 scores (Cronbach’s α-coefficients ≥ 0.70). Test–retest reliability was demonstrated for all symptom VAS (Intra-Class Correlation coefficients > 0.75). Construct validity was acceptable between individual and composite VAS and SNOT-22 total score (r = 0.461–0.598) and between individual symptom VAS and corresponding SNOT-22 items (r = 0.560–0.780), based upon pre-specified ranges. Known-groups validity assessment demonstrated generally acceptable validity based on factors associated with respiratory health, with all VAS responsive to change. Mepolizumab treatment was associated with significantly increased odds of meeting or exceeding meaningful within-patient change thresholds, derived for this very severe cohort using six anchor groups for individual VAS (odds ratio [OR] 2.19–2.68) at Weeks 49–52, and SNOT-22 (OR 1.61–2.96) throughout the study.ConclusionsSymptoms VAS and SNOT-22 had acceptable psychometric properties for use in very severe CRSwNP. Mepolizumab provided meaningful within-patient improvements in symptom severity and health-related quality of life versus placebo, indicating mepolizumab provides substantial clinical benefits in very severe CRSwNP.

Scale Mixture of Maxwell-Boltzmann Distribution

This paper presents a new distribution, the product of the mixture between Maxwell-Boltzmann and a particular case of the generalized gamma distributions. The resulting distribution, called the Scale Mixture Maxwell-Boltzmann, presents greater kurtosis than the recently introduced slash Maxwell-Boltzmann distribution. We obtained closed-form expressions for its probability density and cumulative distribution functions. We studied some of its properties and moments, as well as its skewness and kurtosis coefficients. Parameters were estimated by the moments and maximum likelihood methods, via the Expectation-Maximization algorithm for the latter case. A simulation study was performed to illustrate the parameter recovery. The results of an application to a real data set indicate that the new model performs very well in the presence of outliers compared with other alternatives in the literature.

Velocity Vector Estimation of Two-Dimensional Flow Field Based on STIV

As an important part of hydrometry, river discharge monitoring plays an irreplaceable role in the planning and management of water resources and is an essential element and necessary means of river management. Due to its benefits of simplicity, efficiency and safety, Space-Time Image Velocimetry (STIV) has attracted attention from all around the world. The most crucial component of the STIV is the detection of the Main Orientation of Texture (MOT), and the precision of detection directly affects the results of calculations. However, due to the complicated river flow characteristics and the harsh testing environment in the field, a large amount of noise and interfering textures show up in the space-time images, which affects the detection results of the MOT. In response to the shortage of noise and interference texture, a new non-contact image analysis method is developed. Firstly, Multi-scale Retinex (MSR) is proposed to pre-process the images for contrast enhancement; secondly, a fourth-order Gaussian derivative steerable filter is employed to enhance the structure of the texture; next, based on the probability density distribution function and the orientations of the enhanced images, the noise suppression function and the orientation-filtering function are designed to filter out the noise to highlight the texture. Finally, the Fourier Maximum Angle Analysis (FMAA) is used to filter out the noise further and obtain the clear orientations to achieve the measurement of velocity and discharge. The experimental results show that, compared with the widely used image velocimetry measurements, the accuracy of our method in the average velocity and flow discharge is significantly improved, and the real-time performance is excellent.

Annual cycle of hygroscopic properties and mixing state of the suburban aerosol in Athens, Greece

Abstract. The hygroscopic properties of atmospheric aerosol were investigated at a suburban environment in Athens, Greece, from August 2016 to July 2017. The growth factor distribution probability density function (GF-PDF) and mixing state were determined with a hygroscopicity tandem differential mobility analyser (HTDMA). Four dry particle sizes (D0) were selected to be analysed in terms of their hygroscopic properties at 90 % relative humidity. The annual mean GFs for D0 = 30, 50, 80 and 250 nm were found to be equal to 1.28, 1.11, 1.13 and 1.22, respectively. The hygroscopic growth spectra were divided into two distinct hygroscopic ranges: a non- and/or slightly hygroscopic mode (GF < 1.12) and a moderately hygroscopic mode (GF > 1.12), which are representative of a suburban environment influenced by local/urban emissions and background aerosol. The standard deviation σ of the GF-PDF was employed as a measure of the mixing state of ambient aerosol. The 30 nm particles were mostly internally mixed, whereas larger particles were found to be externally mixed, either with a distinct bimodal structure or with partly overlapping modes. Cluster analysis on the hourly dry number size distributions was performed to identify the link between aerosol hygroscopicity and aerosol emission sources and formation processes. The size distributions were classified into five groups, with the “mixed urban and regional background” aerosol (67 %) and the “fresh traffic-related particles” from the neighbourhood urban area (15 %) accounting for more than 80 % of the results. The hygroscopic properties for 50 and 80 nm were found to be similar in all cases, indicating particles of similar nature and origin across these sizes. This was also confirmed through the modal analysis of the average number size distributions for each cluster; the 50 and 80 nm particles were found to belong to the same Aitken mode in most cases. The 250 nm particles (i.e. accumulation mode) were generally more hygroscopic than Aitken particles but less hygroscopic than the 30 nm particles (nuclei mode).

Determining the Accuracy of the Triangular and PERT Distributions

The Triangular and PERT (Program Evaluation Review Technique) distribution probability density functions are commonly used in decision and risk analyses. These distributions are popular because they are each specified by only three points (two support bounds and the mode) that are believed to be easy to assess from experts or data. In this paper, we carefully analyze how close the Triangular and PERT distributions are to other distributions sharing the same support and mode and show that the errors induced by the Triangular and PERT distributions are significant. We further show that distributions that are characterized by the median tend to provide a better fit than do those that are characterized by the mode. Funding: This research was supported by the Equinor Fellows Program and the Operating System 2.0 research program developed by the Construction Industry Institute.

It takes two to know one: computing accurate one-point PDF covariances from effective two-point PDF models

One-point probability distribution functions (PDFs) of the cosmic matter density are powerful cosmological probes that extract non-Gaussian properties of the matter distribution and complement two-point statistics. Computing the covariance of one-point PDFs is key for building a robust galaxy survey analysis for upcoming surveys like Euclid and the Rubin Observatory LSST and requires good models for the two-point PDFs characterising spatial correlations. In this work, we obtain accurate PDF covariances using effective shifted lognormal two-point PDF models for the mildly non-Gaussian weak lensing convergence and validate our predictions against large sets of Gaussian and non-Gaussian maps. We show how the dominant effects in the covariance matrix capturing super-sample covariance arise from a large-separation expansion of the two-point PDF and discuss differences between the covariances obtained from small patches and full sky maps. Finally, we describe how our formalism can be extended to characterise the PDF covariance for 3D-dimensional spectroscopic fields using the 3D matter PDF as an example. We describe how covariances from simulated boxes with fixed overall density can be supplemented with the missing super-sample covariance effect by relying on theoretical predictions validated against separate-universe style simulations.

Weighted power Maxwell distribution: Statistical inference and COVID-19 applications.

During the course of this research, we came up with a brand new distribution that is superior; we then presented and analysed the mathematical properties of this distribution; finally, we assessed its fuzzy reliability function. Because the novel distribution provides a number of advantages, like the reality that its cumulative distribution function and probability density function both have a closed form, it is very useful in a wide range of disciplines that are related to data science. One of these fields is machine learning, which is a sub field of data science. We used both traditional methods and Bayesian methodologies in order to generate a large number of different estimates. A test setup might have been carried out to assess the effectiveness of both the classical and the Bayesian estimators. At last, three different sets of Covid-19 death analysis were done so that the effectiveness of the new model could be demonstrated.

An Improved Generic Hybrid Prognostic Method for RUL Prediction Based on PF-LSTM Learning

Accurate estimation and prediction of the State-of-Health (SOH) and Remaining Useful Life (RUL) are fundamental to optimal maintenance strategies formulation for Prognostics and Health Management (PHM) of degraded equipment. However, the performance assessment of health state prognostics and RUL prediction is strongly dependent on the errors and uncertainties in physical measurements, and heterogeneous degradation of equipment in time-varying operating conditions. The objective of the paper is to provide a hybrid prognostic framework that integrates a two-phases clustering scheme and a PF-LSTM learning algorithm based on Particle Filter (PF) and Long Short-Term Memory (LSTM) networks for dynamic classification of SOH and long-term RUL prediction in the absence of future observations. The proposed generic hybrid PF-LSTM prognostic approach is demonstrated and compared with other adaptive learning and machine learning methods such as Unscented Particle Filter (UPF) and Radial Basis Function Network (RBFN) on the degradation modeling and RUL prediction for Lithium-ion batteries. The comparison results show that robust prediction performance can be obtained by the hybrid PF-LSTM prognostic approach with accurate characterization of equipment degradation states based on the integrated subtractive-fuzzy clustering analysis. The more accuracy on prognostic estimations in Probability Density Function (PDF) of prior and posterior distributions of battery capacity and RUL that are achieved by particle filtering can gain extensive insights to predictive maintenance action guide.

Introduction to Symbolic 2-Plithogenic Probability Theory

In this paper we present for the first time the concept of symbolic plithogenic random variables and study its properties including expected value and variance. We build the plithogenic formal form of two important distributions that are exponential and uniform distributions. We find its probability density function and cumulative distribution function in its plithogenic form. We also derived its expected values and variance and the formulas of its random numbers generating. We finally present the fundamental form of plithogenic probability density and cumulative distribution functions. All the theorems were proved depending on algebraic approach using isomorphisms. This paper can be considered the base of symbolic plithogenic probability theory.

Probability Distribution of SARS-Cov-2 (COVID) Infectivity Following Onset of Symptoms: Analysis from First Principles

The phasing out of protective measures by governments and public health agencies, despite continued seriousness of the coronavirus pandemic, leaves individuals who are concerned for their health with two basic options over which they have control: 1) minimize risk of infection by being vaccinated and by wearing a face mask when appropriate, and 2) minimize risk of transmission upon infection by self-isolating. For the latter to be effective, it is essential to have an accurate sense of the probability of infectivity as a function of time following the onset of symptoms. Epidemiological considerations suggest that the period of infectivity follows a lognormal distribution. This proposition is tested empirically by construction of the lognormal probability density function and cumulative distribution function based on quantiles of infectivity reported by several independent investigations. A comprehensive examination of a prototypical ideal clinical study, based on general statistical principles (the Principle of Maximum Entropy and the Central Limit Theorem) reveals that the probability of infectivity is a lognormal random variable. Subsequent evolution of new variants may change the parameters of the distribution, which can be updated by the methods in this paper, but the form of the probability function is expected to remain lognormal as this is the most probable distribution consistent with mathematical requirements and available information.

An Uncertainty Analysis and Reliability-Based Multidisciplinary Design Optimization Method Using Fourth-Moment Saddlepoint Approximation

In uncertainty analysis and reliability-based multidisciplinary design and optimization (RBMDO) of engineering structures, the saddlepoint approximation (SA) method can be utilized to enhance the accuracy and efficiency of reliability evaluation. However, the random variables involved in SA should be easy to handle. Additionally, the corresponding saddlepoint equation should not be complicated. Both of them limit the application of SA for engineering problems. The moment method can construct an approximate cumulative distribution function of the performance function based on the first few statistical moments. However, the traditional moment matching method is not very accurate generally. In order to take advantage of the SA method and the moment matching method to enhance the efficiency of design and optimization, a fourth-moment saddlepoint approximation (FMSA) method is introduced into RBMDO. In FMSA, the approximate cumulative generating functions are constructed based on the first four moments of the limit state function. The probability density function and cumulative distribution function are estimated based on this approximate cumulative generating function. Furthermore, the FMSA method is introduced and combined into RBMDO within the framework of sequence optimization and reliability assessment, which is based on the performance measure approach strategy. Two engineering examples are introduced to verify the effectiveness of proposed method.