Skewed Probability Density Function Research Articles

ARCSINE RAYLIEGH PARETO DISTRIBUTION: PROPERTIES AND APPLICATION TO CARBON FIBERS DATA SETS

In this paper, we introduce a new modified distribution called arcsine Rayliegh Pareto (ASRP) Distribution. We derived its mathematical and statistical properties, including survival function, hazard function, entropy, moment, moment generating function, and order statistics. We also used maximum likelihood estimation for estimating the parameters of the distribution. The plots of the cdf, pdf, hazard rate function, and survival function were illustrated with right skewed probability density function, cumulative distribution function with monotone increasing function, which converge at one. The value of some goodness of fit measure (i.e. AIC, AICc, and BIC) were computed, as well as the KS, A, and W statistic. Finally, we suggested that the new modified model outperform better than the other standard distribution using Carbon fibers data sets.

Coupling particle image velocimetry and digital image analysis to characterize cluster dynamics in a fast fluidized bed

Particle clusters affect interphase interaction and dominate the overall performance of gas-solid fluidized beds. In this work, the particle image velocimetry, particle tracking velocimetry and digital image analysis were coupled to characterize clustering dynamics in a lab-scale riser. The clusters were identified by the Otsu algorithm based on grayscale distribution, and the cluster properties such as equivalent size, solids concentration and velocity distribution were extracted and analyzed under various operating conditions. It was found macroscopic operating conditions affect cluster characteristics significantly whereas the effect of particle composition is secondary. A bimodal or skewed probability density function was observed for not only the time-averaged hydrodynamic velocity but also the particle velocity around the cluster interface, indicating the coexistence of dilute and dense phases and the breaking of equilibrium assumption. The nonuniform distributions for laminar and Reynolds-stress-like granular temperatures were further discussed, which show remarkable anisotropy at both the microscopic and mesoscopic scales.

Stochastic characterisation of unstable lean hydrogen–air annular premixed flames

This study is devoted to the characterisation of lean premixed hydrogen–air laminar flames stabilised on an annular burner. The flames studied are chosen to lie close to the lean flammability limit, spanning a range of Damköhler and Reynolds numbers, and correspond to situations where instabilities arise. The experimental characterisation is achieved by examining the statistical moments of the flame hydroxyl radical (OH*) chemiluminescence and schlieren. The standard deviation and the skewness, that are needed to fully characterise the measured probability density functions, are shown to exhibit a non-monotonic behaviour with the distance from the burner surface, first increasing, then decreasing downstream the flame tip. An exponentially modified Gaussian distribution model is proposed to describe the skewed schlieren fluctuations probability density function. This model is closed by developing relations for the probability density function parameters based on the Damköhler and Reynolds numbers, and then used to describe the OH* stochastic behaviour. The relation between the distribution parameters and the transport equation of a reactive scalar gradient is addressed also.

Statistical Inference of Sine Inverse Rayleigh Distribution

We study in this manuscript a new one-parameter model called sine inverse Rayleigh (SIR) model that is a new extension of the classical inverse Rayleigh model. The sine inverse Rayleigh model is aiming to provide more fitting for real data sets of purposes. The proposed extension is more flexible than the original inverse Rayleigh (IR) model and it hasmany applications in physics and medicine. The sine inverse Rayleigh distribution can havea uni-model and right skewed probability density function (PDF). The hazard rate function (HRF) of sine inverse Rayleigh distribution can be increasing and J-shaped. Several of thenew model’s fundamental characteristics, namely quantile function, moments, incompletemoments, Lorenz and Bonferroni Curves are studied. Four classical estimation methods forthe population parameters, namely least squares (LS), weighted least squares (WLS), maximum likelihood (ML), and percentile (PC) methods are discussed, and the performanceof the four estimators (namely LS, WLS, ML and PC estimators) are also compared bynumerical implementations. Finally, three sets of real data are utilized to compare the behavior of the four employed methods for finding an optimal estimation of the new distribution.

Low-rank Gaussian mixture modeling of space-snapshot representation of microphone array measurements for acoustic imaging in a complex noisy environment

The microphone array can simultaneously obtain the multi-dimensional information (space-time-frequency) of sound sources, which is recognized as a fundamental and powerful tool in acoustic imaging. Acoustic Beamforming is one of the widely used methods in acoustic imaging. However, most of the applications of beamforming are based on the Gaussian noise assumption, which is not always accurate in on-site measurements. For example, shock noise with a skewed probability density function (PDF) may appear on the signal record when the turbulent eddies are not controlled. Thus, in this paper, the conventional Gaussian noise model is extended to a Gaussian mixture noise model, which can approximate any probability distribution of the noise in theory. The space-snapshot representation of microphone array measurements is further modeled as a combination of the low-rank matrix part (measurements from the sound sources) and a Gaussian mixture matrix part (measurement noise). The signal from the sources of interest is finally recovered by the Expectation–maximization algorithm, which iterates between the low-rank approximation of the sound sources and the estimation of the parameter of the Gaussian mixture model. The proposed method is further investigated with simulations and compared with robust principal component analysis (RPCA) and Gaussian-based probabilistic factor analysis (PFA). It is concluded that the proposed method outperforms the state-of-the-art denoising methods.

On the variability of motor-evoked potentials: experimental results and mathematical model.

The purpose of this study was to determine the form of the relation between the mean amplitude and variance of motor-evoked potentials (MEP). To this end, single-pulse transcranial magnetic stimulation (TMS) was applied over the motor cortex of seventeen neurologically normal adult human subjects. The coil was positioned at a locus on the scalp that elicited an MEP in the first dorsal interosseous (FDI) at the lowest stimulus intensity. The subjects were instructed to maintain tonic activity in the FDI of 5 or 10% of the maximum voluntary contraction (MVC). The relation between MEP variance and amplitude was found to have an inverted parabolic shape, with maximal variance occurring near the half-maximal MEP amplitude. The coefficient of variation [Formula: see text] of MEPs decreased approximately as a rectangular hyperbolic function of MEP amplitude (i.e. ~ 1/MEP). A probabilistic model is proposed to explain the inverted parabolic relation between MEP variance and MEP amplitude, as well as the sigmoid shape of the MEP input-output relation (i.e. stimulus-response curve). The model is based on a description of α-motoneurons as binary threshold units, with unit thresholds distributed according to a positively skewed probability density function. The units are driven by noisy synaptic input currents having a Gaussian distribution. The model predicts an inverse parabolic relation between MEP variance and amplitude and a sigmoid input-output relation, as experimentally observed. Furthermore, increasing model motoneuron excitability by increasing the background synaptic drive increases MEP variability independently of MEP size, a surprising prediction. The model also explains the approximately rectangular hyperbolic relation between [Formula: see text] and MEP amplitude. The implications of these results for the interpretation of neurophysiological experiments and the statistical analysis of MEPs are discussed.

On the use of non-Gaussian models for statistical description of road micro-surface profiles

When analysing vehicle-road interaction, probability density function (PDF) of random micro-surface is required. Since the asperity tops are polished by tyres stronger than the valley bottoms, the surface height profiles become asymmetrical. As a result, the PDFs of micro-surface signals are often different from the Gaussian model and one needs a non-Gaussian PDF model operating with skewness and kurtosis. Previous solutions by the Pearson and Johnson distributions do not lend themselves for further implementation in analytical form. To overcome this difficulty, a non-Gaussian PDF can be constructed from a few Gaussian sections with different mean values and standard deviations. To use such a piecewise-Gaussian model for analytical derivations, it is simply necessary to apply the classic Gaussian equation several times. An example of skewed PDF of micro-surface of an asphaltic concrete highway measured by a laser scanning system was adequately approximated by the tetra-Gaussian model consisting of four Gaussian sections.

Symmetric Fractional Diffusion and Entropy Production

The discovery of the entropy production paradox (Hoffmann et al., 1998) raised basic questions about the nature of irreversibility in the regime between diffusion and waves. First studied in the form of spatial movements of moments of H functions, pseudo propagation is the pre-limit propagation-like movements of skewed probability density function (PDFs) in the domain between the wave and diffusion equations that goes over to classical partial differential equation propagation of characteristics in the wave limit. Many of the strange properties that occur in this extraordinary regime were thought to be connected in some manner to this form of proto-movement. This paper eliminates pseudo propagation by employing a similar evolution equation that imposes spatial unimodal symmetry on evolving PDFs. Contrary to initial expectations, familiar peculiarities emerge despite the imposed symmetry, but they have a distinct character.

Diversity, Nonlinearity, Seasonality, and Memory Effect in ENSO Simulation and Prediction Using Empirical Model Reduction

Abstract A suite of empirical model experiments under the empirical model reduction framework are conducted to advance the understanding of ENSO diversity, nonlinearity, seasonality, and the memory effect in the simulation and prediction of tropical Pacific sea surface temperature (SST) anomalies. The model training and evaluation are carried out using 4000-yr preindustrial control simulation data from the coupled model GFDL CM2.1. The results show that multivariate models with tropical Pacific subsurface information and multilevel models with SST history information both improve the prediction skill dramatically. These two types of models represent the ENSO memory effect based on either the recharge oscillator or the time-delayed oscillator viewpoint. Multilevel SST models are a bit more efficient, requiring fewer model coefficients. Nonlinearity is found necessary to reproduce the ENSO diversity feature for extreme events. The nonlinear models reconstruct the skewed probability density function of SST anomalies and improve the prediction of the skewed amplitude, though the role of nonlinearity may be slightly overestimated given the strong nonlinear ENSO in GFDL CM2.1. The models with periodic terms reproduce the SST seasonal phase locking but do not improve the prediction appreciably. The models with multiple ingredients capture several ENSO characteristics simultaneously and exhibit overall better prediction skill for more diverse target patterns. In particular, they alleviate the spring/autumn prediction barrier and reduce the tendency for predicted values to lag the target month value.

Lagrangian Stochastic Modelling of Dispersion in the Convective Boundary Layer with Skewed Turbulence Conditions and a Vertical Density Gradient: Formulation and Implementation in the FLEXPART Model

A correction for the vertical gradient of air density has been incorporated into a skewed probability density function formulation for turbulence in the convective boundary layer. The related drift term for Lagrangian stochastic dispersion modelling has been derived based on the well-mixed condition. Furthermore, the formulation has been extended to include unsteady turbulence statistics and the related additional component of the drift term obtained. These formulations are an extension of the drift formulation reported by Luhar et al. (Atmos Environ 30:1407–1418, 1996) following the well-mixed condition proposed by Thomson (J Fluid Mech 180:529–556, 1987). Comprehensive tests were carried out to validate the formulations including consistency between forward and backward simulations and preservation of a well-mixed state with unsteady conditions. The stationary state CBL drift term with density correction was incorporated into the FLEXPART and FLEXPART-WRF Lagrangian models, and included the use of an ad hoc transition function that modulates the third moment of the vertical velocity based on stability parameters. Due to the current implementation of the FLEXPART models, only a steady-state horizontally homogeneous drift term could be included. To avoid numerical instability, in the presence of non-stationary and horizontally inhomogeneous conditions, a re-initialization procedure for particle velocity was used. The criteria for re-initialization and resulting errors were assessed for the case of non-stationary conditions by comparing a reference numerical solution in simplified unsteady conditions, obtained using the non-stationary drift term, and a solution based on the steady drift with re-initialization. Two examples of “real-world” numerical simulations were performed under different convective conditions to demonstrate the effect of the vertical gradient in density on the particle dispersion in the CBL.

Source Separation for Wideband Energy Emissions Using Complex Independent Component Analysis

Complex independent component analysis is formulated for ultrawideband (UWB) incoherent fields, based on maximum-likelihood estimation with iterative natural gradient searching. It can be applied to measurements of complex power separated in contributions by propagating and reactive near-fields. A set of novel nonlinear asymmetric score functions is derived that are asymptotically matched to skewed probability density functions of incoherent energy sources departing from ideal chi-square ensemble distributions. The source decomposition is applied to two analog and digital integrated circuits. Based on the measured intensity of the emitted magnetic field, the technique is shown to enable nonlinear extraction of a set of independent spatially separated UWB sources, without knowledge of neither the detailed circuit geometry nor its on-board signals. Overcompleteness of the data set is found to have a relatively small effect on the accuracy of the estimated magnitude and location of the emission hot spots, but has a major influence on the global spatial maps and nature of the extracted sources.

Kalman filter variants in the closed skew normal setting

The filtering problem (or the dynamic data assimilation problem) is studied for linear and nonlinear systems with continuous state space and over discrete time steps. Filtering approaches based on the conjugate closed skewed normal probability density function are presented. This distribution allows additional flexibility over the usual Gaussian approximations. With linear dynamic systems the filtering problem can be solved in analytical form using expressions for the closed skew normal distribution. With nonlinear dynamic systems an ensemble-based version is proposed for fitting a closed skew normal distribution at each updating step. Numerical examples discuss various special cases of the methods.

On the crest factor of noise from supersonic jets

An important consideration in characterizing noise from heated, supersonic jets is the crest factor (CF). The large CF in high-speed jet noise is the result of a positively skewed probability density function for the waveform, which translates into infrequently occurring, large-amplitude positive peak pressures. Sufficient system headroom is required in the data acquisition system to provide an accurate representation of these peak pressures and thus avoid clipping or microphone saturation/distortion. But the question remains as to the importance of capturing the single largest pressure out of potentially millions of waveform samples or if a percentile-based CF is adequate. Measurements near a static tactical aircraft reveal CF increases with engine power, with the maximum CF directed upstream of the overall sound pressure level, and a maximum CF of 20 dB at full afterburner. Second, clipping of measured waveforms at different thresholds reveals that a CF definition based on the 99.99 percentile is sufficient to represent overall and band pressure levels to within 0.1 dB and waveform and time-derivative skewnesses to within ~1%. If an estimate of the time-derivative kurtosis is needed within 1% accuracy, then the 99.999 percentile CF is required for headroom estimates.

Stochastic simulation of fluid flow in porous media by the complex variable expression method

A stochastic simulation of fluid flow in porous media using a complex variable expression method (SFCM) is presented in this paper. Hydraulic conductivity is considered as a random variable and is then expressed in complex variable form, the real part of which is a deterministic value and the imaginary part is a variable value. The stochastic seepage flow is simulated with the SFCM and is compared with the results calculated with the Monte Carlo stochastic finite element method. In using the Monte Carlo method to simulate the stochastic seepage flow field, the hydraulic conductivity is assumed in three different probability distributions using random sampling method. The obtained seepage flow field is examined through skewness analysis, and the skewed distribution probability density function is given. The head mode value and the head comprehensive standard deviation are used to represent the statistics of calculation results obtained by the Monte Carlo method. The stochastic seepage flow field simulated by the SFCM is confirmed to be similar to that given by the Monte Carlo method from numerical aspects. The range of coefficient of variation of hydraulic conductivity in SFCM is larger than used previously in stochastic seepage flow field simulations, and the computation time is short. The results proved that the SFCM is a convenient calculating method for solving the complex problems.

Quantitative characterization of marine oil slick by satellite near-infrared imagery and oil drift modelling: the Fun Shai Hai case study

Near-infrared (NIR) satellite images of the oil spill event caused by the Fu Shan Hai wreck on 31 May 2003 in the waters between Sweden and Denmark were compared with numerical simulations provided by the MIKE 21 oil drift model. Assuming a skewed probability density function (pdf) of oil parcel thicknesses, a model of the NIR image oil–water contrast reflectance was developed to characterize the expected oil slick distribution in terms of average and maximum oil slick thickness. Since MIKE 21 Spill Analysis (SA) also allows non-uniform distribution of oil volume within the oil slick, both distributions were thus compared by coincidence of the Moderate Resolution Imaging Spectroradiometer (MODIS/Aqua) acquisition, which imaged the oil slick 3 days after the oil spill started. Results showed an excellent agreement in the numerical values of both the expected average and the maximum thickness. In addition, repartition of the oil volume within the slick in the usual thin (sheen) and thick (brown) parts resulted, consistent with the empirical rule of 20% and 80% of the total oil volume, respectively.

The lossy farfield pressure impulse response for a rectangular piston

The impulse response of the velocity potential is useful for computing transient pressures in lossless media, especially for calculations in the nearfield region. Closed form expressions for the lossless impulse response of the velocity potential in the nearfield are available for circular and rectangular transducers and for several other geometries. A closed form lossless farfield expression is also available for rectangular transducers. Typically, when the effects of attenuation are introduced, the numerical calculation is performed in the frequency domain, and the time response is obtained with an inverse fast Fourier transform. To derive an equivalent analytical result directly in the time domain, all path lengths that appear in the denominator scaling term of the lossy diffraction integral are treated as constants, and a binomial expansion is applied to the path length that appears in the time delay term. The resulting analytical expression, which describes the lossy farfield pressure impulse response, is directly expressed in terms of maximally skewed stable probability density functions and cumulative distribution functions. Results are compared with the Rayleigh Sommerfeld integral, and excellent agreement is achieved in the farfield region. [supported in part by NIH Grant R01 EB012079.]

Analytical and numerical approximations for the lossy on-axis impulse response of a circular piston

In biological tissues, the frequency dependence of attenuation and speed of sound for ultrasound propagation is described by the power law wave equation, which is a partial differential equation with fractional time derivatives. As demonstrated previously, the time domain Green's function for the power law wave equation combined with the Rayleigh-Sommerfeld integral is an effective reference for calculations of the lossy on-axis impulse response of a circular piston. Using the result obtained from this reference, two different approximations to the lossy on-axis impulse response are evaluated. The first approximation is an analytical expression that is proportional to the difference between two cumulative distribution functions for maximally skewed stable probability densities. The second approximation numerically convolves the lossless impulse response with a maximally skewed stable probability density function. The results show that both approximations achieve relatively small errors. Furthermore, the analytical approximation provides an excellent estimate for the arrival time of the lossy impulse response, whereas the departure time of the lossy impulse response is more difficult to characterize due to the heavy tail of the maximally skewed stable probability density function. Both approximations are rapidly calculated with the STABLE toolbox. [supported in part by NIH Grant R01 EB012079.]

Statistics of surface renewal of passive scalars in free-surface turbulence

We perform direct numerical simulation to study the transport of gas and heat as passive scalars in free-surface turbulence. Our analysis focuses on the surface age of surface fluid particles, i.e. the time elapsed since the last surface renewal they experienced. Using Lagrangian tracing of fluid particles combined with heat diffusion analysis, we are able to directly quantify surface age to illustrate scalar characteristics at different stages of interfacial transfer. Results show that at the early stage of surface renewal, vertical advection associated with upwellings greatly enhances surface gas flux; random surface renewal model does not apply at this stage when most of the interfacial gas transfer occurs. After a fluid particle leaves the upwelling region, it may enter a nearby downwelling region immediately, where the gas flux is sharply reduced but the variation in surface temperature is small; alternatively, the fluid particle may travel along the surface for some time before it is absorbed by a downwelling, where the surface temperature has changed significantly due to long duration of diffusion and the gas flux is also reduced. To gain further insight into the relationships between surface velocity and scalar quantities, we perform a statistical analysis of upwellings using clustering and nonlinear regression. With this analysis, we are able to provide qualitative and quantitative descriptions of the skewed probability density functions associated with the surface divergence, temperature and gas flux that support our physics-based investigation of surface renewal and surface age.

Current status of sharps waste management in the lower-level health facilities in Tanzania

Sharps waste is part of infectious medical waste, management of which is a critical problem in Tanzanian health facilities. This study aimed at assessing the current status of sharps waste management in lower level health facilities (LLHFs) in Ilala Municipality in Tanzania. In this study a sample of 135 LLHFs (103 dispensaries, 13 clinics, 11 laboratories, and 8 health centers) was involved. The average number of workers per facility was 10, with positively skewed probability density function (up to 80 workers). The average patient-to-workers ratio was 5.87. About 59% of the LLHFs improvised sharps waste containers (SWCs). Sharps waste was transported by hands in 77% of LLHFs leading to high risks of exposure to needle stick injuries. Boots, aprons and masks were among the personal protective equipment (PPE) missing in most LLHFs, while latex gloves that cannot protect workers from injuries caused by sharps waste were readily available. Most facilities stored sharps waste for about 72 hours (before treatment), which is beyond the recommended maximum storage time of 24 hours. About 39.3% of LLHFs utilized on-site single-chamber incinerators for sharps waste treatment, which are of poor design, have rusted mechanical parts, short and rusted chimneys, and without automatic flame ignition burners. It is concluded that sharps waste management in LLHFs is poor, which puts workers, the public and the environment at risk of exposure to blood-borne pathogens. It is, therefore, important that the municipality should establish a waste processing center which will collect and incinerate all sharps waste.

Numerical simulations of compressively driven interstellar turbulence

Context. Supersonic turbulence is ubiquitous in the interstellar me dium and plays an important role in contemporary star formation. Aims. To perform a high-resolution numerical simulation of supersonic isothermal turbulence driven by compressive large-scale forcing and to analyse various statistical properties. Methods. The compressible Euler equations with an external stochastic force field dominated by rotation-free modes are solved wi th the piecewise parabolic method. Both a static grid and adaptive mesh refinement is used with an effective resolution N = 768 3 . Results. After a transient phase dominated by shocks, turbulence evolves into a steady state with root mean square Mach number ≈ 2.2... 2.5, in which cloud-like structures of over-dense gas are surr ounded by highly rarefied gas. The index of the turbulence ene rgy spectrum functionβ≈ 2.0 in the shock-dominated phase. As the flow approaches statis tical equilibrium, the spectrum flattens, with β≈ 1.9. For the scaling exponent of the root mean square velocity fl uctuation, we obtainγ≈ 0.43 from the velocity structure functions of second order. These results are well within the range of observed scaling properties for the velocity dispersion in molecular clouds. Calculating structure functions of order p = 1,..., 5, we find for all scaling exponents significant deviations from the Kolmogor ovBurgers model proposed by Boldyrev. Our results are very well described by a general log-Poisson model with a higher degree of intermittency, which implies an influence of the forcing on t he scaling properties. The spectral index of the quadratic l ogarithmic density fluctuation βδ≈ 1.8. Contrary to previous numerical results for isothermal turbulence, we obtain a skewed probability density function of the mass density fluctuations that is not consist ent with log-normal statistics and entails a substantially higher fraction of mass in the density peaks than implied by the Padoan-Nordlund relation between the variance of the density fluctuation s and the Mach number. Conclusions. Even putting aside further complexity due to magnetic fields , gravity or thermal processes, we question the notion that Larson-type relations are a consequence of universal supersonic turbulence scaling. For a genuine understanding, it seems necessary to account for the production mechanism of turbulence in the ISM.