The Probability Density Function Related to Shallow Cumulus Entrainment Rate and Its Influencing Factors in a Large-Eddy Simulation

The process of entrainment-mixing between cumulus clouds and the ambient air is important for the development of cumulus clouds.Accurately obtaining the entrainment rate(λ)is particularly important for its parameterization within the overall cumulus parameterization scheme.In this study,an improved bulk-plume method is proposed by solving the equations of two conserved variables simultaneously to calculateλof cumulus clouds in a large-eddy simulation.The results demonstrate that the improved bulk-plume method is more reliable than the traditional bulk-plume method,becauseλ,as calculated from the improved method,falls within the range ofλvalues obtained from the traditional method using different conserved variables.The probability density functions ofλfor all data,different times,and different heights can be well-fitted by a log-normal distribution,which supports the assumed stochastic entrainment process in previous studies.Further analysis demonstrate that the relationship betweenλand the vertical velocity is better than other thermodynamic/dynamical properties;thus,the vertical velocity is recommended as the primary influencing factor for the parameterization ofλin the future.The results of this study enhance the theoretical understanding ofλand its influencing factors and shed new light on the development ofλparameterization.

Prediction and Output Estimation of Pattern Moving in Non-Newtonian Mechanical Systems Based on Probability Density Evolution

A prediction framework based on the evolution of pattern motion probability density is proposed for the output prediction and estimation problem of non-Newtonian mechanical systems,assuming that the system satisfies the generalized Lipschitz condition.As a complex nonlinear system primarily governed by statistical laws rather than Newtonian mechanics,the output of non-Newtonian mechanics systems is difficult to describe through deterministic variables such as state variables,which poses difficulties in predicting and estimating the system’s output.In this article,the temporal variation of the system is described by constructing pattern category variables,which are non-deterministic variables.Since pattern category variables have statistical attributes but not operational attributes,operational attributes are assigned to them by posterior probability density,and a method for analyzing their motion laws using probability density evolution is proposed.Furthermore,a data-driven form of pattern motion probabilistic density evolution prediction method is designed by combining pseudo partial derivative(PPD),achieving prediction of the probability density satisfying the system’s output uncertainty.Based on this,the final prediction estimation of the system’s output value is realized by minimum variance unbiased estimation.Finally,a corresponding PPD estimation algorithm is designed using an extended state observer(ESO)to estimate the parameters to be estimated in the proposed prediction method.The effectiveness of the parameter estimation algorithm and prediction method is demonstrated through theoretical analysis,and the accuracy of the algorithm is verified by two numerical simulation examples.

High-dimensional uncertainty quantification of projectile motion in the barrel of a truck-mounted howitzer based on probability density evolution method

This paper proposed an efficient research method for high-dimensional uncertainty quantification of projectile motion in the barrel of a truck-mounted howitzer.Firstly,the dynamic model of projectile motion is established considering the flexible deformation of the barrel and the interaction between the projectile and the barrel.Subsequently,the accuracy of the dynamic model is verified based on the external ballistic projectile attitude test platform.Furthermore,the probability density evolution method(PDEM)is developed to high-dimensional uncertainty quantification of projectile motion.The engineering example highlights the results of the proposed method are consistent with the results obtained by the Monte Carlo Simulation(MCS).Finally,the influence of parameter uncertainty on the projectile disturbance at muzzle under different working conditions is analyzed.The results show that the disturbance of the pitch angular,pitch angular velocity and pitch angular of velocity decreases with the increase of launching angle,and the random parameter ranges of both the projectile and coupling model have similar influence on the disturbance of projectile angular motion at muzzle.

Joint probability density function of the stochastic responses of nonlinear structures

The joint probability density fimction （PDF） of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution method, which is successfully developed to capture the instantaneous PDF of an arbitrary single response of interest, is extended to evaluate the joint PDF of any two responses. A two-dimensional partial differential equation in terms of the joint PDF is established. The strategy of selecting representative points via the number theoretical method and sieved by a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed. The free vibration of an SDOF system is examined to verify the proposed method, and a flame structure exhibiting hysteresis subjected to stochastic ground motion is investigated. It is pointed out that the correlation of different responses results from the fact that randomness of different responses comes from the same set of basic random parameters involved. In other words, the essence of the probabilistic correlation is a physical correlation.

On the Probability Density Function of the Real and Imaginary Parts in WFRFT Signals

Recently a Hybrid Carrier （HC） scheme based on Weighted-type Fractional Fourier Transform （WFRFT） was proposed and developed, which contains Single Carrier （SC） and Multi-Carrier （MC） synergetie transmission. The wide interest is primarily due to its appealing characteristics, such as the robust performances in different types of selective fading channels and a great deal of potential for secure communications. According to the literatures, the HC signal and SC or MC signal probability distributions are different. In particular, some benefits of this HC scheme are brought by the quasi-Gaussian distribution of WFRFT signals. However, until now researchers have only presented statistic properties through computer simulations, and the accurate expressions of signals are not derived yet. In this paper, we derive the accu- rate and rigorously established closed-form expressions of Probability Density Function （PDF） of WFRFT signal real and imaginary parts with a large number of QPSK subcarriers, and this PDF can describe the behavior of data modulated by WFRFT, avoiding the complex computation for extensive computer simulations. Furthermore, the components of PDF expression are described and analyzed, and it is revealed that the tendency of signal quasi-Gaussian changes with the increasing of the parameter a （a in （0,1]）. To validate the analytical results, extensive simulations have been conducted, showing a very good match between the analytical results and the real situations. The contribution of this paper may be useful to deduce the closed form expressions of Bit Error Ratio （BER）, the Complementary Cumulative Distribution Function （CCDF） of Peak to Average Power Ratio （PAPR）, and other analytical studies which adopt the PDF.

Shape control on probability density function in stochastic systems

A novel strategy of probability density function （PDF） shape control is proposed in stochastic systems. The control er is designed whose parameters are optimal y obtained through the improved particle swarm optimization algorithm. The parameters of the control er are viewed as the space position of a particle in particle swarm optimization algorithm and updated continual y until the control er makes the PDF of the state variable as close as possible to the expected PDF. The proposed PDF shape control technique is compared with the equivalent linearization technique through simulation experiments. The results show the superiority and the effectiveness of the proposed method. The control er is excellent in making the state PDF fol ow the expected PDF and has the very smal error between the state PDF and the expected PDF, solving the control problem of the PDF shape in stochastic systems effectively.

Probability Density Function Method for Observing Reconstructed Attractor Structure

Probability density function (PDF) method is proposed for analysing the structure of the reconstructed attractor in computing the correlation dimensions of RR intervals of ten normal old men. PDF contains important information about the spatial distribution of the phase points in the reconstructed attractor. To the best of our knowledge, it is the first time that the PDF method is put forward for the analysis of the reconstructed attractor structure. Numerical simulations demonstrate that the cardiac systems of healthy old men are about 6-6.5 dimensional complex dynamical systems. It is found that PDF is not symmetrically distributed when time delay is small, while PDF satisfies Gaussian distribution when time delay is big enough. A cluster effect mechanism is presented to explain this phenomenon. By studying the shape of PDFs, that the roles played by time delay are more important than embedding dimension in the reconstruction is clearly indicated. Results have demonstrated that the PDF method represents a promising numerical approach for the observation of the reconstructed attractor structure and may provide more information and new diagnostic potential of the analyzed cardiac system.

Probability Density Analysis of Nonlinear Random Ship Rolling

Ship rolling in random waves is a complicated nonlinear motion that contributes substantially to ship instability and capsizing.The finite element method(FEM)is employed in this paper to solve the Fokker Planck(FP)equations numerically for homoclinic and heteroclinic ship rolling under random waves described as periodic and Gaussian white noise excitations.The transient joint probability density functions(PDFs)and marginal PDFs of the rolling responses are also obtained.The effects of stimulation strength on ship rolling are further investigated from a probabilistic standpoint.The homoclinic ship rolling has two rolling states,the connection between the two peaks of the PDF is observed when the periodic excitation amplitude or the noise intensity is large,and the PDF is remarkably distributed in phase space.These phenomena increase the possibility of a random jump in ship motion states and the uncertainty of ship rolling,and the ship may lose stability due to unforeseeable facts or conditions.Meanwhile,only one rolling state is observed when the ship is in heteroclinic rolling.As the periodic excitation amplitude grows,the PDF concentration increases and drifts away from the beginning location,suggesting that the ship rolling substantially changes in a cycle and its stability is low.The PDF becomes increasingly uniform and covers a large region as the noise intensity increases,reducing the certainty of ship rolling and navigation safety.The current numerical solutions and analyses may be applied to evaluate the stability of a rolling ship in irregular waves and capsize mechanisms.

Linearized Controller Design for the Output Probability Density Functions of Non-Gaussian Stochastic Systems

This paper presents a linearized approach for the controller design of the shape of output probability density functions for general stochastic systems. A square root approximation to an output probability density function is realized by a set of B-spline functions. This generally produces a nonlinear state space model for the weights of the B-spline approximation. A linearized model is therefore obtained and embedded into a performance function that measures the tracking error of the output probability density function with respect to a given distribution. By using this performance function as a Lyapunov function for the closed loop system, a feedback control input has been obtained which guarantees closed loop stability and realizes perfect tracking. The algorithm described in this paper has been tested on a simulated example and desired results have been achieved.

Probabilistic Response and Short-Term Extreme Load Estimation of Offshore Monopile Wind Turbine Towers by Probability Density Evolution Method

A new analysis framework based on probability density evolution method(PDEM)and its Chebyshev collocation solution are introduced to predict the dynamic response and short-term extreme load of offshore wind turbine(OWT)towers subjected to random sea state.With regard to the stochastic responses,random function method is employed to generate samples of sea elevation,the probability density evolution equation(PDEE)is solved to calculate time-variant probability density functions of structural responses.For the probabilistic load estimation,a FAST model of NREL 5MW offshore turbine is established to obtain samples of bending moment at the tower base.The equivalent extreme event theory is used to construct a virtual stochastic process(VSP)to assess the short-term extreme load.The results indicate that the proposed approach can predict time-variant probability density functions of the structural responses,and shows good agreement with Monte Carlo simulations.Additionally,the predicted short-term extreme load can capture the fluctuation at the tail of the extreme value distribution,thus is more rational than results from the typical distribution models.Overall,the proposed method shows good adaptation,precision and efficiency for the dynamic response analysis and load estimation of OWT towers.

Steady-state probability density function in wave turbulence under large volume limit

We investigate the possibility for two-mode probability density function （PDF） to have a non-zero flux steady state solution. We take the large volume limit so that the space of modes becomes continuous. It is shown that in this limit all the steady-state twoor higher-mode PDFs are the product of one-mode PDFs. The flux of this steady-state solution turns out to be zero for any finite mode PDF.

ON DOUBLE PEAK PROBABILITY DENSITY FUNCTIONS OF DUFFING OSCILLATOR TO COMBINED DETERMINISTIC AND RANDOM EXCITATIONS

The principal resonance of Duffing random external excitation was investigated. oscillator to combined deterministic and The random excitation was taken to be white noise or harmonic with separable random amplitude and phase. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The one peak probability density function of each of the two stable stationary solutions was calculated by the linearization method. These two one-peak-density functions were combined using the probability of realization of the two stable stationary solutions to obtain the double peak probability density function. The theoretical analysis are verified by numerical results.

Probability Density and Statistical Properties for a Three-State Markovian Noise and Escape of Particles for a System Driven by This Noise

A three-state Markovian noise is investigated.Its probability density and statistical properties are obtained. Escape of particles for a system with potential barrier only driven by this noise is investigated.It is shown that,in some circumstances,this noise can make the particles escape over the potential barrier;but in other circumstances,it cannot. Resonant activation phenomenon appears for the system considered by us.

Controlling the probability density of quantum systems

A new method to design a quantum controller which directly controls the probability density function（PDF） of quantum systems is proposed.Based on the quantum model from the PDF perspective,two specific control algorithms are proposed with uniform and non-uniform fields,respectively.Then a detailed control algorithm with convergence analysis is given for the small error case.By appropriately estimating the selected Lyapunov function,more accurate control effect is achieved.The proposed scheme provides a constructive method to find appropriate parameters for controller design.

Probability density function and estimation for error of digitized map coordinates in GIS

Traditionally, it is widely accepted that measurement error usually obeys the normal distribution. However, in this paper a new idea is proposed that the error in digitized data which is a major derived data source in GIS does not obey the normal distribution but the p-norm distribution with a determinate parameter. Assuming that the error is random and has the same statistical properties, the probability density function of the normal distribution, Laplace distribution and p-norm distribution are derived based on the arithmetic mean axiom, median axiom and p-median axiom, which means that the normal distribution is only one of these distributions but not the least one. Based on this ideal distribution fitness tests such as Skewness and Kurtosis coefficient test, Pearson chi-square chi(2) test and Kolmogorov test for digitized data are conducted. The results show that the error in map digitization obeys the p-norm distribution whose parameter is close to 1.60. A least p-norm estimation and the least square estimation of digitized data are further analyzed, showing that the least p-norm adjustment is better than the least square adjustment for digitized data processing in GIS.

Developing theory of probability density function for stochastic modeling of turbulent gas-particle flows

Turbulent gas-particle flows are studied by a kinetic description using a prob- ability density function （PDF）. Unlike other investigators deriving the particle Reynolds stress equations using the PDF equations, the particle PDF transport equations are di- rectly solved either using a finite-difference method for two-dimensional （2D） problems or using a Monte-Carlo （MC） method for three-dimensional （3D） problems. The proposed differential stress model together with the PDF （DSM-PDF） is used to simulate turbulent swirling gas-particle flows. The simulation results are compared with the experimental results and the second-order moment （SOM） two-phase modeling results. All of these simulation results are in agreement with the experimental results, implying that the PDF approach validates the SOM two-phase turbulence modeling. The PDF model with the SOM-MC method is used to simulate evaporating gas-droplet flows, and the simulation results are in good agreement with the experimental results.

A Lagrangian-based flame index for the transported probability density function method

We propose a new flame index for the transported probability density function(PDF) method. The flame index uses mixing flux projections of Lagrangian particles on mixture fraction and progress variable directions as the metrics to identify the combustion mode, with the Burke-Schumann solution as a reference. A priori validation of the flame index is conducted with a series of constructed turbulent partially premixed reactors. It indicates that the proposed flame index is able to identify the combustion mode based on the subgrid mixing information. The flame index is then applied the large eddy simulation/PDF datasets of turbulent partially premixed jet flames. Results show that the flame index separate different combustion modes and extinction correctly. The proposed flame index provides a promising tool to analyze and model the partially premixed flames adaptively.

Reconstruction of Probability Density Function for Gamma Distribution

The probability distributions of small sample data are difficult to determine,while a large proportion of samples occur in the early failure period,so it is particularly important to make full use of these data in the statistical analysis.Based on gamma distribution,four methods of probability density function(PDF)reconstruction with early failure data are proposed,and then the mean time between failures(MTBF)evaluation expressions are concluded from the reconstructed PDFs.Both theory analysis and an example show that method 2 is the best evaluation method in dealing with early-failure-small-sample data.The reconstruction methods of PDF also have certain guiding significance for other distribution types.

Probability density analysis of SINR in massive MIMO systems with matched filter beamformer

This paper derives an approximate formula for probability density function(PDF) of received signal-to-interference-and-noise ratio(SINR) at user terminal when matched filter(MF) is adopted at a base station(BS).This distribution of SINR can be used to make an analysis of average sum-rate,outage probability,and symbol error rate of massive MIMO downlink with MF at BS.From simulation,it is found that the derived approximate analytical expression for PDF of SINR is consistent with the simulated exact PDF from the definition of SINR in medium-scale and large-scale MIMO systems.

Bounding box based on probability density for the pedestrian tracking

This paper proposes a pedestrian tracking approach using bounding box based on probability densities.It is generally a difficult task to track features like corner points in outdoor images due to complex environment.To solve this problem,the feature points are projected along X and Y direction separately,and a histogram is constructed for each projection,with horizontal axis as positions and vertical axis as the number of feature points that lie on each position.Finally,the vertical axis is normalized for expression as probability.After histogram is constructed,the probability of each feature point is checked with a threshold.A feature point will be ignored if its probability is lower than a threshold,while the remaining feature points are grouped,based on which a bounding box is made.Kanade-Lucas Tomasi(KLT)algorithm is adopted as the tracking algorithm because it is able to track local features in images robustly.The efficiency of the tracking results using this method is verified in real environment test.