Formula For Probability Density Function Research Articles

Probability distribution of market clearing solution in peer-to-peer energy market

This paper investigates stochastic market clearing solutions in peer-to-peer (P2P) energy markets as two parameters of prosumers’ cost functions are flexibly and randomly chosen in certain intervals to compensate uncertainties and guarantee their energy preferences. For tractability, the scenarios in which one parameter is varied while the other is fixed are considered. It is then shown that a few distributions, namely normal, Cauchy, and gamma distributions, are invariant for P2P energy markets, i.e., if prosumer cost function parameters follow one of those distributions, then market clearing solutions will follow the same type of distribution. Explicit and analytical formulas for probability density functions of market clearing price and trading powers in P2P energy markets are derived for each type of the above-mentioned distributions. As such, variations on P2P energy market clearing solutions can be assessed and predicted when parameters in the stochastic distribution of prosumer cost function parameters are changed. Simulations are carried out for a modified IEEE European Low Voltage Test Feeder system whose results validate the obtained theoretical characterizations.

Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitions.

Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity or the Hurst exponent. However, recent single-particle tracking experiments in biological cells revealed highly complicated anomalous diffusion phenomena that cannot be attributed to a class of self-similar random processes. Inspired by these observations, we here study the process that preserves the properties of the fractional Brownian motion at a single trajectory level; however, the Hurst index randomly changes from trajectory to trajectory. We provide a general mathematical framework for analytical, numerical, and statistical analysis of the fractional Brownian motion with the random Hurst exponent. The explicit formulas for probability density function, mean-squared displacement, and autocovariance function of the increments are presented for three generic distributions of the Hurst exponent, namely, two-point, uniform, and beta distributions. The important features of the process studied here are accelerating diffusion and persistence transition, which we demonstrate analytically and numerically.

Pricing foreign equity option under stochastic volatility tempered stable Lévy processes

Considering that financial assets returns exhibit leptokurtosis, asymmetry properties as well as clustering and heteroskedasticity effect, this paper substitutes the logarithm normal jumps in Heston stochastic volatility model by the classical tempered stable (CTS) distribution and normal tempered stable (NTS) distribution to construct stochastic volatility tempered stable Lévy processes (TSSV) model. The TSSV model framework permits infinite activity jump behaviors of return dynamics and time varying volatility consistently observed in financial markets through subordinating tempered stable process to stochastic volatility process, capturing leptokurtosis, fat tailedness and asymmetry features of returns. By employing the analytical characteristic function and fast Fourier transform (FFT) technique, the formula for probability density function (PDF) of TSSV returns is derived, making the analytical formula for foreign equity option (FEO) pricing available. High frequency financial returns data are employed to verify the effectiveness of proposed models in reflecting the stylized facts of financial markets. Numerical analysis is performed to investigate the relationship between the corresponding parameters and the implied volatility of foreign equity option.

Probabilistic Optical and Radar Initial Orbit Determination

Future space surveillance requires dealing with uncertainties directly in the initial orbit determination phase. An approach based on Taylor differential algebra is proposed to both solve the initial orbit determination problem and to map uncertainties from the observables space into the orbital elements space. This is achieved by approximating in Taylor series the general formula for probability density function mapping through nonlinear transformations. In this way, the mapping is obtained in an elegant and general fashion. The proposed approach is applied to both angles-only and two position vectors initial orbit determination for objects in low Earth orbit and geostationary equatorial orbit.

Measuring financial risk and portfolio reversion with time changed tempered stable Lévy processes

Given that underlying assets in financial markets exhibit stylized facts such as leptokurtosis, asymmetry, clustering properties and heteroskedasticity effect, this paper applies the stochastic volatility models driven by tempered stable Lévy processes to construct time changed tempered stable Lévy processes (TSSV) for financial risk measurement and portfolio reversion. The TSSV model framework permits infinite activity jump behaviors of returns dynamics and time varying volatility consistently observed in financial markets by introducing time changing volatility into tempered stable processes which specially refer to normal tempered stable (NTS) distribution as well as classical tempered stable (CTS) distribution, capturing leptokurtosis, fat tailedness and asymmetry features of returns in addition to volatility clustering effect in stochastic volatility. Through employing the analytical characteristic function and fast Fourier transform (FFT) technique, the closed form formulas for probability density function (PDF) of returns, value at risk (VaR) and conditional value at risk (CVaR) can be derived. Finally, in order to forecast extreme events and volatile market, we perform empirical researches on Hangseng index to measure risks and construct portfolio based on risk adjusted reward risk stock selection criteria employing TSSV models, with the stochastic volatility normal tempered stable (NTSSV) model producing superior performances relative to others.

Methodology of Calculation the Terminal Settling Velocity Distribution of Irregular Particles for High Values of the Reynold’s Number/Metodologia Wyliczania Rozkładu Granicznej Prędkości Opadania Ziaren Nieregularnych Dla Wysokich Wartości Liczb Reynoldsa

Abstract Settling velocity of particles, which is the main parameter of jig separation, is affected by physical (density) and the geometrical properties (size and shape) of particles. The authors worked out a calculation algorithm of particles settling velocity distribution for irregular particles assuming that the density of particles, their size and shape constitute independent random variables of fixed distributions. Applying theorems of probability, concerning distributions function of random variables, the authors present general formula of probability density function of settling velocity irregular particles for the turbulent motion. The distributions of settling velocity of irregular particles were calculated utilizing industrial sample. The measurements were executed and the histograms of distributions of volume and dynamic shape coefficient, were drawn. The separation accuracy was measured by the change of process imperfection of irregular particles in relation to spherical ones, resulting from the distribution of particles settling velocity.

METHODOLOGY OF CALCULATION THE TERMINAL SETTLING VELOCITY DISTRIBUTION OF SPHERICAL PARTICLES FOR HIGH VALUES OF THE REYNOLD’S NUMBER

Abstract The particle settling velocity is the feature of separation in such processes as flowing classification and jigging. It characterizes material forwarded to the separation process and belongs to the so-called complex features because it is the function of particle density and size. i.e. the function of two simple features. The affiliation to a given subset is determined by the values of two properties and the distribution of such feature in a sample is the function of distributions of particle density and size. The knowledge about distribution of particle settling velocity in jigging process is as much important factor as knowledge about particle size distribution in screening or particle density distribution in dense media beneficiation. The paper will present a method of determining the distribution of settling velocity in the sample of spherical particles for the turbulent particle motion in which the settling velocity is expressed by the Newton formula. Because it depends on density and size of particle which are random variable of certain distributions, the settling velocity is a random variable. Applying theorems of probability, concerning distributions function of random variables, the authors present general formula of probability density function of settling velocity for the turbulent motion and particularly calculate probability density function for Weibull’s forms of frequency functions of particle size and density. Distribution of settling velocity will calculate numerically and perform in graphical form. The paper presents the simulation of calculation of settling velocity distribution on the basis of real distributions of density and projective diameter of particles assuming that particles are spherical.

Some Explicit Formulae for the Hull and White Stochastic Volatility Model

An explicit formula for the transition probability density function of the Hull and White stochastic volatility model in presence of nonzero correlation between the stochastic differentials of the Wiener processes on the right hand side of the model equations is presented. This formula gives the transition probability density function as a two dimensional integral of an explicitly known integrand. Previously an explicit formula for this probability density function was known only in the case of zero correlation. In the case of nonzero correlation from the formula for the transition probability density function we deduce formulae (expressed by integrals) for the price of European call and put options and closed form formulae (that do not involve integrals) for the moments of the asset price logarithm. These formulae are based on recent results on the Whittaker functions [1] and generalize similar formulae for the SABR and multiscale SABR models [2]. Using the option pricing formulae derived and the least squares method a calibration problem for the Hull and White model is formulated and solved numerically. The calibration problem uses as data a set of option prices. Experiments with real data are presented. The real data studied are those belonging to a time series of the USA S&P 500 index and of the prices of its European call and put options. The quality of the model and of the calibration procedure is established comparing the forecast option prices obtained using the calibrated model with the option prices actually observed in the financial market. The website: http://www.econ.univpm.it/recchioni/finance/w17 contains some auxiliary material including animations and interactive applications that helps the understanding of this paper. More general references to the work of the authors and of their coauthors in mathematical finance are available in the website: http://www.econ.univpm.it/recchioni/finance.

Analyses of turbulence in a wind tunnel by a multifractal theory for probability density functions

The probability density functions (PDFs) for energy dissipation rates, created from time-series data of grid turbulence in a wind tunnel, are analyzed at a high precision by the theoretical formulae for PDFs within multifractal PDF theory which is constructed under the assumption that there are two main elements constituting fully developed turbulence, i.e. coherent and incoherent elements. The tail part of the PDF, representing intermittent coherent motion, is determined by a Tsallis-type PDF for singularity exponents essentially with one parameter with the help of a new scaling relation whose validity is checked for the case of the grid turbulence. For the central part PDF representing both contributions from the coherent motion and the fluctuating incoherent motion surrounding the former, we introduced a trial function specified by three adjustable parameters which amazingly represent scaling behaviors in a much wider area not restricted to the inertial range. From the investigation of the difference between two difference formulae approximating the velocity time derivative, it is revealed that the connection point between the central and tail parts of the PDF extracted by theoretical analyses of PDFs is actually the boundary of the two kinds of instabilities associated respectively with coherent and incoherent elements.

An Explicit Family of Probability Measures for Passive Scalar Diffusion in a Random Flow

We explore the evolution of the probability density function (PDF) for an initially deterministic passive scalar diffusing in the presence of a uni-directional, white-noise Gaussian velocity field. For a spatially Gaussian initial profile we derive an exact spatio-temporal PDF for the scalar field renormalized by its spatial maximum. We use this problem as a test-bed for validating a numerical reconstruction procedure for the PDF via an inverse Laplace transform and orthogonal polynomial expansion. With the full PDF for a single Gaussian initial profile available, the orthogonal polynomial reconstruction procedure is carefully benchmarked, with special attentions to the singularities and the convergence criteria developed from the asymptotic study of the expansion coefficients, to motivate the use of different expansion schemes. Lastly, Monte-Carlo simulations stringently tested by the exact formulas for PDF’s and moments offer complete pictures of the spatio-temporal evolution of the scalar PDF’s for different initial data. Through these analyses, we identify how the random advection smooths the scalar PDF from an initial Dirac mass, to a measure with algebraic singularities at the extrema. Furthermore, the Peclet number is shown to be decisive in establishing the transition in the singularity structure of the PDF, from only one algebraic singularity at unit scalar values (small Peclet), to two algebraic singularities at both unit and zero scalar values (large Peclet).

Application of generalized thermostatistics to fully developed turbulence

We apply the formalism of non-extensive statistical mechanics to fully developed turbulent flows. We obtain analytical formulas for probability density functions of velocity differences depending on distance r/ η and Reynolds number R λ , which are in very good agreement with turbulence experiments. The skewness is taken into account by analysing universal contributions due to higher-order correlations of the chaotic dynamics. We also calculate the moment scaling exponents ζ m , which also agree well with experimental data.

Simple general formula for PDF of angle of arrival in large cell operational environments

To enhance the flexibility of a geometrical single reflection statistical model to describe more accurately different large cell mobile communications operational scenarios, the general case of scatterers distributed within an ellipse is considered. Based on this model, a simple formula for the probability density function (PDF) of the angle of arrival is presented. The PDF for the case of scatterers distributed within a circle is also derived as a special case of the presented PDF.

Assessment of system penalties induced by polarization mode dispersion in a 5 Gb/s optically amplified transoceanic link

The system penalties induced by polarization mode dispersion (PMD) in a 5 Gb/s optically amplified link are analyzed. A relation between differential group delays and system penalties is obtained from numerical simulations and used to derive analytic formulas for probability density functions of system penalties. Cumulated probabilities show that PMDs higher than 20 ps can induce significant fluctuations of the bit-error-rate (BER). Characterization of a 6200 km optically amplified link indicates that the relative contribution of PMD to these fluctuations is not significant when PMD is limited to 0.1 ps//spl radic/km.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>