Analysis of the ratio of the mean square deviations of the kernel probability density estimation in the conditions of independent and dependent random variables

2021 ◽  
pp. 9-14
Author(s):  
Aleksandr V. Lapko ◽  
Vasiliy A. Lapko
Keyword(s):  
Probability Density ◽  
Statistical Data ◽  
Random Variables ◽  
Mean Square ◽  
Two Dimensional ◽  
Approximation Properties ◽  
Dependent Random Variables ◽  
Probability Densities ◽  
Distribution Laws ◽  
The Mean

The influence on the approximation properties of a nonparametric probability density estimate of Rosenblatt-Parzen type of the information on the dependence of random variables is determined. The ratio of the asymptotic expressions of the mean square deviations of independent and dependent random variables is obtained. This relation for a two-dimensional random variable is considered as a quantitative assessment of the influence of information about their dependence on the approximation properties of the kernel probability density estimate. The established ratio is determined by the kind of probability density and the volumes of the initial statistical data that are used in estimating the probability densities of dependent and independent random variables. The general results obtained are considered in detail for two-dimensional linearly dependent random variables with normal distribution laws. The functional dependence of the ratio of the mean square deviations of the independent and dependent two-dimensional random variables on the correlation coefficient is determined. The dependence of the considered ratio on the volume of statistical data is analyzed. A method for estimating the functional of the second derivatives of two-dimensional random variables with normal distribution laws is developed. The results obtained are the basis for the development of modifications of “fast” procedures for optimizing kernel estimates of probability densities in conditions of large samples.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

scholarly journals A search for a stochastic archetype of quantum probability

2021 ◽  
Author(s):  
Tim C Jenkins
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Probability Density ◽  
Random Variables ◽  
Quantum Probability ◽  
Interference Term ◽  
Mathematical Foundation ◽  
Quantum Process ◽  
Hidden Variable ◽  
Probability Densities

Abstract Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory. In recent years, Man’ko and coauthors have successfully reconciled quantum and classic probability using a symplectic tomographic model. Nevertheless, there remains an unexplained coincidence in quantum mechanics, namely, that mathematically, the interference term in the squared amplitude of superposed wavefunctions gives the squared amplitude the form of a variance of a sum of correlated random variables, and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classic probability directly. The properties that would need to be satisfied for this to be the case are identified, and a generic hidden variable that satisfies them is found that would be present everywhere, transforming into a process-specific variable wherever a quantum process is active. Uncovering this variable confirms the possibility that it could be the stochastic archetype of quantum probability.

scholarly journals A search for a stochastic archetype of quantum probability

2021 ◽  
Author(s):  
Tim C Jenkins
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Probability Density ◽  
Random Variables ◽  
Quantum Probability ◽  
Interference Term ◽  
Mathematical Foundation ◽  
Classical Probability ◽  
Quantum Process ◽  
Probability Densities

Abstract Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory. In recent years Man’ko and co-authors have successfully reconciled quantum and classical probability using a symplectic tomographic model. Nevertheless, there remains an unexplained coincidence in quantum mechanics, namely that mathematically the interference term in the squared amplitude of superposed wavefunctions has the form of a variance of a sum of correlated random variables and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classical probability directly. The properties that would need to be satisfied for this to be the case are identified, and a generic variable that satisfies them is found that would be present everywhere, transforming into a process-specific variable wherever a quantum process is active. This hidden generic variable appears to be such an archetype.

An improved energy envelope stochastic averaging method and its application to a nonlinear oscillator

2016 ◽  
Vol 23 (1) ◽  
pp. 119-130 ◽  
Author(s):  
Yaping Zhao
Keyword(s):  
Steady State ◽  
Probability Density Function ◽  
Probability Density ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
System Response ◽  
Mean Square ◽  
Fpk Equation ◽  
The Mean

An improved stochastic averaging method of the energy envelope is proposed, whose application sphere is extensive and whose implementation is convenient. An oscillating system with both nonlinear damping and stiffness is taken into account. Its averaged Fokker-Planck-Kolmogorov (FPK) equation in respect of the transition probability density function of the energy envelope is deduced by virtue of the method mentioned above. Under the initial and boundary conditions, the joint probability density function as to the displacement and velocity of the system is worked out in closed form after solving the averaged FPK equation by right of a technique based on the integral transformation. With the aid of the special functions, the transient solutions of the probabilistic characteristics of the system response are further derived analytically, including the probability density functions and the mean square values. A simple approach to generate the ideal white noise is drastically ameliorated in order to produce the stationary wide-band stochastic external excitation for the Monte Carlo simulating investigation of the nonlinear system. Both the theoretical solution and the numerical solution of the probabilistic properties of the system response are obtained, which are extremely coincident with each other. The numerical simulation and the theoretical computation all show that the time factor has a certain influence on the probability characteristics of the response. For example, the probabilistic distribution of the displacement tends to be scattered and the mean square displacement trends toward its steady-state value as time goes by. Of course the transient process to reach the steady-state value will obviously be shorter if the damping of the system is greater.

Stochastic Response and Stability Analysis of Two-Point Mooring System

2011 ◽  
Author(s):  
A. K. Banik ◽  
T. K. Datta
Keyword(s):  
Stability Analysis ◽  
Probability Density Function ◽  
Probability Density ◽  
Mooring System ◽  
Mean Square ◽  
Stochastic Response ◽  
Sea State ◽  
Mean Square Response ◽  
Linear Damping ◽  
The Mean

The stochastic response and stability of a two-point mooring system are investigated for random sea state represented by the P-M sea spectrum. The two point mooring system is modeled as a SDOF system having only stiffness nonlinearity; drag nonlinearity is represented by an equivalent linear damping. Since no parametric excitation exists and only the linear damping is assumed to be present in the system, only a local stability analysis is sufficient for the system. This is performed using a perturbation technique and the Infante’s method. The analysis requires the mean square response of the system, which may be obtained in various ways. In the present study, the method using van-der-Pol transformation and F-P-K equation is used to obtain the probability density function of the response under the random wave forces. From the moment of the probability density function, the mean square response is obtained. Stability of the system is represented by an inequality condition expressed as a function of some important parameters. A two point mooring system is analysed as an illustrative example for a water depth of 141.5 m and a sea state represented by PM spectrum with 16 m significant height. It is shown that for certain combinations of parameter values, stability of two point mooring system may not be achieved.

scholarly journals Characterisation of vortex shedding on a hydrofoil using PIV measurements

2021 ◽  
Vol 1 (1) ◽  
Author(s):  
Hervé Bonnard ◽  
Ludovic Chatellier ◽  
Laurent David
Keyword(s):  
Vortex Shedding ◽  
Particle Image ◽  
Statistical Data ◽  
Two Dimensional ◽  
Mean Velocity ◽  
Piv Measurements ◽  
Image Velocimetry ◽  
The Mean ◽  
Flow Configurations

An experimental study of vortex shedding on a hydrofoil Eppler 817 was conducted using two-dimensional two components Particle Image Velocimetry. This foil section’s characteristics are adapted for naval applications but sparsely documented. The characterization of the flow modes was realized based on statistical data such as the mean velocity field and the standard deviation of the vertical velocities. The data were acquired at very low Reynolds number which are not often covered for such hydrofoil and at four angles of attack ranging from 2◦ to 30◦. A map of different characteristic flow modes was made for this space of parameters and was used to identify flow configurations exhibiting particular dynamics.

Estimation of the integral of the square of derivatives of symmetric probability densities of one-dimensional random variables

Metrologiya ◽  
2020 ◽  
pp. 15-27
Author(s):  
Aleksandr V. Lapko ◽  
Vasiliy A. Lapko
Keyword(s):  
Standard Deviation ◽  
Probability Density ◽  
Random Variables ◽  
Random Variable ◽  
Density Estimates ◽  
One Dimensional ◽  
Probability Densities ◽  
The One ◽  
Derivatives Of ◽  
Selection Of

When substantiating the method of fast selection of the bandwidth of kernel probability density estimates, a constant was found that is a functional of the second density derivative. In this paper, the obtained result is generalized to derivatives of symmetric probability densities of different orders. The functional dependences of the constants under study on the coeffi cient of antikurtosis of a random variable are established. The regularities peculiar to them are investigated. Based on the results obtained, a method for estimating functionals from derived probability densities has been developed, which involves the following actions. In the original sample estimated standard deviation of the one-dimensional random variables and the coeffi cient of antikurtosis. Using the reconstructed functional dependences on the antikurtosis coeffi cient, the constants are estimated, which are functionals of the derivatives of the probability density. With known estimates of the standard deviation of the investigated random variable and the considered constant, the values of the functional from the derivative of the probability density of the selected order are calculated. The obtained results are confi rmed by the analysis of the data of computational experiments. It is established that with increasing order of the derivative, the values of the estimates of the studied functionals increase. This fact is explained by the complication of the integrand function in the considered functionals. The proposed method provides objective results for the fi rst three derivatives of the probability density of a random variable. The obtained conclusions are confi rmed by the results of the confi dence estimation of the investigated functionals.

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