Description Of Random Variables Research Articles

POLYNOMIAL ESTIMATION OF DATA MODEL PARAMETERS WITH NEGATIVE KURTOSIS

Context. The paper focuses on the problem of estimating the center of distribution of the random component of experimental data for density models with a negative kurtosis.
 Objective. The goal of this research is to develop methods to improve the efficiency of polynomial estimation of parameters of experimental data with a negative kurtosis coefficient.
 Method. The study applies a relatively new approach to obtaining estimates for the center of the probability distribution from the results of experimental data with a stochastic component. This approach is based on polynomial estimation methods that rely on the mathematical apparatus of Kunchenko's stochastic polynomials and the description of random variables by higher-order statistics (moments or cumulants). A number of probability density distributions with a negative kurtosis coefficient are used as models of the random component.
 As a measure of efficiency, the ratio of variance of the estimates for the center of the distribution found using polynomial and classical methods based on the parameter of amount of information obtained is used.
 The relative accuracy of polynomial estimates in comparison with the estimates of the mean, median and quantile estimates (center of curvature) is researched using the Monte Carlo method for multiple tests.
 Results. Polynomial methods for estimating the distribution center parameter for data models of probability distribution density with a negative kurtosis coefficient have been constructed.
 Conclusions. The research carried out in this paper confirms the potentially high efficiency of polynomial estimates of the coordinates of the center of the experimental data, which are adequately described by model distributions with a negative kurtosis. Statistical modeling has confirmed the effectiveness of the obtained estimates in comparison with the known non-parametric estimates based on the statistics of the mean, median, and quantile, even with small sample sizes.

Polynomial parameter estimation of exponential power distribution data

The paper proposes an original approach to obtain the results of multiple measurements at random errors, which are described by the exponential power (generalized Gaussian) distribution model. The approach is based on the polynomial maximization method (PMM), which is based on Kunchchenko’s mathematical apparatus using stochastic polynomials and a partial description of random variables of high-order statistics (moments or cumulants). The theoretical foundations of PMM are presented in relation to finding the estimates of the informative parameter from an equally distributed random variables sample. There are analytical expressions for finding polynomial estimations. It is shown that r \leqslant 2, then polynomial estimates degenerate in linear arithmetic mean estimates . If the polynomial degree r = 3 then the relative accuracy of polynomial estimations increases. The features of numerical procedures (Newton-Raphson method) for finding the stochastic equation roots are considered. Obtained analytical that describe the dispersion of the PMM estimates for an asymptotic case (for n → ∞). It is shown that the theoretical value of reduction coefficient variance of PMM estimates (in comparison with the linear mean estimates) depends on the magnitude of the random error cumulative coefficients of the 4th and 6th order. Through multiple statistical tests (Monte Carlo method) carried out a comparative accuracy analysis of polynomial estimates with known non-parametric estimates (median, mid-range and mean). It is shown that with increasing size of sample the difference between theoretical and experimental data decreases. The efficiency areas for each method are constructed, depending on the exponential power distribution parameter and sample size. It is shown that the accuracy of the proposed approach can significantly (more than twofold) exceed the classical nonparametric estimation.

Reliability Assessment of Steel-Aluminium Lattice Tower

In the subject of the present study a probabilistic approach to the analysis of steelaluminium lattice tower was used. Structural design parameters are defined as the deterministic values and random variables. Random variables are not correlated. The criterion for structural failure is expressed the limits of functions referring to the serviceability limit state. The description of the limit state of structure implicit forms of the random variables function was used. The study presents a combination of the reliability analysis program with the MES3D external FEM program. The NUMPRESS software, created at the IFTR PAS, was used in the reliability analysis. The Hasofer-Lind reliability index, determined using an iterative procedure of Rackwitz-Fiessler, was used as a reliability measure. The values of reliability index for different cases of the vector of random variables, that is, different descriptions of mathematical model of the structure, were determined. The effect of assumed probability distribution of individual random variables on the value of the reliability index was determined. In the description of random variables, the different types of probability distribution were used and the values of the reliability index for the normal distribution and the distribution chosen according to the kind of a variable were compared. The primary research method is the FORM method. In order to verify the correctness of the calculation Monte Carlo and Importance Sampling methods are used. The relative error of the reliability index was calculated taking the simulation Monte Carlo method as a reference. The effectiveness of the primary research method was performed by comparing the number of calls of the limit state, which is connected with the calculation time. The sensitivity of reliability index to the random variables was defined.

Impact of the Mathematical Model Description on the Assessment of the Reliability of Structural Elements

The study presents a probabilistic approach to the problems of static analysis of structural elements. A steel and reinforced concrete elements were analysed. Structural design parameters were defined as deterministic values and random variables. The latter were not correlated. The criterion of structural failure is expressed by limit functions related to the ultimate limit state. In the performed analyses explicit form of the random variables function were used. The Hasofer-Lind index was used as a reliability measure. In the description of random variables were used the different types of probability distribution appropriate to the nature of the variable. Sensitivity of reliability index to the random variables was defined. If the reliability index sensitivity due to the random variable Xi is low when compared with other variables, it can be stated that the impact of this variable on failure probability is small. Therefore, in successive computations it can be treated as a deterministic parameter. Sensitivity analysis leads to simplify the description of the mathematical model, determine the new limit functions and values of the Hasofer-Lind reliability index. Besides the effect of the assumed level of the variation coefficient of selected random variables on value of the reliability index was determined. The primary research method is the FORM method. In order to verify the correctness of the calculation SORM, Monte Carlo and Importance Sampling methods were used. In the examples of reliability analysis the NUMPRESS program was used. In the considered issues the time was not taken into account explicitly.

The Comparison of Methods of Assessing the Reliability of the Steel Frame

The study presents a probabilistic approach to the problems of static analysis of a steel building. Structural design parameters were defined as deterministic values and random variables. The latter were not correlated. The criterion of structural failure is expressed by limit functions related to the ultimate and serviceability limit state. The description of limit functions by the Mathematica program was generated. The Hasofer-Lind index was used as a reliability measure. In the description of random variables were used the normal distribution and, for comparison, different types of probability distribution appropriate to the nature of the variable. Sensitivity of reliability index to the random variables was defined. If the reliability index sensitivity due to the random variable Xi is low when compared with other variables, it can be stated that the impact of this variable on failure probability is small. Therefore, in successive computations it can be treated as a deterministic parameter. Sensitivity analysis leads to simplify the description of the mathematical model, determine the new limit functions and values of the Hasofer-Lind reliability index. The primary research method is the FORM method. In order to verify the correctness of the calculation SORM, Monte Carlo and Importance Sampling methods were used. In the examples of reliability analysis the STAND program was used.