Reconsturction Of Random Variables/random Fields With Limited Data

In the field of engineering,there are many uncertainties.In most cases,random variables or random fields are used to describe the uncertainties such as the strength or mechanical properties of materials,geometric features of structures or loads.Random field is the expansion of random variable in time or space.In practical problems,people cannot know the real random variables or the probability characteristics of random airports.Only by statistical analysis of observed data can they realize the exact cognition of random variables and random airports.In traditional methods,the probability information covered by the integer moments of the sample is very limited,so a lot of random samples or more high-order moments are needed to accurately describe the random variable.When we want to depict random airports with sample data,Karhunen-Loeve expansion method can reconstruct random airports with low-dimensional data,but it cannot guarantee that the data are statistically independent.For this reason,independent component analysis is also used in the reconstruction of random field.Polynomial Chaos Expansion model is used to approximate each independent component,and approximate Bayesian estimation method is used to estimate the location coefficie nt of Polynomial Chaos Expansion model through a small amount of data.A reconstruction method of random variable based on finite data is proposed.Using the relationship between the Mellin transform and the complex moment,the complex moment description method is realized by the inverse Mellin transform.According to the complex moments of samples,the method can reconstruct the probability density function and characteristic function of any unknown random variable with limited samples.The numerical results show that the probability density and characteristic function reconstructed by complex mom ents agree well with the real solution.A random field expansion method based on independent component analysis is proposed.An approximate Bayesian estimation algorithm which can be used to solve unknown coefficients of Polynomial Chaos Expansion model is presented.Through independent component analysis of random variables in Karhunen-Loeve expansion,statistic independence between random variables and each set of sample data is realized.By using approximate Bayesian estimation algorithm,the posterior distribution of unknown coefficients in Polynomial Chaos Expansion model can be obtained more easily,so as to determine their magnitude.Numerical examples show that independent component analysis can be perfectly applied to random field expansion,and the proposed approximate Bayesian estimation algorithm can accurately estimate unknown coefficients.A nonstandard difference method for solving the dynamic characteristics of fractional-order Chen chaotic systems is proposed.The nonstandard difference method is used to improve the computational efficiency of the solution with the system.The fractional derivative of Riemann-Liouville in fractional Chen chaotic system is replaced.The new type of fractional derivative with variable integral interval is applied to improve the solving efficiency.