Statistical Learning And Inference Of Stochastic Dynamical Systems

With the development of the era of big data,statistical analysis of data has become more and more important.How to effectively model and analyze these data has become our latest challenge.In science and engineering,mathematical models of dynamical systems are widely used.Dynamical systems are divided into deterministic dynamical systems and stochastic dynamical systems.Stochastic dynamical systems take noise into account.There are many kinds of noises,and the commonly used noise models are Brownian motion,fractional Brownian motion and L(?)vy motion.This thesis discusses the statistical learning and inference of several of stochastic dynamical systems by combining the knowledge of data and dynamical systems.After reviewing the research background and relevant theoretical knowledge,the main research work of this thesis is as follows:(1)Detecting the maximum likelihood transition path of the stochastic dynamical system from data.The maximum likelihood transition path is the dynamical feature of the stochastic dynamical system,which is of great significance in practical applications.The Kramers-Moyal formulas are used to establish the relationship between the sample path data and the coefficients of the system,and then the basis function expansion method and the stepwise sparse regression algorithm are used to extract the model parameters from data,and the maximum likelihood transition path is calculated.The numerical results show that our method is effective.We not only can effectively extract the maximum likelihood transition path,but also can obtain the governing equation.(2)Estimating parameters of a class of stochastic differential equations with fractional Brownian motion.Fractional Brownian motion is a non-Markov process,and when the Hurst index H > 1/2,it can be converted into a basic Gaussian martingale through a kernel transformation to obtain a Girsanov-type formula.Under some assumptions of random effects,the parameter estimators of random effects are obtained via using the maximum likelihood estimation method.Then the statistical analysis of the estimators is given,and the theoretical analysis and numerical simulation of the discrete observation are performed.The results show that for the same H,as the amount of data increases,the parameter estimators of random effects are more accurate.When the amount of data is the same,the parameter estimators will be more accurate with the increase of H.(3)Extracting the drift function of stochastic differential equations driven by ?-stable L(?)vy motion.First,the Kullback-Leibler divergence between the transition probabilities of two stochastic differential equations with different drift functions is optimized.By using the Lagrangian multiplier,the variational formula based on the stationary Fokker-Planck equation is constructed.Then combined with the data,the empirical distribution is used to replace the stationary density,and the drift function is estimated non-parametrically.In numerical experiments,the different amounts of data and different ? values are studied.The experimental results show that the estimation result of the drift function depends on both.When the amount of data increases,the estimation result will be more accurate,and when the ? value increases,the estimation result is also more accurate.Finally,the research results of this thesis are summarized and the further research plans are discussed.