Research On Expectation-Maximization Algorithm For Posterior Variance Estimation Of Nonlinear Adjustment And Its Applications

In geodetic data processing,the nonlinear adjustment model is an important tool to reveal the characteristics of data and analyze the essence of object.In recent years,with the diversified development of geodetic observation techniques,the observation data are increasingly multi-source,which puts forward higher requirements for the optimization of nonlinear adjustment model and adjustment method.As the link between observation data and function model,adjustment stochastic model plays an important role in the establishment,optimization and parameter estimation of adjustment function model.Previous studies have solved the problem of variance estimation of observations by providing a prior and empirical stochastic model or revising the stochastic model by means of adjustment.However,there are still some problems that need to be further studied and applied.Based on the framework of expectation maximization(EM)algorithm,this article gives full play to its advantages in the field of incomplete data processing,explores and improves its applications on the posterior variance estimation,and aims to obtain more reasonable,accurate function model and parameter estimation accuracy,and to enrich the theory of nonlinear adjustment posterior variance estimation.The specific research of this thesis is as follows:(1)The EM algorithm and its improved algorithm for non-negative variance estimation based on partial errors-in-variables(EIV)model are studied.In view of the existing variance component estimation(VCE)methods are prone to negative variance estimation risks,this article applies EM algorithm to solve the variance component estimation problem of partial EIV model and analyzes the non-negative properties of the variance component estimator.In order to accelerate the efficiency of EM algorithm,a fast algorithm for non-negative variance component estimation is presented,which can be a feasible substitute for the existing variance component estimation methods.Experimental results show that the two proposed methods can obtain more reasonable and accurate parameter results than the conventional least squares(LS)and total least squares(TLS)solutions,and the variance component estimation is consistent with the existing variance component estimation methods,which indicates the effectiveness of the proposed method.When negative variance occurs,the proposed method verifies statistically that the estimation of variance component is close to the edge of its parameter space,and verifies the rationality of the existing estimation methods of non-negative variance component.(2)A variational EM algorithm for posterior variance estimation of nonlinear self-tuning joint probabilistic model is studied.Since the law of normal distribution requires high data accuracy,a small amount of large observation errors or gross errors may have a destructive effect on the estimation results,which makes the variance component estimation method unable to provide a more accurate stochastic model for parameter estimation.This paper introduces the law of t distribution to describe the characteristics of the observation errors,and a nonlinear self-tuning joint probability model with the multi-source observation data is constructed.In order to further inherit the advantage of variance component estimation of EM algorithm,the inherent prior information of parameters and variance components is introduced to construct a random constraint of parameters to stabilize the parametric solution and avoid the risk of negative variance.The variational EM algorithm is applied to solve the parameter estimation problem of nonlinear self-tuning joint probability model with prior parameter distribution.Volcano experiments show that the proposed method is better than the joint inversion results based on the normal distribution probability model.When the observed data have the characteristic of sharp peak and heavy tail,to a great extent,the proposed method can reduce the loss of useful information of data,which plays a positive role in extending the application of non-normal error law in geodetic joint inversion.(3)The estimation method of posterior variance fluctuation estimation for nonlinear coordinate time series with missing values is studied.This article analyzed the noise characteristics of coordinate time series with missing values,introduced autoregressive and generalized autoregressive conditional heteroscedasticity model(AR-GARCH)model for modelling the stochastic model,and combined with the EM algorithm to estimate the missing values in time series.Therefore,the method of nonlinear coordinate time series model under the AR-GARCH noise model is put forward.The numerical example shows that the EM algorithm and AR-GARCH model can be combined to analyze and solve the coordinate time series with missing value,which can obtain more reasonable parameters,the estimation accuracy of missing values and noise amplitude estimation than the traditional variance component estimation methods.In addition,the influence of periodic correlation signal errors on the estimation accuracy is effectively weakened,which has important reference significance to enrich the theory of coordinate time series analysis.(4)The nonlinear velocity field model and the EM algorithm of robust posterior variance fluctuation estimation are studied.For the coordinate time series of Global Navigation Satellite System(GNSS)is influenced by factors such as all kinds of signals and noises,make the posterior variance estimation problem become very complex,and reduce the modelling accuracy of the function model and stochastic model.In addition,mutual coupling and adapting to the situation among the function model,stochastic model and the observations seriously affect the effect of time series analysis and estimation.In this article,a robust AR-GARCH model considering variance inflation is constructed based on Gaussian mixture model.The EM algorithm and the detection,identification and adaptation(DIA)method based on maximum likelihood test were combined to deal with the abnormal observations and unmodeled signals respectively,and the new time series analysis method and processing flow were obtained.The effects of different methods on the estimation of posterior variance fluctuation,the detection of outliers,the parameter solutions and their precision information were compared and analyzed.The experimental analysis shows that abnormal observations in some extent affected the detection accuracy of periods and offsets in dealing with the abnormal observations and multiple time series modelling signals.Although the proposed method can better improve the accuracy of model recognition and parameter estimation accuracy,it is still a challenging task to detect small unknown offset.By applying different methods to the analysis of real time series data,the order of AR model obtained by modeling is different from the previous research results,which reflects that the conventional AR(1)model is a suboptimal model to describe the noise characteristics of time series.Finally,the test results and the difference of asymptotic root-mean-square error(RMSE)between the proposed method and other methods are calculated,which laterally verifies the existence of abnormal observations in real time series.