Parameter Estimation Of AR Model With Observed Noise

Time series analysis occupies a very important position in statistics,and the AR model is an extremely important and widely used statistical model.The AR model is not only a commonly used econometric model in financial time series analysis,but also in many signal processing,the AR model is usually used to model random signals.In practical applications,the observed data is often polluted by noise,not real data.Therefore,the research on the AR model with noise has theoretical significance and application value,and the parameter estimation of the model is an important part of the actual problem modeling.This paper first uses the EM algorithm to give parameter estimates for the first-order AR model with observation noise,and then gives the parameter estimates for the high-order AR model with observation noise by designing the ABC-SMC regression algorithm.When the first-order AR model has observation noise,it is difficult to directly estimate the maximum likelihood function of the parameters.So in this article,we used the EM algorithm to gradually iterate the parameters to give an approximate maximum likelihood estimate.Numerical simulation shows that the EM algorithm is feasible and the estimation effect is better for the first-order AR model with white noise.The ABC algorithm is mainly aimed at the Bayesian approximation estimation of parameters when the model is complex and the likelihood function is difficult to calculate.The SMC algorithm is based on the Monte Carlo method and introduces a sampling method of sequence importance.Because of The more complicated calculation of the marginal distribution of latent variables in high-order AR models with observational noise,the implementation of the EM algorithm is more difficult,we used the ABC-SMC regression algorithm to estimate the parameters in this paper.Firstly,this article makes use of auto-covariance function of the first p-order samples as a statistic to sample the posterior distribution of parameters of AR(p)model with observation noise.Then,the regression model is used to adjust the samples to obtain new samples of the posterior distribution of the parameters.Finally,the article compares it with the ABC algorithm and ABC-SMC algorithm through numerical simulation.The simulation experiments show that ABC-SMC regression algorithm significantly improves the parameter estimation accuracy of the AR(p)model with observation noise.The innovations of this article mainly include the following two aspects: The first point is to use the EM algorithm to solve the first-order maximum likelihood estimation problem of the AR model with measurement noise;The other hand is to adjust the sampling results of ABC-SMC by combining the ABC and SMC algorithms,and then introduce a regression model,which significantly improves the parameter estimation accuracy of the high-order AR model with observation noise.