Selection Of Order Determination Method And Weak Signal Detection Of Autoregressive Model

Time series analysis plays a very important role in mathematical statistics.The Autoregressive model is the most widely studied model in linear time series analysis and its application in many scientific and engineering fields is very mature,so the use of AR models is of great practical significance for establishing models with existing data.In the practical application of the autoregressive model,it is necessary to determine the number of necessary parameters to describe the behavior of the model,because it is directly affecting the availability of the model and the accuracy of use.Although there are multiple model order determination methods,these methods have their own advantages and disadvantages,and there is no universal formula for the order determination problem.An important but unrecognized problem is how to use the existing data to select the most appropriate one of the ordering methods.Considering the influence of noise standard deviation,sequence length,and characteristic root,we introduce a method for estimating the Signal-to-Noise Ratio(SNR)of a model for low order AR models and uses it as the criterion to evaluate the accuracy of Akaike Information Criterion,Bayesian Information Criterion and Final Prediction Error Criterion.The effects of the model's characteristic root,sequence length,and noise on the order accuracy and SNR are studied through design numerical experiments.The results show that the order accuracy reaches the maximum under the condition of the maximum characteristic root when the characteristic root of the model satisfies|?1|=|?2|=…?|?p|=|?max|.The accuracy is positively associated with the length of the sequence and the distance between the characteristic root and the center of the unit circle,and not with the standard deviation of the noise.Finally,based on these experimental phenomena,we propose a scheme to select the order determination method by using the SNR of the reference model,which provides a new perspective for comparing the different order determination methods.After determining the order of the AR model,the next step is to estimate the parameters.Under the condition that the system is not disturbed by strong noise,many parameter estimation methods are effective.When the parameters of the model are small,the signal strength of the system is weak,even if it is affected by the noise with lower intensity,it will cause some errors in some parameter estimation methods.Therefore,this paper first uses the characteristic value of the stationary AR model as an indicator to judge whether the system behaves as a weak signal feature and further introduces the Non-Stationarity(NS)measurement of the time series to study the parameter estimation problem of the AR model under weak signal conditions.In this paper,the difference between the non-stationarity measure of the sample sequence and the estimated sequence is taken as the criterion to distinguish the two similar AR weak signals.After the initial estimation of the parameters of the model,the parameter is corrected by ?NS to make it converge to a smaller neighborhood of the real parameter value.Especially for the AR(2)model,a distance metric for the AR process is introduced to improve the parameter correction performance.Numerical experiment results show that the accuracy of this method is superior to other parameter estimation methods under Gaussian disturbance.