Reconstruction Method Based On Variable Expansion And Least Squares Approximations With Measurement Noise

Complex network exists widely in our life,and the structure of detection network is of great significance in the research of complex network.However,the structure of the network can not be detected directly or the cost of detection is huge.We can detect the structure of the network through the measured data,which is called network reconstruction or system reconstruction.Network reconfiguration has become one of the important topics in nonlinear science and complex network research.At present,a lot of reconstruction methods have been proposed for the system with only system noise.However,due to the influence of measurement environment and instrument accuracy,sampling data often contains measurement noise.In the presence of measurement noise,many reconstruction methods are basically invalid or have a large reconstruction error.Reconstruction method based on variable expansion and least squares approximations(VELSA)was recently proposed by Shi Rundong et al.This method has a good reconstruction effect under nonlinear,strong system noise and low sampling frequency conditions.Under the condition of measurement noise,the analysis of the reconstruction method is still unknown.In this paper,the VELSA reconstruction method is analyzed in the case of noise measurement,and the following progress has been made:(1)Under the condition of measurement noise,we analyze the VELSA method and the explore influence of smoothing data processing on reducing measuring noise.We analyze the influence of measurement noise on VELSA reconstruction theoretically.Theoretical analysis shows that reconstruction error is related to measurement noise and sampling time interval.When the sampling interval is small,the error caused by measurement noise should be dominant.The higher the measurement noise intensity is,the larger the reconstruction error will be.The reconstruction error is inversely proportional to the sampling interval,the smaller the sampling interval is,the larger the reconstruction error will be.When the sampling interval is large,the error caused by the measurement noise can be ignored,and the error caused by the least squares approximations should be dominant.At this time,the error is proportional to the square approximation of the sampling interval.We analyze the influence of smoothing method on VELSA reconstruction theoretically.Theoretical analysis shows that the reconstruction error can be reduced about 1/w times by smoothing the data,where W is the size of the smoothing window.When the smoothing window W is large enough,the influence of measurement error can be effectively reduced.Based on Lorenz system and Fitzhough Nagumo neural network,the influence of sampling time interval,measurement noise intensity,smoothing window size and other factors on the reconstruction effect is studied.The experimental results verify the correctness of the theoretical analysis.High-order correlation calculations(HOCC)is a reconstruction method that uses differential processing data.The influence of measurement noise on HOCC and VELSA methods is studied by numerical method.When the sampling time interval is small,the measurement noise makes the reconstruction effect of the two methods worse,the noise resistance of VELSA method is slightly higher than that of HOCC method,and the smoothing method has a significant improvement on both VELSA and HOCC reconstruction methods.(2)We extend VELSA method to nonlinear discrete systems.The reconstruction results of VELSA method in nonlinear discrete system are analyzed theoretically,and the influence of measurement noise on reconstruction results is analyzed preliminarily.We give the relationship between reconstruction error and measurement noise intensity.Logistic map is used for numerical simulation,and the simulation results verify the correctness of the theoretical analysis.