Estimating Parameters In Phase Space Reconstruction Of Noisy Chaotic Time Series

The internet has become an important infrastructure. Prediction of network attackfrequencies and network traffic has great significance and practical values. The result ofchaotic time series theory is more precise than traditional methods, for the internet is anonlinear system. Phase space reconstruction is the first step in chaotic time seriesanalysis.People always use delay coordinates method to reconstruct phase space, which isbased on Takens theory. Two parameters have to be determined in delay coordinatesmethod, delay time τ and embedding dimension m. There are several methods toestimate these two parameters, such as multiple autocorrelation function method,mutual information method, average diplacement method and C-C method. The resultof these methods is accurate to noise-free chaotic time series, but time series measuredin practical application is corrupted by noise. The noise of chaotic dynamic systemmakes the result of these methods unaccureta by masking dynamic characteristics of thesystem.First, these methods are analyzed in this paper. The performance of some methodsis not roubost, for example, the result of multiple autocorrelation function method is toolarge for high dimension chaotic system. The result of some methods is not appropriatefor all chaotic systems, such as average diplacement method. Some methods haveenormous computing, such as Cao methd and C-C method. And then the reason is alsodiscussed. Numerical results show that the conclusion is correct.Based on the study above, the method, which is based on detecting correlation ofsystem to estimate delay time, is roubust to noise, indenpent on the length of data andeasy to calculate. But the result of this method is too large. This is because it calculatesthe linear correlation of system, but chaotic system is nonlinear so that this method cannot calculate the correlation of system accurately. This paper presents a new method toestimate delay time by calculating nonlinear correlation of chaotic system, callednonlinear multiple autocorrelation function method. The aim of the method is to find anew method which is accurate, roubust to noise and independent on the length of data.In order to calculate nonlinear correlation of chaotic system acuretaly, nonlinearmultiple autocorrelation function method uses a high order multiple autocorrelationfunction to detect nonlinear correlation, which constructes a multiple polynomial space whose dimension is equal to phase space’s in algebraic geometry. In order to estimatethe optimal delay time of chaotic system, the method finds the first local minimumvalue of nonlinear multiple autocorrelation function as delay time. The numerical resultsshow that, the result of low dimension system by this method is equal to the best delaytime when the noise level is up to80%; the result of high dimension system isappropriate when the noise level is up to60%. In addition, the result of100data pointsis equal to2000data points’ for low dimension system; the results are same when noiselevel being below20%and very close to each other when noise level being above20%for high-dimensional chaotic systems.The time complexity of nonlinear multiple autocorrelation function is low.Experimental results show that as the embedding dimension increases, delay timeestimated by the function owns supremum and infimum. Based on this point, this paperalso presents a new method to estimate delay time and embedding dimension. Thismethod calculates the optimal delay time for every embedding dimension, and thenselects the value corresponding to the first point of the curve tending to be stable asparameters. The numerical results show that delay time and embedding dimension bythe method are appropriate.