Time Series Analysis Based On Diffusion Entropy Theory

In reality, there exist a large number of short time series. Although some are verylong, in the formation process of time series often occur mutations of the dynamicbehavior of complex systems. For a better understanding of local characteristics anddetection of mutations in early signal, we need divide the long series into shortersegments. Short time series analysis, which means extract the intrinsic characteristicsfrom the finite-length series, is a fundamental and urgent task for time series analysis.Most of the present methods are based on the statistics structure of series. However,because of limited length of the series will bring system errors in the process ofstatistics. Therefore, this present paper develops new methods to correct the errors.A successful solution is balanced estimation of diffusion entropy (BEDE), caneffectively modified statistical error (variance) and systematic error (bias) induced byfinite lengths for short time series. In this paper, the method has been applied toevaluate the scale behavior in physiological signals and recognition of coding regionsin DNA sequences. By means of the concept of balanced estimation of diffusionentropy, we evaluate the reliable scale invariance embedded in different sleep stagesand stride records. Segments corresponding to waking, light sleep, rapid eyemovement (REM) sleep, and deep sleep stages are extracted from long-termelectroencephalogram signals. For each stage the scaling exponent value is distributedover a considerably wide range, which tell us that the scaling behavior is subject andcycle dependent. For the stride series, the evolutions of local scaling behavior showthat the physiological states change abruptly and may lose rich information, althoughin the experiments great efforts have been made to keep conditions unchanged. Inaddition, the existence of trends may lead to an unreasonable high value of the scalingexponent and consequent conclusions.A great deal of research show that the elements in DNA coding regions aredistributed in a quasi-random way and random distribution pattern, while the elementsin DNA non-coding regions present typical self-similar structure and long-rangecorrelation. Considering the DNA sequences are very short, these statistical featurescould not be extracted well from identify coding regions and non-coding regions. Themethod of balanced estimation of diffusion entropy is effectively introduced to solvethe problem and used for coding regions identification. In addition, based on fractal market theory, the article also establish portfoliooptimization model with fractal dimension as risk measurement standard. Andcalculate the optimal investment portfolio in risk controllable range by Monte Carlo.