Research Of Randomized And Multi-scale Adaptive Multifractal Analysis Method

Nature is composed of many complex systems,and the study of time series can help us to describe these systems indirectly.Therefore,the study of the multifractal characteristics of real-time series,including genome sequences,financial time series,and human physiological time series,has profound practical value and great significance.With the development of fractal theory,many time series multifractal analysis methods have been proposed,they have their own advantages.This paper proposes a randomized multifractal time-weighted detrend fluctuation analysis(RMFTWDFA)method to study the multifractal characteristics of long-term series.This algorithm brings forth new ideas to the application of random thoughts while dividing different sections.To test the performance of RMFTWDFA,this article applies the method to the simulation sequences and the example sequences.The consistency test shows that there is no significant variation between theh(q)estimated by the random method and the original method,and the running time of the random method is increased by more than10 times.Meanwhile,the results of genome sequences can explain the genetic relationship between species,which shows that RMFTWDFA is highly efficient in analyzing long-term series and can achieve high accuracy while comparing to other methods.On the other hand,in order to analyze the multifractal characteristics of shorter time series under the multi-scale framework more accurately,this article introduces the adaptive fractal analysis(AFA)method into the multi-scale multi-fractal analysis(MMA)method,that is,multi-scale adaptive multi-fractal analysis(MAMFA)method.The results of the simulation sequence show that the MAMFA method has higher accuracy in analyzing shorter time series.Taking the Chinese-US stock index in the financial time series and the R-R interval in the human physiological time series as examples,it shows that the multifractal characteristics of the stock sequences are related to the selection of the scale range.Comparing with the US stock indexes,the Chinese stock indexes have more obvious multifractal features,the Hurst surfaces of the US stock indexes are closer to the plane of h = 1,and have smaller deviation from the EMH.From this,we can see the risk and the development in the stock market.For the R-R interval sequences,we find that,compared with healthy subjects,subjects with abnormal heart rates have obvious shape changes in three regions of the Hurst surface,which can promote the efficiency in distinguishing patients through observing their changes.The two multifractal analysis methods introduced in this article have their own advantages for studying long time series and shorter time series.Both the analysis of simulation series and example series show good results,it also provides a reference for applying time series in biological genes,financial markets,and clinical medicine.