The Application Of Wavelets In The Fractal Singularity Spectrum Estimation And Their Numerical Comparison

We tested three techniques: Multifractal Detrended fluctuation analysis (MFDFA), Wavelet-Based Detrended fluctuation analysis (WB-DFA) and wavelet transform modulus maxima(WTMM), which are used to estimate the singularity spectrum of multifractal signals. The experimental data used in this paper include different types of fractal signals: Fractional Brownian Motion (FBM), the Log-poisson and Log-normal type signals which are generated by the Bezi Model. The main purpose of this paper is to investigate the application of wavelets in fractal analysis and the different performance of those methods on different experimental data, and according the results obtained, we can get some reliable basis when we apply those methods in practical problems.The results from our experiments tell us: the stability and accuracy of MFDFA is satisfying; As a mature techniques, WTMM has its own advantage while it is relatively demanding on data; the newly invented method WT3-DFA is very dependent on data and its stability and accuracy is not so good.It still needs more discussion on its applicability.In addition, in order to investigate the different aspects of those methods in real life problems, we choose the the experimental data from stock markets. MFDFA is used to study the multifracl property of the structure of stock market and the difference between mature markets and emerging markets; The local Holder exponent, obtained from WTMM, can be used to as an index to predict the crash of stock market.