The multifractal detrended fluctuation analysis (MF-DFA) method consists of

Transcription

1 Appendix C Methods and Programs Used in Multifractal Analysis C.1 The MF-DFA Method The multifractal detrended fluctuation analysis (MF-DFA) method consists of five steps [36]. The first three steps are essentially identical to the conventional DFA method [40, 41, 42]. We define the normalized logarithmic returns as g t = logp(t+1) logp(t) σ of length N, where P(t) denotes the daily closing prices of the index and σ is the standard deviation of logarithmic returns. Step 1: Calculate the "profile", Y(i) i [g k < g >],i = 1,...,N, (C.1) k=1 where N is the length of series and < g > is the mean ofg t. Step 2: Divide the profile Y(i) into N s int(n/s) non-overlapping segments of equal length s. Since the length of the series is often not a multiple of the considered time scale s, a short part of the series remains, the same procedure is repeated starting from the opposite end. Thereby, 2N s segments are obtained 109

2 110 METHODS AND PROGRAMS USED IN MULTIFRACTAL ANALYSIS altogether. Step 3: Calculate the local trend for each of the2n s segments by a least-square fit of the time-series. Then determine the variance F 2 (s,ν) 1 s s {Y[(ν 1)s+i] y ν (i)} 2 i=1 (C.2) for each segment ν, ν = 1,...,N s and F 2 (s,ν) 1 s s {Y[N (ν N s )s+i] y ν (i)} 2 i=1 (C.3) for ν = N s +1,...,2N s. Here,y ν (i) is the fitting polynomial in segment ν. Step 4: Average over all segments to obtain the q th order fluctuation function, F q (s) { 1 2N s here, the variable q can take any real value except zero [36]. 2N s [F 2 (s,ν)] q/2 } 1/q (C.4) ν=1 Step 5: Determine the scaling behavior of the fluctuation functions by analyzing log-log plot off q (s) versussfor each value ofq. If the time seriesg t are long-range power-law correlated, F q (s) increases for large value of s, as a power-law F q (s) s h(q). (C.5) The family of scaling exponents h(q) can be obtained by observing the slope of the log-log plot off q (s) versus s. h(q) is the generalization of the Hurst exponent H( h(2)). The monofractal time series are characterized by a single exponent over all time scales i.e. h(q) is independent of q, whereas for multifractal time series, h(q) varies with q. The h(q) obtained from the MF-DFA is directly related to the classical multifractal scaling exponent [36] by

3 THE MF-DFA METHOD 111 τ(q) = qh(q) 1. (C.6) Using the spectrum of generalized Hurst exponents h(q), one can calculate the singularity strength α and the singularity spectrum f(α) by using α = h(q)+qh (q) and f(α) = q[α h(q)]+1. (C.7) A Hölder exponent denotes monofractality, while in the multifractal case, the different parts of the structure are characterized by different values ofα,leading to the existence of the spectrum f(α). Computer programs: The Fraclab Toolbox (version 1.0) in Matlab software has been used for the multifractal analysis of time series. This toolbox is downloaded from on May 22, We have used three programs for the multifractal analysis that are explained as follows: (1) load_data.m: Run the program (given on next page) and input time series in the.dat or.txt format and push the button signal. The profile, probability density function, and time frequency spectrum of signal will be displayed. In addition, a file "parameters.mat" is created in the Matlab. The file contains all the concerned variables in the new window as follows: x: the original signal series X: the amplitude of the original signal representing the x axis of Probability Density Function P(x) t: the time axis of Time Frequency Analysis f: the frequency of Time Frequency Analysis db: the intensity of Time Frequency Spectrum. The signal is transformed to file outf.m for usage in latter applications.

4 112 METHODS AND PROGRAMS USED IN MULTIFRACTAL ANALYSIS

5 THE MF-DFA METHOD 113

6 114 METHODS AND PROGRAMS USED IN MULTIFRACTAL ANALYSIS (2) mfaq.m: To compute the scaling function (F q (s) in step 3 of DFA method discussed above) and Hurst exponent (for q = 2) run this program and provide the following parameters: the detrended order m, multifractal order q, and the starting segment size (Smin) and the ending segment box size (Smax) to do the scaling analysis. Now push the process button, the result will be the log-log plot of the fluctuation function F q (s) vs. s. By pressing enter in the window of Hurst exponent provides the Hurst exponent. (3) MFDFA.m: To do the multifractal analysis of given time series by using MF-DFA, run this program and enter values of following parameters: detrended order m, starting point (Qmin), ending point (Qmax), resolution between Qmin and Qmax, Smin, and Smax as discussed above. Now hit the button load and input the file outf.m then press Compute button. This will provide the multifractal scaling exponents. The programs used for the DFA (mfaq.m) and MF-DFA (MFDFA.m) analysis are based on [36, 41, 75]. Both of these programs are given on the next page.

7 THE MF-DFA METHOD 115 /* Beginning of program mfaq.m */

8 116 METHODS AND PROGRAMS USED IN MULTIFRACTAL ANALYSIS

9 THE MF-DFA METHOD 117 /* the end of program mfaq.m */

10 118 METHODS AND PROGRAMS USED IN MULTIFRACTAL ANALYSIS /* MFDFA.m program starts here /

11 THE MF-DFA METHOD 119

12 120 METHODS AND PROGRAMS USED IN MULTIFRACTAL ANALYSIS

13 THE MF-DFA METHOD 121

14 122 METHODS AND PROGRAMS USED IN MULTIFRACTAL ANALYSIS

15 SHUFFLING AND SURROGATING METHODS 123 C.2 Shuffling and Surrogating Methods Computer programs: To investigate the source of multifractality in the given time series we need to shuffle and surrogate the original time series. We have used the "Chaotic System Toolbox" that was developed by Alexandros Leontitsis for the analysis of chaotic systems. This toolbox is downloaded on 17 July, 2005 from The program shuffle.m from this toolbox is based on [72]. To shuffle the original time series, first of all we import the original time series into the Matlab workspace and save it as a vector x. Then use the shuffle program (given on the next page) as follow: c=10; s=shuffle(x,c); here s is the output shuffled time series. To surrogate the original time series, we use the IAAFT.m program from this toolbox to surrogate the original time series (x) which is based on [71]. This program makes "c" Iterative Amplitude Adjusted Fourier Transformed (IAAFT) surrogates of a given time series x. First of all we import the original time series into the Matlab workspace and save it as a vector x. Then use the IAAFT.m program (given on next to next page) as follow: c=10; [s,iter] = IAAFT(x,1,1); here s is the output surrogated time series.

16 124 METHODS AND PROGRAMS USED IN MULTIFRACTAL ANALYSIS /* Program used for shuffling the time series */

17 SHUFFLING AND SURROGATING METHODS 125 /* Beginning of the program to surrogate the series */

18 126 METHODS AND PROGRAMS USED IN MULTIFRACTAL ANALYSIS C.3 Binomial Multifractal Model The Binomial multifractal model (BMFM) is a simple multifractal model which is used to generate the multifractal series. We have used this multifractal model to test the ability of MF-DFA method. We have verified that the MF-DFA method detects the correct multifractal scaling exponents in the given time series. We fit this multifractal model to the financial time series. The multifractal series in the BMFM model is defined by x k = a n(k 1) (1 a) nmax n(k 1), where0.5 < a < 1 is a parameter withk (k = 1,...,N)andN = 2 nmax. n(k) is the number of digits equal to 1 in the binary representation of indexk, for example, n(19) = 3 because 19 corresponds to binary For comparison with the financial data, the binomial multifractal series is generated forn max = 12 with a = 0.6 [79]. Computer programs: We have written a C program mf.c (given on the next page) to generate the multifractal series of any desired length. First of all we take n max = 12 such that N = 2 nmax = 4096 then we calculated n(k) manually. This series for the length k = 1,...,N is taken as an input in the following program:

19 BINOMIAL MULTIFRACTAL MODEL 127 Program for the Binomial Multifractal Model (BMFM)