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Multifractality of stock markets based on cumulative distribution function and multiscale multifractal analysis
Physica A 447 (2016) 527–534
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Physica A journal homepage: www.elsevier.com/locate/physa
Multifractality of stock markets based on cumulative distribution function and multiscale multifractal analysis Aijing Lin ∗ , Pengjian Shang School of Science, Beijing Jiaotong University, Beijing 100044, PR China
highlights • We analyze Chinese and US stock markets during the period of 1992 to 2012. • The cumulative distribution function of modified Hurst surface is calculated. • Stable structures of multifractal scaling are found by performing CDF-MMA technique.
article
info
Article history: Received 28 July 2015 Received in revised form 29 October 2015 Available online 29 December 2015 Keywords: CDF Multiscale MF-DFA MMA Stock market
abstract Considering the diverse application of multifractal techniques in natural scientific disciplines, this work underscores the versatility of multiscale multifractal detrended fluctuation analysis (MMA) method to investigate artificial and real-world data sets. The modified MMA method based on cumulative distribution function is proposed with the objective of quantifying the scaling exponent and multifractality of nonstationary time series. It is demonstrated that our approach can provide a more stable and faithful description of multifractal properties in comprehensive range rather than fixing the window length and slide length. Our analyzes based on CDF-MMA method reveal significant differences in the multifractal characteristics in the temporal dynamics between US and Chinese stock markets, suggesting that these two stock markets might be regulated by very different mechanism. The CDF-MMA method is important for evidencing the stable and fine structure of multiscale and multifractal scaling behaviors and can be useful to deepen and broaden our understanding of scaling exponents and multifractal characteristics. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Finance is an active research area, and a number of studies have used statistical mechanics to investigate the dynamics of financial markets due to the complex structures. Application of the idea gained from fractal theory to study the dynamical properties has been one of the most exciting areas of research in recent times [1–6]. Various complex systems have considered to be well represented by multifractality [7–21]. In previous study, it is widely accepted that financial market illustrates strong sign of complexity, power-law and multifractality [22–28]. The characteristics of the correlation in complex system can be investigated by extracting its scaling exponents. While some processes (monofractal) can be quantified by a single scaling exponent, others (multifractal) require the spectrum of exponents to characterize. The detrended fluctuation analysis (DFA) invented by Peng et al. [12] has been established as an important tool for the determination of fractal scaling properties and detection of long-range power-law correlations in signals. Multifractal
∗
Corresponding author. E-mail address: [email protected] (A. Lin).
http://dx.doi.org/10.1016/j.physa.2015.12.012 0378-4371/© 2015 Elsevier B.V. All rights reserved.
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detrended fluctuation analysis (MF-DFA) method [29] is an universal tool to investigate multifractality, which is a multifractal generalization of the DFA method. In the previous studies, the MF-DFA exponents were estimated as the slope of a MF-DFA function plotted versus the resolution s on a log–log scale. The scaling range for determining the scaling exponents cannot be defined precisely, usually determined by visual inspection of plot. However, it is not adequate to describe the dynamical behaviors of time series by using single or two scaling exponents. Then Gierałtowski et al. [30] generated a new method called multiscale multifractal detrended fluctuation analysis (MMA) for solving the above problem. This new technology allows us to investigate not only the multifractal properties but also dependence of these properties on the time scale. MMA has successfully been applied to diverse field such as heart rate dynamics, economics and traffic systems [31–33]. Hence, previous studies on multiscale DFA inspire us to propose cumulative distribution function (CDF) statistics of Hurst surface h(q, s) based on MMA method, which is called CDF-MMA method in our paper. In order to avoid the individual blind spot, the ensemble group averages of CDF values corresponding to individuals are calculated to explore the scaling behaviors. The present study shows more stable and robust results, allowing the assessment of multifractality and permitting individual discrimination. The organization of this paper is as follows. First, the artificial and real-word data sets used in our work are given in Section 2. Section 3 presents methods employed in study. Section 4 is devoted to show the results by employing our modified CDF-MMA approach. Finally, the important conclusions drawn from this study are provided in Section 5. 2. Data 2.1. Artificial data In order to test validity of the proposed method, we present results for several artificial series: random series, monofractal noise and binomial multifractal series. We now turn to the artificial processes. (1) Monofractal noise Many empirical data sets are characterized by long-range power-law auto-correlations. In the present study we generate artificial monofractal noise data sets of N = 215 samples by using modified Fourier filtering method [34] with scaling exponents α = 0.2, α = 0.4, α = 0.5 and α = 0.7 respectively. The artificial monofractal noises are plotted in Fig. 1. (2) Binomial multifractal model The artificial multifractal data sets considered in this paper are generated by applying binomial multifractal model [29]. In the binomial multifractal model, a series of N = 2nmax numbers k with k = 1, 2, . . . , N is defined by X (k) = β n(k−1) (1 − β)nmax −n(k−1)
(1)
where 0.5 < β < 1 is a parameter and n(k) is the number of digits equal to 1 in the binary representation of the index k, e.g. n(13) = 3, since 13 corresponds to binary 1101. In our study, we consider binomial multifractal series with parameters β = [0.54, 0.56, 0.58, 0.6] and nmax = 15. The time series generated by the binomial multifractal models with these parameters are shown in Fig. 2. 2.2. Real-world data To show how the method applies to real data, we study different real-world financial series, which we consider the outputs of the complex systems-daily closing values of six stock indices. The data sets are obtained from Yahoo Finance covering 5533 days from May 12, 1992 to May 8, 2012. The original stock market indices are shown in Fig. 3. For simplicity, DJI, NYSE, SP500, HSI, ShangZheng and ShenCheng are used for representing six stock markets respectively. Looking at the closing indices of every day in which there is negotiation, we consider the normalized log-returns. Let Pi (t ) be the index of the stock market i = 1, 2, . . . , N at time t and t = 0, 1, . . . , T . The absolute return is calculated as Ri (t ) = ln(Pi (t )) − ln(Pi (t − 1)). 3. Methodology 3.1. MF-DFA method The multifractal detrended fluctuation analysis (MF-DFA) method was developed by Kantelhardt et al. [29] for the multifractal characterization of non-stationary time series. MF-DFA is a generalization of the detrended fluctuation analysis (DFA) method. MF-DFA method can be described as follows. Let us suppose that xt is a series of length N, and this series is of compact support, i.e. xt = 0 for an insignificant fraction of the values only. The corresponding profile Y (i) is computed by integration as Y (i) =
i (xt − ⟨x⟩), t =1
i = 1, 2, . . . , N .
(2)
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Fig. 1. Artificial monofractal data sets with different scaling exponents. In each panel, data set is generated with the same length of N = 215 . (a) α = 0.2, (b) α = 0.4, (c) α = 0.5, (d) α = 0.7.
Fig. 2. The artificial multifractal data sets generated by binomial multifractal model with the same length of N = 215 . (a) β = 0.54, (b) β = 0.56, (c) β = 0.58, (d) β = 0.6.
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Fig. 3. The daily closing prices of six stock markets from May 12, 1992 to May 8, 2012. US stock markets: DJI, NYSE, SP500; Chinese stock markets: HSI, ShangZheng, ShenCheng.
Cut the profile Y (i) into Ns ≡ [N /s] non-overlapping segments of equal length s. Since the record length N need not be a multiple of the considered time scale s, a short part at the end of the profile will remain in most cases. In order not to disregard this part of the record, the same procedure is repeated starting from the other end of the record. Now the local trend yν (i) for each 2Ns segments is determined by the least square fit and then the variance is calculated using F 2 (s, v) =
s 1
s i=1
{Y [(ν − 1)s + i] − yν (i)}2
(3)
for ν = 1, 2, . . . , Ns and F 2 (s, v) =
s 1
s i=1
{Y [N − (ν − Ns )s + i] − yν (i)}2
(4)
for ν = Ns + 1, Ns + 2, . . . , 2Ns , where yν (i) is the fitting polynomial in segment ν th. Because different order MF-DFAs differ in the capability of eliminating trends in the series, linear (MF-DFA1), quadratic (MF-DFA2), cubic (MF-DFA3), or higher order polynomials can be considered in the fitting procedure. Now average over all segments to obtain the q-order fluctuation function
Fq (s) =
2Ns 1
2Ns v=1
[F (s, ν)] 2
q 2
1q .
(5)
The index variable q can take any real non-zero value. For q = 0, we calculate the fluctuation function as given below
F0 (s) = exp
2Ns 1
4Ns v=1
ln[F (s, ν)] . 2
(6)
Finally, determine the scaling behavior of the fluctuation function through analyzing the log–log plot of Fq (s) versus s Fq (s) ∼ sh(q) .
(7)
Here h(q) is known as the generalized Hurst exponent and h(2) is the well-known Hurst exponent H. In general, if the time series is monofractal, h(q) is independent of q, and if the time series is multifractal, h(q) depends on q. For positive value of q, h(q) describes the behavior of segments with large fluctuations while for negative value of q, h(q) describes the behavior of segments with small fluctuations. In general, h(q) is a monotonic function of q for a stationary time series which means that relatively small fluctuations happen more often in the series than relatively large ones. If h(q) > 0.5, the fluctuation related to q is auto-correlated, if h(q) < 0.5, the fluctuation related to q is anti-persistently auto-correlated and if h(q) = 0.5, the fluctuation related to q displays a random walk behavior. 3.2. Modification of MMA Multiscale multifractal analysis (MMA) is a data analysis approach, proposed to describe scaling properties of fluctuations within the signal analyzed. The main result of this procedure is the so-called Hurst surface h(q, s), where q is the multifractal parameter and s is the scale. MMA allows to analyze the time series in many scale ranges simultaneously by using a moving fitting window sweeping through all range of the scales s along the fluctuation function F (q, s). The Second-Order MF-DFA
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Fig. 4. The MF-DFA and MMA plots of DJI. (a) MF-DFA; (b) MMA. The MF-DFA functions are obtained by changing q values from −10 to 10 with step of 1. The MMA plot is determined by the MF-DFA function and with moving window length of 8 and 4 overlap. The Hurst surface shows additional information, which may be hidden by MF-DFA. Where the color code is shown in the vertical bar. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
is performed in this paper. Here we introduce a modification of MMA method: the use of changing window WL and SL instead of the fixed WL and SL. Q , S, WL (Moving window length) and SL (Slide length of each moving window) are four key parameters in MMA model. In the following context four main parameters are defined by equations:
• • • •
Q = [qMin : qStep : qMax ], S = [sMin : sMax , sNumber ], WL = [WLMin : WLStep : WLMax ] (Moving window length) SL = [SLMin : SLStep : SLMax ] (Slide length of each moving window).
For instance, S = [10 : 1000, 30] means generating a scale matrix in logarithm with length of 30 from 10 to 1000. The log–log plots of MF-DFA functions of DJI with parameters Q = [−10 : 1 : 10] and S = [10 : 1000, 30] are shown in Fig. 4(a). The MMA results of DJI with parameters Q = [−10 : 1 : 10], S = [10 : 1000, 30], WL = 8 and SL = 4 are shown in Fig. 4(b). It is observed from Fig. 4 that the DJI exhibits very strong multifractality for positive values of q, this is due to the presence of very strong long range correlations of large fluctuations. The cumulative distribution function (CDF) of local Hurst exponent without any bins, which is called CDF-MMA in this paper, is considered in the following discussion. First, the Hurst surface h(q, s) for all cases with parameters WL = [WLMin : WLStep : WLMax ] and SL = [SLMin : SLStep : SLMax ] are computed. Next the CDF for each Hurst surface h(q, s) is performed ∞ to investigate the multifractal characteristic. The CDF Pi (h) is defined as Pi (h) = h Pi (r )dr, where Pi (h) is the probability density function for the occurrence of a given case WL and SL with a scaling exponent h. We calculate the CDF for each case with parameters WL and SL. Finally, the group results of CDF for variable WL and SL are plotted in one figure to observe more reliable results. 4. Results To illustrate the potential applicability of the proposed method, we start with study of the multifractal behaviors of three types of artificial data sets mentioned in above section. For the purpose of simplicity, Random, Noise, and Multifractal are used for representing three classes of artificial series respectively. In the original MF-DFA measure, for very small scale s < 10 may result in an arithmetic underflow. Therefore, we set the usable range of scales in this paper to be S ∈ [10 : 1000]. First, we analyze the random data sets with length of N = 215 by using the CDF-MMA method. We performed both the individual CDF and group average results which will be shown in the following text. The CDF plots of random series with various parameters Q = [−10 : 1 : 10], S = [10 : 1000, 30], WL = [4 : 2 : 30] and SL = [2 : 2 : 20] (14 × 10 individuals) are displayed in Fig. 5. The black dash lines in Fig. 5 are corresponding to H = 0.5. The right panel of Fig. 5 demonstrates the standard deviation of all individual results. It is shown that different WL and SL values have no obvious influence on the group average CDF values. The group averages of CDFs for all the constructed random data sets are nearly overlapping in all cases, and there are no significant differences among them. In the following experiments, D(H ) is used to evaluate the width of non-zero and non-one continuous segment of ensemble average CDF. D(H ) reflects the complexity and evolution of multifractal characteristics. All the artificial random data sets possess a characteristic CDF with same D(H ) = 0.05. Group average CDFs in Fig. 5 converge to a constant shape regardless of the varying of parameters. The apparent and stable declines at H = 0.5 are found in Fig. 5. The standard deviations for all individual CDFs with parameters WL = [4 : 2 : 30] and SL = [2 : 2 : 20] are all lower than 0.1, which are very small. Next, we use monofractal noise with N = 215 samples for α = 0.2, α = 0.4, α = 0.5 and α = 0.7 respectively. The standard deviations in Fig. 6 illustrate that there are no obvious differences among individual CDFs.
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Fig. 5. The group average results of CDFs and corresponding standard deviations for random time series with different parameters.
Fig. 6. The group average CDFs and corresponding standard deviations for monofractal noise with different parameters. The results show that CDF of Hurst surface h(q, s) can effectively distinguish between the monofractal noise with different scaling exponents.
It can be seen from Fig. 6 that CDF curves start from the constant value CDF = 1 at small H and suddenly drop at H = 0.2, H = 0.4, H = 0.5 and H = 0.7, then decrease to stabilizing CDF = 0. It is worth noting that these points are exactly the real scaling exponents of the artificial monofractal noise. Furthermore, the CDF curves of monofractal data sets exhibit the similar shape, and parallel to each other, which may be the implication of the similar multifractal structure. Moreover, the D(H ) values are about 0.1 for all the monofractal noises. It is worth noting that the results of individual CDFs and group average CDFs imply no obvious distinctions by considering Fig. 6. The above two types of generated data sets are both monofractal time series. However, it is important to note that the performance of CDF-MMA method on Multifractal time series should be investigated to identify the effectiveness of the present study. Multifractal time series with parameters β = [0.54, 0.56, 0.58, 0.6] and nmax = 15 are considered in discussing of CDF structure based on CDF-MMA approach. The CDF plots of multifractal time series with various parameters Q , S, WL and SL are plotted in Fig. 7. As shown in Fig. 7, there are obscure fallings around H ∈ [0.8, 0.9] for all cases. Note that the intercept points of the CDF curves for all individuals are H = 1. The beginnings and tails of ensemble averages of CDF level off around H = 1 and H = 0 respectively. The degree of multifractality can be related to the width of D(h) value mentioned above. All the artificial multifractal series possess characteristic different CDF curves and D(H ) values in relation to its generated parameters β , permitting individual identification. By looking at Fig. 7, corresponding to the cases of series with parameters β = [0.54, 0.56, 0.58, 0.6], it can be observed that all the ensemble group averages of CDF are well separated. Furthermore, the increase of the D(H ) values indicates increase in the degree of multifractality. The curve of the ensemble group average of CDF can be realized as a distinct characteristic of time series. The variations in ensemble group average of CDF principally account for the underlying fractal organization in complex signal. To show how the present method works on real-world data, we investigate six stock markets from US and China. The group averages of CDF are shown in Fig. 8. We observe that the CDF plots of three US stock markets DJI, NYSE and SP500 are similar with each other, declining at H = 0.65. However, the Chinese stock markets possess characteristically different CDF plots and D(H ) values. As can be seen from Fig. 8, The steep descents of HSI, ShangZheng and Shencheng are H = 0.5, H = 0.4 and H = 0.5. We also find that the D(H ) values are smaller for US stock markets, which represent weaker multifractality compared to Chinese stock markets. For Chinese markets, ShangZheng exhibits stronger multifractality than HSI and ShenCheng.
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Fig. 7. The group average CDFs and corresponding standard deviations for multifractal time series with different parameters. The CDF curves of artificial multifractal time series are related to β .
Fig. 8. The group average CDFs and standard deviations for stocks. The CDF plots of three US stock markets are nearly equal to each other, while the three Chinese stock markets are significantly different.
5. Conclusions The purpose of this paper is to investigate the multifractal structures of nonstationary time series. The implementation of the method is based on the modified MMA method. The MMA-based CDF statistic is applied to artificial and real-world data sets to characterize the multiscale multifractal property. Furthermore, D(H ) is used to evaluate the width of declining segment of ensemble average CDF. We first illustrate the potential applicability of the proposed method for three types of artificial data sets: random series, monofractal noise and binomial multifractal series. It is found that the ensemble average of CDF based on Hurst surface h(q, s) can effectively distinguish monofractal noise with different scaling exponents. Moreover, the measured D(H ) values for all the monofractal noises are about 0.1. The group average CDFs for constructed random data sets are nearly overlapping and possess the characteristic CDF with same D(H ) = 0.05. Next, performance of CDF-MMA method on multifractal time series is investigated to identify the effectiveness of the present study. Four series generated by binomial multifractal model are considered in this paper. All the CDF curves of artificial multifractal series exhibit different characteristics, on the flip side, the results demonstrate D(H ) values are related to its generation parameters β . A bigger value of D(H ) indicates stronger multifractality. The application of CDF-MMA method through stock markets shows that the US markets are nearly equal to each other in CDF plots and D(H ) values, however, the Chinese markets depict distinct properties, suggesting individual signature. The low values D(H ) of US stock markets indicate relatively reduced multifractality as compared to the Chinese stock markets. For Chinese markets, ShangZheng exhibits stronger multifractality than HSI and ShenCheng. Our findings lead us to consider that the technique might also be suitable to other fields. In future work, we will apply the CDF-MMA method in diverse fields. Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant No. 61304145), the Research Fund for the Doctoral Program of Higher Education (Grant No. 20130009120016) and the Fundamental Research Funds for the Central Universities (Grant Nos. 2014RC014, 2014-Ia-038).
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Physica A journal homepage: www.elsevier.com/locate/physa
Multifractality of stock markets based on cumulative distribution function and multiscale multifractal analysis Aijing Lin ∗ , Pengjian Shang School of Science, Beijing Jiaotong University, Beijing 100044, PR China
highlights • We analyze Chinese and US stock markets during the period of 1992 to 2012. • The cumulative distribution function of modified Hurst surface is calculated. • Stable structures of multifractal scaling are found by performing CDF-MMA technique.
article
info
Article history: Received 28 July 2015 Received in revised form 29 October 2015 Available online 29 December 2015 Keywords: CDF Multiscale MF-DFA MMA Stock market
abstract Considering the diverse application of multifractal techniques in natural scientific disciplines, this work underscores the versatility of multiscale multifractal detrended fluctuation analysis (MMA) method to investigate artificial and real-world data sets. The modified MMA method based on cumulative distribution function is proposed with the objective of quantifying the scaling exponent and multifractality of nonstationary time series. It is demonstrated that our approach can provide a more stable and faithful description of multifractal properties in comprehensive range rather than fixing the window length and slide length. Our analyzes based on CDF-MMA method reveal significant differences in the multifractal characteristics in the temporal dynamics between US and Chinese stock markets, suggesting that these two stock markets might be regulated by very different mechanism. The CDF-MMA method is important for evidencing the stable and fine structure of multiscale and multifractal scaling behaviors and can be useful to deepen and broaden our understanding of scaling exponents and multifractal characteristics. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Finance is an active research area, and a number of studies have used statistical mechanics to investigate the dynamics of financial markets due to the complex structures. Application of the idea gained from fractal theory to study the dynamical properties has been one of the most exciting areas of research in recent times [1–6]. Various complex systems have considered to be well represented by multifractality [7–21]. In previous study, it is widely accepted that financial market illustrates strong sign of complexity, power-law and multifractality [22–28]. The characteristics of the correlation in complex system can be investigated by extracting its scaling exponents. While some processes (monofractal) can be quantified by a single scaling exponent, others (multifractal) require the spectrum of exponents to characterize. The detrended fluctuation analysis (DFA) invented by Peng et al. [12] has been established as an important tool for the determination of fractal scaling properties and detection of long-range power-law correlations in signals. Multifractal
∗
Corresponding author. E-mail address: [email protected] (A. Lin).
http://dx.doi.org/10.1016/j.physa.2015.12.012 0378-4371/© 2015 Elsevier B.V. All rights reserved.
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detrended fluctuation analysis (MF-DFA) method [29] is an universal tool to investigate multifractality, which is a multifractal generalization of the DFA method. In the previous studies, the MF-DFA exponents were estimated as the slope of a MF-DFA function plotted versus the resolution s on a log–log scale. The scaling range for determining the scaling exponents cannot be defined precisely, usually determined by visual inspection of plot. However, it is not adequate to describe the dynamical behaviors of time series by using single or two scaling exponents. Then Gierałtowski et al. [30] generated a new method called multiscale multifractal detrended fluctuation analysis (MMA) for solving the above problem. This new technology allows us to investigate not only the multifractal properties but also dependence of these properties on the time scale. MMA has successfully been applied to diverse field such as heart rate dynamics, economics and traffic systems [31–33]. Hence, previous studies on multiscale DFA inspire us to propose cumulative distribution function (CDF) statistics of Hurst surface h(q, s) based on MMA method, which is called CDF-MMA method in our paper. In order to avoid the individual blind spot, the ensemble group averages of CDF values corresponding to individuals are calculated to explore the scaling behaviors. The present study shows more stable and robust results, allowing the assessment of multifractality and permitting individual discrimination. The organization of this paper is as follows. First, the artificial and real-word data sets used in our work are given in Section 2. Section 3 presents methods employed in study. Section 4 is devoted to show the results by employing our modified CDF-MMA approach. Finally, the important conclusions drawn from this study are provided in Section 5. 2. Data 2.1. Artificial data In order to test validity of the proposed method, we present results for several artificial series: random series, monofractal noise and binomial multifractal series. We now turn to the artificial processes. (1) Monofractal noise Many empirical data sets are characterized by long-range power-law auto-correlations. In the present study we generate artificial monofractal noise data sets of N = 215 samples by using modified Fourier filtering method [34] with scaling exponents α = 0.2, α = 0.4, α = 0.5 and α = 0.7 respectively. The artificial monofractal noises are plotted in Fig. 1. (2) Binomial multifractal model The artificial multifractal data sets considered in this paper are generated by applying binomial multifractal model [29]. In the binomial multifractal model, a series of N = 2nmax numbers k with k = 1, 2, . . . , N is defined by X (k) = β n(k−1) (1 − β)nmax −n(k−1)
(1)
where 0.5 < β < 1 is a parameter and n(k) is the number of digits equal to 1 in the binary representation of the index k, e.g. n(13) = 3, since 13 corresponds to binary 1101. In our study, we consider binomial multifractal series with parameters β = [0.54, 0.56, 0.58, 0.6] and nmax = 15. The time series generated by the binomial multifractal models with these parameters are shown in Fig. 2. 2.2. Real-world data To show how the method applies to real data, we study different real-world financial series, which we consider the outputs of the complex systems-daily closing values of six stock indices. The data sets are obtained from Yahoo Finance covering 5533 days from May 12, 1992 to May 8, 2012. The original stock market indices are shown in Fig. 3. For simplicity, DJI, NYSE, SP500, HSI, ShangZheng and ShenCheng are used for representing six stock markets respectively. Looking at the closing indices of every day in which there is negotiation, we consider the normalized log-returns. Let Pi (t ) be the index of the stock market i = 1, 2, . . . , N at time t and t = 0, 1, . . . , T . The absolute return is calculated as Ri (t ) = ln(Pi (t )) − ln(Pi (t − 1)). 3. Methodology 3.1. MF-DFA method The multifractal detrended fluctuation analysis (MF-DFA) method was developed by Kantelhardt et al. [29] for the multifractal characterization of non-stationary time series. MF-DFA is a generalization of the detrended fluctuation analysis (DFA) method. MF-DFA method can be described as follows. Let us suppose that xt is a series of length N, and this series is of compact support, i.e. xt = 0 for an insignificant fraction of the values only. The corresponding profile Y (i) is computed by integration as Y (i) =
i (xt − ⟨x⟩), t =1
i = 1, 2, . . . , N .
(2)
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Fig. 1. Artificial monofractal data sets with different scaling exponents. In each panel, data set is generated with the same length of N = 215 . (a) α = 0.2, (b) α = 0.4, (c) α = 0.5, (d) α = 0.7.
Fig. 2. The artificial multifractal data sets generated by binomial multifractal model with the same length of N = 215 . (a) β = 0.54, (b) β = 0.56, (c) β = 0.58, (d) β = 0.6.
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Fig. 3. The daily closing prices of six stock markets from May 12, 1992 to May 8, 2012. US stock markets: DJI, NYSE, SP500; Chinese stock markets: HSI, ShangZheng, ShenCheng.
Cut the profile Y (i) into Ns ≡ [N /s] non-overlapping segments of equal length s. Since the record length N need not be a multiple of the considered time scale s, a short part at the end of the profile will remain in most cases. In order not to disregard this part of the record, the same procedure is repeated starting from the other end of the record. Now the local trend yν (i) for each 2Ns segments is determined by the least square fit and then the variance is calculated using F 2 (s, v) =
s 1
s i=1
{Y [(ν − 1)s + i] − yν (i)}2
(3)
for ν = 1, 2, . . . , Ns and F 2 (s, v) =
s 1
s i=1
{Y [N − (ν − Ns )s + i] − yν (i)}2
(4)
for ν = Ns + 1, Ns + 2, . . . , 2Ns , where yν (i) is the fitting polynomial in segment ν th. Because different order MF-DFAs differ in the capability of eliminating trends in the series, linear (MF-DFA1), quadratic (MF-DFA2), cubic (MF-DFA3), or higher order polynomials can be considered in the fitting procedure. Now average over all segments to obtain the q-order fluctuation function
Fq (s) =
2Ns 1
2Ns v=1
[F (s, ν)] 2
q 2
1q .
(5)
The index variable q can take any real non-zero value. For q = 0, we calculate the fluctuation function as given below
F0 (s) = exp
2Ns 1
4Ns v=1
ln[F (s, ν)] . 2
(6)
Finally, determine the scaling behavior of the fluctuation function through analyzing the log–log plot of Fq (s) versus s Fq (s) ∼ sh(q) .
(7)
Here h(q) is known as the generalized Hurst exponent and h(2) is the well-known Hurst exponent H. In general, if the time series is monofractal, h(q) is independent of q, and if the time series is multifractal, h(q) depends on q. For positive value of q, h(q) describes the behavior of segments with large fluctuations while for negative value of q, h(q) describes the behavior of segments with small fluctuations. In general, h(q) is a monotonic function of q for a stationary time series which means that relatively small fluctuations happen more often in the series than relatively large ones. If h(q) > 0.5, the fluctuation related to q is auto-correlated, if h(q) < 0.5, the fluctuation related to q is anti-persistently auto-correlated and if h(q) = 0.5, the fluctuation related to q displays a random walk behavior. 3.2. Modification of MMA Multiscale multifractal analysis (MMA) is a data analysis approach, proposed to describe scaling properties of fluctuations within the signal analyzed. The main result of this procedure is the so-called Hurst surface h(q, s), where q is the multifractal parameter and s is the scale. MMA allows to analyze the time series in many scale ranges simultaneously by using a moving fitting window sweeping through all range of the scales s along the fluctuation function F (q, s). The Second-Order MF-DFA
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Fig. 4. The MF-DFA and MMA plots of DJI. (a) MF-DFA; (b) MMA. The MF-DFA functions are obtained by changing q values from −10 to 10 with step of 1. The MMA plot is determined by the MF-DFA function and with moving window length of 8 and 4 overlap. The Hurst surface shows additional information, which may be hidden by MF-DFA. Where the color code is shown in the vertical bar. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
is performed in this paper. Here we introduce a modification of MMA method: the use of changing window WL and SL instead of the fixed WL and SL. Q , S, WL (Moving window length) and SL (Slide length of each moving window) are four key parameters in MMA model. In the following context four main parameters are defined by equations:
• • • •
Q = [qMin : qStep : qMax ], S = [sMin : sMax , sNumber ], WL = [WLMin : WLStep : WLMax ] (Moving window length) SL = [SLMin : SLStep : SLMax ] (Slide length of each moving window).
For instance, S = [10 : 1000, 30] means generating a scale matrix in logarithm with length of 30 from 10 to 1000. The log–log plots of MF-DFA functions of DJI with parameters Q = [−10 : 1 : 10] and S = [10 : 1000, 30] are shown in Fig. 4(a). The MMA results of DJI with parameters Q = [−10 : 1 : 10], S = [10 : 1000, 30], WL = 8 and SL = 4 are shown in Fig. 4(b). It is observed from Fig. 4 that the DJI exhibits very strong multifractality for positive values of q, this is due to the presence of very strong long range correlations of large fluctuations. The cumulative distribution function (CDF) of local Hurst exponent without any bins, which is called CDF-MMA in this paper, is considered in the following discussion. First, the Hurst surface h(q, s) for all cases with parameters WL = [WLMin : WLStep : WLMax ] and SL = [SLMin : SLStep : SLMax ] are computed. Next the CDF for each Hurst surface h(q, s) is performed ∞ to investigate the multifractal characteristic. The CDF Pi (h) is defined as Pi (h) = h Pi (r )dr, where Pi (h) is the probability density function for the occurrence of a given case WL and SL with a scaling exponent h. We calculate the CDF for each case with parameters WL and SL. Finally, the group results of CDF for variable WL and SL are plotted in one figure to observe more reliable results. 4. Results To illustrate the potential applicability of the proposed method, we start with study of the multifractal behaviors of three types of artificial data sets mentioned in above section. For the purpose of simplicity, Random, Noise, and Multifractal are used for representing three classes of artificial series respectively. In the original MF-DFA measure, for very small scale s < 10 may result in an arithmetic underflow. Therefore, we set the usable range of scales in this paper to be S ∈ [10 : 1000]. First, we analyze the random data sets with length of N = 215 by using the CDF-MMA method. We performed both the individual CDF and group average results which will be shown in the following text. The CDF plots of random series with various parameters Q = [−10 : 1 : 10], S = [10 : 1000, 30], WL = [4 : 2 : 30] and SL = [2 : 2 : 20] (14 × 10 individuals) are displayed in Fig. 5. The black dash lines in Fig. 5 are corresponding to H = 0.5. The right panel of Fig. 5 demonstrates the standard deviation of all individual results. It is shown that different WL and SL values have no obvious influence on the group average CDF values. The group averages of CDFs for all the constructed random data sets are nearly overlapping in all cases, and there are no significant differences among them. In the following experiments, D(H ) is used to evaluate the width of non-zero and non-one continuous segment of ensemble average CDF. D(H ) reflects the complexity and evolution of multifractal characteristics. All the artificial random data sets possess a characteristic CDF with same D(H ) = 0.05. Group average CDFs in Fig. 5 converge to a constant shape regardless of the varying of parameters. The apparent and stable declines at H = 0.5 are found in Fig. 5. The standard deviations for all individual CDFs with parameters WL = [4 : 2 : 30] and SL = [2 : 2 : 20] are all lower than 0.1, which are very small. Next, we use monofractal noise with N = 215 samples for α = 0.2, α = 0.4, α = 0.5 and α = 0.7 respectively. The standard deviations in Fig. 6 illustrate that there are no obvious differences among individual CDFs.
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Fig. 5. The group average results of CDFs and corresponding standard deviations for random time series with different parameters.
Fig. 6. The group average CDFs and corresponding standard deviations for monofractal noise with different parameters. The results show that CDF of Hurst surface h(q, s) can effectively distinguish between the monofractal noise with different scaling exponents.
It can be seen from Fig. 6 that CDF curves start from the constant value CDF = 1 at small H and suddenly drop at H = 0.2, H = 0.4, H = 0.5 and H = 0.7, then decrease to stabilizing CDF = 0. It is worth noting that these points are exactly the real scaling exponents of the artificial monofractal noise. Furthermore, the CDF curves of monofractal data sets exhibit the similar shape, and parallel to each other, which may be the implication of the similar multifractal structure. Moreover, the D(H ) values are about 0.1 for all the monofractal noises. It is worth noting that the results of individual CDFs and group average CDFs imply no obvious distinctions by considering Fig. 6. The above two types of generated data sets are both monofractal time series. However, it is important to note that the performance of CDF-MMA method on Multifractal time series should be investigated to identify the effectiveness of the present study. Multifractal time series with parameters β = [0.54, 0.56, 0.58, 0.6] and nmax = 15 are considered in discussing of CDF structure based on CDF-MMA approach. The CDF plots of multifractal time series with various parameters Q , S, WL and SL are plotted in Fig. 7. As shown in Fig. 7, there are obscure fallings around H ∈ [0.8, 0.9] for all cases. Note that the intercept points of the CDF curves for all individuals are H = 1. The beginnings and tails of ensemble averages of CDF level off around H = 1 and H = 0 respectively. The degree of multifractality can be related to the width of D(h) value mentioned above. All the artificial multifractal series possess characteristic different CDF curves and D(H ) values in relation to its generated parameters β , permitting individual identification. By looking at Fig. 7, corresponding to the cases of series with parameters β = [0.54, 0.56, 0.58, 0.6], it can be observed that all the ensemble group averages of CDF are well separated. Furthermore, the increase of the D(H ) values indicates increase in the degree of multifractality. The curve of the ensemble group average of CDF can be realized as a distinct characteristic of time series. The variations in ensemble group average of CDF principally account for the underlying fractal organization in complex signal. To show how the present method works on real-world data, we investigate six stock markets from US and China. The group averages of CDF are shown in Fig. 8. We observe that the CDF plots of three US stock markets DJI, NYSE and SP500 are similar with each other, declining at H = 0.65. However, the Chinese stock markets possess characteristically different CDF plots and D(H ) values. As can be seen from Fig. 8, The steep descents of HSI, ShangZheng and Shencheng are H = 0.5, H = 0.4 and H = 0.5. We also find that the D(H ) values are smaller for US stock markets, which represent weaker multifractality compared to Chinese stock markets. For Chinese markets, ShangZheng exhibits stronger multifractality than HSI and ShenCheng.
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Fig. 7. The group average CDFs and corresponding standard deviations for multifractal time series with different parameters. The CDF curves of artificial multifractal time series are related to β .
Fig. 8. The group average CDFs and standard deviations for stocks. The CDF plots of three US stock markets are nearly equal to each other, while the three Chinese stock markets are significantly different.
5. Conclusions The purpose of this paper is to investigate the multifractal structures of nonstationary time series. The implementation of the method is based on the modified MMA method. The MMA-based CDF statistic is applied to artificial and real-world data sets to characterize the multiscale multifractal property. Furthermore, D(H ) is used to evaluate the width of declining segment of ensemble average CDF. We first illustrate the potential applicability of the proposed method for three types of artificial data sets: random series, monofractal noise and binomial multifractal series. It is found that the ensemble average of CDF based on Hurst surface h(q, s) can effectively distinguish monofractal noise with different scaling exponents. Moreover, the measured D(H ) values for all the monofractal noises are about 0.1. The group average CDFs for constructed random data sets are nearly overlapping and possess the characteristic CDF with same D(H ) = 0.05. Next, performance of CDF-MMA method on multifractal time series is investigated to identify the effectiveness of the present study. Four series generated by binomial multifractal model are considered in this paper. All the CDF curves of artificial multifractal series exhibit different characteristics, on the flip side, the results demonstrate D(H ) values are related to its generation parameters β . A bigger value of D(H ) indicates stronger multifractality. The application of CDF-MMA method through stock markets shows that the US markets are nearly equal to each other in CDF plots and D(H ) values, however, the Chinese markets depict distinct properties, suggesting individual signature. The low values D(H ) of US stock markets indicate relatively reduced multifractality as compared to the Chinese stock markets. For Chinese markets, ShangZheng exhibits stronger multifractality than HSI and ShenCheng. Our findings lead us to consider that the technique might also be suitable to other fields. In future work, we will apply the CDF-MMA method in diverse fields. Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant No. 61304145), the Research Fund for the Doctoral Program of Higher Education (Grant No. 20130009120016) and the Fundamental Research Funds for the Central Universities (Grant Nos. 2014RC014, 2014-Ia-038).
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