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PHYSICAL REVIEW E 84, 016106 (2011)

Multifractal detrending moving-average cross-correlation analysis

Zhi-Qiang Jiang (

1

)1,2 and Wei-Xing Zhou (

)1,2,3,*

School of Business, East China University of Science and Technology, Shanghai 200237, China 2 Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237, China 3 School of Science, East China University of Science and Technology, Shanghai 200237, China (Received 7 May 2011; published 21 July 2011) There are a number of situations in which several signals are simultaneously recorded in complex systems, which exhibit long-term power-law cross correlations. The multifractal detrended cross-correlation analysis (MFDCCA) approaches can be used to quantify such cross correlations, such as the MFDCCA based on the detrended ?uctuation analysis (MFXDFA) method. We develop in this work a class of MFDCCA algorithms based on the detrending moving-average analysis, called MFXDMA. The performances of the proposed MFXDMA algorithms are compared with the MFXDFA method by extensive numerical experiments on pairs of time series generated from bivariate fractional Brownian motions, two-component autoregressive fractionally integrated moving-average processes, and binomial measures, which have theoretical expressions of the multifractal nature. In all cases, the scaling exponents hxy extracted from the MFXDMA and MFXDFA algorithms are very close to the theoretical values. For bivariate fractional Brownian motions, the scaling exponent of the cross correlation is independent of the cross-correlation coef?cient between two time series, and the MFXDFA and centered MFXDMA algorithms have comparative performances, which outperform the forward and backward MFXDMA algorithms. For two-component autoregressive fractionally integrated moving-average processes, we also ?nd that the MFXDFA and centered MFXDMA algorithms have comparative performances, while the forward and backward MFXDMA algorithms perform slightly worse. For binomial measures, the forward MFXDMA algorithm exhibits the best performance, the centered MFXDMA algorithms performs worst, and the backward MFXDMA algorithm outperforms the MFXDFA algorithm when the moment order q < 0 and underperforms when q > 0. We apply these algorithms to the return time series of two stock market indexes and to their volatilities. For the returns, the centered MFXDMA algorithm gives the best estimates of hxy (q) since its hxy (2) is closest to 0.5, as expected, and the MFXDFA algorithm has the second best performance. For the volatilities, the forward and backward MFXDMA algorithms give similar results, while the centered MFXDMA and the MFXDFA algorithms fail to extract rational multifractal nature. DOI: 10.1103/PhysRevE.84.016106 PACS number(s): 89.75.Da, 05.45.Tp, 05.45.Df, 05.40.?a

I. INTRODUCTION

Natural and socioeconomic systems are usually complex systems from which macroscopic statistical laws emerge. These macroscopic laws are the outcomes of self-organization and interactions among constituents, which cannot be explained by the sum of the microscopic behaviors of individuals. Statistical laws can be extracted from time series, which is the most usual recorded form of observable quantities in real world. The fractal and multifractal nature of time series has been extensively studied for different systems [1]. For a nonstationary time series, the detrended ?uctuation analysis (DFA) can be adopted to explore its long-range autocorrelations [2–6] and multifractal features [7–9]. Alternatively, the detrending moving-average (DMA) method can also be used for fractal analysis [10–14] or multifractal analysis [15,16]. Numerical experiments on monofractal time series unveil that the performance of the DMA method is comparable to the DFA method with slightly different priorities under different situations [17–19]. However, for multifractal time series, the multifractal detrending moving average (MFDMA) performs better than the multifractal detrended ?uctuation analysis (MFDFA) [15]. In addition, we note that both the DFA and DMA algorithms can be extended from one dimension

to higher dimensions for fractal and multifractal analysis [15,20–22]. A complex system usually contains several observable variables that exhibit long-range dependence or multifractal nature. In turbulent ?ows, the velocity, temperature, and concentration ?elds are embedded in the same space as joint multifractal measures [23–27], in which the scaling behavior of the joint moments of two joint multifractal measures μ1 and μ2 is investigated: J (s) = [μ1 (s)]p [μ2 (s)]q , (1)

where s is the box size. This framework has also been applied to study the joint multifractal nature between topographic indices and crop yield in agronomy [28,29], trading volume and volatility in stock markets [30], nitrogen dioxide and groundlevel ozone [31], heart rate variability and brain activity of healthy humans [32], and wind patterns and land surface air temperature [33]. For two stationary time series {x(i)} and {y(i)} of the same length, the time-lagged cross correlation or covariance provides another example [34–37], C(s) = x(t)y(t + s) .

*

wxzhou@ecust.edu.cn 016106-1

(2)

1539-3755/2011/84(1)/016106(11)

?2011 American Physical Society

ZHI-QIANG JIANG AND WEI-XING ZHOU

PHYSICAL REVIEW E 84, 016106 (2011)

For two nonstationary time series {X(i)} and {Y (i)} of the same length, one can study the following cross-correlation function between two detrended series [38]: Cxy (s) = [X(t) ? X(t)][Y (t + s) ? Y (t + s)] , (3)

where X(t) and Y (t) are certain trend functions of X(t) and Y (t), respectively. The detrended cross-correlation analysis (DCCA) was introduced to investigate the long-range powerlaw cross correlations between two nonstationary time series [39–41]: Fxy (s) = [X(t) ? X(t)][Y (t) ? Y (t)] , (4)

analytical expressions [bivariate fractional Brownian motions, two-component autoregressive fractionally integrated moving average (ARFIMA) processes, and binomial measures] to investigate the performance of the algorithms. In Sec. IV, we apply the algorithms to daily stock index returns and volatilities. We discuss and summarize our ?ndings in Sec. V.

II. MFXDMA AND MFXDFA

where X(t) and Y (t) are certain trend functions of X(t) and Y (t) speci?c to moving windows of size s, respectively. The DFA method is a special case of this DCCA method when X(t) = Y (t). The DCCA method studies the temporal (not the cross-sectional) properties of two nonstationary time series, which is similar to the instant cross correlations [42,43]. The signi?cance of the cross correlation can be assessed by statistical tests [44,45]. Podobnik and Stanley pointed out that, when Fxy (s) versus s ?uctuates around zero, there are no power-law cross correlations with a unique exponent: either no cross correlations or only short-range cross correlations exist between X(t) and Y (t) [40]. The DCCA method has been applied to study volume change and price change of the Standard and Poor’s (S&P) 500 Index [46], volatilities of the Brazilian agrarian commodities and stocks [47], traf?c ?ows [48], and self-af?ne time series of taxi accidents [49]. Alternatively, the temporal cross-correlation property can also be investigated using the cross-spectral density function or the coherence function based on wavelet transform [50,51]. More generally, the multifractal detrended cross-correlation analysis was introduced to investigate the multifractal nature in the long-range power-law cross correlations between two nonstationary time series [52], which recovers the MFDFA method when X(t) = Y (t). We call this method the MFXDFA for reasons that will be clear in Sec. II. Note that the MFXDFA method is relevant to the multifractal height cross-correlation analysis with differences [53]. The MFXDFA method has been applied to temporal and spatial seismic data [54], sunspot numbers and river ?ow ?uctuations [55], stock index prices [56,57], price-volume relationships in agricultural commodity futures markets [58], prices of the Chinese and US agricultural futures [59], spot and futures markets of West Texas Intermediate (WTI) crude oil [60], and traf?c signals [61]. In this work, we introduce a variant of the MFXDFA algorithm, termed multifractal detrending moving-average cross-correlation analysis (MFXDMA), which combines the ideas of MFDMA and DCCA. The main difference between MFXDFA and MFXDMA is that the latter adopts local moving average as the trend function. Since the MFDMA algorithm outperforms the MFDFA algorithm for multifractal time series, we expect that the MFXDMA algorithm will show advantages over the MFXDFA algorithm. Our numerical experiments and real-work data analysis con?rm this conjecture. The paper is organized as follows. Section II describes a uni?ed framework of the MFXDFA and MFXDMA algorithms. Section III gives extensive numerical experiments using fractal and multifractal time series with known

Consider two stationary time series {x(i)} and {y(i)} of the same length M, where i = 1,2, . . . ,M. Without loss of generality, we can assume that these two time series have zero means. Each time series is covered with Ms = [M/s] nonoverlapping boxes of size s. The pro?les within the vth box [lv + 1,lv + s], where lv = (v ? 1)s, are determined to be

k k

Xv (k) =

j =1

x(lv + j ), Yv (k) =

j =1

y(lv + j ),

(5)

where k = 1, . . . ,s. Assume that the local trending functions of {Xv (k)} and {Yv (k)} are {Xv (k)} and {Yv (k)}, respectively. The cross correlation for each box is calculated as follows: Fv (s) = 1 s

s

[Xv (k) ? Xv (k)][Yv (k) ? Yv (k)].

k=1

(6)

The qth order cross correlation is calculated as follows: Fxy (q,s) = when q = 0, and Fxy (0,s) = exp 1 2m

m

1 m

m

1/q

|Fv (s)|q/2

v=1

(7)

ln |Fv (s)| .

v=1

(8)

We then expect the following scaling relation: Fxy (q,s) ? s hxy (q) . (9)

According to the standard multifractal formalism, the multifractal scaling exponent τ (q) can be used to characterize the multifractal nature, which reads τxy (q) = qhxy (q) ? Df , (10)

where Df is the fractal dimension of the geometric support of the multifractal measure [9]. For time series analysis, we have Df = 1. If the scaling exponent function τ (q) is a nonlinear function of q, the signal has a multifractal nature. It is easy to obtain the singularity strength function α(q) and the multifractal spectrum f (α) via the Legendre transform [62] αxy (q) = dτxy (q)/dq fxy (q) = qαxy ? τxy (q). (11)

There are many different methods for the determination of Xv and Yv . The local detrending functions could be polynomials [2,3], which recovers the MFDXA method [52]. The local detrending function could also be the moving averages [10,11], in which case the algorithm is called MFXDMA. To be more clear, we rename the MFDXA algorithm as MFXDFA, and all multifractal analysis algorithms

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PHYSICAL REVIEW E 84, 016106 (2011)

for cross correlations based on local detrending are termed multifractal detrending or detrended cross-correlation analysis (or MFDCCA). MFXDFA is an MFDCCA method based on DFA, and MFXDMA is an MFDCCA method based on DMA. When X = Y , MFXDFA in Ref. [52] reduces to MFDFA in Ref. [9], and MFXDMA reduces to MFDMA in Ref. [15]. We note that the extension of the MFDCCA algorithms to high dimensions is straightforward [52]. The moving-average function Z(t) of Z ∈ {X,Y } in a moving window [13] can be calculated as follows: Z(t) = 1 n k=?

(n?1)(1?θ)

Z(t ? k),

(n?1)θ

(12)

where n is the window size, g is the largest integer smaller than g, g is the smallest integer larger than g, and θ is the position parameter with the value varying in the range [0,1]. Hence, the moving-average function considers (n ? 1)(1 ? θ ) data points in the past and (n ? 1)θ points in the future. We consider three special cases in this paper. The ?rst case, θ = 0, refers to the backward moving average [18], in which the moving-average function Z(t) is calculated over all the past n ? 1 data points of the signal. The second case, θ = 0.5, corresponds to the centered moving average [18], where Z(t) contains half past and half future information in each window. The third case, θ = 1, is called the forward moving average, where Z(t) considers the trend of n ? 1 data points in the future. Usually, one chooses n = s to obtain better results [15].

III. NUMERICAL EXPERIMENTS

This property allows us to investigate the performances of the proposed algorithms on a solid foundation. An ef?cient simulation technique for univariate fractional Brownian motions (FBMs) relies upon the embedding of the covariance matrix into a circulant matrix, whose square root can be easily obtained by the discrete Fourier transform [65]. This method is an exact simulation algorithm provided that the circulant matrix is positive de?nite. This algorithm can be generalized to simulate bivariate FBMs, which embeds the circulant of a block Toeplitz covariance matrix and uses the fast Fourier transform to diagonalize the block circulant matrix [66]. A detailed description of the simulation procedure can be found in Refs. [35,36]. In the simulation algorithm, the two Hurst indexes Hxx and Hyy of the two univariate FBMs and their cross-correlation coef?cient ρ are input arguments. We have generated a huge number of BFBMs, where Hxx , Hyy , and ρ all vary from 0.1 to 0.9 with a spacing of 0.1. For a given triple of (Hxx ,Hyy ,ρ), 100 repeated simulations are conducted, and 100 BFBMs with a length of 216 are generated. In most cases, the positive-de?niteness condition is not ful?lled. We then perform MFXDMA and MFXDFA on each BFBM to obtain the scaling exponent hxy . The average over 100 repeated simulations is calculated. We have observed for each BFBM and each algorithm that hxy = (hxx + hyy )/2. (14)

In order to investigate the validity and performance of the proposed MFDCCA algorithms, we perform extensive numerical experiments using bivariate fractional Brownian motions (BFBMs) [34–36], two-component ARFIMA processes [40, 63], and binomial measures generated from the multiplicative p model [64]. By de?nition, there is no multifractality in BFBMs and two-component ARFIMA processes. Therefore, the hxy (q) function is independent of q, and the τxy (q) function is linear. In contrast, binomial measures are expected to possess a multifractal nature. These three classes of time series are adopted to test the performance of the algorithms since the theoretical expressions of Hxx (q) are known for individual time series and we know the theoretical expressions of Hxy (q) for the ?rst two classes and the numerical expression for the third class. Note that we have used H for theoretical values and h for estimated values below.

A. Bivariate fractional Brownian motions

A bivariate fractional Brownian motion [x(t),y(t)] with parameters {Hxx ,Hyy } ∈ (0,1)2 is a self-similar Gaussian process with stationary increments, where x(t) and y(t) are two univariate fractional Brownian motions with Hurst indices Hxx and Hyy and are the two components of the BFBM [34–36]. The basic properties of multivariate fractional Brownian motions have been extensively studied [34–36]. Particularly, it has been proven that the Hurst index Hxy of the cross correlation between the two components is [34–36,38]: Hxy = (Hxx + Hyy )/2. (13)

Our main ?ndings are the following: (1) The exponent hxy is independent of the cross-correlation coef?cient ρ. (2) The hxy (q) functions are independent of q, indicating that the BFBMs are monofractals. (3) All four algorithms give nice estimates hxy of the scaling exponents, which are very close to the corresponding theoretical Hxy values. (4) The centered MFXDMA algorithm (θ = 0.5) and the MFXDFA algorithms have comparative performance and perform better than the backward and forward MFXDMA algorithms (θ = 0 and θ = 1). Since there are too many results to present in a concise way, we present a part of the results to manifest these ?ndings. A typical example of the BFBM with Hxx = 0.1, Hyy = 0.5, and ρ = 0.3 is illustrated in Fig. 1(a), and the corresponding power-law dependence of the ?uctuation functions Fxy (q,s) with respect to the scale s for the four algorithms is shown in Fig. 1(b). For MFXDMA algorithms, s should not be too large due to the ?nite-size effect. The scaling ranges span over two orders of magnitude for the MFXDMA algorithms and three orders of magnitude for the MFXDFA algorithm. In the determination of the scaling exponents hxy , we have used the same scaling ranges as in Fig. 1(b) for all the BFBMs, and nice power-law relationships are observed. Figure 1(c) shows the hxy (q) functions for Hxx = Hyy = 0.8 and ρ = 0.5 (top panel) and for Hxx = 0.1, Hyy = 0.5, and ρ = 0.3 (bottom panel). Although there is a decreasing trend in each function, the theoretical functions Hxy (q) = 0.8 and Hxy (q) = 0.3 are well within the error bars, indicating that the hxy (q) functions are independent of the order q. Hence, the four algorithms are able to correctly capture the monofractal nature of the BFBMs. We focus on q = 2 below.

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4 2 0 q=0 q=4 ?2 ?4

θ=1 Fxy DFA Fxy

120 (a) 80 40 Hxx = 0.1, Hyy = 0.5, ρ = 0.3 0 6 3 0 ?3 0 1000 2000 3000 4000 5000

10

hxy

y

θ=0 Fxy Hxx = 0.1, Hyy = 0.5, ρ = 0.3 θ=0.5 Fxy

0.85 0.8 0.75 0.32

Hxx = 0.8, H yy = 0.8, ρ = 0.5

Fxy (q, s)

10 10 10 10

q = ?4

hxy

5

0.3 0.28 (c) Hxx = 0.1, Hyy = 0.5, ρ = 0.3 ?5 ?3 ?1 1 3 5

x

(b)

t

10

0

10

1

10

2

s

10

3

10

4

10

q

0.84

0.8 0.76 0.32

Hxx = 0.8, Hyy = 0.8 hθ=0 xy hθ=0.5 xy hθ=1 xy DFA hxy Hxy

0.3 0.28 0

(d)

Hxx = 0.1, Hyy = 0.5

0.2

0.4

ρ

0.6

0.8

1

ΔhDFA Δhθ=0 Δhθ=0.5 Δhθ=1 11 xy xy xy xy 3 Hxx = 0.1, ρ = 0.5 ?5 5 0 ?5 Hxx = 0.1, ρ = 0.3 5 0 ?5 (e) Hxx = 0.1, ρ = 0.1 0 0.2 0.4 0.6 0.8 1

Hyy

7 0 ρ = 0.9 ?7 7 0 ρ = 0.5 ?7 7 0 ρ = 0.1 ?7 (f) 0 0.2 0.4

Δhxy × 103

hxy

Δhxy × 103

hxy

Hxx = Hyy

0.6

0.8

1

FIG. 1. (Color online) Multifractal detrended cross-correlation analysis of bivariate fractional Brownian motions. Comparisons are performed among three MFXDMA algorithms with θ = 0, 0.5, and 1 and the MFXDFA algorithm. The results in (c)–(f) are averaged over 100 repeated simulations. (a) A typical example of BFBM with Hxx = 0.1, Hyy = 0.5, and ρ = 0.3. (b) Power-law dependence of the ?uctuation functions Fxy (q,s) of the BFBM shown in (a) with respect to the scale s for q = ?4, q = 0, and q = 4. The straight lines are the best power-law ?ts to the data. The results have been translated vertically for better visibility. (c) Scaling exponents hxy (q) with the theoretical values as a dashed line for Hxx = Hyy = 0.8 and ρ = 0.5 (top) and Hxx = 0.1, Hyy = 0.5, and ρ = 0.3 (bottom). (d) Independence of the scaling exponents hxy with respect to the cross-correlation coef?cient ρ for Hxx = Hyy = 0.8 (top) and Hxx = 0.1 and Hyy = 0.5 (bottom). (e) Differences hxy (q) between the estimated scaling exponents hxy and the theoretical exponents Hxy for BFBMs, where Hxx = 0.1 is ?xed, Hyy varies from 0.1 to 0.9, and ρ takes different values. (f) Differences hxy (q) between hxy and Hxy for BFBMs with Hxx = Hyy varying from 0.1 to 0.9 and different ρ values.

Figure 1(d) shows the dependence of the scaling exponents hxy with respect to the cross-correlation coef?cient ρ for Hxx = Hyy = 0.8 (top panel) and for Hxx = 0.1 and Hyy = 0.5 (bottom panel). We ?nd that the hxy functions (hθ=0 , xy hθ=0.5 , hθ=1 and hDFA ) for the four algorithms are independent xy xy xy of ρ. This ?nding is very important since it distinguishes the temporal cross correlations quanti?ed by MFDCCA algorithms and the cross-sectional correlation quanti?ed by ρ. Two uncorrelated time series may exhibit long-term power-law cross correlation. In addition, the centered MFXDMA and the MFXDFA give similarly very accurate estimates of the scaling exponents with hθ=0.5 ≈ hDFA , Hxy = 0.8 for the top panel and xy xy Hxy = 0.3 for the bottom panel. In contrast, Fig. 1(d) shows that the backward and forward MFXDMA algorithms perform slightly worse and hθ=0 ≈ hθ=1 . xy xy In order to compare the performance of the four algorithms, we calculate the difference between the estimated exponent hxy and the theoretical exponent Hxy : hxy = hxy ? Hxy . (15)

are less than 0.01, implying that all the four algorithms give good estimates. It is interesting to observe that hθ=0 ≈ hθ=1 xy xy for all the cases. In addition, the centered MFXDMA and the MFXDFA algorithms outperform the backward and forward MFXDMA algorithms. We note that these conclusions also hold for other BFBMs. The relative performances between the centered MFXDMA and the MFXDFA algorithms are a little bit complicated. When Hxx = Hyy , as shown in Fig. 1(e), the two algorithms have comparable performance since hθ=0.5 ≈ xy hDFA ≈ 0. When Hxx = Hyy , as shown in Fig. 1(f), the cenxy tered MFXDMA algorithm slightly outperforms the MFXDFA algorithm. In summary, the centered MFXDMA algorithm (θ = 0.5) is recommended for analyzing bivariate fractional Brownian motions.

B. Two-component ARFIMA processes

The power-law autocorrelations in stochastic variables can be modeled by an ARFIMA process [67]: z(t) = Z(d,t) + (t), (16)

Figure 1(e) shows the dependence of hxy with respect to Hyy with a ?xed Hxx = 0.1 for ρ = 0.5 (top panel), ρ = 0.3 (middle panel), and ρ = 0.1 (bottom panel), while Fig. 1(f) shows the dependence of hxy with respect to Hxx = Hyy for ρ = 0.9 (top panel), ρ = 0.5 (middle panel), and ρ = 0.1 (bottom panel). All the hxy values in Figs. 1(e) and 1(f)

where d ∈ (0,0.5) is a memory parameter, z is an independent and identically distributed Gaussian variable, and

∞

Z(d,t) =

n=1

an (d)z(t ? n),

(17)

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θ=0 Fxy θ=0.5 Fxy

PHYSICAL REVIEW E 84, 016106 (2011)

10 10

0.91

(a) (b)

0.04 0.02

Δhθ=0 xy Δhθ=1 xy

Δhθ=0.5 xy ΔhDFA xy

(c)

Fxy (q, s)

hxy (q)

10 10 10 10 10 10

0.89 0.88

Δhxy (q)

hθ=0 xy hθ=0.5 xy hθ=1 xy hDFA xy

q = ?4

0.9

q=0

θ=1 Fxy DFA Fxy

0 ?0.02

10

8 5 2 ?1 ?4

0

10

1

10

2

s

10

3

10

4

10

(d)

5

0.87 ?4

0.77 0.76 0.75 0.74 ?4

?2

q

hθ=0.5 xy

0

2

4

?0.04

0

0.1

0.2

d

0.3

0.4

0.5

θ=0 Fxy θ=0.5 Fxy

0.04

hθ=0 xy hθ=1 xy hDFA xy

(e)

Δhθ=0 xy Δhθ=1 xy

Δhθ=0.5 xy ΔhDFA xy

(f)

Fxy (q, s)

hxy (q)

10 10 10 10

Δhxy (q)

q = ?4

0.02 0 ?0.02

q=0

θ=1 Fxy DFA Fxy

q=4 ?7

10

0

10

1

10

2

s

10

3

10

4

10

5

?2

q

0

2

4

?0.04

0

0.1

0.2

d

0.3

0.4

0.5

FIG. 2. (Color online) Multifractal detrended cross-correlation analysis of two-component ARFIMA processes. Comparisons are performed among three MFXDMA algorithms with θ = 0, 0.5, and 1 and the MFXDFA method. (a) Power-law dependence of the ?uctuation functions Fxy (q,s) with respect to the scale s for q = ?4, q = 0, and q = 4 for the process in Eq. (20) with d1 = d2 = 0.4. The straight lines are the best power-law ?ts to the data. The results have been translated vertically for better visibility. (b) Scaling exponents hxy (q) for the process in Eq. (20) with d1 = d2 = 0.4. (c) Differences hxy between the estimated scaling exponents hxy and the theoretical exponents Hxy for the process in Eq. (20) with different d values where d1 = d2 = d. (d) Power-law dependence of the ?uctuation functions Fxy (q,s) with respect to the scale s for q = ?4, q = 0, and q = 4 for the process in Eq. (22) with d1 = 0.1 and d2 = 0.4. (e) Scaling exponents hxy (q) for the process in Eq. (22) with d1 = 0.1 and d2 = 0.4. (f) Differences hxy between hxy and Hxy for the process in Eq. (22) with different d values where d1 = d2 = d.

in which an (d) is the weight, an (d) = d (n ? d)/[ (1 ? d) (n + 1)]. (18)

When W = 1 and x (t) = y (t) = (t), the two-component ARFIMA process becomes [40] x(t) = X(d1 ,t) + (t) y(t) = Y (d2 ,t) + (t) . (22)

The Hurst index Hzz is related to the memory parameter d by [40,68] Hzz = 0.5 + d. (19)

For the two-component ARFIMA processes discussed below, we take Z = X or Y . The two-component ARFIMA process is de?ned as follows [63]: x(t) = W X(d1 ,t) + (1 ? W )Y (d2 ,t) + y(t) = (1 ? W )X(d1 ,t) + W Y (d2 ,t) +

x (t) y (t),

(20)

where W ∈ [0.5,1] quanti?es the coupling strength between the two processes x(t) and y(t). When W = 1, x(t) and y(t) are fully decoupled and become two separate ARFIMA processes, as de?ned in Eq. (16). The cross correlation between x(t) and y(t) increases when W decreases from 1 to 0.5 [63]. To our knowledge, no general expression has been analytically derived for Hxy . When d1 > d2 , the Hurst index Hxx of x(t) decreases from 0.5 + d1 to certain value greater than 0.5 + d2 when W decreases from 1 to 0.5 [63]. In other words, Hxx locates within the interval [0.5 + d2 ,0.5 + d1 ]. When d1 = d2 = d, i.e., d1 → d2 , we obtain Hxx = Hyy = 0.5 + d, which does not depend on the value of W . (21)

If x and y are long-range power-law cross-correlated, it has been analytically derived that Eq. (13) holds [44]. Figures 2(a)–2(c) show the results for the process in Eq. (20). Figure 2(a) illustrates in log-log scale the dependence of the ?uctuation functions Fxy (q,s) with respect to the scale s for q = ?4, q = 0, and q = 4 for the process in Eq. (20) with d1 = d2 = 0.4. Nice power-law relations are observed, which are also evident for other (d1 ,d2 ) pairs. Figure 2(b) shows the corresponding scaling exponents hxy (q) for the process in Eq. (20) with d1 = d2 = 0.4. We note that the equation hxy (q) = [hxx (q) + hyy (q)]/2 holds for all the four curves. For the four algorithms, hxy (q) is close to the horizontal line H = 0.9, indicating that all the four algorithms correctly unveil the fractal nature of the two-component ARFIMA process. For q = 2, Fig. 2(b) shows that the MFXDFA gives the best estimate of hxy . Figure 2(c) depicts the differences hxy between hxy and Hxy with q = 2 for the process in Eq. (20) with different d values where d1 = d2 = d. It is found that (1) the two MFXDMA algorithms with θ = 0 and θ = 1 have the same performance, (2) the two MFXDMA algorithms with θ = 0 and θ = 1 perform better than the MFXDFA and the MFXDMA with θ = 0.5 for relatively small d values,

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PHYSICAL REVIEW E 84, 016106 (2011)

and (3) the two MFXDMA algorithms with θ = 0 and θ = 1 perform worse for large d values. Figures 2(d)–2(f) show the results for the process in Eq. (22). Figure 2(d) illustrates in log-log scale the dependence of the ?uctuation functions Fxy (q,s) with respect to the scale s for q = ?4, q = 0, and q = 4 for the process in Eq. (22) with d1 = 0.1 and d2 = 0.4. Nice power-law relations are observed, which are also evident for other (d1 ,d2 ) pairs. Figure 2(e) shows the corresponding scaling exponents hxy (q) for the process in Eq. (22) with d1 = 0.1 and d2 = 0.4. Again, the equation hxy (q) = [hxx (q) + hyy (q)]/2 holds for all the four curves. For the four algorithms, hxy (q) is close to the horizontal line H = 0.75, indicating that all the four algorithms correctly unveil the fractal nature of the two-component ARFIMA process. For q = 2, Fig. 2(e) shows that the MFXDMA algorithms with θ = 0 and θ = 1 give the best estimate of hxy . Figure 2(f) shows the differences between hxy and Hxy with q = 2 for the process in Eq. (22) with different d values where d1 = d2 = d. It is found that the MFXDFA and the MFXDMA with θ = 0.5 outperform the other two algorithms and give comparably nice estimates. He and Chen have investigated the two-component ARFIMA process de?ned in Eq. (20) of different lengths utilizing the DCCA method (the MFXDFA method with q = 2) and the DMCA method (the MFXDMA method with θ = 0 and q = 2) and found that the DMCA method performs better in most cases and performs worse in a few cases [69]. However, both methods are prone to underestimate the exponents hxy [69]. Our results shown in Fig. 2(c) are consistent with their results for d = 0.15, 0.25, and 0.35 in the sense that the MFXDMA method with θ = 0 outperforms the MFXDFA method. However, we have obtained better estimates for hxy , and there is no systematic underestimation. For instance, the three MFXDMA methods give hxy ≈ 0.85 or hxy ≈ 0 when d = 0.35, as shown in Fig. 2(c). There are two subtle issues that might worsen the estimation of hxy . Podobnik and Stanley [40] introduced a cutoff length M = 104 in their numerical simulations and let the sum run from 1 to M, i.e., they set aj = 0 for j > M. Our numerical experiments show that this cutoff seems optimal and a smaller or larger cutoff will worsen the estimation of the exponents. This ?nding applies for both MFXDMA and MFXDFA. In addition, we stress that the upper bound of the scaling range for the MFXDMA algorithms should not be too large because each moving average is calculated within a window of size s. Let us take the MFXDMA algorithm with θ = 0 as an example. In this case, the moving averages of the ?rst s ? 1 data points are not well de?ned. The bias becomes more signi?cant for large window size s. When s is large, the Fxy (q,s) function bends downward, and the overall slope ?attens. Similar arguments apply for other MFXDMA algorithms with different θ values. In contrast, the MFXDFA algorithm does not suffer from this ?nite-size effect.

C. Multifractal binomial measures

We start with the zeroth iteration k = 0, where the data set z(i) consists of one value, z(0) (1) = 1. In the kth iteration, the data set {z(k) (i) : i = 1,2, . . . ,2k } is obtained from z(k) (2i ? 1) = pz z(k?1) (i) z(k) (2i) = (1 ? pz )z(k?1) (i) (23)

for i = 1,2, . . . ,2k?1 . We notice that there are typos in the formula in Ref. [52]. When k → ∞, z(k) (i) approaches a binomial measure, whose scaling exponent function Hzz (q) has an analytic form [62,64],

q Hzz (q) = 1/q ? log2 pz + (1 ? pz )q /q.

(24)

According to Eq. (10), we have

q Tzz (q) = ?log2 pz + (1 ? pz )q .

(25)

In our simulation, we have performed k = 16 iterations with px = 0.3 for x(i) and py = 0.4 for y(i). The analytic scaling exponent functions Hxx (q) and Hyy (q) of x and y are expressed in Eq. (24). The two time series x and y are strongly correlated with a coef?cient of 0.82, which originates from the fact that the two sequences are constructed according to the same rules. The results are depicted in Fig. 3. Figure 3(a) illustrates the power-law dependence of Fxy (q,s) against s for the four algorithms. Since the time series is not very long, we investigate ?4 q 4 to ensure the convergence of the qth moments [70,71]. For the MFXDMA algorithms, there is a ?nite-size effect since the moving averages at the ends of the time series are ill de?ned. This effect becomes signi?cant, and the estimation of Fxy for large scales s deteriorates. The scaling range is chosen as [24 ,211 ] for the three MFXDMA methods. In contrast, the MFXDFA method performs poorly if the same scaling range is adopted. We use [28 ,215 ] for the MFXDFA method, which seems optimal. The algorithm-speci?c selection of the scaling range reveals the difference in the applicability of the two types of methods. Figure 3(a) shows that the power-law scaling is excellent for both positive and negative q values. The power-law scaling exponents (hDFA , hθ=0 , hθ=0.5 , and xy xy xy θ=1 hxy ) are presented in Fig. 3(b), while the mass scaling expoDFA θ=0 θ=0.5 θ=1 nents (τxy , τxy , τxy , and τxy ) and the multifractal specDFA θ=0 θ=0.5 θ=1 tra (fxy , fxy , fxy , and fxy ) are illustrated in Figs. 3(c) and 3(d), respectively. It is evident that the MFXDMA method with θ = 0.5 fails in a large part to correctly estimate the exponents, while the other three methods work much better. This ?nding is consistent with the conclusion that the MFDMA method with θ = 0.5 performs much worse than the MFDFA method and the MFDMA methods with θ = 0 and θ = 1 [15]. The insets of Fig. 3(b) show the interesting feature for all the four algorithms that hxy (q) = [hxx (q) + hyy (q)]/2 (26)

We construct two binomial measures {x(i) : i = 1,2, . . . ,2k } and {y(i) : i = 1,2, . . . ,2k } from the p model with known analytic multifractal properties as the third example [64]. Each multifractal signal is obtained in an iterative way.

no matter how accurate the estimates of an algorithm is. Similar relationships hold for individual monofractal ARFIMA signals [40] and individual binomial measures [15]. Hence, we can give the “theoretical” expression of Hxy (q) as follows: Hxy (q) = [Hxx (q) + Hyy (q)]/2, (27)

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PHYSICAL REVIEW E 84, 016106 (2011)

10 10

3

θ=0 Fxy θ=0.5 Fxy

1.4

θ=1 Fxy DFA Fxy

hθ=0 xy

hθ=0.5 xy

hθ=1 xy

hDFA xy

Hxy

4

(c)

0

hxy (q)

Fxy (q, s)

10 10 10

?3 q=0

?6

1 0.8 ?4

0.02

(e) (b)

τxy (q)

q = ?4

1.2

0 ?4 ?8 ?4

0.07

(f)

?9

q=4 0

(a)

1

θ=0 τxy θ=0.5 τxy θ=1 τxy DFA τxy Txy

10

10

10

2

s

10

3

10

4

10

5

?2

0 q

2

4

?2

0 q

Δτxy θ=0 Δτxy θ=1 DFA Δτxy

2

4

1 (d)

fxy (α)

Δhxy (q)

0.6 0.4 0.6

fxy (α) fxy (α)θ=0.5 fxy (α)θ=1 fxy (α)DFA Fxy (α) 0.9

θ=0

0 ?0.01 Δhθ=0 xy Δhθ=1 xy ΔhDFA xy ?2 0 q 2 4

Δτxy (q)

0.8

0.01

0.05 0.03 0.01 ?0.01 ?4 ?2

αxy

1.2

1.5

?0.02 ?4

0 q

2

4

FIG. 3. (Color online) Multifractal detrended cross-correlation analysis of two cross-correlated binomial measures generated from the p model with px = 0.3 and py = 0.4. Comparisons are performed among three MFXDMA algorithms with θ = 0, 0.5, and 1 and the MFXDFA method. (a) Power-law dependence of the ?uctuation functions Fxy (q,s) with respect to the scale s for q = ?4, q = 0, and q = 4. The straight lines are the best power-law ?ts to the data. The results have been translated vertically for better visibility. (b) Scaling exponents hxy (q) with the theoretical values as a dashed line. The insets show the hxy (q) curves and the corresponding [hxx (q) + hyy (q)]/2 curves, verifying the relation hxy (q) = [hxx (q) + hyy (q)]/2. (c) Multifractal mass exponents τ (q) obtained from the MFXDMA and MFXDFA methods, with the theoretical curve shown as a dashed line. (d) Multifractal spectra f (α) with respect to the singularity strength α for the four methods. The dashed curve is the theoretical multifractal spectrum. (e) Differences hxy (q) between the estimated mass exponents and their theoretical values for the three algorithms: MFXDFA, MFXDMA with θ = 0, and MFXDMA with θ = 1. (f) Differences τ (q) between the estimated mass exponents and their theoretical values for the three algorithms: MFXDFA, MFXDMA with θ = 0, and MFXDMA with θ = 1.

where Hxx (q) and Hyy (q) are given in Eq. (24). The theoretical line is plotted in Fig. 3(b) as a dashed line. According to Eq. (10), we obtain Txy (q) = [Txx (q) + Tyy (q)]/2, (28)

MFXDMA method (θ = 0) has the best performance and is thus recommended.

IV. APPLICATION TO STOCK MARKET INDEX RETURNS AND VOLATILITIES

which is shown in Fig. 3(c) for comparison. Similarly, we have Fxy (α) = [Fxx (α) + Fyy (α)]/2, (29)

which is illustrated in Fig. 3(d) as a dashed curve. In order to further assess the performance of the MFXDFA method and the two MFXDMA methods with θ = 0 and θ = 1, we compare the empirical estimates of hxy (q) and τxy (q) with the theoretical values of Hxy (q) and Txy (q) by calculating the relative errors: hxy (q) = hxy (q) ? Hxy (q) and τxy (q) = τxy (q) ? Txy (q), (31) (30)

which are shown in Figs. 3(e) and 3(f), respectively. Roughly speaking, the MFXDMA algorithm with θ = 1 performs best and the MFXDFA algorithm performs worst for most negative q values, and the MFXDMA method with θ = 0 performs best and the MFXDMA method with θ = 1 performs worst for most negative q values. On average, the backward

We now apply the MFXDMA algorithms to investigate the temporal cross correlations of the daily return and volatility time series of the Dow Jones Industrial Average (DJIA) and the National Association of Securities Dealers Automated Quotations (NASDAQ) index. The power-law cross correlations between the DJIA volatility and the NASDAQ volatility have been studied using the DCCA method [40] and the MFXDFA method [52]. The closing prices of the DJIA and the NASDAQ from 5 February 1971 to 25 January 2011 have been retrieved. The length of the time series is 10084 trading days. The return is de?ned as the daily difference of the logarithmic closing prices, and the volatility is de?ned as the absolute value of the return. Figure 4(a) illustrates in log-log scale the dependence of the ?uctuation functions Fxy (q,s) with respect to the scale s for q = ?4, q = 0, and q = 4 for the returns. Excellent power laws are observed spanning over two orders of magnitude for all the four algorithms. The resulting hxy functions are shown in Fig. 4(b). All four hxy functions are monotonically decreasing, indicating that the cross correlations between

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PHYSICAL REVIEW E 84, 016106 (2011)

10 10

5 2 ?1 ?4 ?7 ?10 q=0 q=4

θ=0 Fxy θ=0.5 Fxy

0.75 0.65 0.55 0.45 ?4 1.1 1.05

(b)

Fxy (q, s)

10 10 10 10

fxy (α)

q = ?4

hxy (q)

hθ=0 xy hθ=0.5 xy hθ=1 xy hDFA xy

1

(c)

0.8 0.6 0.4 0.3 1

(f)

(a)

0

θ=1 Fxy DFA Fxy

fxy (α)θ=0 fxy (α)θ=0.5 fxy (α)θ=1 fxy (α)DFA 0.4 0.5

10

10

θ=0 Fxy θ=0.5 Fxy

1

10

2

s

10

3

10

4

?2

q

0

2

hθ=0 xy hθ=1 xy

4

αxy

0.6

0.7

0.8

10 10 10 10

6

Fxy (q, s)

q=0 ?4 q=4 ?9

θ=1 Fxy DFA Fxy

1 0.95

(e)

fxy (αxy )

1

q = ?4

0.8 0.6 0.4 f (α)hθ=0 xy f (α)hθ=1 xy 0.8 0.9

(d)

0

hxy (q)

10

10

1

10

2

s

10

3

10

4

0.9 ?4

?2

q

0

2

4

αxy

1

1.1

FIG. 4. (Color online) Multifractal detrended cross-correlation analysis of (a–c) the return time series and (d–f) the volatility time series for the DJIA index and the NASDAQ index. Comparisons are performed among three MFXDMA algorithms with θ = 0, 0.5, and 1 and the MFXDFA method. (a, d) Dependence of the ?uctuation functions Fxy (q,s) with respect to the scale s for q = ?4, q = 0, and q = 4. The straight lines are the best power-law ?ts to the data. The results have been translated vertically for better visibility. (b, e) Scaling exponents hxy (q) with respect to q. (c, f) Multifractal spectra fxy (α) with respect to the singularity strength α.

the index returns exhibit a multifractal nature. We also ?nd that hθ=0 (q) > hθ=1 (q) > hDFA (q) > hθ=0.5 (q) xy xy xy xy (32) for ?4 q 4. When q = 2, all the hxy values are less than 0.6 and particularly hθ=0.5 ≈ 0.5. This means that there is no xy signi?cant linear long-term memory in the cross correlations of the returns. The multifractal spectra fxy (α) are plotted in Fig. 4(c). The singularity widths are all signi?cantly greater than 0, con?rming that the cross correlations possess a multifractal nature. Figure 4(d) illustrates in log-log scale the dependence of the ?uctuation functions Fxy (q,s) with respect to the scale s for q = ?4, q = 0, and q = 4 for the volatilities. Excellent power-law scaling is observed in the ?uctuation functions for the MFXDMA algorithms with θ = 0 and 1. However, for the MFXDFA algorithm and the MFXDMA algorithm with θ = 0.5, there is a crossover in each curve. If we treat each curve with two scaling ranges and perform analysis on each scaling range, the resulting hxy (q) functions are not monotonically decreasing, and the multifractal spectra fxy (α) are not concave. We thus focus on the two MFXDMA algorithms with θ = 0 and 1, which lead to one scaling range. The two hxy (q) functions are depicted in Fig. 4(e). The two functions are monotonically decreasing and close to each other. When q = 2, hxy is close to 0.98, showing a very strong linear long-term memory in the cross correlations between volatilities. We note that the relation hxy (q) = [hxx (q) + hyy (q)]/2 does not hold, which is consistent with previous work [52]. Figure 4(f) plots the two multifractal spectra. The large singularity width means

that the cross correlations between the two index volatilities exhibit a multifractal nature. Our results for the volatility seem different from those in Refs. [40,52]. First of all, the DJIA and NASDAQ time series are much longer in the current work. More importantly, the power-law scaling in the previous works exhibits signi?cant ?uctuations [40,52], which makes it dif?cult to determine a proper scaling range. According to Fig. 4(d), it is evident that the MFXDMA algorithms with θ = 0 and 1 signi?cantly outperform the MFXDFA algorithm.

V. CONCLUSION AND DISCUSSION

In this work, we have developed a class of MFDCCA algorithms based on the detrending moving-average analysis. The performances of the MFXDMA algorithms are compared with the MFXDFA method by extensive numerical experiments on pairs of time series generated from bivariate fractional Brownian motions, two-component autoregressive fractionally integrated moving-average processes, and binomial measures, which have theoretical expressions of the multifractal nature. In all cases, the scaling exponents hxy extracted from the MFXDMA and MFXDFA algorithms are very close to the theoretical values. For bivariate fractional Brownian motions, the scaling exponent hxy of the cross correlation is found to be independent of the cross-correlation coef?cient ρ between two time series. The MFXDFA and centered MFXDMA algorithms outperform the forward and backward MFXDMA algorithms. When Hxx = Hyy , the MFXDFA and centered MFXDMA

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PHYSICAL REVIEW E 84, 016106 (2011) APEENDIX: HIGHER-DIMENSIONAL MFXDMA

algorithms show comparable performance. When Hxx = Hyy , the centered MFXDMA algorithm performs slightly better than the MFXDFA algorithm. Our numerical experiments veri?ed the validity of the BFBM generating algorithm [35,36, 66]. For two-component autoregressive fractionally integrated moving-average processes, we also found that the MFXDFA and centered MFXDMA algorithms have comparative performances, while the forward and backward MFXDMA algorithms perform slightly worse. All four algorithms are able to correctly unveil the monofractal nature in the cross correlations between the components of BFBMs and twocomponent ARFIMA processes. For binomial measures, the forward MFXDMA algorithm exhibits the best performance, the centered MFXDMA algorithms performs worst, and the backward MFXDMA algorithm outperforms the MFXDFA algorithm when the moment order q < 0 and underperforms when q > 0. In all three mathematical models, the relation hxy = (hxx + hyy )/2 has been con?rmed for all four algorithms, where hxy , hxx , and hyy are estimated scaling exponents. Previous works have shown that the MFDFA and MFDMA algorithms are able to give nice estimates for univariate signals, that is, hxx ≈ Hxx and hyy ≈ Hyy . It follows immediately that hxy ≈ (Hxx + Hyy )/2. Combining the theoretical fact that Hxy ≈ (Hxx + Hyy )/2, we obtain Hxy ≈ hxy . For monofractal time series, extensive numerical experiments unveiled that the performance of the DMA algorithms is comparable to the DFA algorithm and the centered DMA algorithm performs slightly better than DFA under certain situations [17–19]. This explains our numerical results for BFBMs and twocomponent ARFIMA processes. For multifractal measures generated from the p model, the backward MFDMA algorithm performs best [15], which explains our ?ndings for MFDCCA algorithms. We applied these algorithms to the returns and volatilities of two US stock market indexes. For the returns, the centered MFXDMA algorithm gives the best estimates of hxy (q) since its hxy (2) is closest to 0.5, and the MFXDFA algorithm has the second best performance. For the volatilities, the forward and backward MFXDMA algorithms give similar results, while the centered MFXDMA and the MFXDFA algorithms fail to extract a rational multifractal nature. These two applications are interesting since they showed that the choice of algorithms is automatic, although we did not know which one should be used before the analysis. The key message of our work is that we should use all four algorithms and compare the results to make a choice. We note that the MFXDMA algorithms are easy to implement (see the Supplemental Material [72]).

In this work, we have focused on time series analysis. It is easy to generalize the one-dimensional MFXDMA algorithms to higher dimensions. The higher-dimensional MFXDMA algorithms are closely related to the MFXDFA algorithms [52], the DMA algorithms [21], and the MFDMA algorithms [15] in higher dimensions. Consider two physical quantities in d dimensions: {X(i1 , . . . , id )} and {Y (i1 , . . . , id )}, where ij = 1,2, . . . , Nj for j = 1,2, . . . , d. Before proceeding, we need to construct the difference matrices {x(i1 , . . . , id )} and {y(i1 , . . . , id )} of X and Y . For simplicity, we denote Z ∈ {X,Y } and z as the corresponding difference matrix, which are related by the following equation:

i1 id

Z(i1 , . . . , id ) =

j1 =1

···

jd =1

z(i1 , . . . , id ).

(A1)

The matrix z is expressed as a square block matrix of size 2d , whose block is z(i1 , . . . ,id ), where the intervals Ij = ij or [1 : ij ? 1] for j = 1,2, . . . ,d. In real-world applications, we can focus on d = 2 and d = 3. For the two-dimensional case d = 2, the four blocks are z(i1 ,i2 ), z(i1 ? 1,i2 ), z(i1 ,1 : i2 ? 1), and z(1 : i1 ? 1,1 : i2 ? 1). According to Eq. (A1), we have

i1 ?1

Z(i1 ,i2 ) = z(i1 ,i2 ) +

j =1 i2 ?1

z(j,i2 )

i1 ?1 i2 ?1

+

j =1

z(i2 ,j ) +

j1 =1 j2 =1

z(j1 ,j2 ).

(A2)

Since

i1 ?1 i2 ?1

z(j1 ,j2 ) = Z(i1 ? 1,i2 ? 1),

j1 =1 j2 =1 i1 ?1

z(j,i2 ) = Z(i1 ? 1,i2 ) ? Z(i1 ? 1,i2 ? 1),

j =1 i2 ?1

z(i2 ,j ) = Z(i1 ,i2 ? 1) ? Z(i1 ? 1,i2 ? 1),

j =1

it follows that z(i1 ,i2 ) = Z(i1 ,i2 ) + Z(i1 ? 1,i2 ? 1) ? Z(i1 ? 1,i2 ) ? Z(i1 ,i2 ? 1), (A3) where Z(i,j ) 0 if i × j = 0. For the three-dimensional case d = 3, the eight blocks are z(i1 ,i2 ,i3 ), z(1 : i1 ? 1,i2 ,i3 ), z(i1 ,1 : i2 ? 1,i3 ), z(i1 ,i2 ,1 : i3 ? 1), z(1 : i1 ? 1,1 : i2 ? 1,i3 ), z(1 : i1 ? 1,i2 ,1 : i3 ? 1), z(i1 ,1 : i2 ? 1,1 : i3 ? 1), and z(1 : i1 ? 1,1 : i2 ? 1,1 : i3 ? 1). We can derive similar to the two-dimensional case that z(i1 ,i2 ,i3 ) = Z(i1 ,i2 ,i3 ) ? Z(i1 ? 1,i2 ? 1,i3 ? 1) + Z(i1 ? 1,i2 ? 1,i3 ) + Z(i1 ? 1,i2 ,i3 ? 1) + Z(i1 ,i2 ? 1,i3 ? 1) ? Z(i1 ? 1,i2 ,i3 ) ? Z(i1 ,i2 ? 1,i3 ) ? Z(i1 ,i2 ,i3 ? 1), (A4)

ACKNOWLEDGMENTS

We are grateful to Jean-Francois Coeurjolly for providing ? the R code to generate bivariate fractional Brownian motions and to Anna Carbone, Gao-Feng Gu, Boris Podobnik, and Qun-Zhi Zhang for invaluable discussions. This work was partially supported by the National Natural Science Foundation of China under Grant No. 11075054 and the Fundamental Research Funds for the Central Universities.

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PHYSICAL REVIEW E 84, 016106 (2011)

where Z(i,j,k) 0 if i × j × k = 0. When i3 = 1, Eq. (A4) reduces to Eq. (A3). The algorithm of d-dimensional multifractal detrending moving-average cross-correlation analysis is described as follows. Step 1. For each quantity z = x or z = y, determine the moving averages Z(i1 , . . . , id ), where sj ij Nj ? (sj ? 1)θ1 and {θj } are the position parameters with the values varying in the range [0,1] for j = 1,2, . . . , d. For each point located at (i1 , . . . , id ) in the d-dimensional space, we extract a submatrix z(k1 , . . . , kd ) with size s1 × · · · × sd from the matrix z, where kj ∈ [ij ? (sj ? 1)(1 ? θj ) ,ij + (sj ? 1)θj ] [mj,1 ,mj,2 ]. We calculate the cumulative sums Z (k1 , . . . ,kd ) of the points within the box:

k1 k2

Step 4. Each residual matrix z (i1 , . . . ,id ) is partitioned into Ns1 × · · · × Nsd disjoint boxes of the same size s1 × · · · × sd , where Nsj = [Nj ? sj (1 + θj )]/sj . Each box z can be denoted by v1 ,...,vd for vj = 1, . . . ,Nsj such that z z (lv1 + k1 , . . . ,lvd + kd ) for 1 kj v1 ,...,vd (k1 , . . . ,kd ) = sj , where lvj = vj sj . The cross correlation between X and Y in each box is calculated as follows: Fv1 ,...,vd = 1 s1 · · · sd ×

s1 sd

···

x v1 ,...,vd (k1 , . . . ,kd )

k1 =1 kd =1 y v1 ,...,vd (k1 , . . . ,kd ).

(A9)

Step 5. The qth order overall detrending cross-correlation function Fq (n) is calculated as follows: [Fq (s)]q = 1 Ns1 · · ·Nsd

Ns1 Nsd

Z (k1 , . . . ,kd ) =

1 =m1,1

···

1 =md,1

z( 1 , . . . , d ),

(A5)

···

v1 =1 vd =1

|Fv1 ,...,vd |q/2 ,

(A10)

and the moving average Z(i1 , . . . ,id ) at location (i1 , . . . ,id ) is calculated as follows: 1 Z= s1 · · ·sd

m1,2 md,2

2 where s 2 = d=1 sj /d and q can take any real values except j for q = 0. When q = 0, we have

···

k1 =m1,1 k1 =md,1

Z (k1 , . . . ,kd ).

(A6)

1 ln[F0 (s)] = Ns1 · · ·Nsd

Ns1

Nsd

···

v1 =1 vd =1

ln |Fv1 ,...,vd |,

(A11)

Step 2. For each quantity, calculate the cumulative sums Q(i1 , . . . ,id ) in a sliding window with size s1 × · · · × sd , where sj ij Nj ? (sj ? 1)θj . For each point located at (i1 , . . . ,id ), we have

i1 id

according to L’H? pital’s rule. o Step 6. Varying the box sizes sj , we are able to determine the power-law relation between the ?uctuation function Fq (s) and the scale s, Fq (s) ? s h(q) . (A12) In real-world applications, one usually uses s1 = · · · = sd = s. When Nsj = [Nj ? sj (1 + θj )]/sj , one needs to start from different directions as for the DFA algorithm [73] or uses a random algorithm [74]. In addition, the box-by-box procedure is crucial for multifractal analysis, which was shown for high-dimensional MFDFA [20] and MFDMA [15]. However, the “traditional” procedure works well for highdimensional fractals [21].

Q=

k1 =i1 ?s1 +1

···

kd =id ?sd +1

z(k1 , . . . ,kd ).

(A7)

Step 3. Detrend the matrix by removing the movingaverage function Z(i1 , . . . ,id ) from Q(i1 , . . . ,id ), and obtain the residual matrix z (i1 ,i2 ) as follows:

z

(i1 , . . . ,id ) = Q(i1 , . . . ,id ) ? Z(i1 , . . . ,id ), ij Nj ? (sj ? 1)θj .

(A8)

where sj

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016106-11

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