This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

CHAOS 24, 022102 (2014)

Modified multidimensional scaling approach to analyze financial markets Yi Yina) and Pengjian Shangb) Department of Mathematics, School of Science, Beijing Jiaotong University, No. 3 of Shangyuan Residence, Haidian District, Beijing 100044, People’s Republic of China

(Received 11 February 2014; accepted 15 April 2014; published online 24 April 2014) Detrended cross-correlation coefficient (rDCCA) and dynamic time warping (DTW) are introduced as the dissimilarity measures, respectively, while multidimensional scaling (MDS) is employed to translate the dissimilarities between daily price returns of 24 stock markets. We first propose MDS based on rDCCA dissimilarity and MDS based on DTW dissimilarity creatively, while MDS based on Euclidean dissimilarity is also employed to provide a reference for comparisons. We apply these methods in order to further visualize the clustering between stock markets. Moreover, we decide to confront MDS with an alternative visualization method, “Unweighed Average” clustering method, for comparison. The MDS analysis and “Unweighed Average” clustering method are employed based on the same dissimilarity. Through the results, we find that MDS gives us a more intuitive mapping for observing stable or emerging clusters of stock markets with similar behavior, while the MDS analysis based on rDCCA dissimilarity can provide more clear, detailed, and accurate information on the classification of the stock markets than the MDS analysis based on Euclidean dissimilarity. The MDS analysis based on DTW dissimilarity indicates more knowledge about the correlations between stock markets particularly and interestingly. Meanwhile, it reflects more abundant results on the clustering of stock markets and is much more intensive than the MDS analysis based on Euclidean dissimilarity. In addition, the graphs, originated from applying MDS methods based on rDCCA dissimilarity and DTW dissimilarity, may also guide C 2014 AIP Publishing LLC. the construction of multivariate econometric models. V [http://dx.doi.org/10.1063/1.4873523] Recently, stock markets have become active areas and attracted much attention. Stock market indices are important measures of financial and economical performance and are normally used to benchmark the performance of stock portfolios. It is of great interest to analyze the correlations embedded in international stock markets. This study mainly proposes the multidimensional scaling (MDS) methods based on detrended crosscorrelation analysis (DCCA) coefficient (rDCCA) and dynamic time warping (DTW) as the dissimilarity measures, while the more traditional Euclidean dissimilarity is also employed to MDS to provide a reference. Then these methods are illustrated on daily price returns of 24 stock markets in order to further visualize the clustering between stock markets. This study also confronts MDS with the “Unweighed Average” clustering method based on the same dissimilarity, for comparison. The results suggests that MDS gives us a more intuitive mapping for observing stable or emerging clusters of stock markets with similar behavior, while the MDS analysis based on rDCCA dissimilarity can provide more clear, detailed, and accurate information on the classification of the stock markets than the MDS analysis based on Euclidean dissimilarity. The MDS analysis based on DTW dissimilarity indicates more knowledge about the correlation between stock markets particularly and interestingly. Meanwhile, it reflects more abundant results on the clustering of a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]. Tel.:þ86-10-51684410 and þ86-15210586330. b) [email protected] 1054-1500/2014/24(2)/022102/15/$30.00

stock markets and is much more intensive than the MDS analysis based on Euclidean dissimilarity. In addition, the graphs originated from applying MDS methods based on rDCCA dissimilarity and DTW dissimilarity may also guide the construction of multivariate econometric models. It will be helpful to understand the correlations between stock markets and to master the inner mechanism and dynamics of the financial markets, which can be beneficial to design good portfolios further.

I. INTRODUCTION

Recently, stock markets have become active areas and attracted much attention. Stock market indices are important measures of financial and economical performance and are normally used to benchmark the performance of stock portfolios. The study of the international stock markets is attractive for both fundamental and practical researches. On the fundamental side, a financial market has been referred to as an example of a complex system consisting of many interacting components. An abrupt variant on a certain component can be spatially affected by the others as well as itself temporally. Statistical motivations are to visualize correlations in order to suggest some potentially plausible parameter relations and restrictions. On the practical side, economic motivations are to identify the main factors which affect the behavior of stock markets across different exchanges and countries. Moreover, it is important for evaluating the risk of an investment in the stock market. The understanding of

24, 022102-1

C 2014 AIP Publishing LLC V

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

022102-2

Y. Yin and P. Shang

such relations would be helpful to design good portfolios.1,2 It is therefore of great interest to analyze the correlations embedded in international stock markets. In the previous studies, there are many different methods, which have been used to the study of the international stock markets. Considerable research efforts during the last years demonstrated that these signals have a chaotic nature and require sophisticated mathematical tools for analyzing their characteristics. Classical methods, such as the Fourier transform, reveal considerable limitations in discriminating different periods of time. The paper3 analyzes the dynamical properties of financial data series from worldwide stock market indices. Meanwhile, the paper4 proposes a graphical method using MDS and dendograms to observe stable or emerging clusters of stock markets with similar behavior. Moreover, some papers have applied MDS and Fourier Transform to analyze different periods of the business cycle5 and the dynamical properties of financial data series6 and visualize possible time-varying correlations between stock market values.7 Our paper follows these works and is in the sequel of these previous papers.3–7 In this paper, the MDS visualization tool is adopted. Multidimensional scaling is a statistical technique originating in psychometrics. The data used for MDS are dissimilarities between pairs of objects. The main objective of MDS is to represent these dissimilarities as distances between points in a low dimensional space such that the distances correspond as closely as possible to the dissimilarities. Summarily, MDS is a technique that translates the dissimilarities between pairs of objects into a map where distances between the points match the dissimilarities as well as possible. The use of MDS is not limited to psychology but has applications in a wide area of disciplines such as sociology, economics, biology, chemistry, and archaeology. Often, it is used as a technique for exploring the data. In addition, it can be used as a technique for dimension reduction. The term dissimilarity is used to indicate the degree of unlikeness between two objects. Generally, MDS uses the Euclidean distance as a measure of the dissimilarity between two time series. However, we propose the rDCCA18 to obtain the dissimilarity measure and further apply it to the MDS procedure. The DCCA method, recently proposed by Podobnik and Stanley8 for the analysis of power law cross-correlations between simultaneously recorded non-stationary time series, represents a strong candidate as a tool for cross-correlation studies. Thus, DCCA method has been applied to investigate the cross-correlation in many previous studies.9–16 The scaling exponent k obtained by DCCA quantifies the long-range power-law cross-correlations and also identifies seasonality,17 but k does not quantify the level of cross-correlations. To quantify the level of cross-correlation, we can apply the DCCA cross-correlation coefficient.18 Using the dissimilarity measure obtained by rDCCA method on international financial indices, we can simply compare original data on the same day from around the world. There may be regional seasonal affects inherent in the raw data. For instance, public holidays occur on a variety of dates and will affect values. Also there are year-by-year changes. For the MDS combined

Chaos 24, 022102 (2014)

with rDCCA, the data have been pre-processed to days on which all financial markets considered were in operation. It is for this reason that DTW is employed to find a best match between a pair of time series. DTW is a well-known technique to find an optimal alignment between two given (time-dependent) sequences under certain restrictions. Intuitively, the sequences are “warped” non-linearly in the time dimension to determine a measure of their similarity independent of certain non-linear variations in the time dimension. This sequence alignment method is often used in time series classification. Originally, DTW has been successfully applied to video, audio, and graphics—indeed, any data, which can be turned into a linear representation, can be analyzed with DTW. A well known application is automatic speech recognition, to cope with different speaking speeds. Other applications include speaker recognition and online signature recognition. Also it is seen that it can be used in partial shape matching application. Thus, the dissimilarity measure obtained by DTW is employed in the MDS procedure. Bearing these ideas in mind, our paper is organized as follows. Section II briefly introduces the fundamental methods adopted in the study, including the MDS based on Euclidean dissimilarity, MDS based on rDCCA dissimilarity, and MDS based on DTW dissimilarity, which are the core of our paper. In Sec. III, we apply these methods for daily data on 24 stock markets, including major American, Asian/Pacific, and European stock markets. In Sec. IV, we present the MDS analyses based on Euclidean dissimilarity, rDCCA dissimilarity, and DTW dissimilarity. Finally, Sec. V gives a brief conclusion and provides some potential topics for our further research. II. METHODOLOGIES A. MDS based on Euclidean dissimilarity

Multidimensional scaling has become more and more popular as a technique for both multivariate and exploratory data analyses. MDS techniques develop spatial representations of psychological stimuli or other complex objects about which people make judgments (e.g., preference, relatedness), that is they represent each object as a point in a m-dimensional space. What distinguishes MDS from other similar techniques (e.g., factor analysis, cluster analysis) is that in MDS there are no preconceptions about which factors might drive each dimension. Thus, the raw data entering into a MDS analysis are typically a measure of the global similarity or dissimilarity of the stimuli or objects under investigation. The result of a MDS analysis is the transformation of the data into similarity measures which can be represented by Euclidean distances in a space of unknown dimensions.19 After having the distances between all the objects, the MDS techniques analyze how well they can be fitted by spaces of different dimensions. Then we can get a spatial configuration, in which the objects are represented as points. The points in this spatial representation are arranged in such a way that their distances correspond to the similarities of the objects: similar object are represented by points that are close to each other, dissimilar objects by points that are far

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

022102-3

Y. Yin and P. Shang

apart and the greater the similarity of two objects, the closer they are in the m-dimensional space. The analysis is normally made by gradually increasing the number of dimensions until the quality of fit (measured for example by the correlation between the data and the distance) is little improved with the addition of a new dimension (i.e., the formation of an “elbow” shape curve). In practice, a good result is normally reached well before the number of dimensions theoretically needed to a perfectly fit is reached.20–23 In the MDS method, a small distance between two points corresponds to a high correlation between two stock markets and a large distance corresponds to low or even negative correlation.24 One correlation should lead to zero distance between the points representing perfectly correlated stock markets. MDS tries to estimate the distances for all pairs of stock markets to match the correlations as close as possible. Hence, MDS may be seen as an exploratory technique without any distributional assumptions on the data. The distances between the points in the MDS maps are generally not difficult to interpret and thus may be used to formulate more specific models or hypotheses. Besides, the distance between two points should be interpreted as being the distance conditional on all the other distances. Note that, as distances do not change under rotation, a rotation of the plot does not affect the interpretation. Similarly, a translation of the solution (that is, a shift of all coordinates by a fixed value per dimension) does not change the distances either, nor does a reflection of one or both of the axes. As a result, the obtained representation of points is not unique.25 One possibility to obtain such an approximate solution is given by minimizing the stress function. The MDS procedure can be described as follow. Let n be the number of different objects and let the dissimilarity for objects i and j be given by dij. The coordinates are gathered in an n p matrix X, where p is the dimensionality of the solution to be specified in advance. Thus, row ir from X gives the coordinates for object i on dimension r. Let dij be the Euclidean distance between rows i and j of X defined as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Pp 2 dij ¼ r¼1 ðxir xjr Þ that is the length of the shortest line connecting points i and j on dimension r. The objective of MDS is to find a matrix X such that dij matches dij as closely as possible. This objective can be formulated in a variety of ways but here we use the Kruskal’s Stress-1 r1, ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ sP n Pi1 2 i¼2 j¼1 wij ðdij dij Þ r1 ¼ proposed by Kruskal,26 who Pn Pi1 2 d w i¼2 j¼1 ij ij was the first one to propose a formal measure for doing MDS, where wij is a defined weight that must be non-negative. Note that due to the symmetry of the dissimilarities and the distances, the summation only involves the pairs i, j where i > j. For example, many MDS programs implicitly choose wij ¼ 0 for dissimilarities that are missing. The advantage of this measure is that its value is independent of the scale and the number of dissimilarities. The minimization of r1 is a complex problem. Therefore, MDS programs use iterative numerical algorithms to find a matrix X for which r1 is a minimum. In addition to the Kruskal’s Stress-1 measure, there exist other measures for doing stress. One of them is simply the square of the Kruskal’s Stress-1. The second measure is Kruskal’s Stress-2,

Chaos 24, 022102 (2014)

which is similar to Stress-1 except that the denominator is based on the variance of the distances instead of the sum of squares. Another measure that seems reasonably popular is called Young’s S-Stress and it measures the sum of squared errors between squared distances and squared dissimilarities.27 In order to assess the quality of the MDS solution, we can study the differences between the MDS solution and the original data. One convenient way to do this is by inspecting the so-called Shepard diagram.28 A Shepard diagram displays both the transformation and the error. Let pij denote the proximity between objects i and j. Then, a Shepard diagram plots simultaneously the pairs (pij, dij) and (pij, dij). By connecting the points (pij, dij), a line is obtained representing the relationship between the proximities and the disparities. The vertical distances between the points (pij, dij) and (pij, dij) are equal to dij dij, showing the errors of representation for each pair of objects. Thus, the Shepard diagram can be applied to reveal the transformation and residuals of the MDS solution. MDS based on Euclidean dissimilarity employs the Euclidean distance P (Euc) which is generally used for MDS, where Eucðx; yÞ ¼ ni¼1 ðxi yi Þ2 , as the dissimilarity and further obtain the dissimilarity matrix E. In matrix E, each cell represents the Euclidean distance between a pair of indexes. MDS based on Euclidean dissimilarity means applying MDS based on the Euclidean dissimilarity matrix. B. MDS based on rDCCA dissimilarity

MDS based on rDCCA dissimilarity is the combination of MDS and rDCCA coefficient. We first introduce the algorithm of DCCA cross-correlation coefficient18 method. Detrended fluctuation analysis (DFA)29 is one of the most frequently cited methods to analyze time series of complex problems. This method provides a relationship between FDFA(s) (root mean square fluctuation) and the scale s. DFA method has been very efficient at detecting long-range autocorrelations embedded in a patch landscape and also avoiding spurious detection of apparent long-range auto-correlations. This fact can be proved by a great number of applications and citations.29–38 However, if we have two time series, xk and yk, the analysis of cross-correlation can be applied, like in Refs. 39–47. In this paper, we propose to analyze and quantify cross-correlation for global stock market data. We adopt the recently proposition implemented by Zebende,18 based on the DCCA method.8 The DCCA method is a generalization of the DFA method and is based on detrended covariance. This method is designed to investigate power-law cross-correlations between different simultaneously recorded time series in the presence of nonstationarity. The DCCA procedure consists of five steps and can be briefly described as follows. Step 1: Consider two time series {xk, k ¼ 1, 2, …, N} and {yk, k ¼ 1, 2, …, N} where N is the equal length of the time series. Then, we determine the profile {Xi} and {Yi} Xi ¼

i X ½xk x; k¼1

Yi ¼

i X ½yk y

(1)

k¼1

for i ¼ 1, 2, …, N.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

022102-4

Y. Yin and P. Shang

Chaos 24, 022102 (2014)

Step 2: Divide the profile {Xi} and {Yi} into Ns ¼ int(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 series, the same procedure is repeated starting from the other end of the profile series. Thus, 2Ns segments are obtained altogether. We set 10 < s < N/4. Step 3: Calculate the local trends for each of the 2Ns segments by a least-square fit of each series. Then we calculate the difference between the original time series and the fitting polynomial 2 ðs; vÞ ¼ fDCCA

s 1X ½X½ðv1Þsþi X~v;i ½Y½ðv1Þsþi Y~v;i (2) s i¼1

for v ¼ 1, 2,…, Ns and 2 ðs; vÞ ¼ fDCCA

s 1X ½X½NðvNs Þsþi X~v;i ½Y½NðvNs Þsþi Y~v;i s i¼1

(3) for v ¼ Ns þ 1,…, 2Ns. X~v;i and Y~v;i are computed from quadratic polynomial fit. Step 4: Average over all segments to get the fluctuation function ( FDCCA ðs; vÞ ¼

2Ns 1 X f 2 ðs; vÞ 2Ns v¼1 DCCA

)1=2 :

(4)

Step 5: Determine the scaling behavior of the fluctuation function by analyzing log–log plot of FDCCA(s) versus s: FDCCA ðsÞ ðsÞk . The scaling exponent k represents the degree of the cross-correlation between the two time series {xk, k ¼ 1, 2, …, N} and {yk, k ¼ 1, 2, …, N}. The value k ¼ 0.5 indicates the absence of cross-correlation. k > 0.5 indicates persistent long-range cross-correlation, meaning that a large value in one time series is more likely to be followed by a large value in another series, k < 0.5 indicates anti-persistent cross-correlation, meaning that a large value in one time series is more likely to be followed by a small value in another series and vice versa. The k exponent quantifies the long-range power-law cross-correlations and also identifies seasonality,17 but k does not quantify the level of cross-correlations. To quantify the level of cross-correlation, we can apply the DCCA crosscorrelation coefficient,18 defined as the ratio between the detrended covariance function F2DCCA and the detrended varF2

iance function FDFA, i.e., rDCCA ¼ FDFAxDCCA FDFAy . This equak

k

tion leads us to a new scale of cross-correlation in nonstationary time series. The rDCCA ranges from 1 to þ1, where the value 1 indicates complete anti-cross-correlation between the series, the value 1 indicates perfect crosscorrelation, and the value 0 corresponds to the absence of cross-correlation. We calculate these DCCA cross-correlation coefficients between different time series and then obtain their rDCCA curves. We can observe whether these curves fluctuate

around their averages and their standard deviations are very small generally. Moreover, all these DCCA cross-correlation coefficients range from 0 to 1, which indicates the crosscorrelations between these series. Thus, we decide to compute the averages of the DCCA cross-correlation coefficients to be representatives of the dissimilarities among the series in order to obtain a matrix and then apply MDS on the rDCCA dissimilarity matrix if these curves fluctuate around their averages and their standard deviations are very small. In this presentation, points represent the time series. In matrix, each cell represents the average rDCCA between a pair of time series. C. MDS based on DTW dissimilarity

MDS based on DTW dissimilarity is based on the algorithms of DTW and MDS. DTW is first introduced by Bellman and Kalaba48 and is designed to compare two sequences of points and to find a best match between a pair of time series. A cost function is defined between any pair of observations, with one selected from each series. The cost function most usually adopted is the square of the Euclidean distance between the observations. The total cost of comparing the two sequences is then the sum of the cost for each pair of corresponding observations defined by the warping of time. We wish to find the best warping that gives the lowest cost for comparing the two sequences of observations. Since DTW assumes continuity and monotonicity of the time, the minimum cost can be computed recursively by computing the minimum cost between two sub-sequences starting at the origin, then stepping forward in time, either on one sequence or the other or both and computing the minimum cost between the new sequences by adding the cost of the newly matched points. We are only interested in the final cost, which is the value of the dissimilarity between the two point sequences. More formally, DTW is a dynamic programming method that allows an optimal match to be found between two given time series with arbitrary lengths. The series are warped nonlinearly in the time dimension to determine a measure of their dissimilarity, independent of any nonlinear variations in the time dimension. For a useful review, see Ref. 49 in which some of the following introductory discussion is loosely based. As a general approach, time series of different lengths are assumed. Here, the first series is denoted by xi: i ¼ 1,2, …, n, it is indexed by i and of length n, the second time series is yj: j ¼ 1,2, …, n, which is indexed by j and of length m. The procedure is a time series association algorithm; it relates two time series by warping the time axis of one series onto the other. Because DTW procedure is a dynamic programming technique, the problem is divided into a series of sub-problems, each of which contributes to calculating the cumulative distance. By DTW(x1, …, xi, y1, …, yj), we denote the minimum time warped cost between (x1, …, xi) and (y1, …, yj) written, for brevity, as DTW(i, j). The recursion scheme that governs the computation is DTWði; jÞ ¼ dðxi ; yj Þ þ minðDTWði 1; jÞ; DTWði; j 1Þ; DTWði 1; j 1ÞÞ:

(5)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

022102-5

Y. Yin and P. Shang

Chaos 24, 022102 (2014)

The new minimum cost is related to the minimum of those at a previous stage. Here, d(xi, yj) is the distance between the two observations xi and yj. In this case, dðxi ; yj Þ ¼ ðxi yj Þ2 was used, to provide a natural comparison to the Euclidean dissimilarity, which is employed below. An alternative would be to utilize dðxi ; yj Þ ¼ jxi yj j. The DTW algorithm can be briefly summarized as follow: DTW ð0; 0Þ ¼ 0; DTWði; 0Þ ¼ 1;

i ¼ 1; …; n;

DTWð0; jÞ ¼ 1;

j ¼ 1; …; m;

DTWði; jÞ ¼ dðxi ; yj Þ þ minðDTWði 1; jÞ; DTWði 1; j 1ÞÞ;

i ¼ 1; …; n;

DTWði; j 1Þ;

j ¼ 1; …; m:

The aim of the algorithm is to find DTW(n, m), the total minimum cost. MDS based on DTW dissimilarity method utilizes the DTW value as the dissimilarity between a pair of series to obtain a DTW dissimilarity matrix, and then apply the MDS method on the DTW dissimilarity matrix and get the stress and Shepard plots.

length, while MDS methods based on DTW dissimilarity can be employed to time series of different lengths. Because these stock markets have the different opening dates, we exclude the asynchronous datum and then reconnect the remaining parts of the original series to obtain the same length time series when employing the MDS based on Euclidean dissimilarity and rDCCA dissimilarity. As a result, the pre-processed data consist of 2132 daily closing prices. The daily closing prices of the 24 stock markets are illustrated in Fig. 1. Let xt denote the closing price of stock market on day t. The daily price return, rt, is calculated as its logarithmic difference, rt ¼ logðxt Þ logðxt1 Þ. Assuming that stock market indexes are random variables, one of the most important parameter for the analysis of stock market indexes is the volatility.50 Volatility measures variability or dispersion about a central tendency and is normally defined as the deviation from their mean. The historical volatility is the volatility of a series of indexes where we look back over the historical index. The historical volatility estiqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ph 1 Þ2 mate, for each index i, is given by w ¼ h1 t¼1 ðrt r where h is the number of the data and r is the arithmetic average of the rt. The parameter volatility values w for the 24 indexes are shown in Table I. IV. ANALYSIS AND RESULTS

III. DATA

We choose daily closing data of 24 stock markets, including eight American markets, eight European markets, and 16 Asian/Pacific markets from January 2, 2001 to December 31, 2012. The stock markets are listed in Table I. MDS based on Euclidean dissimilarity and rDCCA dissimilarity on international financial indices restrict the time series to be of equal

A. MDS analysis based on Euclidean dissimilarity

We first employ MDS based on Euclidean dissimilarity to reveal possible relationships between the stock market indexes. In this perspective, several MDS criteria are tested. The Sammon criterion reveals good results3,5,51 and is adopted in this work. Figs. 2 and 3 depict the stress and the Shepard

TABLE I. Twenty-four stock markets and value of volatility values. i 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Stock market index All Ordinaries equities markets Australian S&P200 index EURONEXT BEL-20 San Paulo (Brazil) Stock Cotation Assisee en Continu Deutscher Aktien Index Dow Jones Industrial Average Footsie S&P/TSX Composite Index Stock Market Index in Hong Kong Iberia Index Bombay Stock Exchange Index Stock market index of South Korea Italian Bourse Bolsa Mexicana de Valores Nikkei 225 Nasdaq Composite Index Russian Trading System Stock Exchange Swiss Market Index Standard & Poor’s Straits Times Index Shanghai Stock Exchange Shenzhen Stock Exchange Taiwan Stock Exchange

Abbreviation AORD ASP200 BFX BVSP CAC DAX DJI FTSE GSPTSE HIS IBEX BSE KOSPI MIB MXX N225 NQCI RTS SMI SP500 STI SSEC SZSEC TWII

Country

Volatility (w)

Australia Australia Belgium Brazil France Germany USA UK Canada HongKong Spain India South Korea Italy Mexico Japan USA Russia Switzerland USA Singapore China China Taiwan

0.240998 0.24684 0.32551 0.4409568 0.370558 0.385895 0.289314 0.301683 0.2753277 0.358564 0.371182 0.387743 0.374106 0.382907 0.324514 0.363697 0.378303 0.555219 0.295158 0.3083888 0.309339 0.392525 0.439736 0.353545

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

022102-6

Y. Yin and P. Shang

Chaos 24, 022102 (2014)

FIG. 1. The daily closing prices of the 24 stock markets from January 2, 2001 to December 31, 2012.

plots for the MDS, respectively. The stress plot, as function of the dimension of the representation space, reveals that a three dimensional space describes the “map” of the 24 signal indexes with a reasonable accuracy. Moreover, the Shepard plot shows that a good distribution of points around the 45 line is obtained. Fig. 4 shows the 3D locus of each index positioning in the perspective of Euclidean dissimilarity. By observing Fig. 4, the indexes seem to be organized according

to their characteristics on the three dimensional MDS suggesting that we may group the 24 indexes into five clusters: (i) cluster A: SSEC, SZSEC; (ii) cluster B: RTS; (iii) cluster C: BVSP; (iv) cluster D: AORD, ASP200, KOSPI, TWII, HSI, BSE, N225, STI; and (v) cluster E: BFX, CAC, DAX, DJI, FTSE, GSPTSE, IBEX, MIB, MXX, NQCI, SMI, SP500. It is decided to confront MDS with an alternative visualization method for comparison. For this purpose,

FIG. 2. Stress plot of 3D MDS representation of the 24 indexes vs number of dimension using Euclidean dissimilarity.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

022102-7

Y. Yin and P. Shang

Chaos 24, 022102 (2014)

FIG. 3. Shepard plot of 3D MDS representation of the 42 indexes vs number of dimension using Euclidean dissimilarity.

dendograms using the same information matrices of the MDS case are adopted. To generate the dendograms, we selected the MultiDendograms hierarchical clustering package, configured for the “Unweighed Average” clustering method.52,53 Several other methods (Single Linkage,

Complete Linkage, Weighted Average, Unweighed Centroid, Weighted Centroid, Join Between-Within) are tested leading to dendograms qualitatively of the same type. The dendogram of Fig. 5 shows the hierarchical clustering of the 24 indexes with matrix E based on Euclidean

FIG. 4. Three dimensional MDS graph for the 24 indexes using Euclidean dissimilarity.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

022102-8

Y. Yin and P. Shang

Chaos 24, 022102 (2014)

FIG. 5. Dendogram for the 24 indexes using Euclidean dissimilarity. The number of variable at the vertical axis, i.e., VAR000i is corresponding to the stock market index i in Table I.

dissimilarity. The number of variables, i.e., VAR000i at the vertical axis in Fig. 5, are corresponding to the stock market index “i” in Table I. We observe that the two visualization techniques give the same conclusions on the clustering. We explore each cluster: The cluster A includes the two Chinese mainland stock markets: SSEC and SZSEC. It may be caused by the major influence of the regulation and control imposed by government on the stock markets, and the special mechanism of the Chinese mainland market. The indexes grouped on clusters B and C (i.e., RTS and BVSP) are the ones with the highest and the second highest volatilities (i.e., 0.555 and 0.441). The reason why the clusters B and C emerge may be that investors have a different behavior in volatile markets. Besides, the standard financial theory suggests that there is a negative relation between volatility and expected return.54,55 Hence, when the value of portfolios moves more violently, some investors worry and may make irrational responses or at least decisions which are not aligned with the normal practices. The cluster D is composed of the two stock markets from Australia and the stock markets from Asia except SSEC and SZSEC, while the cluster E is made up of the

stock markets belonging to Europe and America. The clusters D and E group the majority of the indexes and may be considered to represent the norm. Moreover, MDS gives us a more intuitive mapping which makes the cross-correlation between them understandably clear. B. MDS analysis based on rDCCA dissimilarity

In this subsection, we plan to apply MDS based on rDCCA dissimilarity for the selected stock markets. For all the 24 markets, we consider their rDCCA between the daily price returns. We first calculate these DCCA crosscorrelation coefficients between the 24 markets and then obtain their rDCCA curves. Although we do not show these rDCCA curves in order to shorten this paper, we find that these curves fluctuate around their averages and their standard deviations are very small generally. Moreover, all these DCCA cross-correlation coefficients range from 0 to 1, which indicates the cross-correlations between these markets. Thus, we decide to compute the averages of the DCCA cross-correlation coefficients to be representatives of the

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

022102-9

Y. Yin and P. Shang

Chaos 24, 022102 (2014)

FIG. 6. Stress plot of 3D MDS representation of the 24 indexes vs number of dimension using rDCCA dissimilarity.

dissimilarities among the 24 stock markets in order to obtain a 24 24 matrix and then apply MDS. In this presentation, points represent the stock markets. In matrix, each cell represents the average rDCCA between a pair of indexes. Thus,

we apply MDS based on rDCCA dissimilarity to analyze these 24 stock indexes. Fig. 6 depicts the stress as function of the dimension of the representation space based on rDCCA dissimilarity, revealing that a three dimensional

FIG. 7. Shepard plot of 3D MDS representation of the 24 indexes vs number of dimension using rDCCA dissimilarity.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

022102-10

Y. Yin and P. Shang

Chaos 24, 022102 (2014)

FIG. 8. Three dimensional MDS graph for the 24 indexes using rDCCA dissimilarity.

FIG. 9. Dendogram for the 24 indexes using rDCCA dissimilarity. The number of variable at the vertical axis, i.e., VAR000i is corresponding to the stock market index i in Table I.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

022102-11

Y. Yin and P. Shang

space describes with reasonable accuracy the “map” of the 24 indexes similar to that seen in the MDS based on Euclidean dissimilarity, and Fig. 7 shows the Shepard plot for the MDS. Moreover, the resulting Shepard plot also shows that a good distribution of points around the 45 line is obtained for the average of rDCCA. Fig. 8 displays the 3D locus of each index positioning in the perspective of the average of rDCCA. Similarly, we obtain the dendogram of Fig. 8 which shows the hierarchical clustering of the 24 indexes, using the “Unweighed Average” clustering method based on rDCCA dissimilarity matrix in Fig. 9. As a result, the 24 indexes are grouped into six clusters by the MDS analysis based on rDCCA dissimilarity: (i) cluster A: SSEC, SZSEC; (ii) cluster B: RTS; (iii) cluster C: BVSP; (iv) cluster D: AORD, ASP200; (v) cluster E: HSI, STI, N225, BSE, TWII, KOSPI, GSPTSE, MXX; (vi) cluster F: BFX, CAC, DAX, FTSE, IBEX, MIB, SMI, SP500, DJI, NQCI. Meanwhile, the grouping result of dendogram represents: (i) cluster A: SSEC, SZSEC; (ii) cluster B: RTS; (iii) cluster C: AORD, ASP200; (iv) cluster D: HSI, STI, N225, BSE, TWII, KOSPI, GSPTSE, MXX, BVSP; (v) cluster E: BFX, CAC, DAX, FTSE, IBEX, MIB, SMI, SP500, DJI, NQCI. The conclusions on the grouping received by the two visualization techniques are similar, while the MDS analysis points out the BVSP as one cluster comparing with the hierarchical clustering, indicating the more accurate and detailed grouping obtained by MDS. The reasons why the clusters A, B, C, and F emerge are same with the reasons why the clusters A, B, C, and E emerge in Subsection IV A by the MDS analysis based on Euclidean dissimilarity respectively. The MDS analysis based on rDCCA dissimilarity further divides the cluster D obtained by the MDS analysis based on Euclidean dissimilarity into cluster D and E in this subsection. The cluster D is composed of the two stock markets from Australia, while the cluster E is made up of the stock markets from Asia except SSEC and SZSEC and two stock

Chaos 24, 022102 (2014)

markets from America: GSPTSE, MXX. Hence, the MDS analysis based on rDCCA dissimilarity can provide more clear, detailed, and accurate information on the classification of the stock markets than the MDS analysis based on Euclidean dissimilarity. C. MDS analysis based on DTW dissimilarity

DTW is designed to compare two sequences of points and to find a best match between a pair of time series. We utilize the DTW value as the dissimilarity between a pair of indexes. Hence, DTW is employed to these stock market indexes in order to provide a DTW dissimilarity matrix. Then based on the DTW dissimilarity matrix, we apply the MDS method and get the stress and Shepard plots in Figs. 10 and 11, respectively. By means of Figs. 10 and 11, we find that 3D locus of each index positioning in Fig. 12 is the appropriate visualization. What is more, the hierarchical clustering of these indexes, using the “Unweighed Average” clustering method based on DTW dissimilarity matrix, is shown in Fig. 13. It is obvious that the three dimensional MDS graph using DTW dissimilarity suggests the seven clusters of the 24 indexes: (i) cluster A: SSEC, SZSEC; (ii) cluster B: RTS; (iii) cluster C: BVSP; (iv) cluster D: NQCI; (v) cluster E: FTSE, SMI, BFX, IBEX, MIB, CAC, DAX; (vi) cluster F: BSE; (vii) cluster G: AORD, ASP200, GSPTSE, MXX, DJI, SP500, STI, KOSPI, TWII, HSI, N225. In addition, the hierarchical clustering based on the same DTW dissimilarity matrix also presents seven clusters: (i) cluster A: SSEC, SZSEC; (ii) cluster B: RTS; (iii) cluster C: BVSP; (iv) cluster D: NQCI; (v) cluster E: FTSE, SMI, BFX, IBEX, MIB, CAC, DAX; (vi) cluster F: AORD, ASP200, GSPTSE, MXX, DJI, SP500, STI; (vii) cluster G: BSE, KOSPI, TWII, HSI, N225. The clustering obtained by MDS analysis based on DTW dissimilarity separates BSE from the cluster G in the

FIG. 10. Stress plot of 3D MDS representation of the 24 indexes vs number of dimension using DTW dissimilarity.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

022102-12

Y. Yin and P. Shang

Chaos 24, 022102 (2014)

FIG. 11. Shepard plot of 3D MDS representation of the 24 indexes vs number of dimension using DTW dissimilarity.

hierarchical clustering based on the same DTW dissimilarity to be the cluster F. After RTS, BVSP, SZSEC, and SSEC, BSE possesses the fifth highest volatility. It may be the reason why it can be a cluster. Besides, MDS analysis based on DTW dissimilarity combines cluster F and the rest of cluster G in the hierarchical clustering to be the cluster G.

Compared with the MDS analyses based on Euclidean dissimilarity and rDCCA dissimilarity, the MDS analysis based on DTW dissimilarity points out the cluster D: NQCI and cluster F: BSE and groups the stock markets from Europe into the cluster E, indicating more knowledge about the correlation between stock markets particularly and interestingly.

FIG. 12. Three dimensional MDS graph for the 24 indexes using DTW dissimilarity.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

022102-13

Y. Yin and P. Shang

Chaos 24, 022102 (2014)

FIG. 13. Dendogram for the 24 indexes using DTW dissimilarity. The number of variable at the vertical axis, i.e., VAR000i is corresponding to the stock market index i in Table I.

In the previous study,56 we have known that NQCI is quite different to the SP500 and DJI indices because it is much more comprehensive than they are. As a result, NQCI is selected individually and further distinct from the other stock markets from America, Europe, and Australia in the cluster G of the MDS analysis based on DTW dissimilarity. The MDS analysis based on DTW dissimilarity reflects more abundant results on the clustering of stock markets and is much more intensive than the MDS analysis based on Euclidean dissimilarity. V. CONCLUSIONS

In this paper, DCCA cross-correlation coefficient and dynamic time warping are introduced as the dissimilarity measures, respectively, while multidimensional scaling is employed to translate the dissimilarities between daily price returns of 24 stock markets. We first propose MDS based on rDCCA dissimilarity and MDS based on DTW dissimilarity creatively, while MDS based on Euclidean dissimilarity is also be employed to provide a reference for comparisons.

We apply these methods in order to further visualize the clustering between stock markets. Moreover, we decide to confront MDS with an alternative visualization method, “Unweighed Average” clustering method, for comparison. The MDS analysis and “Unweighed Average” clustering method are employed based on the same dissimilarity. The two visualization techniques based on Euclidean dissimilarity give the same conclusions on the clustering. The cluster A includes the two Chinese mainland stock markets: SSEC and SZSEC. It may be caused by the major influence of the regulation and control imposed by government on the stock markets and the special mechanism of the Chinese mainland market. The indexes grouped on clusters B and C (i.e., RTS and BVSP) are the ones with the highest and the second highest volatilities (i.e., 0.555 and 0.441). The reason why the clusters B and C emerge may be that investors have a different behavior in volatile markets. The clusters D and E group the majority of the indexes and may be considered to represent the norm. Moreover, MDS gives us a more intuitive mapping for observing stable or emerging clusters of stock markets with similar behavior. The clustering conclusions by

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

022102-14

Y. Yin and P. Shang

the two visualization techniques based on rDCCA dissimilarity are similar, while the MDS analysis points out the BVSP as one cluster comparing with the hierarchical clustering, indicating the more accurate and detailed grouping obtained by MDS. The reasons why the clusters A, B, C, and F emerge are same with the reasons why the clusters A, B, C, and E emerge by the MDS analysis based on Euclidean dissimilarity, respectively. The MDS analysis based on rDCCA dissimilarity further divides the cluster D obtained by the MDS analysis based on Euclidean dissimilarity into clusters D and E in this subsection, which separates the two Australian stock markets from these markets. Hence, the MDS analysis based on rDCCA dissimilarity can provide more clear, detailed, and accurate information on the classification of the stock markets than the MDS analysis based on Euclidean dissimilarity. The clustering obtained by MDS analysis based on DTW dissimilarity separates BSE from the cluster G in the hierarchical clustering based on the same DTW dissimilarity to be the cluster F. After RTS, BVSP, SZSEC, and SSEC, BSE possesses the fifth highest volatility. It may be the reason why it can be a cluster. Besides, MDS analysis based on DTW dissimilarity combines cluster F and the rest of cluster G in the hierarchical clustering to be the cluster G. Compared with the MDS analyses based on Euclidean dissimilarity and rDCCA dissimilarity, the MDS analysis based on DTW dissimilarity points out the cluster D: NQCI and cluster F: BSE and groups the stock markets from Europe into the cluster E, indicating more knowledge about the correlation between stock markets particularly and interestingly. Perhaps this may be explained by the fact that NQCI is quite different to the SP500 and DJI indices because it is much more comprehensive than they are, let alone correlation to the other stock markets from America, Europe, and Australia in the cluster G of the MDS analysis based on DTW dissimilarity. As a result, NQCI is selected individually. The MDS analysis based on DTW dissimilarity reflects more abundant results on the clustering of stock markets and is much more intensive than the MDS analysis based on Euclidean dissimilarity. In addition, the graphs, originated from applying MDS methods based on rDCCA dissimilarity and DTW dissimilarity, may also guide the construction of multivariate econometric models. There are several issues relevant for further research. A first issue concerns applying the two MDS methods based rDCCA dissimilarity and DTW dissimilarity to alternative data sets, with perhaps different sampling frequencies or returns and absolute returns, to see how informative these methods can be in other cases. A second issue concerns taking the graphical evidence seriously and incorporating it in an econometric time series model to see if it can improve empirical specification strategies. At last, the two MDS methods based rDCCA dissimilarity and DTW dissimilarity can be used to other fields, e.g., transportation and physiology, to explore and get some interesting and abundant results. ACKNOWLEDGMENTS

The financial supports from the funds of the China National Science (Nos. 61071142 and 61371130), the

Chaos 24, 022102 (2014)

Beijing National Science (No. 4122059), and the National High Technology Research Development Program of China (863 Program) (No. 2011AA110306) are gratefully acknowledged. 1

R. R. Nigmatullin, “Universal distribution function for the stronglycorrelated fluctuations: General way for description of different random sequences,” Commun. Nonlinear Sci. Numer. Simul. 15, 637–647 (2010). 2 V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Amaral, and H. E. Stanley, “Econophysics: Financial time series from a statistical physics point of view,” Physica A 279, 443–456 (2000). 3 F. B. Duarte, J. T. Machado, and G. M. Duarte, “Dynamics of the Dow jones and the NASDAQ stock indexes,” Nonlinear Dyn. 61, 691–705 (2010). 4 J. T. Machado, F. B. Duarte, and G. M. Duarte, “Analysis of stock market indices through multidimensional scaling,” Commun. Nonlinear Sci. Numer. Simul. 16, 4610–4618 (2011). 5 J. T. Machado, G. M. Duarte, and F. B. Duarte, “Identifying economic periods and crisis with the multidimensional scaling,” Nonlinear Dyn. 63, 611–622 (2011). 6 J. T. Machado, F. B. Duarte, and G. M. Duarte, “Analysis of financial data series using fractional Fourier transform and multidimensional scaling,” Nonlinear Dyn. 65, 235–245 (2011). 7 G. M. Duarte, J. T. Machado, and F. B. Duarte, “Multidimensional scaling analysis of stock market indexes,” in Nonlinear Complex Dynamics (Springer, 2011), pp. 307–321. 8 B. Podobnik and H. E. Stanley, “Detrended cross-correlation analysis: A new method for analyzing two non-stationary time series,” Phys. Rev. Lett. 100, 084102 (2008). 9 A. Lin, P. Shang, and X. Zhao, “The cross-correlations of stock markets based on DCCA and time-delay DCCA,” Nonlinear Dyn. 67, 425–435 (2012). 10 E. L. Siqueira, T. Stosic, L. Bejan, and B. Stosic, “Correlations and crosscorrelations in the Brazilian agrarian commodities and stocks,” Physica A 389, 2739–2743 (2010). 11 I. Gvozdanovic, B. Podobnik, D. Wang, and H. E. Stanley, “1/f behavior in cross-correlations between absolute returns in a US market,” Physica A 391, 2860–2866 (2012). 12 Y. Wang, Y. Wei, and C. Wu, “Detrended fluctuation analysis on spot and futures markets of West Texas Intermediate crude oil,” Physica A 390, 864–875 (2011). 13 B. Podobnik, D. Horvatic, A. Petersen, and H. E. Stanley, “Cross-correlations between volume change and price change,” Proc. Natl. Acad. Sci. U.S.A. 106, 22079–22084 (2009). 14 B. Podobnik, Z.-Q. Jiang, W.-X. Zhou, and H. E. Stanley, “Statistical tests for power-law cross-correlated processes,” Phys. Rev. E 84, 066118 (2011). 15 D. Horvatic, H. E. Stanley, and B. Podobnik, “Detrended cross-correlation analysis for non-stationary time series with periodic trends,” EPL 94, 18007 (2011). 16 B. Podobnik, I. Grosse, D. Horvatic, S. Ilic, P. Ch. Ivanov, and H. E. Stanley, “Quantifying cross-correlations using local and global detrending approaches,” Eur. Phys. J. B 71, 243–250 (2009). 17 G. F. Zebende and A. M. Filho, “Cross-correlation between time series of vehicles and passengers,” Physica A 388, 4863–4866 (2009). 18 G. F. Zebende, “DCCA cross-correlation coefficient: Quantifying level of cross-correlation,” Physica A 390, 614–618 (2011). 19 I. Borg and P. Groenen, Modern Multidimensional Scaling: Theory and Applications (Springer, New York, 2005). 20 T. Cox and M. Cox, Multidimensional Scaling (Chapman & Hall/CRC, New York, 2001). 21 J. Kruskal and M. Wish, Multidimensional Scaling (Sage Publications, Inc., Newbury Park, CA, 1978). 22 J. O. Ramsay, “Some small sample results for maximum likelihood estimation in multidimensional scaling,” Psychometrika 45, 139–144 (1980). 23 J. Woelfel and G. A. Barnett, “Multidimensional scaling in Riemann space,” Qual. Quant. 16, 469–491 (1982). 24 S. Nirenberg and P. E. Latham, “Decoding neuronal spike trains: How important are correlations,” Proc. Natl. Acad. Sci. U.S.A. 100, 7348–7353 (2003). 25 A. Buja, D. Swayne, M. Littman, N. Dean, H. Hofmann, and L. Chen, “Data visualization with multidimensional scaling,” J. Comput. Graph. Stat. 17, 444–472 (2008).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33

022102-15 26

Y. Yin and P. Shang

J. Kruskal, “Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis,” Psychometrika 29, 1–27 (1964). 27 Y. Takane, F. W. Young, and J. D. Leeuw, “Nonmetric individual differences multidimensional scaling: An alternating least-squares method with optimal scaling features,” Psychometrika 42, 7–67 (1977). 28 R. Shepard, “The analysis of proximities: Multidimensional scaling with an unknown distance function,” Psychometrika I 27, 219–246 (1962). 29 C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, “Mosaic organization of DNA nucleotides,” Phys. Rev. E 49, 1685–1689 (1994). 30 A. Witt, J. Kurths, and A. Pikovsky, “Testing stationarity in time series,” Phys. Rev. E 58, 1800–1810 (1998). 31 Z. Chen, P. Ch. Ivanov, K. Hu, and H. E. Stanley, “Effect of nonstationarities on detrended fluctuation analysis,” Phys. Rev. E 65, 041107 (2002). 32 C.-K. Peng, S. Havlin, H. E. Stanley, and A. L. Goldberger, “Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series,” Chaos 5, 82–87 (1995). 33 K. Hu, P. Ch. Ivanov, Z. Chen, P. Carpena, and H. E. Stanley, “Effect of trends on detrended fluctuation analysis,” Phys. Rev. E 64, 011114 (2001). 34 S. V. Buldyrev, A. L. Goldberger, S. Havlin, R. N. Mantegna, M. E. Matsa, C.-K. Peng, M. Simons, and H. E. Stanley, “Long-range correlation properties of coding and noncoding DNA sequences: GenBank analysis,” Phys. Rev. E 51, 5084–5091 (1995). 35 M. A. Moret, G. F. Zebende, Jr., E. Nogueira, and M. G. Pereira, “Fluctuation analysis of stellar x-ray binary systems,” Phys. Rev. E 68, 041104 (2003). 36 G. F. Zebende, M. G. Pereira, E. Nogueira, Jr., and M. A. Moret, “Universal persistence in astrophysical sources,” Physica A 349, 452–458 (2005). 37 G. F. Zebende, M. V. S. da Silva, Jr., A. C. P. Rosa, A. S. Alves, J. C. O. de Jesus, and M. A. Moret, “Studying long-range correlations in a liquidvapor-phase transition,” Physica A 342, 322–328 (2004). 38 G.-F. Gu and W.-X. Zhou, “Detrending moving average algorithm for multifractals,” Phys. Rev. E 82, 011136 (2010). 39 R. N. Mantegna, “Hierarchical structure in financial markets,” Eur. Phys. J. B 11, 193–197 (1999). 40 V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Amaral, and H. E. Stanley, “Universal and non-universal properties of cross-correlations in financial time series,” Phys. Rev. Lett. 83, 1471–1474 (1999).

Chaos 24, 022102 (2014) 41

B. LeBaron, W. B. Arthur, and R. Palmer, “Time series properties of an artificial stock market,” J. Econom. Dynam. Control 23, 1487–1516 (1999). L. Kullmann, J. Kertesz, and K. Kaski, “Time-dependent crosscorrelations between different stock returns: A directed network of influence,” Phys. Rev. E 66, 026125 (2002). 43 T. Mizunoa, H. Takayasu, and M. Takayasu, “Correlation networks among currencies,” Physica A 364, 336–342 (2006). 44 B. Podobnik, D. Fu, H. E. Stanley, and P. Ch. Ivanov, “Power-law autocorrelated stochastic processes with long-range cross-correlations,” Eur. Phys. J. B 56, 47–52 (2007). 45 B. Podobnik, D. Horvatic, A. Lam, H. E. Stanley, and P. Ch. Ivanov, “Modeling long-range cross-correlations in two-component ARFIMA and FIARCH processes,” Physica A 387, 3954–3959 (2008). 46 B. Podobnik, D. Wang, D. Horvatic, I. Grosse, and H. E. Stanley, “Timelag cross-correlations in collective phenomena,” Europhys. Lett. 90, 68001 (2010). 47 Z.-Q. Jiang and W.-X. Zhou, “Multifractal detrending moving average cross-correlation analysis,” Phys. Rev. E 84, 016106 (2011). 48 R. Bellman and R. Kalaba, “On adaptive control processes,” IRE Trans. Automat. Control 4, 1–9 (1959). 49 G. Al-Naymat, S. Chawla, and J. Taheri, “SparseDTW: A novel approach to speed up dynamic time warping,” Australasian Data Mining 101, 117–127 (2009). 50 F. Black and M. Scholes, “The pricing of options and corporate liabilities,” J. Polit. Econ. 81, 637–654 (1973). 51 B. Ahrens, “Distance in spatial interpolation of daily rain gauge data,” Hydrol. Earth Syst. Sci. 10, 197–208 (2006). 52 K. Benabdeslem and Y. Bennani, “Dendogram-based SVM for multi-class classification,” J. Comput. Inf. Technol. 14, 283–289 (2006). 53 A. Fernandez and S. G omez, “Solving non-uniqueness in agglomerative hierarchical clustering using multidendograms,” J. Classif. 25, 43–65 (2008). 54 J. Y. Campbell, “Stock returns and the term structure,” J. Finance Econ. 18, 373–400 (1987). 55 E. F. Fama and G. W. Schwert, “Asset returns and inflation,” J. Finance Econ. 5, 115–146 (1977). 56 Y. Yin and P. Shang, “Modified DFA and DCCA approach for quantifying the multiscale correlation structure of financial markets,” Physica A 392, 6442–6457 (2013). 42

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 139.184.14.159 On: Fri, 16 Jan 2015 16:48:33