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# Detrended Cross-Correlation Analysis Consistently Extended to Multifractality

## Author

Listed:
• Pawe{l} O'swic{e}cimka
• Stanis{l}aw Dro.zd.z
• Marcin Forczek
• Jaros{l}aw Kwapie'n

## Abstract

We propose a novel algorithm - Multifractal Cross-Correlation Analysis (MFCCA) - that constitutes a consistent extension of the Detrended Cross-Correlation Analysis (DCCA) and is able to properly identify and quantify subtle characteristics of multifractal cross-correlations between two time series. Our motivation for introducing this algorithm is that the already existing methods like MF-DXA have at best serious limitations for most of the signals describing complex natural processes and often indicate multifractal cross-correlations when there are none. The principal component of the present extension is proper incorporation of the sign of fluctuations to their generalized moments. Furthermore, we present a broad analysis of the model fractal stochastic processes as well as of the real-world signals and show that MFCCA is a robust and selective tool at the same time, and therefore allows for a reliable quantification of the cross-correlative structure of analyzed processes. In particular, it allows one to identify the boundaries of the multifractal scaling and to analyze a relation between the generalized Hurst exponent and the multifractal scaling parameter $\lambda_q$. This relation provides information about character of potential multifractality in cross-correlations and thus enables a deeper insight into dynamics of the analyzed processes than allowed by any other related method available so far. By using examples of time series from stock market, we show that financial fluctuations typically cross-correlate multifractally only for relatively large fluctuations, whereas small fluctuations remain mutually independent even at maximum of such cross-correlations. Finally, we indicate possible utility of MFCCA to study effects of the time-lagged cross-correlations.

## Suggested Citation

• Pawe{l} O'swic{e}cimka & Stanis{l}aw Dro.zd.z & Marcin Forczek & Stanis{l}aw Jadach & Jaros{l}aw Kwapie'n, 2013. "Detrended Cross-Correlation Analysis Consistently Extended to Multifractality," Papers 1308.6148, arXiv.org, revised Feb 2014.
• Handle: RePEc:arx:papers:1308.6148
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File URL: http://arxiv.org/pdf/1308.6148

## References listed on IDEAS

as
1. He, Ling-Yun & Chen, Shu-Peng, 2011. "Nonlinear bivariate dependency of price–volume relationships in agricultural commodity futures markets: A perspective from Multifractal Detrended Cross-Correlation Analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(2), pages 297-308.
2. Linares, L. Oriana & Li, MinMin & Shrout, Patrick E., 2012. "Child training for physical aggression?," Children and Youth Services Review, Elsevier, vol. 34(12), pages 2416-2422.
Full references (including those not matched with items on IDEAS)

## Citations

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Cited by:

1. repec:eee:phsmap:v:505:y:2018:i:c:p:374-384 is not listed on IDEAS
2. Lu, Xinsheng & Sun, Xinxin & Ge, Jintian, 2017. "Dynamic relationship between Japanese Yen exchange rates and market anxiety: A new perspective based on MF-DCCA," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 474(C), pages 144-161.
3. repec:eee:phsmap:v:494:y:2018:i:c:p:454-464 is not listed on IDEAS
4. Ruan, Qingsong & Yang, Bingchan & Ma, Guofeng, 2017. "Detrended cross-correlation analysis on RMB exchange rate and Hang Seng China Enterprises Index," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 91-108.
5. Ladislav Kristoufek, 2018. "Power-law cross-correlations: Issues, solutions and future challenges," Papers 1806.01616, arXiv.org.
6. repec:eee:phsmap:v:502:y:2018:i:c:p:228-235 is not listed on IDEAS
7. Shen, Chenhua, 2017. "A comparison of principal components using TPCA and nonstationary principal component analysis on daily air-pollutant concentration series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 467(C), pages 453-464.
8. repec:eee:phsmap:v:482:y:2017:i:c:p:127-146 is not listed on IDEAS
9. Zhai, Lu-Sheng & Zong, Yan-Bo & Wang, Hong-Mei & Yan, Cong & Gao, Zhong-Ke & Jin, Ning-De, 2017. "Characterization of flow pattern transitions for horizontal liquid–liquid pipe flows by using multi-scale distribution entropy in coupled 3D phase space," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 469(C), pages 136-147.
10. repec:eee:phsmap:v:505:y:2018:i:c:p:316-327 is not listed on IDEAS
11. Yao, Can-Zhong & Lin, Ji-Nan & Zheng, Xu-Zhou, 2017. "Coupling detrended fluctuation analysis for multiple warehouse-out behavioral sequences," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 465(C), pages 75-90.
12. repec:eee:phsmap:v:490:y:2018:i:c:p:504-512 is not listed on IDEAS
13. Shen, Chen-hua & Li, Cao-ling, 2016. "An analysis of the intrinsic cross-correlations between API and meteorological elements using DPCCA," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 446(C), pages 100-109.
14. Rotundo, G. & Ausloos, M. & Herteliu, C. & Ileanu, B., 2015. "Hurst exponent of very long birth time series in XX century Romania. Social and religious aspects," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 429(C), pages 109-117.
15. Sun, Xinxin & Lu, Xinsheng & Yue, Gongzheng & Li, Jianfeng, 2017. "Cross-correlations between the US monetary policy, US dollar index and crude oil market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 467(C), pages 326-344.
16. Cao, Guangxi & Han, Yan & Li, Qingchen & Xu, Wei, 2017. "Asymmetric MF-DCCA method based on risk conduction and its application in the Chinese and foreign stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 119-130.
17. Wang, Fang & Yang, Zhaohui & Wang, Lin, 2016. "Detecting and quantifying cross-correlations by analogous multifractal height cross-correlation analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 954-962.
18. Shen, Chen-hua & Li, Chao-ling & Si, Ya-li, 2015. "A detrended cross-correlation analysis of meteorological and API data in Nanjing, China," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 419(C), pages 417-428.
19. Fan, Qingju & Li, Dan, 2015. "Multifractal cross-correlation analysis in electricity spot market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 429(C), pages 17-27.
20. Longfeng Zhao & Wei Li & Andrea Fenu & Boris Podobnik & Yougui Wang & H. Eugene Stanley, 2017. "The q-dependent detrended cross-correlation analysis of stock market," Papers 1705.01406, arXiv.org, revised Jun 2017.
21. repec:eee:phsmap:v:490:y:2018:i:c:p:171-184 is not listed on IDEAS

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