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Effects of polynomial trends on detrending moving average analysis

Author

Listed:
  • Ying-Hui Shao

    (ECUST)

  • Gao-Feng Gu

    (ECUST)

  • Zhi-Qiang Jiang

    (ECUST)

  • Wei-Xing Zhou

    (ECUST)

Abstract

The detrending moving average (DMA) algorithm is one of the best performing methods to quantify the long-term correlations in nonstationary time series. Many long-term correlated time series in real systems contain various trends. We investigate the effects of polynomial trends on the scaling behaviors and the performances of three widely used DMA methods including backward algorithm (BDMA), centered algorithm (CDMA) and forward algorithm (FDMA). We derive a general framework for polynomial trends and obtain analytical results for constant shifts and linear trends. We find that the behavior of the CDMA method is not influenced by constant shifts. In contrast, linear trends cause a crossover in the CDMA fluctuation functions. We also find that constant shifts and linear trends cause crossovers in the fluctuation functions obtained from the BDMA and FDMA methods. When a crossover exists, the scaling behavior at small scales comes from the intrinsic time series while that at large scales is dominated by the constant shifts or linear trends. We also derive analytically the expressions of crossover scales and show that the crossover scale depends on the strength of the polynomial trend, the Hurst index, and in some cases (linear trends for BDMA and FDMA) the length of the time series. In all cases, the BDMA and the FDMA behave almost the same under the influence of constant shifts or linear trends. Extensive numerical experiments confirm excellently the analytical derivations. We conclude that the CDMA method outperforms the BDMA and FDMA methods in the presence of polynomial trends.

Suggested Citation

  • Ying-Hui Shao & Gao-Feng Gu & Zhi-Qiang Jiang & Wei-Xing Zhou, 2015. "Effects of polynomial trends on detrending moving average analysis," Papers 1505.02750, arXiv.org.
  • Handle: RePEc:arx:papers:1505.02750
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    Cited by:

    1. Dong, Xiaofeng & Fan, Qingju & Li, Dan, 2023. "Detrending moving-average cross-correlation based principal component analysis of air pollutant time series," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    2. Xi, Caiping & Zhang, Shuning & Xiong, Gang & Zhao, Huichang & Yang, Yonghong, 2017. "The application of the multifractal cross-correlation analysis methods in radar target detection within sea clutter," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 839-854.
    3. Gao-Feng Gu & Xiong Xiong & Hai-Chuan Xu & Wei Zhang & Yongjie Zhang & Wei Chen & Wei-Xing Zhou, 2021. "An empirical behavioral order-driven model with price limit rules," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 7(1), pages 1-24, December.
    4. Gao, Xing-Lu & Shao, Ying-Hui & Yang, Yan-Hong & Zhou, Wei-Xing, 2022. "Do the global grain spot markets exhibit multifractal nature?," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    5. Sikora, Grzegorz, 2018. "Statistical test for fractional Brownian motion based on detrending moving average algorithm," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 54-62.
    6. Shao Ying-Hui & Liu Ying-Lin & Yang Yan-Hong, 2022. "The short-term effect of COVID-19 pandemic on China's crude oil futures market: A study based on multifractal analysis," Papers 2204.05199, arXiv.org.
    7. Yang, Yan-Hong & Shao, Ying-Hui & Shao, Hao-Lin & Stanley, H. Eugene, 2019. "Revisiting the weak-form efficiency of the EUR/CHF exchange rate market: Evidence from episodes of different Swiss franc regimes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 734-746.
    8. Wang, Fang & Han, Guosheng, 2023. "Coupling correlation adaptive detrended analysis for multiple nonstationary series," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    9. Xi, Caiping & Zhang, Shunning & Xiong, Gang & Zhao, Huichang, 2016. "A comparative study of two-dimensional multifractal detrended fluctuation analysis and two-dimensional multifractal detrended moving average algorithm to estimate the multifractal spectrum," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 454(C), pages 34-50.
    10. Yue-Hua Dai & Wei-Xing Zhou, 2017. "Temporal and spatial correlation patterns of air pollutants in Chinese cities," PLOS ONE, Public Library of Science, vol. 12(8), pages 1-24, August.
    11. Xi, Caiping & Zhang, Shuning & Xiong, Gang & Zhao, Huichang & Yang, Yonghong, 2017. "Two-dimensional multifractal cross-correlation analysis," Chaos, Solitons & Fractals, Elsevier, vol. 96(C), pages 59-69.
    12. Chatterjee, Sucharita & Ghosh, Dipak, 2021. "Impact of Global Warming on SENSEX fluctuations — A study based on Multifractal detrended cross correlation analysis between the temperature anomalies and the SENSEX fluctuations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 571(C).

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