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多元时间序列因果关系分析研究综述

任伟杰 韩敏

任伟杰, 韩敏. 多元时间序列因果关系分析研究综述. 自动化学报, 2021, 47(1): 64−78 doi: 10.16383/j.aas.c180189
引用本文: 任伟杰, 韩敏. 多元时间序列因果关系分析研究综述. 自动化学报, 2021, 47(1): 64−78 doi: 10.16383/j.aas.c180189
Ren Wei-Jie, Han Min. Survey on causality analysis of multivariate time series. Acta Automatica Sinica, 2021, 47(1): 64−78 doi: 10.16383/j.aas.c180189
Citation: Ren Wei-Jie, Han Min. Survey on causality analysis of multivariate time series. Acta Automatica Sinica, 2021, 47(1): 64−78 doi: 10.16383/j.aas.c180189

多元时间序列因果关系分析研究综述

doi: 10.16383/j.aas.c180189
基金项目: 国家自然科学基金(61773087), 中央高校基本科研业务费(DUT18RC(6)005)资助
详细信息
    作者简介:

    任伟杰:大连理工大学电子信息与电气工程学部博士研究生. 主要研究方向为时间序列分析和特征选择.E-mail: renweijie@mail.dlut.edu.cn

    韩敏:大连理工大学电子信息与电气工程学部教授. 主要研究方向为模式识别, 复杂系统建模及时间序列预测. 本文通信作者.E-mail: minhan@dlut.edu.cn

Survey on Causality Analysis of Multivariate Time Series

Funds: Supported by National Natural Science Foundation of China (61773087) and Fundamental Research Funds for the Central Universities (DUT18RC(6)005)
  • 摘要:

    多元时间序列的因果关系分析是数据挖掘领域的研究热点. 时间序列数据包含着与时间动态有关的、未知的、有价值的信息, 因此若能挖掘出这些知识进而对时间序列未来趋势进行预测或干预, 具有重要的现实意义. 为此, 本文综述了多元时间序列因果关系分析的研究进展、应用与展望. 首先, 本文归纳了主要的因果分析方法, 包括Granger因果关系分析、基于信息理论的因果分析和基于状态空间的因果分析; 然后, 总结了不同方法的优缺点、适用范围和发展方向, 并概述了其在不同领域的典型应用; 最后, 讨论了多元时间序列因果分析方法待解决的问题和未来研究趋势.

  • 图  1  收敛交叉映射基本原理示意图

    Fig.  1  Schematic diagram of the basic principle of convergence cross mapping

    表  1  Granger因果关系分析及其改进方法

    Table  1  Granger causality analysis and its improvement methods

    类别 研究者 发表年份 方法名称 文献
    Granger因果模型 Granger 1969 Granger 因果指数 (GCI) [15]
    条件Granger因果模型 Geweke 1982 条件 Granger 因果指数 (CGCI) [23]
    Chen 等 2004 条件扩展 Granger 因果指数 (CEGCI) [24]
    Siggiridou 等 2016 限制条件 Granger 因果指数 (RCGCI) [25]
    Lasso-Granger因果模型 Arnold 等 2007 Lasso-Granger 因果模型 [26]
    Shojaie 等 2010 截断 Lasso-Granger 因果模型 [27]
    Bolstad 等 2011 Grouped-Lasso-Granger 因果模型 [28]
    Yang 等 2017 Grouped-Lasso 非线性条件 Granger 因果模型 [29]
    非线性Granger因果模型 Ancona 等 2004 RBF-Granger 因果模型 [30]
    Marinazzo 等 2008 Kernel-Granger 因果模型 [31-32]
    Wu 等 2011 KCCA-Granger 因果模型 [33]
    Hu 等 2014 Copula-Granger 因果模型 [34]
    Montalto 等 2015 NN-Granger 因果模型 [35]
    频域Granger因果模型 Geweke 1982 Spectral-Granger 因果模型 [23]
    Baccalá 等 2001 偏定向相干性 (PDC) [36]
    Kamiński 等 2001 直接传递函数 (DTF) [37]
    下载: 导出CSV

    表  2  基于信息理论的因果关系分析方法

    Table  2  Causality analysis methods based on information theory

    类别 研究者 发表年份 方法名称 文献
    转移熵 Schreiber 2000 转移熵 (TE) [40]
    Staniek 等 2008 符号转移熵 (STE) [42]
    Kugiumtzis 2013 偏符号转移熵 (PSTE) [43]
    条件熵 Faes 等 2011 条件熵 (CE) [44]
    条件互信息 Frenzel 等 2007 偏互信息 (PMI) [45]
    Kugiumtzis 2013 基于混合嵌入的偏互信息 (PMIME) [46]
    下载: 导出CSV

    表  3  因果分析方法应用范围比较

    Table  3  Comparison of application range of causality analysis methods

    研究者 方法名称 非线性 多变量 非平稳 文献
    Granger Granger 因果指数 [15]
    Geweke 条件 Granger 因果指数 [23]
    Chen 等 条件扩展 Granger 因果指数 [24]
    Siggiridou 等 限制条件 Granger 因果指数 [25]
    Arnold 等 Lasso-Granger 因果模型 [26]
    Shojaie 等 截断 Lasso-Granger 因果模型 [27]
    Bolstad 等 Grouped-Lasso-Granger 因果模型 [28]
    Yang 等 Grouped-Lasso 非线性条件 Granger 因果模型 [29]
    Ancona 等 RBF-Granger 因果模型 [30]
    Marinazzo 等 Kernel-Granger 因果模型 [31-32]
    Wu 等 KCCA-Granger 因果模型 [33]
    Hu 等 Copula-Granger 因果模型 [34]
    Montalto 等 NN-Granger 因果模型 [35]
    Geweke Spectral-Granger 因果模型 [23]
    Baccalá 等 偏定向相干性 [36]
    Kamiński 等 直接传递函数 [37]
    Schreiber 转移熵 [40]
    Staniek 等 符号转移熵 [42]
    Kugiumtzis 偏符号转移熵 [43]
    Faes 等 条件熵 [44]
    Frenzel 等 偏互信息 [45]
    Kugiumtzis 基于混合嵌入的偏互信息 [46]
    Arnhold 等 非线性相互依赖指标 SH [61]
    Quiroga 等 非线性相互依赖指标 N [62]
    Andrzejak 等 非线性相互依赖指标 M [63]
    Chicharro 等 非线性相互依赖指标 L [64]
    Sugihara 等 收敛交叉映射 [65]
    下载: 导出CSV
  • [1] 高月, 宿翀, 李宏光. 一类基于非线性PCA和深度置信网络的混合分类器及其在PM2.5浓度预测和影响因素诊断中的应用. 自动化学报, 2018, 44(2): 318−329

    Gao Yue, Su Chong, Li Hong-Guang. A kind of deep belief networks based on nonlinear features extraction with application to PM2.5 concentration prediction and diagnosis. Acta Automatica Sinica, 2018, 44(2): 318−329
    [2] Han M, Liu X X. Feature selection techniques with class separability for multivariate time series. Neurocomputing, 2013, 110: 29−34 doi: 10.1016/j.neucom.2012.12.006
    [3] He J Y, Shang P J. Comparison of transfer entropy methods for financial time series. Physica A: Statistical Mechanics and Its Applications, 2017, 482: 772−785 doi: 10.1016/j.physa.2017.04.089
    [4] Baek S, Kim D Y. Empirical sensitivity analysis of discretization parameters for fault pattern extraction from multivariate time series data. IEEE Transactions on Cybernetics, 2017, 47(5): 1198−1209 doi: 10.1109/TCYB.2016.2540657
    [5] 周平, 刘记平. 基于数据驱动多输出ARMAX建模的高炉十字测温中心温度在线估计. 自动化学报, 2018, 44(3): 552−561

    Zhou Ping, Liu Ji-Ping. Data-driven multi-output ARMAX modeling for online estimation of central temperatures for cross temperature measuring in blast furnace ironmaking. Acta Automatica Sinica, 2018, 44(3): 552−561
    [6] Fu T C. A review on time series data mining. Engineering Applications of Artificial Intelligence, 2011, 24(1): 164−181 doi: 10.1016/j.engappai.2010.09.007
    [7] Esling P, Agon C. Time-series data mining. ACM Computing Surveys, 2012, 45(1): 12
    [8] 刘强, 秦泗钊. 过程工业大数据建模研究展望. 自动化学报, 2016, 42(2): 161−171

    Liu Qiang, Qin S Joe. Perspectives on big data modeling of process industries. Acta Automatica Sinica, 2016, 42(2): 161−171
    [9] Hardoon D R, Szedmak S, Shawe-Taylor J. Canonical correlation analysis: An overview with application to learning methods. Neural Computation, 2004, 16(12): 2639−2664 doi: 10.1162/0899766042321814
    [10] Han M, Ren W J. Global mutual information-based feature selection approach using single-objective and multi-objective optimization. Neurocomputing, 2015, 168: 47−54 doi: 10.1016/j.neucom.2015.06.016
    [11] Reshef D N, Reshef Y A, Finucane H K, Grossman S R, McVean G, Turnbaugh P J, et al. Detecting novel associations in large data sets. Science, 2011, 334(6062): 1518−1524 doi: 10.1126/science.1205438
    [12] Shi J, Ding Z H, Lee W J, Yang Y P, Liu Y Q, Zhang M M. Hybrid forecasting model for very-short term wind power forecasting based on grey relational analysis and wind speed distribution features. IEEE Transactions on Smart Grid, 2014, 5(1): 521−526 doi: 10.1109/TSG.2013.2283269
    [13] Liebscher E. Copula-based dependence measures. Dependence Modeling, 2014, 2(1): 49−64
    [14] Sun Y Q, Li J Y, Liu J X, Chow C W, Sun B Y, Wang R J. Using causal discovery for feature selection in multivariate numerical time series. Machine Learning, 2015, 101(1-3): 377−395 doi: 10.1007/s10994-014-5460-1
    [15] Granger C W J. Investigating causal relations by econometric models and cross-spectral methods. Econometrica, 1969, 37(3): 424−438 doi: 10.2307/1912791
    [16] Barnett L, Seth A K. The MVGC multivariate Granger causality toolbox: A new approach to Granger-causal inference. Journal of Neuroscience Methods, 2014, 223: 50−68 doi: 10.1016/j.jneumeth.2013.10.018
    [17] Hlaváčková-Schindler K, Paluš M, Vejmelka M, Bhattacharya J. Causality detection based on information-theoretic approaches in time series analysis. Physics Reports, 2007, 441(1): 1−46 doi: 10.1016/j.physrep.2006.12.004
    [18] Cummins B, Gedeon T, Spendlove K. On the efficacy of state space reconstruction methods in determining causality. SIAM Journal on Applied Dynamical Systems, 2015, 14(1): 335−381 doi: 10.1137/130946344
    [19] Zou C L, Feng J F. Granger causality vs. dynamic Bayesian network inference: A comparative study. BMC Bioinformatics, 2009, 10(1): 122−122 doi: 10.1186/1471-2105-10-122
    [20] Kleinberg S, Hripcsak G. A review of causal inference for biomedical informatics. Journal of Biomedical Informatics, 2011, 44(6): 1102−1112 doi: 10.1016/j.jbi.2011.07.001
    [21] Porta A, Faes L. Wiener-Granger causality in network physiology with applications to cardiovascular control and neuroscience. Proceedings of the IEEE, 2016, 104(2): 282−309 doi: 10.1109/JPROC.2015.2476824
    [22] Seth A K, Barrett A B, Barnett L. Granger causality analysis in neuroscience and neuroimaging. The Journal of Neuroscience, 2015, 35(8): 3293−3297 doi: 10.1523/JNEUROSCI.4399-14.2015
    [23] Geweke J. Measurement of linear dependence and feedback between multiple time series. Journal of the American Statistical Association, 1982, 77(378): 304−313 doi: 10.1080/01621459.1982.10477803
    [24] Chen Y H, Rangarajan G, Feng J F, Ding M Z. Analyzing multiple nonlinear time series with extended Granger causality. Physics Letters A, 2004, 324(1): 26−35 doi: 10.1016/j.physleta.2004.02.032
    [25] Siggiridou E, Kugiumtzis D. Granger causality in multivariate time series using a time-ordered restricted vector autoregressive model. IEEE Transactions on Signal Processing, 2016, 64(7): 1759−1773 doi: 10.1109/TSP.2015.2500893
    [26] Arnold A, Liu Y, Abe N. Temporal causal modeling with graphical granger methods. In: Proceedings of the 13th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. San Jose, California, USA: ACM, 2007. 66−75
    [27] Shojaie A, Michailidis G. Discovering graphical Granger causality using the truncating lasso penalty. Bioinformatics, 2010, 26(18): i517−i523 doi: 10.1093/bioinformatics/btq377
    [28] Bolstad A, Van Veen B D, Nowak R. Causal network inference via group sparse regularization. IEEE Transactions on Signal Processing, 2011, 59(6): 2628−2641 doi: 10.1109/TSP.2011.2129515
    [29] Yang G X, Wang L, Wang X F. Reconstruction of complex directional networks with group lasso nonlinear conditional Granger causality. Scientific Reports, 2017, 7(1): 2991 doi: 10.1038/s41598-017-02762-5
    [30] Ancona N, Marinazzo D, Stramaglia S. Radial basis function approach to nonlinear Granger causality of time series. Physical Review E, 2004, 70(5): 056221 doi: 10.1103/PhysRevE.70.056221
    [31] Marinazzo D, Pellicoro M, Stramaglia S. Kernel method for nonlinear Granger causality. Physical Review Letters, 2008, 100(14): 144103 doi: 10.1103/PhysRevLett.100.144103
    [32] Marinazzo D, Pellicoro M, Stramaglia S. Kernel-Granger causality and the analysis of dynamical networks. Physical Review E, 2008, 77(5): 056215 doi: 10.1103/PhysRevE.77.056215
    [33] Wu G R, Duan X J, Liao W, Gao Q, Chen H F. Kernel canonical-correlation Granger causality for multiple time series. Physical Review E, 2011, 83(4): 041921 doi: 10.1103/PhysRevE.83.041921
    [34] Hu M, Liang H L. A copula approach to assessing Granger causality. NeuroImage, 2014, 100: 125−134 doi: 10.1016/j.neuroimage.2014.06.013
    [35] Montalto A, Stramaglia S, Faes L, Tessitore G, Prevete R, Marinazzo D. Neural networks with non-uniform embedding and explicit validation phase to assess Granger causality. Neural Networks, 2015, 71: 159−171 doi: 10.1016/j.neunet.2015.08.003
    [36] Baccalá L A, Sameshima K. Partial directed coherence: A new concept in neural structure determination. Biological Cybernetics, 2001, 84(6): 463−474 doi: 10.1007/PL00007990
    [37] Kamiński M, Ding M Z, Truccolo W A, Bressler S L. Evaluating causal relations in neural systems: Granger causality, directed transfer function and statistical assessment of significance. Biological Cybernetics, 2001, 85(2): 145−157 doi: 10.1007/s004220000235
    [38] Stokes P A, Purdon P L. A study of problems encountered in Granger causality analysis from a neuroscience perspective. Proceedings of the National Academy of Sciences, 2017, 114(34): E7063−E7072 doi: 10.1073/pnas.1704663114
    [39] Barrett A B, Barnett L, Seth A K. Multivariate Granger causality and generalized variance. Physical Review E, 2010, 81(4): 041907 doi: 10.1103/PhysRevE.81.041907
    [40] Schreiber T. Measuring information transfer. Physical Review Letters, 2000, 85(2): 461−464 doi: 10.1103/PhysRevLett.85.461
    [41] Barnett L, Barrett A B, Seth A K. Granger causality and transfer entropy are equivalent for Gaussian variables. Physical Review Letters, 2009, 103(23): 238701 doi: 10.1103/PhysRevLett.103.238701
    [42] Staniek M, Lehnertz K. Symbolic transfer entropy. Physical Review Letters, 2008, 100(15): 158101 doi: 10.1103/PhysRevLett.100.158101
    [43] Kugiumtzis D. Partial transfer entropy on rank vectors. The European Physical Journal Special Topics, 2013, 222(2): 401−420 doi: 10.1140/epjst/e2013-01849-4
    [44] Faes L, Nollo G, Porta A. Information-based detection of nonlinear Granger causality in multivariate processes via a nonuniform embedding technique. Physical Review E, 2011, 83(5): 051112 doi: 10.1103/PhysRevE.83.051112
    [45] Frenzel S, Pompe B. Partial mutual information for coupling analysis of multivariate time series. Physical Review Letters, 2007, 99(20): 204101 doi: 10.1103/PhysRevLett.99.204101
    [46] Kugiumtzis D. Direct-coupling information measure from nonuniform embedding. Physical Review E, 2013, 87(6): 062918 doi: 10.1103/PhysRevE.87.062918
    [47] Vlachos I, Kugiumtzis D. Nonuniform state-space reconstruction and coupling detection. Physical Review E, 2010, 82(1): 016207 doi: 10.1103/PhysRevE.82.016207
    [48] Runge J, Heitzig J, Petoukhov V, Kurths J. Escaping the curse of dimensionality in estimating multivariate transfer entropy. Physical Review Letters, 2012, 108(25): 258701 doi: 10.1103/PhysRevLett.108.258701
    [49] Takens F. Detecting strange attractors in turbulence. Dynamical Systems and Turbulence. Heidelberg, Germany: Springer-Verlag, 1981. 366−381
    [50] Kalman R E. A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 1960, 82(1): 35−45 doi: 10.1115/1.3662552
    [51] Solo V. State-space analysis of Granger-Geweke causality measures with application to fMRI. Neural Computation, 2016, 28(5): 914−949 doi: 10.1162/NECO_a_00828
    [52] Jinno K, Xu S G, Berndtsson R, Kawamura A, Matsumoto M. Prediction of unspots using reconstructed chaotic system equations. Journal of Geophysical Research: Space Physics, 1995, 100(A8): 14773−14781 doi: 10.1029/95JA01167
    [53] Hong M, Wang D, Wang Y K, Zeng X K, Ge S S, Yan H Q, Singh V P. Mid-and long-term runoff predictions by an improved phase-space reconstruction model. Environmental Research, 2016, 148: 560−573 doi: 10.1016/j.envres.2015.11.024
    [54] 殷礼胜, 何怡刚, 董学平, 鲁照权. 交通流量VNNTF神经网络模型多步预测研究. 自动化学报, 2014, 40(9): 2066−2072

    Yin Li-Sheng, He Yi-Gang, Dong Xue-Ping, Lu Zhao-Quan. Research on the multi-step prediction of Volterra neural network for traffic flow. Acta Automatica Sinica, 2014, 40(9): 2066−2072
    [55] Luo S H, Gao C H, Zeng J S, Huang J. Blast furnace system modeling by multivariate phase space reconstruction and neural networks. Asian Journal of Control, 2013, 15(2): 553−561 doi: 10.1002/asjc.574
    [56] Cao L. Practical method for determining the minimum embedding dimension of a scalar time series. Physica D: Nonlinear Phenomena, 1997, 110(1): 43−50
    [57] Molkov Y I, Mukhin D N, Loskutov E M, Feigin A M, Fidelin G A. Using the minimum description length principle for global reconstruction of dynamic systems from noisy time series. Physical Review E, 2009, 80(4): 046207 doi: 10.1103/PhysRevE.80.046207
    [58] Kugiumtzis D. State space reconstruction parameters in the analysis of chaotic time series-the role of the time window length. Physica D: Nonlinear Phenomena, 1996, 95(1): 13−28 doi: 10.1016/0167-2789(96)00054-1
    [59] Kim H, Eykholt R, Salas J D. Nonlinear dynamics, delay times, and embedding windows. Physica D: Nonlinear Phenomena, 1999, 127(1−2): 48−60 doi: 10.1016/S0167-2789(98)00240-1
    [60] Shen M, Chen W N, Zhang J, Chung H S H, Kaynak O. Optimal selection of parameters for nonuniform embedding of chaotic time series using ant colony optimization. IEEE Transactions on Cybernetics, 2013, 43(2): 790−802 doi: 10.1109/TSMCB.2012.2219859
    [61] Arnhold J, Grassberger P, Lehnertz K, Elger C E. A robust method for detecting interdependences: application to intracranially recorded EEG. Physica D: Nonlinear Phenomena, 1999, 134(4): 419−430 doi: 10.1016/S0167-2789(99)00140-2
    [62] Quiroga R Q, Arnhold J, Grassberger P. Learning driver-response relationships from synchronization patterns. Physical Review E, 2000, 61(5): 5142 doi: 10.1103/PhysRevE.61.5142
    [63] Andrzejak R G, Kraskov A, Stögbauer H, Mormann F, Kreuz T. Bivariate surrogate techniques: Necessity, strengths, and caveats. Physical Review E, 2003, 68(6): 066202 doi: 10.1103/PhysRevE.68.066202
    [64] Chicharro D, Andrzejak R G. Reliable detection of directional couplings using rank statistics. Physical Review E, 2009, 80(2): 026217 doi: 10.1103/PhysRevE.80.026217
    [65] Sugihara G, May R, Ye H, Hsieh C H, Deyle E, Fogarty M, Munch S. Detecting causality in complex ecosystems. Science, 2012, 338(6106): 496−500 doi: 10.1126/science.1227079
    [66] Schäck T, Muma M, Feng M L, Guan C T, Zoubir A M. Robust nonlinear causality analysis of nonstationary multivariate physiological time series. IEEE Transactions on Biomedical Engineering, 2017, 65(6): 1213−1225
    [67] Montalto A, Faes L, Marinazzo D. MuTE: A MATLAB toolbox to compare established and novel estimators of the multivariate transfer entropy. PloS One, 2014, 9(10): e109462 doi: 10.1371/journal.pone.0109462
    [68] Ma H F, Aihara K, Chen L N. Detecting causality from nonlinear dynamics with short-term time series. Scientific Reports, 2014, 4: 7464
    [69] Clark A T, Ye H, Isbell F, Deyle E R, Cowles J, Tilman G D, Sugihara G. Spatial convergent cross mapping to detect causal relationships from short time. Ecology, 2015, 96(5): 1174−1181 doi: 10.1890/14-1479.1
    [70] Mønster D, Fusaroli R, Tylén K, Roepstorff A, Sherson J F. Causal inference from noisy time-series data—testing the convergent cross-mapping algorithm in the presence of noise and external influence. Future Generation Computer Systems, 2017, 73: 52−62 doi: 10.1016/j.future.2016.12.009
    [71] Zhu J Y, Zhang C, Zhang H C, Zhi S, Li V O K, Han J W, Zheng Y. pg-Causality: Identifying spatiotemporal causal pathways for air pollutants with urban big data. IEEE Transactions on Big Data, 2018, 4(4): 571−585 doi: 10.1109/TBDATA.2017.2723899
    [72] Liang X S. Unraveling the cause-effect relation between time series. Physical Review E, 2014, 90(5): 052150 doi: 10.1103/PhysRevE.90.052150
    [73] Faybishenko B. Detecting dynamic causal inference in nonlinear two-phase fracture flow. Advances in Water Resources, 2017, 106: 111−120 doi: 10.1016/j.advwatres.2017.02.011
    [74] Zhu J Y, Sun C, Li V O K. An extended spatio-temporal Granger causality model for air quality estimation with heterogeneous urban big data. IEEE Transactions on Big Data, 2017, 3(3): 307−319 doi: 10.1109/TBDATA.2017.2651898
    [75] Chen Z Y, Cai J, Gao B B, Xu B, Dai S, He B, Xie X M. Detecting the causality influence of individual meteorological factors on local PM 2.5 concentration in the Jing-Jin-Ji region. Scientific Reports, 2017, 7: 40735 doi: 10.1038/srep40735
    [76] Hu S Q, Dai G J, Worrell G A, Dai Q H, Liang H L. Causality analysis of neural connectivity: Critical examination of existing methods and advances of new methods. IEEE Transactions on Neural Networks, 2011, 22(6): 829−844 doi: 10.1109/TNN.2011.2123917
    [77] Dhamala M, Rangarajan G, Ding M Z. Analyzing information flow in brain networks with nonparametric Granger causality. NeuroImage, 2008, 41(2): 354−362 doi: 10.1016/j.neuroimage.2008.02.020
    [78] Wu G R, Chen F Y, Kang D Z, Zhang X Y, Marinazzo D, Chen H F. Multiscale causal connectivity analysis by canonical correlation: Theory and application to epileptic brain. IEEE Transactions on Biomedical Engineering, 2011, 58(11): 3088−3096 doi: 10.1109/TBME.2011.2162669
    [79] Li P Y, Huang X Y, Li F L, Wang X R, Zhou W W, Liu H, et al. Robust Granger analysis in Lp norm space for directed EEG network analysis. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 2017, 25(11): 1959−1969 doi: 10.1109/TNSRE.2017.2711264
    [80] Hu M, Li W, Liang H L. A copula-based Granger causality measure for the analysis of neural spike train data. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 2018, 15(2): 562−569 doi: 10.1109/TCBB.2014.2388311
    [81] Faes L, Marinazzo D, Montalto A, Nollo G. Lag-specific transfer entropy as a tool to assess cardiovascular and cardiorespiratory information transfer. IEEE Transactions on Biomedical Engineering, 2014, 61(10): 2556−2568 doi: 10.1109/TBME.2014.2323131
    [82] Wang Z, Alahmadi A, Zhu D C, Li T T. Causality analysis of fMRI data based on the directed information theory framework. IEEE Transactions on Biomedical Engineering, 2016, 63(5): 1002−1015 doi: 10.1109/TBME.2015.2481723
    [83] Heskamp L, Meel-van den Abeelen A S, Lagro J, Claassen J A. Convergent cross mapping: A promising technique for cerebral autoregulation estimation. International Journal of Clinical Neurosciences and Mental Health, 2014, 1(1): S20
    [84] Wang S, Li Q, Fang C, Zhou C. The relationship between economic growth, energy consumption, and CO2 emissions: Empirical evidence from China. Science of the Total Environment, 2016, 542: 360−371 doi: 10.1016/j.scitotenv.2015.10.027
    [85] Zhou C S, Wang S J, Feng K S. Examining the socioeconomic determinants of CO2 emissions in China: A historical and prospective analysis. Resources, Conservation and Recycling, 2018, 130: 1−11 doi: 10.1016/j.resconrec.2017.11.007
    [86] Rafindadi A A, Ozturk I. Impacts of renewable energy consumption on the German economic growth: Evidence from combined cointegration test. Renewable and Sustainable Energy Reviews, 2017, 75: 1130−1141 doi: 10.1016/j.rser.2016.11.093
    [87] Tiwari A K. Causality between wholesale price and consumer price indices in India: An empirical investigation in the frequency domain. Indian Growth and Development Review, 2012, 5(2): 151−172 doi: 10.1108/17538251211268071
    [88] Bekiros S, Nguyen D K, Junior L S, Uddin G S. Information diffusion, cluster formation and entropy-based network dynamics in equity and commodity markets. European Journal of Operational Research, 2017, 256(3): 945−961 doi: 10.1016/j.ejor.2016.06.052
    [89] Papana A, Kyrtsou C, Kugiumtzis D, Diks C. Detecting causality in non-stationary time series using partial symbolic transfer entropy: evidence in financial data. Computational Economics, 2016, 47(3): 341−365 doi: 10.1007/s10614-015-9491-x
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出版历程
  • 收稿日期:  2018-04-02
  • 录用日期:  2018-11-22
  • 网络出版日期:  2021-01-29
  • 刊出日期:  2021-01-29

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