Distinguishing cause from effect is a scientific challenge resisting solutions from mathematics, statistics, information theory and computer science. Compression-Complexity Causality (CCC) is a recently proposed interventional measure of causality, inspired by Wiener-Granger's idea. It estimates causality based on change in dynamical compression-complexity (or compressibility) of the effect variable, given the cause variable. CCC works with minimal assumptions on given data and is robust to irregular-sampling, missing-data and finite-length effects. However, it only works for one-dimensional time series. We propose an ordinal pattern symbolization scheme to encode multidimensional patterns into one-dimensional symbolic sequences, and thus introduce the Permutation CCC (PCCC). We demonstrate that PCCC retains all advantages of the original CCC and can be applied to data from multidimensional systems with potentially unobserved variables which can be reconstructed using the embedding theorem. PCCC is tested on numerical simulations and applied to paleoclimate data characterized by irregular and uncertain sampling and limited numbers of samples.

Department of Complex Systems Institute of Computer Science of the Czech Academy of Sciences 182 07 Prague Czech Republic

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Pearl J, Mackenzie D. The Book of Why: The New Science of Cause and Effect. Basic Books; 2018.

Kathpalia A, Nagaraj N. Measuring causality. Resonance. 2021;26:191. doi: 10.1007/s12045-021-1119-y. DOI

Wiener N. The theory of prediction. Mod. Math. Eng. 1956;1:125–139.

Granger C. Investigating causal relations by econometric models and cross-spectral methods. Econometrica. 1969;37:424–438. doi: 10.2307/1912791. DOI

Geweke J. Inference and causality in economic time series models. Handb. Econom. 1984;2:1101–1144.

Hiemstra C, Jones JD. Testing for linear and nonlinear granger causality in the stock price-volume relation. J. Financ. 1994;49:1639–1664.

Chiou-Wei SZ, Chen C-F, Zhu Z. Economic growth and energy consumption revisited: Evidence from linear and nonlinear granger causality. Energy Econ. 2008;30:3063–3076. doi: 10.1016/j.eneco.2008.02.002. DOI

Seth AK, Barrett AB, Barnett L. Granger causality analysis in neuroscience and neuroimaging. J. Neurosci. 2015;35:3293–3297. doi: 10.1523/JNEUROSCI.4399-14.2015. PubMed DOI PMC

Mosedale TJ, Stephenson DB, Collins M, Mills TC. Granger causality of coupled climate processes: Ocean feedback on the north Atlantic oscillation. J. Clim. 2006;19:1182–1194. doi: 10.1175/JCLI3653.1. DOI

Tirabassi G, Masoller C, Barreiro M. A study of the air–sea interaction in the south Atlantic convergence zone through granger causality. Int. J. Climatol. 2015;35:3440–3453. doi: 10.1002/joc.4218. DOI

Runge J, et al. Inferring causation from time series in earth system sciences. Nat. Commun. 2019;10:1–13. doi: 10.1038/s41467-019-10105-3. PubMed DOI PMC

Bell D, Kay J, Malley J. A non-parametric approach to non-linear causality testing. Econ. Lett. 1996;51:7–18. doi: 10.1016/0165-1765(95)00791-1. DOI

Chen Y, Rangarajan G, Feng J, Ding M. Analyzing multiple nonlinear time series with extended granger causality. Phys. Lett. A. 2004;324:26–35. doi: 10.1016/j.physleta.2004.02.032. DOI

Schiff SJ, So P, Chang T, Burke RE, Sauer T. Detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble. Phys. Rev. E. 1996;54:6708. doi: 10.1103/PhysRevE.54.6708. PubMed DOI

Le Van Quyen, M., Martinerie, J., Adam, C. & Varela, F. J. Nonlinear analyses of interictal EEG map the brain interdependences in human focal epilepsy. Physica D Nonlinear Phenomena127, 250–266 (1999).

Marinazzo D, Pellicoro M, Stramaglia S. Kernel method for nonlinear granger causality. Phys. Rev. Lett. 2008;100:144103. doi: 10.1103/PhysRevLett.100.144103. PubMed DOI

Baccalá LA, Sameshima K. Partial directed coherence: A new concept in neural structure determination. Biol. Cybern. 2001;84:463–474. doi: 10.1007/PL00007990. PubMed DOI

Kamiński M, Ding M, Truccolo WA, Bressler SL. Evaluating causal relations in neural systems: Granger causality, directed transfer function and statistical assessment of significance. Biol. Cybern. 2001;85:145–157. doi: 10.1007/s004220000235. PubMed DOI

Korzeniewska A, Mańczak M, Kamiński M, Blinowska KJ, Kasicki S. Determination of information flow direction among brain structures by a modified directed transfer function (dDTF) method. J. Neurosci. Methods. 2003;125:195–207. doi: 10.1016/S0165-0270(03)00052-9. PubMed DOI

Schreiber T. Measuring information transfer. Phys. Rev. Lett. 2000;85:461–464. doi: 10.1103/PhysRevLett.85.461. PubMed DOI

Paluš M, Komárek V, Hrnčíř Z, Štěrbová K. Synchronization as adjustment of information rates: Detection from bivariate time series. Phys. Rev. E. 2001;63:046211. doi: 10.1103/PhysRevE.63.046211. PubMed DOI

Paluš M, Vejmelka M. Directionality of coupling from bivariate time series: How to avoid false causalities and missed connections. Phys. Rev. E. 2007;75:056211. doi: 10.1103/PhysRevE.75.056211. PubMed DOI

Vicente R, Wibral M, Lindner M, Pipa G. Transfer entropy: A model-free measure of effective connectivity for the neurosciences. J. Comput. Neurosci. 2011;30:45–67. doi: 10.1007/s10827-010-0262-3. PubMed DOI PMC

Bauer M, Cox JW, Caveness MH, Downs JJ, Thornhill NF. Finding the direction of disturbance propagation in a chemical process using transfer entropy. IEEE Trans. Control Syst. Technol. 2007;15:12–21. doi: 10.1109/TCST.2006.883234. DOI

Dimpfl T, Peter FJ. Using transfer entropy to measure information flows between financial markets. Stud. Nonlinear Dyn. Econom. 2013;17:85–102.

Paluš M. Multiscale atmospheric dynamics: Cross-frequency phase-amplitude coupling in the air temperature. Phys. Rev. Lett. 2014;112:078702. doi: 10.1103/PhysRevLett.112.078702. PubMed DOI

Jajcay N, Kravtsov S, Sugihara G, Tsonis AA, Paluš M. Synchronization and causality across time scales in el niño southern oscillation. NPJ Climate Atmos. Sci. 2018;1:1–8. doi: 10.1038/s41612-017-0007-3. DOI

Takens, F. Detecting strange attractors in turbulence. In Dynamical Systems and Turbulence, Warwick 1980, 366–381 (Springer, 1981).

Fraser AM, Swinney HL. Independent coordinates for strange attractors from mutual information. Phys. Rev. A. 1986;33:1134. doi: 10.1103/PhysRevA.33.1134. PubMed DOI

Wibral M, et al. Measuring information-transfer delays. PloS one. 2013;8:e55809. doi: 10.1371/journal.pone.0055809. PubMed DOI PMC

Sugihara G, May R, Ye H, Hsieh C, Deyle E. Detecting causality in complex ecosystems. Science. 2012;338:496–500. doi: 10.1126/science.1227079. PubMed DOI

Harnack D, Laminski E, Schünemann M, Pawelzik KR. Topological causality in dynamical systems. Phys. Rev. Lett. 2017;119:098301. doi: 10.1103/PhysRevLett.119.098301. PubMed DOI

Krakovská A, Hanzely F. Testing for causality in reconstructed state spaces by an optimized mixed prediction method. Phys. Rev. E. 2016;94:052203. doi: 10.1103/PhysRevE.94.052203. PubMed DOI

Barrios A, Trincado G, Garreaud R. Alternative approaches for estimating missing climate data: Application to monthly precipitation records in south-central Chile. For. Ecosyst. 2018;5:1–10. doi: 10.1186/s40663-018-0147-x. DOI

Anderson CI, Gough WA. Accounting for missing data in monthly temperature series: Testing rule-of-thumb omission of months with missing values. Int. J. Climatol. 2018;38:4990–5002. doi: 10.1002/joc.5801. DOI

DiCesare, G. Imputation, estimation and missing data in finance. Ph.D. Thesis, University of Waterloo (2006).

John C, Ekpenyong EJ, Nworu CC. Imputation of missing values in economic and financial time series data using five principal component analysis approaches. CBN J. Appl. Stat. (JAS) 2019;10:3.

Gyimah S. Missing data in quantitative social research. PSC Discuss. Papers Ser. 2001;15:1.

Kulp C, Tracy E. The application of the transfer entropy to Gappy time series. Phys. Lett. A. 2009;373:1261–1267. doi: 10.1016/j.physleta.2009.02.009. DOI

Smirnov D, Bezruchko B. Spurious causalities due to low temporal resolution: Towards detection of bidirectional coupling from time series. Europhys. Lett. 2012;100:10005. doi: 10.1209/0295-5075/100/10005. DOI

Kathpalia A, Nagaraj N. Data based intervention approach for complexity-causality measure. PeerJ Comput. Sci. 2019;e196:5. PubMed PMC

Kathpalia, A. Theoretical and Experimental Investigations into Causality, its Measures and Applications. Ph.D. Thesis, NIAS (2021).

Nagaraj N, Balasubramanian K. Dynamical complexity of short and noisy time series. Eur. Phys. J. Special Top. 2017;226:1–14. doi: 10.1140/epjst/e2017-02677-8. DOI

Staniek M, Lehnertz K. Symbolic transfer entropy. Phys. Rev. Lett. 2008;100:158101. doi: 10.1103/PhysRevLett.100.158101. PubMed DOI

Staniek M, Lehnertz K. Symbolic transfer entropy: Inferring directionality in biosignals. Biomed. Tech. 2009;54:323–328. doi: 10.1515/BMT.2009.040. PubMed DOI

Kugiumtzis D. Partial transfer entropy on rank vectors. Eur. Phys. J. Special Top. 2013;222:401–420. doi: 10.1140/epjst/e2013-01849-4. DOI

Papana A, Kyrtsou C, Kugiumtzis D, Diks C. Simulation study of direct causality measures in multivariate time series. Entropy. 2013;15:2635–2661. doi: 10.3390/e15072635. DOI

Li X, Ouyang G. Estimating coupling direction between neuronal populations with permutation conditional mutual information. Neuroimage. 2010;52:497–507. doi: 10.1016/j.neuroimage.2010.05.003. PubMed DOI

Wen D, et al. Estimating coupling strength between multivariate neural series with multivariate permutation conditional mutual information. Neural Netw. 2019;110:159–169. doi: 10.1016/j.neunet.2018.11.006. PubMed DOI

Bandt C, Pompe B. Permutation entropy: A natural complexity measure for time series. Phys. Rev. Lett. 2002;88:174102. doi: 10.1103/PhysRevLett.88.174102. PubMed DOI

Fadlallah B, Chen B, Keil A, Principe J. Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information. Phys. Rev. E. 2013;87:022911. doi: 10.1103/PhysRevE.87.022911. PubMed DOI

Amigó, J. Permutation Complexity in Dynamical Systems: Ordinal Patterns, Permutation Entropy and All That (Springer Science & Business Media, 2010).

Zanin M, Zunino L, Rosso OA, Papo D. Permutation entropy and its main biomedical and econophysics applications: A review. Entropy. 2012;14:1553–1577. doi: 10.3390/e14081553. DOI

Keller K, Unakafov AM, Unakafova VA. Ordinal patterns, entropy, and EEG. Entropy. 2014;16:6212–6239. doi: 10.3390/e16126212. DOI

Zanin M, Olivares F. Ordinal patterns-based methodologies for distinguishing chaos from noise in discrete time series. Commun. Phys. 2021;4:1–14. doi: 10.1038/s42005-021-00696-z. DOI

McCullough M, Small M, Stemler T, Iu HH-C. Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems. Chaos Interdiscip. J. Nonlinear Sci. 2015;25:053101. doi: 10.1063/1.4919075. PubMed DOI

Bandt C, Keller G, Pompe B. Entropy of interval maps via permutations. Nonlinearity. 2002;15:1595. doi: 10.1088/0951-7715/15/5/312. DOI

Amigó JM, Kennel MB, Kocarev L. The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems. Physica D. 2005;210:77–95. doi: 10.1016/j.physd.2005.07.006. DOI

Solomon S, Manning M, Marquis M, Qin D, et al. Climate Change 2007-The Physical Science Basis: Working Group I Contribution to the Fourth Assessment Report of the IPCC. Cambridge University Press; 2007.

Press WH, Flannery BP, Teukolsky SA, Vetterling WT, Kramer PB. Numerical recipes: The art of scientific computing. Phys. Today. 1987;40:120. doi: 10.1063/1.2820230. DOI

Theiler J, Eubank S, Longtin A, Galdrikian B, Farmer JD. Testing for nonlinearity in time series: The method of surrogate data. Physica D. 1992;58:77–94. doi: 10.1016/0167-2789(92)90102-S. DOI

Mills BJ, et al. Modelling the long-term carbon cycle, atmospheric co2, and earth surface temperature from late neoproterozoic to present day. Gondwana Res. 2019;67:172–186. doi: 10.1016/j.gr.2018.12.001. DOI

Wong TE, Cui Y, Royer DL, Keller K. A tighter constraint on earth-system sensitivity from long-term temperature and carbon-cycle observations. Nat. Commun. 2021;12:1–8. doi: 10.1038/s41467-020-20314-w. PubMed DOI PMC

Past Interglacials Working Group of PAGES. Interglacials of the last 800,000 years. Rev. Geophys.54, 162–219 (2016).

Lüthi D, et al. High-resolution carbon dioxide concentration record 650,000–800,000 years before present. Nature. 2008;453:379–382. doi: 10.1038/nature06949. PubMed DOI

Bereiter B, et al. Revision of the Epica dome c co2 record from 800 to 600 kyr before present. Geophys. Res. Lett. 2015;42:542–549. doi: 10.1002/2014GL061957. DOI

Loulergue L, et al. Orbital and millennial-scale features of atmospheric ch 4 over the past 800,000 years. Nature. 2008;453:383–386. doi: 10.1038/nature06950. PubMed DOI

Bazin L, et al. An optimized multi-proxy, multi-site Antarctic ice and gas orbital chronology (aicc2012): 120–800 ka. Climate Past. 2013;9:1715–1731. doi: 10.5194/cp-9-1715-2013. DOI

Elderfield H, et al. Evolution of ocean temperature and ice volume through the mid-pleistocene climate transition. Science. 2012;337:704–709. doi: 10.1126/science.1221294. PubMed DOI

Lawrimore JH, et al. An overview of the global historical climatology network monthly mean temperature data set, version 3. J. Geophys. Res. Atmos. 2011 doi: 10.1029/2011JD016187. DOI

Li J, et al. Interdecadal modulation of el niño amplitude during the past millennium. Nat. Clim. Chang. 2011;1:114–118. doi: 10.1038/nclimate1086. DOI

Shi F, Li J, Wilson RJ. A tree-ring reconstruction of the south Asian summer monsoon index over the past millennium. Sci. Rep. 2014;4:1–8. doi: 10.1038/srep05230. PubMed DOI PMC

Rayner N, et al. Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century. J. Geophys. Res. Atmos. 2003 doi: 10.1029/2002JD002670. DOI

Luterbacher J, Schmutz C, Gyalistras D, Xoplaki E, Wanner H. Reconstruction of monthly nao and eu indices back to ad 1675. Geophys. Res. Lett. 1999;26:2745–2748. doi: 10.1029/1999GL900576. DOI

Luterbacher J, et al. Extending north Atlantic oscillation reconstructions back to 1500. Atmos. Sci. Lett. 2001;2:114–124. doi: 10.1006/asle.2001.0044. DOI

Trenberth KE, Paolino DA., Jr The northern hemisphere sea-level pressure data set: Trends, errors and discontinuities. Mon. Weather Rev. 1980;108:855–872. doi: 10.1175/1520-0493(1980)108<0855:TNHSLP>2.0.CO;2. DOI

Dobrovolnỳ P, et al. Monthly, seasonal and annual temperature reconstructions for central Europe derived from documentary evidence and instrumental records since ad 1500. Clim. Change. 2010;101:69–107. doi: 10.1007/s10584-009-9724-x. DOI

Barnston AG, Livezey RE. Classification, seasonality and persistence of low-frequency atmospheric circulation patterns. Mon. Weather Rev. 1987;115:1083–1126. doi: 10.1175/1520-0493(1987)115<1083:CSAPOL>2.0.CO;2. DOI

Chen WY, Van den Dool H. Sensitivity of teleconnection patterns to the sign of their primary action center. Mon. Weather Rev. 2003;131:2885–2899. doi: 10.1175/1520-0493(2003)131<2885:SOTPTT>2.0.CO;2. DOI

Van den Dool H, Saha S, Johansson A. Empirical orthogonal teleconnections. J. Clim. 2000;13:1421–1435. doi: 10.1175/1520-0442(2000)013<1421:EOT>2.0.CO;2. DOI

Klein Tank A, et al. Daily dataset of 20th-century surface air temperature and precipitation series for the European climate assessment. Int. J. Climatol. J. R. Meteorol. Soc. 2002;22:1441–1453. doi: 10.1002/joc.773. DOI

Politis DN, Romano JP. The stationary bootstrap. J. Am. Stat. Assoc. 1994;89:1303–1313. doi: 10.1080/01621459.1994.10476870. DOI

Foote, E. Art. xxxi.–circumstances affecting the heat of the sun’s rays. American Journal of Science and Arts (1820-1879)22, 382 (1856).

Arrhenius, S. Xxxi. on the influence of carbonic acid in the air upon the temperature of the ground. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science41, 237–276 (1896).

Kodra E, Chatterjee S, Ganguly AR. Exploring granger causality between global average observed time series of carbon dioxide and temperature. Theoret. Appl. Climatol. 2011;104:325–335. doi: 10.1007/s00704-010-0342-3. DOI

Attanasio A. Testing for linear granger causality from natural/anthropogenic forcings to global temperature anomalies. Theoret. Appl. Climatol. 2012;110:281–289. doi: 10.1007/s00704-012-0634-x. DOI

Stern DI, Kaufmann RK. Anthropogenic and natural causes of climate change. Clim. Change. 2014;122:257–269. doi: 10.1007/s10584-013-1007-x. DOI

Kang J, Larsson R. What is the link between temperature and carbon dioxide levels? A granger causality analysis based on ice core data. Theoret. Appl. Climatol. 2014;116:537–548. doi: 10.1007/s00704-013-0960-7. DOI

Triacca U. On the use of granger causality to investigate the human influence on climate. Theoret. Appl. Climatol. 2001;69:137–138. doi: 10.1007/s007040170019. DOI

Triacca U. Is granger causality analysis appropriate to investigate the relationship between atmospheric concentration of carbon dioxide and global surface air temperature? Theoret. Appl. Climatol. 2005;81:133–135. doi: 10.1007/s00704-004-0112-1. DOI

Stips A, Macias D, Coughlan C, Garcia-Gorriz E, San Liang X. On the causal structure between co2 and global temperature. Sci. Rep. 2016;6:1–9. doi: 10.1038/srep21691. PubMed DOI PMC

Goulet Coulombe P, Göbel M. On spurious causality, co2, and global temperature. Econometrics. 2021;9:33. doi: 10.3390/econometrics9030033. DOI

Van Nes EH, et al. Causal feedbacks in climate change. Nat. Clim. Chang. 2015;5:445–448. doi: 10.1038/nclimate2568. DOI

Koutsoyiannis D, Kundzewicz ZW. Atmospheric temperature and co2: Hen-or-egg causality? Sci. 2020;2:83. doi: 10.3390/sci2040083. DOI

Mønster D, Fusaroli R, Tylén K, Roepstorff A, Sherson JF. Causal inference from noisy time-series data—Testing the convergent cross-mapping algorithm in the presence of noise and external influence. Futur. Gener. Comput. Syst. 2017;73:52–62. doi: 10.1016/j.future.2016.12.009. DOI

Schiecke, K., Pester, B., Feucht, M., Leistritz, L. & Witte, H. Convergent cross mapping: Basic concept, influence of estimation parameters and practical application. In 2015 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), 7418–7421 (IEEE, 2015). PubMed

Janse RJ, et al. Conducting correlation analysis: Important limitations and pitfalls. Clin. Kidney J. 2021;14:2337. doi: 10.1093/ckj/sfab085. PubMed DOI PMC

Brook EJ, Sowers T, Orchardo J. Rapid variations in atmospheric methane concentration during the past 110,000 years. Science. 1996;273:1087–1091. doi: 10.1126/science.273.5278.1087. PubMed DOI

Thirumalai K, Clemens SC, Partin JW. Methane, monsoons, and modulation of millennial-scale climate. Geophys. Res. Lett. 2020;47:e2020GL087613. doi: 10.1029/2020GL087613. DOI

Kripalani RH, Kulkarni A. Rainfall variability over south-east Asia-connections with Indian monsoon and enso extremes: New perspectives. Int. J. Climatol. J. R. Meteorol. Soc. 1997;17:1155–1168. doi: 10.1002/(SICI)1097-0088(199709)17:11<1155::AID-JOC188>3.0.CO;2-B. DOI

Kumar KK, Rajagopalan B, Cane MA. On the weakening relationship between the Indian monsoon and enso. Science. 1999;284:2156–2159. doi: 10.1126/science.284.5423.2156. PubMed DOI

Krishnamurthy V, Goswami BN. Indian monsoon-enso relationship on interdecadal timescale. J. Clim. 2000;13:579–595. doi: 10.1175/1520-0442(2000)013<0579:IMEROI>2.0.CO;2. DOI

Sarkar S, Singh RP, Kafatos M. Further evidences for the weakening relationship of Indian rainfall and enso over India. Geophys. Res. Lett. 2004 doi: 10.1029/2004GL020259. DOI

Maraun D, Kurths J. Epochs of phase coherence between el nino/southern oscillation and Indian monsoon. Geophys. Res. Lett. 2005 doi: 10.1029/2005GL023225. DOI

Zubair L, Ropelewski CF. The strengthening relationship between ENSO and northeast monsoon rainfall over Sri Lanka and southern India. J. Clim. 2006;19:1567–1575. doi: 10.1175/JCLI3670.1. DOI

Mokhov II, et al. Alternating mutual influence of el-niño/southern oscillation and Indian monsoon. Geophys. Res. Lett. 2011 doi: 10.1029/2010GL045932. DOI

Mokhov I, Smirnov D, Nakonechny P, Kozlenko S, Kurths J. Relationship between el-nino/southern oscillation and the Indian monsoon. Izv. Atmos. Ocean. Phys. 2012;48:47–56. doi: 10.1134/S0001433812010082. DOI

Le T, Ha K-J, Bae D-H, Kim S-H. Causal effects of Indian ocean dipole on el niño-southern oscillation during 1950–2014 based on high-resolution models and reanalysis data. Environ. Res. Lett. 2020;15:1040b6. doi: 10.1088/1748-9326/abb96d. DOI

Wanner H, et al. North Atlantic oscillation-concepts and studies. Surv. Geophys. 2001;22:321–381. doi: 10.1023/A:1014217317898. DOI

Hurrell JW, Deser C. North Atlantic climate variability: The role of the north Atlantic oscillation. J. Mar. Syst. 2010;79:231–244. doi: 10.1016/j.jmarsys.2009.11.002. DOI

Deser C, Hurrell JW, Phillips AS. The role of the north Atlantic oscillation in European climate projections. Clim. Dyn. 2017;49:3141–3157. doi: 10.1007/s00382-016-3502-z. DOI

Wang W, Anderson BT, Kaufmann RK, Myneni RB. The relation between the north Atlantic oscillation and SSTS in the north Atlantic basin. J. Clim. 2004;17:4752–4759. doi: 10.1175/JCLI-3186.1. DOI

Wang G, Zhang N, Fan K, Palus M. Central European air temperature: Driving force analysis and causal influence of NAO. Theoret. Appl. Climatol. 2019;137:1421–1427. doi: 10.1007/s00704-018-2676-1. DOI

Hlinka J, Jajcay N, Hartman D, Paluš M. Smooth information flow in temperature climate network reflects mass transport. Chaos Interdiscip. J. Nonlinear Sci. 2017;27:035811. doi: 10.1063/1.4978028. PubMed DOI

Nagaraj N, Balasubramanian K, Dey S. A new complexity measure for time series analysis and classification. Eur. Phys. J. Special Top. 2013;222:847–860. doi: 10.1140/epjst/e2013-01888-9. DOI

Causes of extreme events revealed by Rényi information transfer