Quantification of relations between measured variables of interest by statistical measures of dependence is a common step in analysis of climate data. The choice of dependence measure is key for the results of the subsequent analysis and interpretation. The use of linear Pearson’s correlation coefficient is widespread and convenient. On the other side, as the climate is widely acknowledged to be a nonlinear system, nonlinear dependence quantification methods, such as those based on information-theoretical concepts, are increasingly used for this purpose. In this paper we outline an approach that enables well informed choice of dependence measure for a given type of data, improving the subsequent interpretation of the results. The presented multi-step approach includes statistical testing, quantification of the specific non-linear contribution to the interaction information, localization of areas with strongest nonlinear contribution and assessment of the role of specific temporal patterns, including signal nonstationarities. In detail we study the consequences of the choice of a general nonlinear dependence measure, namely mutual information, focusing on its relevance and potential alterations in the discovered dependence structure. We document the method by applying it to monthly mean temperature data from the NCEP/NCAR reanalysis dataset as well as the ERA dataset. We have been able to identify main sources of observed non-linearity in inter-node couplings. Detailed analysis suggested an important role of several sources of nonstationarity within the climate data. The quantitative role of genuine nonlinear coupling at monthly scale has proven to be almost negligible, providing quantitative support for the use of linear methods for monthly temperature data.
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An S, Jin F (2004) Nonlinearity and asymmetry of ENSO. J Clim 17(12):2399–2412
Boucharel J, Dewitte B, du Penhoat Y, Garel B, Yeh SW, Kug JS (2011) ENSO nonlinearity in a warming climate. Clim Dyn 37(9–10):2045–2065
Cellucci C, Albano A, Rapp P (2005) Statistical validation of mutual information calculations: comparison of alternative numerical algorithms. Phys Rev E 71(6, Part 2):066,208
Dee DP, Uppala SM, Simmons AJ, Berrisford P, Poli P, Kobayashi S, Andrae U, Balmaseda MA, Balsamo G, Bauer P, Bechtold P, Beljaars ACM, van de Berg L, Bidlot J, Bormann N, Delsol C, Dragani R, Fuentes M, Geer AJ, Haimberger L, Healy SB, Hersbach H, Holm EV, Isaksen L, Kallberg P, Koehler M, Matricardi M, McNally AP, Monge-Sanz BM, Morcrette JJ, Park BK, Peubey C, de Rosnay P, Tavolato C, Thepaut JN, Vitart F (2011) The ERA-Interim reanalysis: configuration and performance of the data assimilation system. Q J R Meteorol Soc 137(656, Part a):553–597
Diks C, Mudelsee M (2000) Redundancies in the earth’s climatological time series. Phys Lett A 275(5–6):407–414
Donges JF, Zou Y, Marwan N, Kurths J (2009) The backbone of the climate network. EPL 87(4):48,007
Donges JF, Schultz HCH, Marwan N, Zou Y, Kurths J (2011) Investigating the topology of interacting networks theory and application to coupled climate subnetworks. Eur Phys J B 84(4):635–651
Feliks Y, Ghil M, Robertson AW (2010) Oscillatory climate modes in the eastern mediterranean and their synchronization with the north atlantic oscillation. J Clim 23(15):4060–4079
Hannachi A, Stephenson D, Sperber K (2003) Probability-based methods for quantifying nonlinearity in the ENSO. Clim Dyn 20(2–3):241–256
Hannachi A, Jolliffe IT, Stephenson DB (2007) Empirical orthogonal functions and related techniques in atmospheric science: a review. Int J Climatol 27(9):1119–1152
Hartman D, Hlinka J, Paluš M, Mantini D, Corbetta M (2011) The role of nonlinearity in computing graph-theoretical properties of resting-state functional magnetic resonance imaging brain networks. Chaos 21(1):013,119
Hlinka J, Palus M, Vejmelka M, Mantini D, Corbetta M (2011) Functional connectivity in resting-state fMRI: is linear correlation sufficient?. NeuroImage 54:2218–2225
Hlinka J, Hartman D, Palus M (2012) Small-world topology of functional connectivity in randomly connected dynamical systems. Chaos 22(3):033,107
Hsieh W, Wu A, Shabbar A (2006) Nonlinear atmospheric teleconnections. Geophys Res Lett 33(7):L07,714
Kalnay E, Kanamitsu M, Kistler R, Collins W, Deaven D, Gandin L, Iredell M, Saha S, White G, Woollen J, Zhu Y, Chelliah M, Ebisuzaki W, Higgins W, Janowiak J, Mo K, Ropelewski C, Wang J, Leetmaa A, Reynolds R, Jenne R, Joseph D (1996) The NCEP/NCAR 40-year reanalysis project. Bull Am Meteorol Soc 77(3):437–471
Kendall M (1938) A new measure of rank correlation. Biometrika 30(Part 1/2):81–93
Khan S, Bandyopadhyay S, Ganguly AR, Saigal S, Erickson DJ III, Protopopescu V, Ostrouchov G (2007) Relative performance of mutual information estimation methods for quantifying the dependence among short and noisy data. Phys Rev E 76(2, Part 2):026,209
Kistler R, Kalnay E, Collins W, Saha S, White G, Woollen J, Chelliah M, Ebisuzaki W, Kanamitsu M, Kousky V, van den Dool H, Jenne R, Fiorino M (2001) The NCEP-NCAR 50-year reanalysis: monthly means CD-ROM and documentation. Bull Am Meteorol Soc 82(2):247–267
Kraskov A, Stogbauer H, Grassberger P (2004) Estimating mutual information. Phys Rev E 69(6, Part 2):066,138
Palus M (1997) Detecting phase synchronization in noisy systems. Phys Lett A 235(4):341–351
Palus M, Novotna D (1994) Testing for nonlinearity in weather records. Phys Lett A 193(1):67–74
Palus M, Novotna D (2004) Enhanced monte carlo singular system analysis and detection of period 7.8 years oscillatory modes in the monthly NAO index and temperature records. Nonlinear Process Geophys 11(5–6):721–729
Palus M, Novotna D (2006) Quasi-biennial oscillations extracted from the monthly nao index and temperature records are phase-synchronized. Nonlinear Process Geophys 13(3):287–296
Palus M, Novotna D (2009) Phase-coherent oscillatory modes in solar and geomagnetic activity and climate variability. J Atmos Solar Terr Phys 71(8–9):923–930
Palus M, Novotna D (2011) Northern hemisphere patterns of phase coherence between solar/geomagnetic activity and ncep/ncar and era40 near-surface air temperature in period 7–8 years oscillatory modes. Nonlinear Process Geophys 18(2):251–260
Palus M, Albrecht V, Dvorak I (1993) Information theoretic test for nonlinearity in time series. Phys Lett A 175(3–4):203–209
Palus M, Hartman D, Hlinka J, Vejmelka M (2011) Discerning connectivity from dynamics in climate networks. Nonlinear Process Geophys 18(5):751–763
Paluš M (1995) Testing for nonlinearity using redundancies: quantitative and qualitative aspects. Physica D 80(1–2):186–205
Papana A, Kugiumtzis D (2009) Evaluation of mutual information estimators for time series. Int J Bifurcat Chaos 19:4197–4215
Prichard D, Theiler J (1994) Generating surrogate data for time series with several simultaneously measured variables. Phys Rev Lett 73(7):951
Schreiber T, Schmitz A (1996) Improved surrogate data for nonlinearity tests. Phys Rev Lett 77(4):635–638
Schreiber T, Schmitz A (2000) Surrogate time series. Physica D 142(3–4):346–382
Shannon C (1948) A mathematical theory of communication. Bell Syst Tech J 27(3):379–423
Spearman C (1904) The proof and measurement of association between two things. Am J Psychol 15:72–101
Theiler J, Eubank S, Longtin A, Galdrikian B, Farmer JD (1992) Testing for nonlinearity in time series: the method of surrogate data. Physica D 58(1–4):77–94
Trenberth K (1997) The definition of el nino. Bull Am Meteorol Soc 78(12):2771–2777
Tsonis A, Roebber P (2004) The architecture of the climate network. Physica A 333:497–504
Tsonis AA, Wang G, Swanson KL, Rodrigues FA, Costa LdF (2011) Community structure and dynamics in climate networks. Clim Dyn 37(5–6):933–940
Uppala S, Kallberg P, Simmons A, Andrae U, Bechtold V, Fiorino M, Gibson J, Haseler J, Hernandez A, Kelly G, Li X, Onogi K, Saarinen S, Sokka N, Allan R, Andersson E, Arpe K, Balmaseda M, Beljaars A, Van De Berg L, Bidlot J, Bormann N, Caires S, Chevallier F, Dethof A, Dragosavac M, Fisher M, Fuentes M, Hagemann S, Holm E, Hoskins B, Isaksen L, Janssen P, Jenne R, McNally A, Mahfouf J, Morcrette J, Rayner N, Saunders R, Simon P, Sterl A, Trenberth K, Untch A, Vasiljevic D, Viterbo P, Woollen J (2005) The ERA-40 re-analysis. Q J R Meteorol Soc 131(612, Part b):2961–3012
Venema V, Bachner S, Rust HW, Simmer C (2006) Statistical characteristics of surrogate data based on geophysical measurements. Nonlinear Process Geophys 13(4):449–466
This study is supported by the Czech Science Foundation, Project No. P103/11/J068.
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Hlinka, J., Hartman, D., Vejmelka, M. et al. Non-linear dependence and teleconnections in climate data: sources, relevance, nonstationarity. Clim Dyn 42, 1873–1886 (2014). https://doi.org/10.1007/s00382-013-1780-2
- Climate networks
- Mutual information
- Seasonality in variance