- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Linear-inverse-modeling approach to estimating Liang-Kleeman information flow in a cyclostationary process under memoryless and persistent noise

Justin Lien*

Hiroyasu Ando

Yong-Yub Kim

Tomoki Tozuka

  • *Contact author: lien.justin.t8@dc.tohoku.ac.jp
PDF

Phys. Rev. E 112, 054101 – Published 3 November, 2025

DOI: https://doi.org/10.1103/5cd4-5cb4

Abstract

The Liang-Kleeman (LK) information flow provides a powerful framework for quantifying causality among variables, while linear inverse modeling (LIM) offers an effective empirical approach to studying the dynamical evolution of system states from input data. In this study, we unify these two concepts by proposing the LIM-LK framework—a data-driven method for estimating LK information flow from input data using LIM. Beyond capturing causality among state variables, the proposed framework establishes a direct connection between causality and system dynamics, and also enables the quantification of entropy transfer from the ambient environment to the system through both memoryless and persistent stochastic forcing. The effectiveness of this unified approach is demonstrated through an application to the interaction between the Pacific and Indian Oceans, offering both causal and dynamical insight into ocean variability and its seasonal modulation.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (88)

  1. R. Guo, L. Cheng, J. Li, P. R. Hahn, and H. Liu, A survey of learning causality with data: Problems and methods, ACM Comput. Surv. 53, 1 (2020).
  2. X. S. Liang, Unraveling the cause-effect relation between time series, Phys. Rev. E 90, 052150 (2014).
  3. A. Alper and O. Oguz, The role of renewable energy consumption in economic growth: Evidence from asymmetric causality, Renew. Sustain. Energy Rev. 60, 953 (2016).
  4. A. Shojaie and E. B. Fox, Granger causality: A review and recent advances, Annu. Rev. Stat. Applic. 9, 289 (2022).
  5. K. Yu, X. Guo, L. Liu, J. Li, H. Wang, Z. Ling, and X. Wu, Causality-based feature selection: Methods and evaluations, ACM Comput. Surv. 53, 1 (2020).
  6. S. Shimizu, P. O. Hoyer, A. Hyvärinen, and A. Kerminen, A linear non-Gaussian acyclic model for causal discovery, J. Mach. Learn. Res. 7, 2003 (2006).
  7. A. Hyvärinen, K. Zhang, S. Shimizu, and P. O. Hoyer, Estimation of a structural vector autoregression model using non-Gaussianity, J. Mach. Learn. Res. 11, 1709 (2010).
  8. H. Majdi, M. Azarnoosh, M. Ghoshuni, and V. R. Sabzevari, Direct lingam and visibility graphs for analyzing brain connectivity in BCI, Med. Biolog. Eng. Comput. 62, 2117 (2024).
  9. S. Shimizu, T. Inazumi, Y. Sogawa, A. Hyvärinen, Y. Kawahara, T. Washio, P. O. Hoyer, and K. A. Bollen, DirectLiNGAM: A direct method for learning a linear non-Gaussian structural equation model, J. Mach. Learn. Res. 12, 1225 (2011).
  10. H. Suzuki, LayeredLiNGAM: A practical and fast method for learning a linear non-Gaussian structural equation model, in Machine Learning and Knowledge Discovery in Databases. Research Track: European Conference, ECML PKDD 2024, Proceedings, Part VI (Springer-Verlag, Berlin, 2024), pp. 217–233.
  11. L.-Y. Leung and G. R. North, Information theory and climate prediction, J. Clim. 3, 5 (1990).
  12. X. S. Liang and R. Kleeman, Information transfer between dynamical system components, Phys. Rev. Lett. 95, 244101 (2005).
  13. A. J. Majda and J. Harlim, Information flow between subspaces of complex dynamical systems, Proc. Natl. Acad. Sci. USA 104, 9558 (2007).
  14. G. Bianconi, Statistical mechanics of multiplex networks: Entropy and overlap, Phys. Rev. E 87, 062806 (2013).
  15. T. DelSole, Predictability and information theory. Part I: Measures of predictability, J. Atmos. Sci. 61, 2425 (2004).
  16. T. DelSole, Predictability and information theory. Part II: Imperfect forecasts, J. Atmos. Sci. 62, 3368 (2005).
  17. R. Kleeman and A. Majda, Predictability in a model of geophysical turbulence, J. Atmos. Sci. 62, 2864 (2005).
  18. R. Kleeman, Information flow in ensemble weather predictions, J. Atmos. Sci. 64, 1005 (2007).
  19. P. Talkner and P. Hänggi, Colloquium: Statistical mechanics and thermodynamics at strong coupling: Quantum and classical, Rev. Mod. Phys. 92, 041002 (2020).
  20. T. Schreiber, Measuring information transfer, Phys. Rev. Lett. 85, 461 (2000).
  21. X. S. Liang, Normalized multivariate time series causality analysis and causal graph reconstruction, Entropy 23, 679 (2021).
  22. D. F. T. Hagan, G. Wang, X. S. Liang, and H. A. J. Dolman, A time-varying causality formalism based on the Liang-Kleeman information flow for analyzing directed interactions in nonstationary climate systems, J. Clim. 32, 7521 (2019).
  23. Y. Takaya, K. K. Komatsu, N. G. Ganeshi, T. Toyoda, and H. Hasumi, A sub-monthly timescale causality between snow cover and surface air temperature in the northern hemisphere inferred by Liang-Kleeman information flow analysis, Climate Dynam. 62, 2735 (2024).
  24. Z. Ye and T. Tozuka, Causal relationship between sea surface temperature and precipitation revealed by information flow, Front. Climate 4, 1024384 (2022).
  25. J. Lien, H. Ando, I. Richter, and S. Kido, Noise persistence in a cyclostationary linear inverse model framework and its impact on ENSO evolution and prediction skills (unpublished).
  26. S. Zhao, F.-F. Jin, M. F. Stuecker, P. R. Thompson, J.-S. Kug, M. J. McPhaden, M. A. Cane, A. T. Wittenberg, and W. Cai, Explainable El Niño predictability from climate mode interactions, Nature (London) 630, 891 (2024).
  27. J. Lien and H. Ando, Nonlinear extensions of linear inverse models under memoryless or persistent random forcing, Phys. Rev. Res. 7, L032038 (2025).
  28. J. Lou, T. J. O'Kane, and N. J. Holbrook, A linear inverse model of tropical and South Pacific seasonal predictability, J. Clim. 33, 4537 (2020).
  29. M. Newman, S.-I. Shin, and M. A. Alexander, Natural variation in ENSO flavors, Geophys. Res. Lett. 38, L14705 (2011).
  30. C. Penland and T. Magorian, Prediction of Niño 3 Sea surface temperatures using linear inverse modeling, J. Clim. 6, 1067 (1993).
  31. W. A. Perkins and G. Hakim, Linear inverse modeling for coupled atmosphere-ocean ensemble climate prediction, J. Adv. Model. Earth Syst. 12, e2019MS001778 (2020).
  32. J. Lien, A Liang-Kleeman causality analysis based on linear inverse modeling, arXiv:2409.06797.
  33. S. Kido, I. Richter, T. Tozuka, and P. Chang, Understanding the interplay between ENSO and related tropical SST variability using linear inverse models, Climate Dynam. 61, 1029 (2023).
  34. J. Lien, Y.-N. Kuo, H. Ando, and S. Kido, Colored linear inverse model: A data-driven method for studying dynamical systems with temporally correlated stochasticity, Phys. Rev. Res. 7, 023042 (2025).
  35. J. Lien, Y.-N. Kuo, H. Ando, and S. Kido, On cyclostationary linear inverse models: A mathematical insight and implication, Front. Complex Syst. 3, 1563687 (2025).
  36. S.-I. Shin, P. D. Sardeshmukh, M. Newman, C. Penland, and M. A. Alexander, Impact of annual cycle on ENSO variability and predictability, J. Clim. 34, 171 (2021).
  37. G. Gradziuk, G. Torregrosa, and C. P. Broedersz, Irreversibility in linear systems with colored noise, Phys. Rev. E 105, 024118 (2022).
  38. X. S. Liang, Information flow within stochastic dynamical systems, Phys. Rev. E 78, 031113 (2008).
  39. X. S. Liang, Normalizing the causality between time series, Phys. Rev. E 92, 022126 (2015).
  40. C. Penland and L. Matrosova, A balance condition for stochastic numerical models with application to the El Niño-southern oscillation, J. Clim. 7, 1352 (1994).
  41. T. DelSole and M. K. Tippett, Predictability: Recent insights from information theory, Rev. Geophys. 45, 2006RG000202 (2007).
  42. W. Cai, A. Santoso, G. Wang, S.-W. Yeh, S.-I. An, K. M. Cobb, M. Collins, E. Guilyardi, F.-F. Jin, J.-S. Kug, M. Lengaigne, M. J. McPhaden, K. Takahashi, A. Timmermann, G. Vecchi, M. Watanabe, and L. Wu, ENSO and greenhouse warming, Nat. Climate Change 5, 849 (2015).
  43. P. J. Ward, B. Jongman, M. Kummu, M. D. Dettinger, F. C. Sperna Weiland, and H. C. Winsemius, Strong influence of El Niño southern oscillation on flood risk around the world, Proc. Natl. Acad. Sci. USA 111, 15659 (2014).
  44. D. MacLeod, E. W. Kolstad, K. Michaelides, and M. B. Singer, Sensitivity of rainfall extremes to unprecedented Indian Ocean dipole events, Geophys. Res. Lett. 51, e2023GL105258 (2024).
  45. N. H. Saji, B. N. Goswami, P. N. Vinayachandran, and T. Yamagata, A dipole mode in the tropical Indian Ocean, Nature (London) 401, 360 (1999).
  46. E. Galanti and E. Tziperman, ENSO's phase locking to the seasonal cycle in the fast-SST, fast-wave, and mixed-mode regimes, J. Atmos. Sci. 57, 2936 (2000).
  47. K.-Y. Kim and C. Chung, On the evolution of the annual cycle in the tropical pacific, J. Clim. 14, 991 (2001).
  48. E. Tziperman, M. A. Cane, S. E. Zebiak, Y. Xue, and B. Blumenthal, Locking of El Niño's peak time to the end of the calendar year in the delayed oscillator picture of ENSO, J. Clim. 11, 2191 (1998).
  49. K. Ashok, Z. Guan, and T. Yamagata, Influence of the Indian Ocean dipole on the Australian winter rainfall, Geophys. Res. Lett. 30, 2003GL017926 (2003).
  50. C.-H. Park and S.-W. Son, Subseasonal variability of ENSO–East Asia teleconnections driven by tropical convection over the Indian Ocean and maritime continent, Geophys. Res. Lett. 51, e2023GL108062 (2024).
  51. J. Yan, W. Zhang, S. Hu, and F. Jiang, Different ENSO impacts on eastern china precipitation patterns in early and late winter associated with seasonally-varying Kuroshio anticyclonic anomalies, Adv. Atmos. Sci. 41, 1691 (2024).
  52. W. Cai, P. Van Rensch, T. Cowan, and H. H. Hendon, Teleconnection pathways of ENSO and the IOD and the mechanisms for impacts on Australian rainfall, J. Clim. 24, 3910 (2011).
  53. T. Doi, S. K. Behera, and T. Yamagata, Wintertime impacts of the 2019 super IOD on East Asia, Geophys. Res. Lett. 47, e2020GL089456 (2020).
  54. P. H. Hrudya, H. Varikoden, and R. Vishnu, A review on the Indian Summer monsoon rainfall, variability and its association with ENSO and IOD, Meteorol. Atmosph. Phys. 133, 1 (2021).
  55. T.-H. Huang, C.-T. A. Chen, W.-Z. Zhang, and X.-F. Zhuang, Varying intensity of Kuroshio intrusion into Southeast Taiwan Strait during ENSO events, Continent. Shelf Res. 103, 79 (2015).
  56. A. Kurniadi, E. Weller, S.-K. Min, and M.-G. Seong, Independent ENSO and IOD impacts on rainfall extremes over Indonesia, Intl. J. Climatol. 41, 3640 (2021).
  57. D. Nguyen-Le, T. Ngo-Duc, and J. Matsumoto, The teleconnection of the two types of ENSO and Indian Ocean dipole on Southeast Asian autumn rainfall anomalies, Climate Dynam. 62, 1 (2024).
  58. H. Annamalai, S. P. Xie, J. P. McCreary, and R. Murtugudde, Impact of Indian Ocean sea surface temperature on developing El Niño, J. Clim. 18, 302 (2005).
  59. F. Jiang, W. Zhang, F.-F. Jin, M. F. Stuecker, and R. Allan, El Niño pacing orchestrates inter-basin Pacific-Indian Ocean interannual connections, Geophys. Res. Lett. 48, e2021GL095242 (2021).
  60. T. Izumo, J. Vialard, M. Lengaigne, C. de Boyer Montegut, S. K. Behera, J.-J. Luo, S. Cravatte, S. Masson, and T. Yamagata, Influence of the state of the Indian Ocean dipole on the following year's El Niño, Nat. Geosci. 3, 168 (2010).
  61. M. F. Stuecker, A. Timmermann, F.-F. Jin, Y. Chikamoto, W. Zhang, A. T. Wittenberg, E. Widiasih, and S. Zhao, Revisiting ENSO/Indian Ocean dipole phase relationships, Geophys. Res. Lett. 44, 2481 (2017).
  62. S. K. Behera, J. J. Luo, S. Masson, S. A. Rao, H. Sakuma, and T. Yamagata, A CGCM study on the interaction between IOD and ENSO, J. Clim. 19, 1688 (2006).
  63. W. Cai, P. van Rensch, T. Cowan, and H. H. Hendon, An asymmetry in the IOD and ENSO teleconnection pathway and its impact on Australian climate, J. Clim. 25, 6318 (2012).
  64. T. Yamagata, S. K. Behera, J.-J. Luo, S. Masson, M. R. Jury, and S. A. Rao, Coupled ocean-atmosphere variability in the tropical Indian Ocean, Earth's Climate (American Geophysical Union, 2004), pp. 189–211.
  65. P. J. Reddy, S. E. Perkins-Kirkpatrick, N. N. Ridder, and J. J. Sharples, Combined role of ENSO and IOD on compound drought and heatwaves in Australia using two CMIP6 large ensembles, Weath. Climate Extrem. 37, 100469 (2022).
  66. A. M. Jenney, D. A. Randall, and E. A. Barnes, Drivers of uncertainty in future projections of Madden–Julian oscillation teleconnections, Weath. Climate Dynam. 2, 653 (2021).
  67. R. A. Madden and P. R. Julian, Observations of the 40–50-day tropical oscillation—A review, Mon. Weather Rev. 122, 814 (1994).
  68. L. Yu, R. A. Weller, and W. T. Liu, Case analysis of a role of ENSO in regulating the generation of westerly wind bursts in the Western Equatorial Pacific, J. Geophys. Res.: Oceans 108, 3128 (2003).
  69. J. R. Albers and M. Newman, A priori identification of skillful extratropical subseasonal forecasts, Geophys. Res. Lett. 46, 12527 (2019).
  70. M. Newman, M. A. Alexander, and J. D. Scott, An empirical model of tropical ocean dynamics, Climate Dynam. 37, 1823 (2011).
  71. B. Huang, P. W. Thorne, V. F. Banzon, T. Boyer, G. Chepurin, J. H. Lawrimore, M. J. Menne, T. M. Smith, R. S. Vose, and H.-M. Zhang, NOAA Extended Reconstructed Sea Surface Temperature (ERSST), ver. 5 (NOAA National Centers for Environmental Information, 2017), sst.mnmean.nc.
  72. NOAA Physical Sciences Laboratory, Dipole Mode Index (DMI) Monthly Time-series, NOAA PSL, 325 Broadway, Boulder, CO, USA (2024), https://psl.noaa.gov/data/timeseries/month/DS/DMI/.
  73. K. Ashok and T. Yamagata, The El Niño with a difference, Nature (London) 461, 481 (2009).
  74. H.-Y. Kao and J.-Y. Yu, Contrasting Eastern-Pacific and Central-Pacific types of ENSO, J. Clim. 22, 615 (2009).
  75. S.-W. Yeh, J.-S. Kug, and S.-I. An, Recent progress on two types of El Niño: Observations, dynamics, and future changes, Asia-Pacific J. Atmosph. Sci. 50, 69 (2014).
  76. S. N. Hameed, D. Jin, and V. Thilakan, A model for super El Niños, Nat. Commun. 9, 2528 (2018).
  77. J. Song, P. J. Klotzbach, and Y. Duan, Statistical linkage between coastal El Niño–southern oscillation and tropical cyclone formation over the Western North Pacific, Atmosph. Sci. Lett. 23, e1071 (2022).
  78. Q. Peng, S.-P. Xie, G. A. Passalacqua, A. Miyamoto, and C. Deser, The 2023 extreme coastal El Niño: Atmospheric and air-sea coupling mechanisms, Sci. Adv. 10, eadk8646 (2024).
  79. D. Rudloff and J. F. Lübbecke, What controls the evolution of pacific coastal Niño events in the CESM large ensemble? Geophys. Res. Lett. 52, e2024GL113706 (2025).
  80. F.-F. Jin, J. Neelin, and M. Ghil, El Niño/southern oscillation and the annual cycle: Subharmonic frequency-locking and aperiodicity, Physica D 98, 442 (1996), nonlinear Phenomena in Ocean Dynamics.
  81. J.-J. Luo, S. Masson, S. Behera, S. Shingu, and T. Yamagata, Seasonal climate predictability in a coupled OAGCM using a different approach for ensemble forecasts, J. Clim. 18, 4474 (2005).
  82. M. Newman and P. D. Sardeshmukh, Are we near the predictability limit of tropical Indo-Pacific sea surface temperatures?, Geophys. Res. Lett. 44, 8520 (2017).
  83. C. Torrence and P. J. Webster, The annual cycle of persistence in the El Niño/southern oscillation, Quarter. J. R. Meteorolog. Soc. 124, 1985 (1998).
  84. C. Martinez-Villalobos, B. Dewitte, R. D. Garreaud, and L. Loyola, Extreme coastal El Niño events are tightly linked to the development of the Pacific meridional modes, npj Climate Atmosph. Sci. 7, 123 (2024).
  85. M. Baldovin, F. Cecconi, A. Provenzale, and A. Vulpiani, Extracting causation from millennial-scale climate fluctuations in the last 800 kyr, Sci. Rep. 12, 15320 (2022).
  86. N. D. B. Keyes, L. T. Giorgini, and J. S. Wettlaufer, Stochastic paleoclimatology: Modeling the EPICA ice core climate records, Chaos: Interdisc. J. Nonlin. Sci. 33, 093132 (2023).
  87. W. Moon and J. S. Wettlaufer, Coupling functions in climate, Philos. Trans. R. Soc. A 377, 20190006 (2019).
  88. N. H. Saji and T. J. C. R. Yamagata, Possible impacts of Indian Ocean dipole mode events on global climate, Climate Res. 25, 151 (2003).

Outline

Information

Phys. Rev. E 112, 054101– Published 3 November, 2025
  • Received 27 June 2025
  • Accepted 22 September 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation