Linear dynamical modes as new variables for data-driven ENSO forecast

  • Andrey Gavrilov
  • Aleksei Seleznev
  • Dmitry Mukhin
  • Evgeny Loskutov
  • Alexander Feigin
  • Juergen Kurths


A new data-driven model for analysis and prediction of spatially distributed time series is proposed. The model is based on a linear dynamical mode (LDM) decomposition of the observed data which is derived from a recently developed nonlinear dimensionality reduction approach. The key point of this approach is its ability to take into account simple dynamical properties of the observed system by means of revealing the system’s dominant time scales. The LDMs are used as new variables for empirical construction of a nonlinear stochastic evolution operator. The method is applied to the sea surface temperature anomaly field in the tropical belt where the El Nino Southern Oscillation (ENSO) is the main mode of variability. The advantage of LDMs versus traditionally used empirical orthogonal function decomposition is demonstrated for this data. Specifically, it is shown that the new model has a competitive ENSO forecast skill in comparison with the other existing ENSO models.


Empirical modeling Data dimensionality reduction Nonlinear stochastic modeling ENSO forecast 



The results (Sects. 34) were obtained and analyzed under support of the Government of Russian Federation (Agreement #14.Z50.31.0033 with the Institute of Applied Physics of RAS). The methods (Sect. 2) were developed under support of the Russian Science Foundation (Grant #16-12-10198).


  1. Alexander MA, Bladé I, Newman M, Lanzante JR, Lau NC, Scott JD (2002) The atmospheric bridge: the influence of ENSO teleconnections on air-sea interaction over the global oceans. J Clim 15(16):2205–2231.<2205:TABTIO>2.0.CO;2 CrossRefGoogle Scholar
  2. Barnston AG, Chelliah M, Goldenberg SB (1997) Documentation of a highly ENSO related sst region in the equatorial Pacific: research note. Atmos Ocean 35(3):367–383. CrossRefGoogle Scholar
  3. Barnston AG, Tippett MK, L’Heureux ML, Li S, Dewitt DG (2012) Skill of real-time seasonal ENSO model predictions during 2002–11: is our capability increasing? Bull Am Meteorol Soc 93(5):631–651. CrossRefGoogle Scholar
  4. Berliner LM, Wikle CK, Cressie N (2000) Long-lead prediction of Pacific SSTs via Bayesian dynamic modeling. J Clim 13(22):3953–3968.<3953:LLPOPS>2.0.CO;2 CrossRefGoogle Scholar
  5. Bjerknes J (1969) Atmospheric teleconnections from the equatorial Pacific. Mon Weather Rev 97(3):163–172.<0163:ATFTEP>2.3.CO;2 CrossRefGoogle Scholar
  6. Chekroun MD, Kondrashov D (2017) Data-adaptive harmonic spectra and multilayer Stuart–Landau models. Chaos Interdiscip J Nonlinear Sci 27(9):093110. CrossRefGoogle Scholar
  7. Coifman RR, Lafon S (2006) Diffusion maps. Appl Comput Harmon Anal 21(1):5–30. CrossRefGoogle Scholar
  8. de la Iglesia MD, Tabak EG (2013) Principal dynamical components. Commun Pure Appl Math 66(1):48–82. arXiv:1012.3963v1 CrossRefGoogle Scholar
  9. DelSole T, Tippett MK (2009b) Average predictability time. Part II: seamless diagnoses of predictability on multiple time scales. J Atmos Sci 66(5):1188–1204. CrossRefGoogle Scholar
  10. DelSole T, Tippett MK (2009a) Average predictability time. Part I: theory. J Atmos Sci 66(5):1172–1187. CrossRefGoogle Scholar
  11. Dong D, McAvoy T (1996) Nonlinear principal component analysis–based on principal curves and neural networks. Comput Chem Eng 20(1):65–78. CrossRefGoogle Scholar
  12. Gámez AJ, Zhou CS, Timmermann A, Kurths J (2004) Nonlinear dimensionality reduction in climate data. Nonlinear Process Geophys 11(3):393–398. CrossRefGoogle Scholar
  13. Gavrilov A, Mukhin D, Loskutov E, Volodin E, Feigin A, Kurths J (2016) Method for reconstructing nonlinear modes with adaptive structure from multidimensional data. Chaos Interdiscip J Nonlinear Sci 26(12):123101. CrossRefGoogle Scholar
  14. Gavrilov A, Loskutov E, Mukhin D (2017) Bayesian optimization of empirical model with state-dependent stochastic forcing. Chaos Solitons Fract 104:372. CrossRefGoogle Scholar
  15. Ghil M, Allen MR, Dettinger MD, Ide K, Kondrashov D, Mann ME, Robertson AW, Saunders A, Tian Y, Varadi F, Yiou P (2002) Advanced spectral methods for climatic time series. Rev Geophys 40(1):1003. CrossRefGoogle Scholar
  16. Grieger B, Latif M (1994) Reconstruction of the El Niño attractor with neural networks. Clim Dyn 10:267–276CrossRefGoogle Scholar
  17. Guckenheimer J, Timmermann A, Dijkstra H, Roberts A (2017) (Un)predictability of strong El Niño events. Dyn Stat Clim Syst. Google Scholar
  18. 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. CrossRefGoogle Scholar
  19. Hasselmann K (1988) PIPs and POPs: the reduction of complex dynamical systems using principal interaction and oscillation patterns. J Geophys Res 93(D9):11015. CrossRefGoogle Scholar
  20. Hastie T (1984) Principal curves and surfaces. Ph.D Dissertation. PhD thesis, Stanford Linear Accelerator Center, Stanford University. Accessed 29 May 2015
  21. Hornik K, Stinchcombe M, White H (1989) Multilayer feedforward networks are universal approximators. Neural Netw 2(5):359–366CrossRefGoogle Scholar
  22. Huang B, Banzon VF, Freeman E, Lawrimore J, Liu W, Peterson TC, Smith TM, Thorne PW, Woodruff SD, Zhang HM, Huang B, Banzon VF, Freeman E, Freeman E, Lawrimore J, Liu W, Peterson TC, Smith TM, Thorne PW, Woodruff SD, Zhang HM (2015) Extended reconstructed sea surface temperature version 4 (ERSST.v4). Part I: upgrades and intercomparisons. J Clim 28(3):911–930. CrossRefGoogle Scholar
  23. Jeffreys H (1998) Theory of probability. Clarendon. Accessed 9 May 2017
  24. Johnson SD, Battisti DS, Sarachik ES (2000) Seasonality in an empirically derived Markov model of tropical Pacific sea surface temperature anomalies. J Clim 13(18):3327–3335.<3327:SIAEDM>2.0.CO;2 CrossRefGoogle Scholar
  25. Jolliffe IT (1986) Principal component analysis. Springer series in statistics, 2nd edn. Springer, New York. CrossRefGoogle Scholar
  26. Kao HY, Yu JY (2009) Contrasting eastern-Pacific and central-Pacific types of ENSO. J Clim 22(3):615–632. CrossRefGoogle Scholar
  27. Kessler WS (2002) Is ENSO a cycle or a series of events? Geophys Res Lett 29(23):40. CrossRefGoogle Scholar
  28. Kondrashov D, Kravtsov D, Robertson AW, Ghil M (2015) A hierarchy of data-based ENSO models. J Clim 18(21):4425–4444. CrossRefGoogle Scholar
  29. Kramer MA (1991) Nonlinear principal component analysis using autoassociative neural networks. AIChE 37(2):233–243. CrossRefGoogle Scholar
  30. Kravtsov S (2012) An empirical model of decadal ENSO variability. Clim Dyn 39(9–10):2377–2391. CrossRefGoogle Scholar
  31. Kravtsov S, Kondrashov D, Ghil M (2005) Multilevel regression modeling of nonlinear processes: derivation and applications to climatic variability. J Clim 18:4404–4424CrossRefGoogle Scholar
  32. Kravtsov S, Kondrashov D, Ghil M (2009) Empirical model reduction and the modeling hierarchy in climate dynamics. In: Palmer T, Williams P (eds) Stochastic physics and climate modelling. Cambridge University Press, Cambridge, pp 35–72Google Scholar
  33. Kwasniok F (1996) The reduction of complex dynamical systems using principal interaction patterns. Phys D Nonlinear Phenom 92(1–2):28–60. CrossRefGoogle Scholar
  34. Kwasniok F (1997) Optimal Galerkin approximations of partial differential equations using principal interaction patterns. Phys Rev 55(5):5365–5375. Google Scholar
  35. Kwasniok F (2007) Reduced atmospheric models using dynamically motivated basis functions. J Atmos Sci 64(10):3452–3474. CrossRefGoogle Scholar
  36. Liu W, Huang B, Thorne PW, Banzon VF, Zhang HM, Freeman E, Lawrimore J, Peterson TC, Smith TM, Woodruff SD (2015) Extended reconstructed sea surface temperature version 4 (ERSST.v4): part II. Parametric and structural uncertainty estimations. J Clim 28(3):931–951CrossRefGoogle Scholar
  37. Loskutov EM, Molkov YI, Mukhin DN, Feigin AM (2008) Markov chain Monte Carlo method in Bayesian reconstruction of dynamical systems from noisy chaotic time series. Phys Rev E Stat Nonlinear Soft Matter Phys 77(6):066214. CrossRefGoogle Scholar
  38. Maimon O, Rokach L (2010) Data mining and knowledge discovery handbook. Springer, New YorkCrossRefGoogle Scholar
  39. Molkov YI, Mukhin DN, Loskutov EM, Feigin AM, Fidelin GA (2009) Using the minimum description length principle for global reconstruction of dynamic systems from noisy time series. Phys Rev E Stat Nonlinear Soft Matter Phys 80(4):046207. CrossRefGoogle Scholar
  40. Molkov YI, Mukhin DN, Loskutov EM, Timushev RI, Feigin AM (2011) Prognosis of qualitative system behavior by noisy, nonstationary, chaotic time series. Phys Rev E 84(3):036215. CrossRefGoogle Scholar
  41. Molkov YI, Loskutov EM, Mukhin DN, Feigin AM (2012) Random dynamical models from time series. Phys Rev E 85(3):036216. CrossRefGoogle Scholar
  42. Mukhin DN, Feigin AM, Loskutov EM, Molkov YI (2006) Modified Bayesian approach for the reconstruction of dynamical systems from time series. Phys Rev E Stat Nonlinear Soft Matter Phys 73(3):036211. CrossRefGoogle Scholar
  43. Mukhin D, Gavrilov A, Feigin A, Loskutov E, Kurths J (2015a) Principal nonlinear dynamical modes of climate variability. Sci Rep 5:15510. CrossRefGoogle Scholar
  44. Mukhin D, Kondrashov D, Loskutov E, Gavrilov A, Feigin A, Ghil M (2015b) Predicting critical transitions in ENSO models. Part II: spatially dependent models. J Clim 28(5):1962–1976. CrossRefGoogle Scholar
  45. Mukhin D, Loskutov E, Mukhina A, Feigin A, Zaliapin I, Ghil M (2015c) Predicting critical transitions in ENSO models. Part I: methodology and simple models with memory. J Clim 28(5):1940–1961. CrossRefGoogle Scholar
  46. Mukhin D, Gavrilov A, Loskutov E, Feigin A, Kurths J (2017) Nonlinear reconstruction of global climate leading modes on decadal scales. Clim Dyn. Google Scholar
  47. Penland C, Magorian T (1993) Prediction of Niño 3 sea surface temperatures using linear inverse modeling. J Clim 6(6):1067–1076.<1067:PONSST>2.0.CO;2 CrossRefGoogle Scholar
  48. Penland C, Sardeshmukh PD (1995) The optimal growth of tropical sea surface temperature anomalies. J Clim 8(8):1999–2024.<1999:TOGOTS>2.0.CO;2 CrossRefGoogle Scholar
  49. Pires CAL, Hannachi A (2017) independent subspace analysis of the sea surface temperature variability: non-Gaussian sources and sensitivity to sampling and dimensionality. Complexity 2017:1–23. CrossRefGoogle Scholar
  50. Pires CAL, Ribeiro AFS (2016) Separation of the atmospheric variability into non-Gaussian multidimensional sources by projection. Clim Dyn 48:1–30. Google Scholar
  51. Preisendorfer R (1988) Principal component analysis in meteorology and oceanography. Elsevier, LondonGoogle Scholar
  52. Rasmusson EM, Carpenter TH (1982) Variations in tropical sea surface temperature and surface wind fields associated with the Southern oscillation/El Niño. Mon Weather Rev 110(5):354–384.<0354:VITSST>2.0.CO;2 CrossRefGoogle Scholar
  53. Rossi V, Vila JP (2006) Bayesian multioutput feedforward neural networks comparison: a conjugate prior approach. IEEE Trans Neural Netw 17(1):35–47. CrossRefGoogle Scholar
  54. Saha S, Moorthi S, Wu X, Wang J, Nadiga S, Tripp P, Behringer D, Hou YT, Chuang HY, Iredell M, Ek M, Meng J, Yang R, Mendez MP, van den Dool H, Zhang Q, Wang W, Chen M, Becker E (2014) The NCEP climate forecast system version 2. J Clim 27(6):2185–2202. CrossRefGoogle Scholar
  55. Scholkopf B, Smola A, Muller KR (1998) Nonlinear component analysis as a Kernel Eigenvalue problem. Neural Comp 10(5):1299–1319. CrossRefGoogle Scholar
  56. Strounine K, Kravtsov S, Kondrashov D, Ghil M (2010) Reduced models of atmospheric low-frequency variability: parameter estimation and comparative performance. Phys D Nonlinear Phenom 239:145–166CrossRefGoogle Scholar
  57. Suarez MJ, Schopf PS (1988) A delayed action oscillator for ENSO. J Atmos Sci 45(21):3283–3287.<3283:ADAOFE>2.0.CO;2 CrossRefGoogle Scholar
  58. Tan S, Mayrovouniotis ML (1995) Reducing data dimensionality through optimizing neural network inputs. AIChE J 41(6):1471–1480. CrossRefGoogle Scholar
  59. Tippett MK, Barnston AG, Li S (2012) Performance of recent multimodel ENSO forecasts. J Appl Meteorol Climatol 51(3):637–654. CrossRefGoogle Scholar
  60. Trenberth KE (1997) The definition of El Niño. Bull Am Meteorol Soc 78(12):2771–2777.<2771:TDOENO>2.0.CO;2 CrossRefGoogle Scholar
  61. Vejmelka M, Pokorná L, Hlinka J, Hartman D, Jajcay N, Paluš M (2015) Non-random correlation structures and dimensionality reduction in multivariate climate data. Clim Dyn 44(9–10):2663–2682. CrossRefGoogle Scholar
  62. Wang C, Deser C, Yu JY, DiNezio P, Clement A (2017) El Niño and southern oscillation (ENSO): a review. Springer, Dordrecht, pp 85–106. Google Scholar
  63. Wu A, Hsieh WW, Tang B (2006) Neural network forecasts of the tropical Pacific sea surface temperatures. Neural Netw 19(2):145–154. CrossRefGoogle Scholar
  64. Wyrtki K (1975) El Niño The dynamic response of the equatorial Pacific ocean to atmospheric forcing. J Phys Oceanogr 5(4):572–584.<0572:ENTDRO>2.0.CO;2 CrossRefGoogle Scholar
  65. Xin X, Gao F, Wei M, Wu T, Fang Y, Zhang J (2017) Decadal prediction skill of BCC-CSM1.1 climate model in East Asia. Int J Climatol 38:584. CrossRefGoogle Scholar
  66. Xue Y, Leetmaa A, Ji M (2000) ENSO prediction with Markov models: the impact of sea level. J Clim 13(4):849–871.<0849:EPWMMT>2.0.CO;2 CrossRefGoogle Scholar
  67. Zebiak SE, Cane MA (1987) A Model El Ni&ntilde southern oscillation. Mon Weather Rev 115(10):2262–2278.<2262:AMENO>2.0.CO;2 CrossRefGoogle Scholar
  68. Zhang G, Kline D (2007) Quarterly time-series forecasting with neural networks. IEEE Trans Neural Netw 18(6):1800–1814.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Applied Physics of RASNizhny NovgorodRussia
  2. 2.Potsdam Institute for Climate Impact ResearchPotsdamGermany

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