References
- Ahn, J. H. and Kim, H. S. 2005. Nonlinear modeling of El Nino/southern oscillation index. J. Hydrol. Eng. 10, 8–15. doi:https://doi.org/10.1061/(ASCE)1084-0699(2005)10:1(8)
- Akaike, H. 1974. A new look at the statistical model identification. IEEE Trans. Automat. Contr. 19, 716–723. doi:https://doi.org/10.1109/TAC.1974.1100705
- An, S. I. 2009. A review of interdecadal changes in the nonlinearity of the El Nino-southern oscillation. Theor. Appl. Climatol. 97, 29–40. doi:https://doi.org/10.1007/s00704-008-0071-z
- An, S. I. and Jin, F.-F. 2004. Nonlinearity and asymmetry of ENSO. J. Clim. 17, 2399–2412. doi:https://doi.org/10.1175/1520-0442(2004)017<2399:NAAOE>2.0.CO;2
- An, S. I. and Wang, B. 2000. Interdecadal change of the structure of the ENSO mode and its impact on the ENSO frequency. J. Clim. 13, 2044–2055. doi:https://doi.org/10.1175/1520-0442(2000)013<2044:ICOTSO>2.0.CO;2
- Ashley, R., Patterson, D. and Hinich, M. 1986. A diagnostic test for nonlinear serial dependence in time series fitting errors. J. Time Ser. Anal. 7, 165–178. doi:https://doi.org/10.1111/j.1467-9892.1986.tb00500.x
- Berner, J., Achatz, U., Batté, L., Bengtsson, L., Cámara, A. D. L. and co-authors. 2017. Stochastic parameterization: toward a new view of weather and 574 climate models. Bull. Am. Meteorol. Soc. 98, 565–588. doi:https://doi.org/10.1175/BAMS-D-15-00268.1
- Berner, J., Sardeshmukh, P. D. and Christensen, H. M. 2018. On the dynamical mechanisms governing El Niño-southern oscillation irregularity. J. Clim. 31, 8401–8419. doi:https://doi.org/10.1175/JCLI-D-18-0243.1
- Bianucci, M., Capotondi, A., Mannella, R. and Merlino, S. 2018. Linear or nonlinear modeling for ENSO dynamics? Atmosphere 9, 435. doi:https://doi.org/10.3390/atmos9110435
- Birkelund, Y. and Hanssen, A. 1999. Multitaper estimators for bispectra. Proc. IEEE Workshop on Higher-Order Statistics, Caesarea, Israel, pp. 207–211.
- Birkelund, Y. and Hanssen, A. 2000. Adaptive bispectral estimation using Thomson’s multitaper approach. Proc. IEEE Adaptive Systems for Signal Processing, Communication and Control Symposium, Lake Louise, Alberta, Canada, pp. 283–288.
- Biswas, M., Chandrasekar, A. and Goswami, B. N. 1995. Bispectra of a tropical coupled ocean–atmosphere system. Curr. Sci. 68, 1236–1243. https://www.jstor.org/stable/24096599
- Boucharel, J., Dewitte, B., Garel, B., Penhoat, Y. and Du. 2009. ENSO’s non-stationary and non-Gaussian character: the role of climate shifts. Nonlin. Process. Geophys. 16, 453–473. doi:https://doi.org/10.5194/npg-16-453-2009
- Brillinger, D. 1965. An introduction to polyspectra. Ann. Math. Statist. 36, 1351–1374. doi:https://doi.org/10.1214/aoms/1177699896
- Brillinger, D. and Rosenblatt, M. 1967. Asymptotic theory of k-th order spectra. In: Spectral Analysis of Time Series (ed. B. Harris). John Wiley, New York, pp. 153–188.
- Burgers, G. and Stephenson, D. B. 1999. The ‘normality’ of ENSO. Geophys. Res. Lett. 26, 1027–1030. doi:https://doi.org/10.1029/1999GL900161
- Chunzai, W. 2018. A review of ENSO theories. Natl. Sci. Rev. 5, 813–825. doi:https://doi.org/10.1093/nsr/nwy104.
- Cox, D. R. 1991. Long-range dependence, nonlinearity and time irreversibility. J. Time Ser. Anal. 12, 329–335. doi:https://doi.org/10.1111/j.1467-9892.1991.tb00087.x
- De Gooijer, J. G. 2017. Elements of Nonlinear Time Series Analysis and Forecasting: Springer Series in Statistics. Springer-Verlag, New York. doi:https://doi.org/https://doi.org/10.1007/978-3-319-43252-6.
- Deser, C., Alexander, M. A., Xie, S. P. and Phillips, A. S. 2010. Sea surface temperature variability: patterns and mechanisms. Ann. Rev. Mar. Sci. 2, 115–143.. doi:https://doi.org/10.1146/annurev-marine-120408-151453
- Dijkstra, H. A., Petersik, P., Hernández-García, E. and López, C. 2019. The application of machine learning techniques to improve El Niño prediction skill. Front. Phys. 7, 153. doi:https://doi.org/10.3389/fphy.2019.00153
- Duan, W. and Wei, C. 2013. The ‘spring predictability barrier’ for ENSO predictions and its possible mechanism: results from a fully coupled model. Int. J. Climatol. 33, 1280–1292. doi:https://doi.org/10.1002/joc.3513
- Falguerolles, A. and Francis, B. 1992. Algorithmic approaches for fitting bilinear models. In: Computational Statistics (eds. Y. Dodge and J. Whittaker). Physica, Heidelberg, pp. 77–82. doi:https://doi.org/https://doi.org/10.1007/978-3-662-26811-7_12
- Fan, H., Huang, B., Yang, S., Li, Z. and Deng, K. 2019. Seasonally-dependent impact of easterly wind bursts on the development of El Niño events. Clim. Dyn. 53, 1527–1546. doi:https://doi.org/10.1007/s00382-019-04688-2
- Fedorov, A. V. and Philander, S. G. 2000. Is El Nino changing? Science 288, 1997–2002. doi:https://doi.org/10.1126/science.288.5473.1997
- Franzke, C., Majda, A. J. and Branstator, G. 2007. The origin of nonlinear signatures of planetary wave dynamics: mean phase space tendencies and contributions from non-Gaussianity. J. Atmos. Sci. 64, 3987–4003. doi:https://doi.org/10.1175/2006JAS2221.1
- Frauen, C., Dommenget, D. and Tyrrell, N. 2014. Analysis of the nonlinearity of El Niño-southern oscillation 33721 teleconnections. J. Clim. doi:https://doi.org/10.1175/JCLI-D-13-00757.1
- Gabr, M. M. 1998. Robust estimation of bilinear time series models. Commun. Stat. Theory Methods 27, 41–53. doi:https://doi.org/10.1080/03610929808832649
- Grahn, T. 1995. A conditional least squares approach to bilinear time series estimation. J. Time Ser. Anal. 16, 509–529. doi:https://doi.org/10.1111/j.1467-9892.1995.tb00251.x
- Granger, C. W. J. and Anderson, A. P. 1978. Introduction to Bilinear Time Series Models. Vandenhoeck & Ruprecht, Gottingen.
- Guegan, D. and Pham, D. T. 1989. A note on the estimation of the parameters of the diagonal bilinear model by the method of least squares. Scand. J. Stat. 16, 129–136.
- Haggan, V. and Oyetunji, O. 1980. On the selection of subset autoregressive time series models. UMIST Tech. Report 124. Dept. of Maths, UMIST.
- Hall, A., Skalin, J., & Teräsvirta, T. 2001. A nonlinear time series model of El Niño. Environmental Modelling & Software, 16(2), 139–146. doi: https://doi.org/10.1016/S1364-8152(00)00077–3
- Ham, Y.-G., Kim, J.-H. and Luo, J.-J. 2019. Deep learning for multi-year ENSO forecasts. Nature 573: 568–572. doi:https://doi.org/10.1038/s41586-019-1559-7
- Hannachi, A. and Iqbal, W. 2019. On the nonlinearity of winter northern hemisphere atmospheric variability. J. Atmos. Sci. 76, 333–356. doi:https://doi.org/10.1175/JAS-D-18-0182.1
- Hannachi, A., Straus, D., Franzke, C., Corti, S. and Woollings, T. 2017. Low-frequency nonlinearity and regime behavior in the Northern Hemisphere extratropical atmosphere. Rev. Geophys. 55, 199–234. doi:https://doi.org/10.1002/2015RG000509
- Hinich, J. and Wolinsky, M. A. 1988. A test for aliasing using bispectral analysis source. J. Am. Stat. Assoc. 83, 499–502. doi:https://doi.org/10.1080/01621459.1988.10478623
- Hinich, M. J. 1982. Testing for Gaussianity and linearity of a stationary time series. J. Time Ser. Anal. 3, 169–175. doi:https://doi.org/10.1111/j.1467-9892.1982.tb00339.x
- Hinich, M. J. and Wolinsky, M. A. 2005. Normalizing bispectra. J. Stat. Plann. Inference 130, 405–411. doi:https://doi.org/10.1016/j.jspi.2003.12.022
- Hocke, K. and Kämpfer, N. 2008. Bispectral analysis of the long-term recording of surface pressure at Jakarta. J. Geophys. Res. 113, D10113. doi:https://doi.org/10.1029/2007JD009356
- Jajcay, N., Kravtsov, S., Sugihara, S. G., Tsonis, A. A. and Paluš, M. 2018. Synchronization and causality across time scales in El Niño Southern Oscillation. Clim. Atmos. Sci. 1, 33. doi:https://doi.org/10.1038/s41612-018-0043-7
- Jenkins, G. M. and Watts, D. G. 1968. Spectral Analysis and its Applications. Holden-Day, San Francisco, xviii p. 525.
- Jiang, N., Neelin, J. D. and Ghil, M. 1995. Quasi-quadrennial and quasi-biennial variability in the equatorial Pacific. Clim. Dyn. 12, 101–112. doi:https://doi.org/10.1007/BF00223723
- Kim, Y. C. and Powers, E. J. 1979. Digital Bispectral Analysis and Its Applications to Nonlinear Wave Interactions. IEEE Trans. Plasma Sci. 7, 120–131. doi:https://doi.org/10.1109/TPS.1979.4317207
- Kim, K. Y. 2002. Investigation of ENSO variability using cyclostationary EOFs of observational data. Meteorol. Atmos. Phys. 81, 149–168. doi:https://doi.org/10.1007/s00703-002-0549-7
- Kim, W. K., Billard, L. and Basawa, I. V. 1990. Estimation of the first order diagonal bilinear time series model. J. Time Ser. Anal. 11, 215–230. doi:https://doi.org/10.1111/j.1467-9892.1990.tb00053.x
- Kondrashov, D., Kravtsov, S., Robertson, A. W. and Ghil, M. 2005. A hierarchy of data-based ENSO models. J. Clim. 18, 4425–4444. doi:https://doi.org/10.1175/JCLI3567.1
- Kovach, C. K., Oya, H. and Kawasaki, H. 2018. The bispectrum and its relationship to phase-amplitude coupling. Neuroimage. 173, 518–539. doi:https://doi.org/10.1016/j.neuroimage.2018.02.033
- Kravtsov, S. 2012. An empirical model of decadal ENSO variability. Clim. Dyn. 39, 2377–2391. doi:https://doi.org/10.1007/s00382-012-1424-y
- Lii, K. S. and Rosenblatt, M. 1982. Deconvolution and estimation of transfer function phase and coefficients for non-Gaussian linear processes. Ann. Statist. 10, 1195–1208. doi:https://doi.org/10.1214/aos/1176345984
- Lovejoy, S. 2018. Spectra, intermittency, and extremes of weather, macroweather and climate. Sci. Rep. 8, 1269.
- Marquardt, D. W. 1963. An algorithm for least squares estimation of nonlinear parameters. J. Soc. Indus. Appl. Math. 11, 431–441. doi:https://doi.org/10.1137/0111030
- Martinez-Villalobos, C., Newman, M., Vimont, D. J., Penland, C. and Neelin, J. D. 2019. Observed El Niño-La Niña asymmetry in a linear model. Geophys. Res. Lett. 46, 9909–9919. doi:https://doi.org/10.1029/2019GL082922
- Monahan, A. H. 2020. Bispectral unfolding of the skewness of correlated additive and multiplicative noise processes. Chaos 30, 023126. doi:https://doi.org/10.1063/1.5125787
- Müller, D. 1987. Bispectra of sea-surface temperature anomalies. J. Phys. Oceanogr. 17, 26–36. doi:https://doi.org/10.1175/1520-0485(1987)017<0026:BOSSTA>2.0.CO;2
- Neelin, J. D., Battisti, D. S., Hirst, A. C., Jin, F. F., Wakata, Y. and co-authors. 1998. ENSO theory. J. Geophys. Res. 103, 14261–14290. doi:https://doi.org/10.1029/97JC03424
- Nikias, C. L. and Raghuveer, M. R. 1987. Bispectrum estimation: a digital signal processing framework. Proc. IEEE 75, 869–891. doi:https://doi.org/10.1109/PROC.1987.13824
- Penland, C. 1996. A stochastic model of Indo-Pacific sea-surface temperature anomalies. Phys. D 98, 534–558. doi:https://doi.org/10.1016/0167-2789(96)00124-8
- Pham, D. T. and Tran, L. T. 1981. On the first order bilinear time series models. J. Appl. Prob. 18, 617–627. doi:https://doi.org/10.2307/3213316
- Pires, C. and 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. doi:https://doi.org/10.1155/2017/3076810
- Pires, C. and Perdigão, R. 2015. Non-Gaussian interaction information: estimation, optimization and diagnostic application of triadic wave resonance. Nonlin. Process. Geophys. 22, 87–108. doi:https://doi.org/10.5194/npg-22-87-2015
- Pires, C. A., Talagrand, O. and Bocquet, M. 2010. Diagnosis and impacts of non-Gaussianity of innovations in data assimilation. Phys. D 239, 1701–1717. doi:https://doi.org/10.1016/j.physd.2010.05.006
- Priestley, M. B. 1981. Spectral Analysis and Time Series 1. Academic, New York.
- Privalsky, V. and Muzylev, S. 2013. An experimental stochastic model of the El Niño-Southern oscillation system at climatic time scales. Univ. J. Geosci. 1, 28–36. http://www.hrpub.org.
- Proistosescu, C., Rhines, A. and Huybers, P. 2016. Identification and interpretation of nonnormality in atmospheric time series. Geophys. Res. Lett. 43, 5425–5434. doi:https://doi.org/10.1002/2016GL068880
- Raghuveer, M. R. and Nikias, C. L. 1986. Bispectrum estimation via AR modeling. Signal Process. 10, 35–48. doi:https://doi.org/10.1016/0165-1684(86)90063-0
- Rao, S. S., Rao, T. S. and Rao, C. R. 2012. Time Series Analysis: Methods and Applications, Vol. 30, Elsevier Science and Technology, North-Holland.
- Rao, T. S. 1981. On the theory of bilinear time series models. J. R. Statist. Soc. B 43, 2, 244–255.
- Rao, T. S. and Gabr, M. M. 1984. Bilinear time series models. In: An Introduction to Bispectral Analysis and Bilinear Time Series Models. Lecture Notes in Statistics, Vol. 24, Springer, New York, NY.
- Rayner, N. A., Parker, D. E., Horton, E. B., Folland, C. K., Alexander, L. V. and co-authors. 2003. Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century. J. Geophys. Res. 108, 4407. 2003. doi:https://doi.org/10.1029/2002JD002670
- Richardson, A. M. and Hodgkiss, W. S. 1994. Bispectral analysis of underwater acoustic data. J. Acoust. Soc. Am. 96, 828–837. doi:https://doi.org/10.1121/1.410321
- Rusticelli, E., Ashley, R. A., Dagum, E. B. and Patterson, D. M. 2008. A new bispectral test for nonlinear serial dependence. Econ. Rev. 28, 279–293. doi:https://doi.org/10.1080/07474930802388090
- Sardeshmukh, P. D. and Sura, P. 2009. Reconciling non-Gaussian climate statistics with linear dynamics. J. Clim. 22, 1193–1207. doi:https://doi.org/10.1175/2008JCLI2358.1
- Schulte, J., Policielli, F. and Zaitchik, B. 2020. A skewed perspective of the Indian rainfall–ENSO relationship. Hydrol. Earth Syst. Sci., 24, 5473–5489. doi:https://doi.org/10.5194/hess-24-5473-2020
- Sesay, S. A. O. and Rao, T. S. 1988. Yule–Walker type difference equations for higher-order moments and cumulants for bilinear time series models. J. Time Ser. Anal. 9, 385–401. doi:https://doi.org/10.1111/j.1467-9892.1988.tb00478.x
- Stein, K., Timmermann, A., Schneider, N., Jin, F.-F. and Stuecker, M. F. 2014. ENSO seasonal synchronization theory. J. Clim. 27, 5285–5310. doi:https://doi.org/10.1175/JCLI-D-13-00525.1
- Stuecker, M. F., Timmermann, A., Jin, F.-F., McGregor, S. and Ren, H.-L. 2013. A combination mode of the annual cycle and the El Niño/Southern oscillation. Nat. Geosci. 6, 540–544. doi:https://doi.org/10.1038/ngeo1826
- Sun, F. and Yu, J.-Y. 2009. A 10–15-yr modulation cycle of ENSO intensity. J. Clim. 22, 1718–1735. doi:https://doi.org/10.1175/2008JCLI2285.1
- Sura, P. and Hannachi, A. 2015. Perspectives of non-Gaussianity in atmospheric synoptic and low-frequency variability. J. Clim. 28, 5091–5114. doi:https://doi.org/10.1175/JCLI-D-14-00572.1
- Sura, P. and Sardeshmukh, P. 2008. A global view of non-Gaussian SST variability. J. Phys. Oceanogr. 38, 639–647. doi:https://doi.org/10.1175/2007JPO3761.1
- Takens, F., Rand, D. A. and Young, D. S., eds. 1981. Dynamical Systems and Turbulence (Warwick 1980), Lecture Notes in Mathematics, Vol. 898, Springer, Berlin, 365 pp.
- Tang, Y., Zhang, R. H., Liu, T., Duan, W., Yang, D. and co-authors. 2018. Progress in ENSO prediction and predictability study. Natl. Sci. Rev. 5, 826–839. doi:https://doi.org/10.1093/nsr/nwy105
- Tang, Z. and Mohler, R. R. 1988. Bilinear time series: theory and applications. In Lecture Notes in Control and Information Sciences (eds. M. Thoma and A. Wyner), 106, pp. 43–58. Springer-Verlag, Berlin.
- Timmermann, A. 2003. Decadal ENSO amplitude modulations: a nonlinear paradigm. Glob Planet Change 37, 135–156. doi:https://doi.org/10.1016/S0921-8181(02)00194-7
- Timmermann, A., An, S.-I., Kug, J.-S., Jin, F.-F., Cai, W. and co-authors. 2018. El Niño-Southern oscillation complexity. Nature 559, 535–545. doi:https://doi.org/https://doi.org/10.1038/s4158 6-018-0252-6
- Timmermann, A., Voss, H. U. and Pasmanter, R. 2001. Empirical dynamical system modeling of ENSO using nonlinear inverse techniques. J. Phys. Oceanogr. 31, 1579–1598. doi:https://doi.org/10.1175/1520-0485(2001)031<1579:EDSMOE>2.0.CO;2
- Ubilava, D. and Helmers, C. G. 2013. Forecasting ENSO with a smooth transition autoregressive model. Environ. Model. Softw. 40, 181–190. doi:https://doi.org/10.1016/j.envsoft.2012.09.008
- Usoro, A. E. 2015. Comparative analysis of linear and bilinear time series models American. J. Math. Stat. 5, 265–271.
- von Storch, H. and Zwiers, F. W. 1999. Statistical Analysis in Climate Research. Cambridge University Press, Cambridge, 484 p. doi: https://doi.org/10.1017/CBO9780511612336
- Wang, C. 2018. A review of ENSO theories. Natl. Sci. Rev. 5, 813–825. doi:https://doi.org/10.1093/nsr/nwy104
- Wang, C., Deser, C., Yu, J.-Y., DiNezio, P., and Clement, A. 2016. El Niño-Southern Oscillation (ENSO): a review. In Coral Reefs of the Eastern Pacific (eds. P. Glymn, D. Manzello, and I. Enochs). Springer Science Publisher, Berlin, pp. 85–106.
- Weiss, G. 1975. Time-reversibility of linear stochastic processes. J. Appl. Prob. 12, 831–836. doi:https://doi.org/10.2307/3212735
- Wilks, D. S. 2011. Statistical Methods in the Atmospheric Sciences. 3rd ed. Elsevier, Academic Press, Amsterdam.
- Woollings, T., Hannachi, A. and Hoskins, B. 2010. Variability of the North Atlantic eddy-driven jet stream. QJR. Meteorol. Soc. 136, 856–868. doi:https://doi.org/10.1002/qj.625
- Wu, A. and Hsieh, W. W. 2003. Nonlinear interdecadal changes of the El Niño-Southern oscillation. Clim. Dyn. 21, 719–730. doi:https://doi.org/10.1007/s00382-003-0361-1
- Yeh, S.-W. and Kirtman, B. P. 2004. Tropical Pacific decadal variability and ENSO amplitude modulation in a CGCM. J. Geophys. Res. 109, C11009. doi:https://doi.org/10.1029/2004JC002442
- Yishuai, J., Zhengyao, L. and Zhengyu, L. 2020. Controls of spring persistence barrier strength in different ENSO regimes and implications for 21st century changes. Geophys. Res. Lett. 47, 11. https://doi.org/10.1029/2020GL088010.
- Zhang, Y., Wallace, J. M. and Battisti, D. S. 1997. ENSO‐like interdecadal variability: 1900–93. J. Clim. 10, 1004–1020. doi:https://doi.org/10.1175/1520-0442(1997)010<1004:ELIV>2.0.CO;2