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Climatic Change

, Volume 107, Issue 3–4, pp 247–265 | Cite as

Long memory in temperature reconstructions

  • William Rea
  • Marco RealeEmail author
  • Jennifer Brown
Article

Abstract

Ever since H. E. Hurst brought the concept of long memory time series to prominence in his study of river flows the origins of the so-called Hurst phenomena have remained elusive. Two sets of competing models have been proposed. The fractional Gaussian noises and their discrete time counter-part, the fractionally integrated processes, possess genuine long memory in the sense that the present state of a system has a temporal dependence on all past states. The alternative to these genuine long memory models are models which are non-stationary in the mean but for physical reasons are constrained to lie in a bounded range, hence on visual inspection appear to be stationary. In these models the long memory is merely an artifact of the method of analysis. There are now a growing number of millenial scale temperature reconstructions available. In this paper we present a new way of looking at long memory in these reconstructions and proxies, which gives support to them being described by the non-stationary models. The implications for climatic change are that the temperature time series are not mean reverting. There is no evidence to support the idea that the observed rise in global temperatures are a natural fluctuation which will reverse in the near future.

Keywords

Structural Break Spectral Estimate Temperature Time Series Time Series Plot Colorado Plateau 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Abry P, Veitch D (1998) Wavelet analysis of long-range-dependent traffic. IEEE Trans Inf Theory 44(1):2–15CrossRefGoogle Scholar
  2. Abry P, Veitch D, Flandrin P (1998) Long-range dependence: revisiting aggregation with wavelets. J Time Ser Anal 19(3):253–266CrossRefGoogle Scholar
  3. Allen MR, Smith LA (1994) Investigating the origin and significance of low-frequency modes of climate variability. Geophys Res Lett 21(10):883–886CrossRefGoogle Scholar
  4. Baillie RT, Chung S-K (2002) Modeling and forecasting from trend-stationary long memory models with applications to climatology. Int J Forecast 18:215–226CrossRefGoogle Scholar
  5. Beran J (1992) A Goodness-of-fit test for time series with long range dependence. J R Stat Soc B 54(3):749–760Google Scholar
  6. Beran J (1994) Statistics for long memory processes. Chapman & Hall/CRC PressGoogle Scholar
  7. Beran J, Terrin N (1996) Testing for a change in the long-memory parameter. Biometrika 83(3):627–638CrossRefGoogle Scholar
  8. Beran J, Terrin N (1999) Testing for a change in the long-memory parameter. Biometrika 86(1):233CrossRefGoogle Scholar
  9. Beran J, Whitcher B, Maechler M (2006) Longmemo: statistics for long-memory processes (Jan Beran)—data and functions. R package version 0.9-3Google Scholar
  10. Bloomfield P (1992) Trends in global temperature. Clim Change 21:1–16CrossRefGoogle Scholar
  11. Bloomfield P, Nychka D (1992) Climate spectra and detecting climate change. Clim Change 21:275–287CrossRefGoogle Scholar
  12. Briffa KR, Jones PD, Wigley TML, Picher JR, Ballie MGL (1992) Fennoscandian summers from 500AD: temperature changes on short and long timescales. Clim Dyn 7:111–119CrossRefGoogle Scholar
  13. Brown RL, Durbin J, Evans J (1975) Techniques for testing the constancy of regression relationships over time. J R Stat Soc Ser B 37(2):149–192Google Scholar
  14. Cappelli C, Penny RN, Rea WS, Reale M (2008) Detecting multiple mean breaks at unknown points with Atheoretical Regression Trees. Math Comput Simul 78(2–3):351–356CrossRefGoogle Scholar
  15. Chuine I, Yiou P, Viovy N, Seguin B, Daux V, Ladurie ELR (2004) Grape ripening as a past climate indicator. Nature 432(7015):289–290CrossRefGoogle Scholar
  16. Cook ER, Buckley BM, D’Arrigo RD, Peterson MJ (2000) Warm-season temperatures since 1600BC reconstructed from Tasmanian tree rings and their relationship to large-scale sea surface temperature anomalies. Clim Dyn 16:79–91CrossRefGoogle Scholar
  17. D’Arrigo R, Wilson R, Jacoby G (2006) On the long-term context for late twentieth century warming. J Geophys Res 111. doi: 10.1029/2005JD006352 Google Scholar
  18. Deo RS, Chen WW (2000) On the integral of the squared periodogram. Stoch Process Their Appl 85:159–176CrossRefGoogle Scholar
  19. Doukhan P, Oppenheim G, Taqqu M (2003) Theory and applications of long-range dependence. BirkhaüserGoogle Scholar
  20. Embrechts P, Maejima M (2002) Selfsimilar processes. Princeton University PressGoogle Scholar
  21. Fisher DA, Koerner RM, Kuiviner K, Clausen HB, Johnsen SJ, Steffensen J-P, Gundestrup N, Hammer CU (1996) Intercomparison of ice core and precipitation records from sites in Canada and Greenland over the last few centuries using EOF techniques. In: Jones PD, Bradley RS, Jouzel J (eds) Climatic variations and forcing mechanisms of the last 2000 years, NATO ASI series, vol 41. SpringerGoogle Scholar
  22. Fraley C, Leisch F, Maechler M, Reisen V, Lemonte A (2006) fracdiff: fractionally differenced ARIMA aka ARFIMA(p,d,q) models. R package version 1.3-0Google Scholar
  23. Geweke J, Porter-Hudak S (1983) The estimation and application of long memory time series models. J Time Ser Anal 4:221–237CrossRefGoogle Scholar
  24. Gil-Alana LA (2005) Statistical modeling of the temperatures in the northern hemisphere using fractional integration techniques. J Climate 18(24):5357–5370CrossRefGoogle Scholar
  25. Gipp MR (2001) Interpretation of climate dynamics from phase space portraits: is the climate strange or just different? Paleoceanography 16(4):335–351CrossRefGoogle Scholar
  26. Granger CWJ, Joyeux R (1980). An introduction to long-range time series models and fractional differencing. J Time Ser Anal 1:15–30CrossRefGoogle Scholar
  27. Hantemirov RM, Shiyatov SG (2002) A continuous multimillenial ring-width chronology in Yamal, northwestern Siberia. The Holocene 12(6):717–726CrossRefGoogle Scholar
  28. Haslett J, Raftery AE (1989) Space-time modelling with long-memory dependence: assessing Ireland’s wind power resource (with discussion). Appl Stat 38(1):1–50CrossRefGoogle Scholar
  29. Higuchi T (1988) Approach to an irregular time series on the basis of fractal theory. Physica D 31:277–283CrossRefGoogle Scholar
  30. Hosking JRM (1981) Fractional differencing. Biometrika 68(1):165–176CrossRefGoogle Scholar
  31. Hurst HE (1951) Long-term storage capacity of reservoirs. Trans Am Soc Civ Eng 116:770–808Google Scholar
  32. Jensen MJ (1999) Using wavelets to obtain a consistent ordinary least square estimator of the long-memory parameter. J Forecast 18:17–32CrossRefGoogle Scholar
  33. Jones PD, Mann ME (2004) Climate over the past Millennia. Rev Geophys 42:1–42 RG2002CrossRefGoogle Scholar
  34. Klemes V (1974) The Hurst phenomenon—a puzzle? Water Resour Res 10(4):675–688CrossRefGoogle Scholar
  35. Mandelbrot BB, van Ness JW (1968) Fractional brownian motions, fractional noises and applications. SIAM Rev 10(4):422–437CrossRefGoogle Scholar
  36. Mandelbrot BB, Wallis JR (1969) Global dependence in geophysical records. Water Resour Res 5:321–340CrossRefGoogle Scholar
  37. Mann ME, Bradley RS, Hughes MK (1998) Global-scale temperature patterns and climate forcing over the past six centuries. Nature 392:779–787CrossRefGoogle Scholar
  38. Mann ME, Lees JM (1996) Robust estimation of background noise and signal detection in climatic time series. Clim Change 33:409–445CrossRefGoogle Scholar
  39. Milhoj A (1981) The test of fit in time series models. Biometrika 68(1):177–188Google Scholar
  40. Mills T (2007) Time series modelling of two millenia of northern hemispere temperatures: long memory or shifting trends? J R Stat Soc Ser A 170:83–94CrossRefGoogle Scholar
  41. Moberg A, Sonechkin DM, Holmgren K, Datsenko NM, Karlen W (2005) Highly variable Northern Hemispere temperatures reconstructed from low- and high-resolution proxy data. Nature 433:613–617CrossRefGoogle Scholar
  42. Moore JJ, Hughen KA, Miller GH, Overpeck JT (2001) Little ice age recorded in summer temperatures from varved sediments of Donard Lake, Baffin Island, Canada. J Paleolimnol 25:503–517CrossRefGoogle Scholar
  43. Ohanissian A, Russell JR, Tsay RS (2008) True or spurious long memory? A new test. J Bus Econ Stat 26(2):161–175CrossRefGoogle Scholar
  44. Overland JE, Percival DB, Mofjeld HO (2006) Regime shifts and red noise in the North Pacific. Deep-sea Res Part 1 53:582–588CrossRefGoogle Scholar
  45. Palma W (2007) Long-memory time series theory and methods. Wiley-InterscienceGoogle Scholar
  46. Peng CK, Buldyrev SV, Simons M, Stanley HE, Goldberger AL (1994) Mosaic organization of DNA nucleotides. Phys Rev E 49:1685–1689CrossRefGoogle Scholar
  47. R Development Core Team (2005) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0Google Scholar
  48. Ripley B (2005) Tree: classification and regression trees. R package version 1.0-19Google Scholar
  49. Robinson PM (2003) Time series with long memory. Oxford University PressGoogle Scholar
  50. Salzer MW, Kipfmueller KF (2005) Reconstructed temperature and precipitation on a millennial timescale from tree-rings in the Southern Colorado Plateau, USA. Clim Change 70:465–487CrossRefGoogle Scholar
  51. Sibbertsen P (2004) Long memory versus structural breaks: an overview. Stat Pap 45(4):465–515CrossRefGoogle Scholar
  52. Smith AD (2005) Level shift and the illusion of long memory in economic time series. J Bus Econ Stat 23(3):321–335CrossRefGoogle Scholar
  53. Smith RL, Wigley TML, Santer BD (2003) A bivariate time series approach to anthropogenic trend detection in hemispheric mean temperatures. J Climate 16(8):1228–1240CrossRefGoogle Scholar
  54. Stephenson DB, Pavan V, Bojariu R (2000) Is the North Atlantic oscillation a random walk? Int J Climatol 20:1–18CrossRefGoogle Scholar
  55. Tan M, Liu TS, Hou J, Qin X, Zhang H, Li T (2001) 2650-year Beijing Stalagmite Layer Thickness and Temperature Reconstruction IGBP PAGES/World Data Center for Paleoclimatology Data Contribution Series No. 2003-050Google Scholar
  56. Taqqu M, Teverovsky V, Willinger W (1995) Estimators for long-range dependence: an empirical study. Fractals 3(4):785–798CrossRefGoogle Scholar
  57. Teverovsky V, Taqqu M (1999) Testing for long-range dependence in the presence of shifting means or a slowly declining trend, using a variance-type estimator. J Time Ser Anal 18(3):279–304CrossRefGoogle Scholar
  58. Thomson DJ (1990) Time series analysis of Holocene climate data. Philos Trans R Soc Lond A 330:601–616CrossRefGoogle Scholar
  59. Veitch D, Abry P (1999) A wavelet-based joint estimator of the parameters of long-range dependence. IEEE Trans Inf Theory 45(3):878–897CrossRefGoogle Scholar
  60. Wuertz D (2005a) fBasics: financial software collection—fBasics. R package version 220.10063Google Scholar
  61. Wuertz D (2005b) fSeries: financial software collection. R package version 220.10063Google Scholar
  62. Zeileis A, Leisch F, Hornik K, Kleiber C (2002) strucchange: an R package for testing for structural change in linear regression models. J Stat Softw 7(2):1–38Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.University of CanterburyChristchurchNew Zealand

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