Theoretical and Applied Climatology

, Volume 134, Issue 1–2, pp 241–249 | Cite as

Modeling fractionally integrated maximum temperature series in India in presence of structural break

  • Ranjit Kumar PaulEmail author
  • Priyanka Anjoy
Original Paper


In this study, the long memory behaviour of monthly maximum temperature of India for the period 1901 to 2007 is investigated. The correlogram of the series reveals a slow hyperbolic decay, a typical shape for time series having the long memory property. Wavelet transformation is applied to decompose the temperature series into time–frequency domain in order to study the local as well as global variation over different scale and time epochs. Significant increasing trend is found in the maximum temperature series in India. The rate of increase in maximum temperature accelerated after 1960s as compared to the earlier period. Here, an attempt is also made to detect the structural break for seasonally adjusted monthly maximum temperature series. It is found that there is a significant break in maximum temperature during July, 1963. Two-stage forecasting (TSF) approach to deal with the coexistence of long memory and structural change in temperature pattern is discussed thoroughly. The forecast performance of the fitted model is assessed on the basis of relative mean absolute prediction error (RMAPE), sum of squared errors (SSE) and mean squared errors (MSE) for different forecast horizons.



We would like to express our sincere thanks to the anonymous reviewers for their valuable suggestions that helped us a lot in improving this manuscript.

Compliance with ethical standards

Conflict of interests

The authors declare that they have no conflict of interests regarding the publication of this paper.


  1. Aggarwal PK (2009) Vulnerability of Indian agriculture to climate change: current state of knowledge, paper presented at the National Workshop – Review of Implementation of Work Programme towards Indian Network of Climate Change Assessment, October 14. Ministry of Environment and Forests, New Delhi Google Scholar
  2. Beran J (1995) Statistics for long memory processes. Chapman & HallGoogle Scholar
  3. Birthal PS, Negi DS, Kumar S, Aggarwal S, Suresh A, Khan T (2014) How sensitive is Indian agriculture to climate change? Indian Journal of Agricultural Economics 69(4):474–487Google Scholar
  4. De Salvo M, Raffael R, Moser R (2013) The impact of climate change on permanent crops in an alpine region: a Ricardian analysis. Agric Syst 118:23–32CrossRefGoogle Scholar
  5. Eichner JF, Koscielny-Bunde E, Bunde A, Havlin S, Schellnhuber HJ (2003) Power-law persistence and trends in the atmosphere: a detailed study of long temperature records. Phys Rev E 68:046133CrossRefGoogle Scholar
  6. Geweke J, Porter-Hudak S (1983) The estimation and application of long-memory time-series models. J Time Ser Anal 4:221–238CrossRefGoogle Scholar
  7. Gil-Alana LA (2005) Statistical modeling of the temperatures in the northern hemisphere using fractional integration techniques. J Clim 18:5357–5369CrossRefGoogle Scholar
  8. Gil-Alana LA (2008) Time trend estimation with breaks in temperature time series. Clim Chang 89:325–337CrossRefGoogle Scholar
  9. Gilbert CG (1953) An aid for forecasting the minimum temperature at Denver, Colo. Mon Weather Rev 81:233–245CrossRefGoogle Scholar
  10. Hurst HE (1951) Long term storage capacity of reservoirs. Trans Am Soc Agric Eng 116:770–799Google Scholar
  11. Huybers P, Curry W (2006) Links between annual, Milankovitch and continuum temperature variability. Nature 441:329–332CrossRefGoogle Scholar
  12. Jensen MJ (1999) Using wavelets to obtain a consistent ordinary least squares estimator of the long-memory parameter. J Forecast 18:17–32CrossRefGoogle Scholar
  13. Kangieser PC (1959) Forecasting minimum temperatures on clear winter nights in an arid region. Mon Weather Rev 87:19–28CrossRefGoogle Scholar
  14. Killick R, Eckley IA (2014) Changepoint: an R package for Changepoint analysis. J Stat Softw 58(3):1–19CrossRefGoogle Scholar
  15. Kothawale DR, Rupa Kumar K (2005) On the recent changes in surface temperature trends over India. Geophys Res Lett 32:L18714CrossRefGoogle Scholar
  16. Kumar KK, Kumar KR, Pant GB (1997) Pre-monsoon maximum and minimum temperatures over India in relation to the summer monsoon rainfall. Int J Climatol 17:1115–1127CrossRefGoogle Scholar
  17. Lennartz S, Bunde A (2009) Trend evaluation in records with long-term memory: application to global warming. Geophys Res Lett 36:L16706CrossRefGoogle Scholar
  18. Mallows CL (1973) Some comments on Cp. Technometrics 15:661–675Google Scholar
  19. Malamud BD and Turcotte DL (1999) Advances in geophysics: long range persistence in geophysical time series, self-affine time series: I. Generation and analysis, Dmowska R and Saltzman B (ed.), pp 1–87. Academic press, San DiegoGoogle Scholar
  20. Mantis HT, Dickey WW (1945) Objective methods of forecasting the daily minimum and maximum temperature. In: Report number 4. Army Air Force, Weather Station, New York University, U.S.Google Scholar
  21. Mendelsohn R, Dinar A, Williams L (2006) The distributional impact of climate change on rich and poor countries. Environ Dev Econ 11:159–178CrossRefGoogle Scholar
  22. Mills CT (2014) Time series modelling of temperatures: an example from Kefalonia. Meteorol Appl 21:578–584CrossRefGoogle Scholar
  23. Monetti RA, Havlin S, Bunde A (2003) Long-term persistence in the sea surface temperature fluctuations. Physica A 320:581–589CrossRefGoogle Scholar
  24. Nagarajan R (2009) Drought assessment. Springer, The NetherlandGoogle Scholar
  25. Papailias F, Dias GF (2015) Forecasting long memory series subject to structural change: a two-stage approach. Int J Forecast 31:1056–1066CrossRefGoogle Scholar
  26. Pattantyús-Ábrahám M, Király A, Jánosi IM (2004) Nonuniversal atmospheric persistence: different scaling of daily minimum and maximum temperatures. Phys Rev E 69:021110CrossRefGoogle Scholar
  27. Paul RK, Birthal PS and Khokhar A. (2014) Structural breaks in mean temperature over agro-climatic zones in India. Sci World J.
  28. Paul RK, Birthal PS, Paul AK, Gurung B (2015a) Temperature trend in different agro-climatic zones in India. Mausam 66(4):841–846Google Scholar
  29. Paul RK, Samanta S, Gurung B (2015b) Monte Carlo simulation for comparison of different estimators of long memory parameter: an application of ARFIMA model for forecasting commodity price. Model Assist Stat Appl 10(2):116–127Google Scholar
  30. Paul RK (2017) Modelling long memory in maximum and minimum temperature series in India. Mausam 68(2):317–326Google Scholar
  31. Pelletier JD (1997) Analysis and modeling of the natural variability of climate. J Clim 10:1331–1342CrossRefGoogle Scholar
  32. Percival DB, Walden AT (2000) Wavelet methods for time-series analysis. Cambridge Univ, Press, U.K.CrossRefGoogle Scholar
  33. Rohini P, Rajeevan M, Srivastava AK (2016) On the variability and increasing trends of heat waves over India. Sci Rep 6:26153CrossRefGoogle Scholar
  34. Sowell FB (1992) Maximum likelihood estimation of stationary univariate fractionally integrated time series models. J Econ 53:165–188CrossRefGoogle Scholar
  35. Spreen WC (1956) Empirically determined distributions of hourly temperatures. J Atmos Sci 13:351–355Google Scholar
  36. Ustaoglu B, Cigizoglub HK, Karaca M (2008) Forecast of daily mean, maximum and minimum temperature time series by three artificial neural network methods. Meteorol Appl 15:431–445CrossRefGoogle Scholar
  37. Van Loon H, Jenne RL (1975) Estimates of seasonal mean temperature, using persistence between seasons. Mon Weather Rev 103:1121–1128CrossRefGoogle Scholar
  38. Vyushin DI, Kushner PJ (2009) Power-law and long-memory characteristics of the atmospheric general circulation. J Clim 22:2890–2904CrossRefGoogle Scholar
  39. Wang CSH, Bauwens L, Hsiao C (2013) Forecasting a long memory process subject to structural breaks. J Econ 177:171–184CrossRefGoogle Scholar
  40. Werner R, Valev D, Danov D, Guineva V (2015) Study of structural break points in global and hemispheric temperature series by piecewise regression. Adv Space Res 56(11):2323–2334CrossRefGoogle Scholar
  41. Yuan N, Fu Z, Liu S (2014) Extracting climate memory using fractional integrated statistical model: a new perspective on climate prediction. Sci Rep 4:6577CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  1. 1.ICAR-Indian Agricultural Statistics Research InstituteNew DelhiIndia

Personalised recommendations