Skip to main content

Time series interval forecast using GM(1,1) and NGBM(1, 1) models


Grey forecast is used for few and uncertain data, and its forecast results have very high accuracy. Although numerous researchers have developed various grey forecasting models, the forecast results of these models are limited to single-point forecast values and cannot provide more valuable information (e.g. possible estimation range) for decision-makers. In order to address this problem, this paper proposes two grey interval forecasting methods: interval GM(1, 1) and interval NGBM(1, 1), for few and uncertain time series data. To evaluate the forecast accuracy of the two grey interval methods, this study took the short-term forecast of the passenger volume of Taiwan High Speed Rail as an example and compared the forecast accuracy of the proposed two methods with that of three current grey forecasting methods. The forecast results showed that the proposed two methods have the highest forecast accuracy among the five grey forecasting methods. The grey interval forecast value provided by the proposed methods can help decision-makers make more accurate judgement within a probable variation range.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


  • Akay D, Atak M (2007) Grey prediction with rolling mechanism for electricity demand forecasting of Turkey. Energy 32:1670–1675

    Article  Google Scholar 

  • Chang TS, Ku CY, Fu HP (2013) Grey theory analysis of online population and online game industry revenue in Taiwan. Technol Forecast Soc Change 80:175–185

  • Chen CI (2008) Application of the novel nonlinear Grey Bernoulli model for forecasting unemployment rate. Chaos Solitons Fractals 37:278–287

    Article  MATH  Google Scholar 

  • Chen CI, Chen HL, Chen SP (2008) Forecasting of foreign exchange rates of Taiwan’s major trading partners by novel nonlinear Grey Bernoulli model NGBM(1, 1). Commun Nonlinear Sci Numer Simul 13:1194–1204

    Article  Google Scholar 

  • Chen CI, Hsin PH, Wu CS (2010) Forecasting Taiwan’s major stock indices by the Nash nonlinear Grey Bernoulli model. Expert Syst Appl 37:7557–7562

  • Chen Z, Wang X (2012) Applying the grey forecasting model to the energy supply management engineering. Syst Eng Proc 5:179–184

    Article  Google Scholar 

  • Deng JL (1982) Control problems of grey systems. Syst Control Lett 5:288–94

    MathSciNet  MATH  Google Scholar 

  • Deng JL (1989) Introduction to grey system theory. J Grey Syst 1:1–24

    MathSciNet  MATH  Google Scholar 

  • Huang YL, Lin CT (2011) Developing an interval forecasting method to predict undulated demand. Qual Quant 45:513–524

    Article  Google Scholar 

  • Lee SC, Shih LH (2011) Forecasting of electricity costs based on an enhanced gray-based learning model: a case study of renewable energy in Taiwan. Technol Forecast Soc Change 78:1242–1253

    Article  Google Scholar 

  • Lei M, Feng Z (2012) A proposed grey model for short-term electricity price forecasting in competitive power markets. Electr Power Energy Syst 43:531–538

    Article  Google Scholar 

  • Lewis EB (1982) Control of body segment differentiation in Drosophila by the Bithorax gene complex, embryonic development, part A: genetics aspects. Alan R. Liss, New York, pp 269–288

    Google Scholar 

  • Li DC, Chang CJ, Chen CC, Chen WC (2012) Forecasting short-term electricity consumption using the adaptive grey-based approach–an Asian case. Omega 40:767–773

    Article  Google Scholar 

  • Li GD, Yamaguchi D, Nagai M (2008) The development of stock exchange simulation prediction modeling by a hybrid grey dynamic model. Int J Adv Manuf Technol 36:195–204

    Article  Google Scholar 

  • Lin CT, Yang SY (2003) Forecast of the output value of Taiwan’s opto-electronics the grey forecasting model. Technol Forecast Soc Change 70:177–186

    Article  Google Scholar 

  • Liu S, Lin Y (2006) Grey information: theory and practical applications. Springer, London

    Google Scholar 

  • Mao M, Chirwa EC (2006) Application of grey model GM(1, 1) to vehicle fatality risk estimation. Technol Forecast Soc Change 73:588–605

    Article  Google Scholar 

  • Tseng GH, Yu HC, Tseng GH (2001) Applied hybrid grey model to forecast seasonal time series. Technol Forecast Soc Change 67(2–3):291–302

    Article  Google Scholar 

  • Wang CH (2004) Predicting tourism demand using fuzzy time series and hybrid grey theory. Tour Manage 25(3):367–374

    MathSciNet  Article  Google Scholar 

  • Wang CH, Hsu LC (2008) Using genetic algorithms grey theory to forecast high technology industrial output. Appl Math Comput 195:256–263

    MathSciNet  MATH  Google Scholar 

  • Yang Y, John R (2012) Grey sets and greyness. Inform Sci 185:249–264

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Hao-Tien Liu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Human participants

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Communicated by V. Loia.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chen, YY., Liu, HT. & Hsieh, HL. Time series interval forecast using GM(1,1) and NGBM(1, 1) models. Soft Comput 23, 1541–1555 (2019).

Download citation

  • Published:

  • Issue Date:

  • DOI:


  • Few data
  • Time series forecast
  • GM(1, 1)
  • NGBM(1, 1)
  • Grey interval forecast