Milestones and developments in classical statistical time series forecasting: a comprehensive review
DOI:
https://doi.org/10.15282/daam.v6i1.12113Keywords:
Time series, Forecasting, Univariate, Methods, ApplicationsAbstract
Time series offers a potent framework for forecasting the future based on past data. Time series forecasting, a cornerstone of predictive analytics, is essential to decision-making in many fields, including finance, economics, weather, and healthcare, necessitating various forecasting methodologies. In the 1950s, statistical tools for forecasting began utilising exponential smoothing methods. These approaches were changed depending on the pattern seen in the data sets and the objective of the analysis. Automated functions were used after simple additive and multiplicative effects to assess the complexity of the data for forecasting purposes. This paper thoroughly reviewed the developments and turning points in classical time series forecasting techniques. Time series analysis has long been built on classical forecasting methods, shaping the field's growth. A thorough examination of the literature and historical data has identified noteworthy advancements that have significantly influenced the area. The review emphasises how classical approaches can still be helpful in today's data-driven society and how they can be combined with cutting-edge data science methodologies. This review also provides insightful analyses into the lengthy past and bright future of time series forecasting by charting the development of conventional approaches.
References
[1] Frost J. Introduction to statistics: An intuitive guide for analyzing data and unlocking discoveries. State College (PA): Statistics by Jim Publishing; 2020.
[2] Chatfield C, Koehler A, Ord K, Snyder R. A new look at models for exponential smoothing. Journal of the Royal Statistical Society: Series D (The Statistician). 2001;50(2):147–159.
[3] Tsay RS. Time series and forecasting: Brief history and future research. Journal of the American Statistical Association. 2000;95(450):638–43.
[4] Montgomery DC, Jennings CL, Kulahci M. Introduction to time series analysis and forecasting. Hoboken (NJ): Wiley; 2008.
[5] Box GEP, Jenkins GM, Reinsel GC, Ljung GM. Time series analysis: forecasting and control. 5th ed. Hoboken (NJ): Wiley; 2016.
[6] Granger CWJ, Newbold P. Forecasting economic time series. 2nd ed. Orlando (FL): Academic Press; 1986.
[7] Faloutsos C, Gasthaus J, Januschowski T, Wang Y. Classical and contemporary approaches to big time series forecasting. In Proceedings of the 2019 international conference on management of data 2019 (pp. 2042-2047).
[8] Hu YC, Wu G, Jiang P. Tourism demand forecasting using nonadditive forecast combinations. Journal of Hospitality & Tourism Research. 2021;47(5):775–799.
[9] Mado I, Soeprijanto A, Suhartono S. Applying of double seasonal ARIMA model for electrical power demand forecasting at PT. PLN Gresik Indonesia. International Journal of Electrical and Computer Engineering. 2018;8(6):4892–4901.
[10] Wawale S, Bisht A, Vyas DS, Narawish C, Ray S. An overview: Modeling and forecasting of time series data using different techniques in reference to human stress. Neuroscience Informatics. 2022;2(3):100052.
[11] Perone G. Comparison of ARIMA, ETS, NNAR, TBATS and hybrid models to forecast the second wave of COVID-19 hospitalizations in Italy. The European Journal of Health Economics. 2022;23:917–940.
[12] Yadav AK, Das K, Das P, Raman R, Kumar J, Das B. Growth trends and forecasting of fish production in Assam, India using ARIMA model. Journal of Applied and Natural Science. 2020;12(3):415–421.
[13] Sharma R, Sharma R. Demand forecasting of engine oil for automotive and industrial lubricant manufacturing company using neural network. Materials Today: Proceedings. 2019;18:2308–2314.
[14] Nor ME, Safuan HM, Md Shab NF, Asrul M, Abdullah A, Nurul Mohamad NAI, Lee MH. Neural network versus classical time series forecasting models. In: AIP Conference Proceedings 2017;1842:030027.
[15] Makridakis S. The M3-Competition: results, conclusions and implications. International Journal of Forecasting. 2000;16(4):451–476.
[16] Makridakis S, Andersen A, Carbone R, Fildes R, Hibon M, Lewandowski R, Newton J, Parzen E, Winkler R. The accuracy of extrapolation (time series) methods: Results of a forecasting competition. Journal of Forecasting. 1982;1(2):111–153.
[17] Makridakis S, Chatfield C, Hibon M, Lawrence M, Mills T, Ord K, Simmons LF. The M2-competition: A real-time judgmentally based forecasting study. International Journal of Forecasting. 1993;9(1):5–22.
[18] Makridakis S, Spiliotis E, Assimakopoulos V. The M4 Competition: Results, findings, conclusion and way forward. International Journal of Forecasting. 2018;34(4):802–808.
[19] Makridakis S, Spiliotis E, Assimakopoulos V. The M5 competition: Background, organization, and implementation. International Journal of Forecasting. 2021;38(4):1325–1336.
[20] Makridakis S, Spiliotis E, Hollyman R, Petropoulos F, Swanson N, Gaba A. The M6 forecasting competition: Bridging the gap between forecasting and investment decisions. International Journal of Forecasting. Forthcoming 2024. Corrected proof.
[21] Dissanayake B, Hemachandra O, Lakshitha N, Haputhanthri D, Wijayasiri A. A comparison of ARIMAX, VAR, and LSTM on multivariate short-term traffic volume forecasting. In: Proceedings of the 28th Conference of the Open Innovations Association FRUCT; 2021 Jan 27–29; Moscow, Russia. Moscow: FRUCT Oy; 2021. pp. 564–570.
[22] Hyndman RJ, Athanasopoulos G. Forecasting: Principles and practice. OTexts; 2021.
[23] Hyndman RJ, Koehler AB, Ord KJ, Snyder RD. Forecasting with exponential smoothing: The State Space Approach. Springer; 2008.
[24] Gardner Jr E. Exponential smoothing: The state of the art—Part II. International Journal of Forecasting. 2006;22(4):637–666
[25] Holt CC. Forecasting seasonals and trends by exponentially weighted moving averages. International journal of forecasting. 2004;20(1):5-10.
[26] Winters PR. Forecasting sales by exponentially weighted moving averages. Management Science Journal of the Institute of Management Science: Application and Theory Series. 1960;6(3):324–342.
[27] Taylor JW. Short-term electricity demand forecasting using double seasonal exponential smoothing. Journal of the Operational Research Society. 2003;54(8):799–805.
[28] Bermúdez J, Segura-Heras JV, Vercher E. Holt-Winters forecasting: An alternative formulation applied to UK air passenger data. Journal of Applied Statistics. 2007;34(9):1075–1090.
[29] Wąsik E, Chmielowski K. The use of Holt–Winters method for forecasting the amount of sewage inflowing into the wastewater treatment plant in Nowy Sącz. Environmental Protection and Natural Resources. 2016;27(2):7–12.
[30] Siregar B, Butar-Butar IA, Rahmat RF, Andayani U, Fahmi F. Comparison of exponential smoothing methods in forecasting palm oil real production. Journal of Physics: Conference Series. 2017;801:012004.
[31] Suppalakpanya K, Booranawong A, Booranawong T, Nikhom R. An evaluation of Holt-Winters methods with different initial trend values for forecasting crude palm oil production and prices in Thailand. Suranaree Journal of Science and Technology. 2019;26(1):13–22.
[32] Nurhamidah, Wan N, Faisol A. Forecasting seasonal time series data using the Holt-Winters exponential smoothing method of additive models. Jurnal Matematika Integratif. 2020;16(2):151–157.
[33] Goodwin P. The Holt-Winters approach to exponential smoothing: 50 years old and going strong. Foresight: The International Journal of Applied Forecasting. 2010:19;30–33.
[34] Muth JF. Optimal properties of exponentially weighted forecasts. Journal of the American Statistical Association. 1960;55(290):299–306.
[35] Gould PG, Koehler AB, Ord JK, Snyder RD, Hyndman RJ, Vahid-Araghi F. Forecasting time series with multiple seasonal patterns. European Journal of Operational Research. 2008;191(1):207–222.
[36] McCormick GP. Communications to the editor—Exponential forecasting: Some new variations. Management Science. 1969;15(5):311–315.
[37] De Gooijer JG, Hyndman RJ. 25 years of time series forecasting. International Journal of Forecasting. 2006;22(3):443–473.
[38] Gardner ES. Exponential smoothing: The state of the art. Journal of Forecasting. 1985;4(1):1–28.
[39] Ord K, Koehler A, Snyder R. Estimation and prediction for a class of dynamic nonlinear statistical models. Journal of the American Statistical Association. 1997;92(440):1621–1629.
[40] Hyndman R, Koehler A, Snyder R, Grose S. A state space framework for automatic forecasting using exponential smoothing methods. International Journal of Forecasting. 2002;18(3):439–454.
[41] Gould PG, Koehler AB, Ord JK, Snyder RD, Hyndman RJ, Vahid-Araghi F. Forecasting time series with multiple seasonal patterns. European Journal of Operational Research. 2008;191(1):207–222.
[42] Akram M, Hyndman RJ, Ord JK. Exponential smoothing and non-negative data. Australian & New Zealand Journal of Statistics. 2009;51(4):415–432.
[43] De Livera AM, Hyndman RJ, Snyder RD. Forecasting time series with complex seasonal patterns using exponential smoothing. Journal of the American Statistical Association. 2011;106:1513–1527.
[44] Ferbar Tratar L. Forecasting method for noisy demand. International Journal of Production Economics. 2015;161:64–73.
[45] Ferbar Tratar L, Mojškerc B, Toman A. Demand forecasting with four-parameter exponential smoothing. International Journal of Production Economics. 2016;181:162–173.
[46] Makridakis S, Hibon M. ARMA models and the Box-Jenkins methodology. Journal of Forecasting. 1997;16(3):147–163.
[47] Box GEP, Jenkins GM. Time series analysis: Forecasting and control. San Francisco: Holden-Day; 1976.
[48] Makridakis S, Winkler RL. Averages of forecasts: Some empirical results. Management Science. 1983;29(9):987–996.
[49] Yule GU. On a method of investigating periodicities in disturbed series, with special reference to Wolfer’s sunspot numbers. In: The Foundations of Econometric Analysis [Internet]. Cambridge: Cambridge University Press; 2012. pp. 267–273.
[50] Majid R, Mir S. Advances in statistical forecasting methods: An overview. Economic Affairs 2018;63(4):815–831.
[51] Somboonsak P. Forecasting dengue fever epidemics using ARIMA model. In: Proceedings of the 2019 2nd Artificial Intelligence and Cloud Computing Conference, New York, NY, USA: Association for Computing Machinery; 2020. p. 144–150.
[52] Taylor JW. Exponential smoothing with a damped multiplicative trend. International Journal of Forecasting 2003;19(4):715–725.
[53] Svetunkov I, Boylan J. State-space ARIMA for supply-chain forecasting. International Journal of Production Research 2019;58(1):1–10.
[54] Petropoulos F, Apiletti D, Assimakopoulos V, Babai MZ, Barrow DK, Ben Taieb S, et al. Forecasting: theory and practice. International Journal of Forecasting 2022;38(3):705–871.
[55] Promprou S, Jaroensutasinee M, Jaroensutasinee K. Forecasting Dengue Haemorrhagic Fever cases in Southern Thailand using ARIMA Models. Dengue Bulletin. 2006;30:99–106.
[56] Gharbi M, Quenel P, Gustave J, Cassadou S, Ruche GL, Girdary L, et al. Time series analysis of dengue incidence in Guadeloupe, French West Indies: forecasting models using climate variables as predictors. BMC Infectious Diseases. 2011;11:166.
[57] Martinez EZ, da Silva EAS. Predicting the number of cases of dengue infection in Ribeirão Preto, São Paulo State, Brazil, using a SARIMA model. Cadernos de Saúde Pública. 2011;27(9):1809–1819.
[58] Suhartono. Time series forecasting by using seasonal autoregressive integrated moving average: Subset, multiplicative or additive model. Journal of Mathematics and Statistics. 2011;7(1):20–27.
[59] Junior PR, Salomon FLR, de Oliveira Pamplona E. ARIMA: An applied time series forecasting model for the Bovespa stock index. Applied Mathematics. 2014;5(21):3383–3391.
[60] Bandyopadhyay G. Gold price forecasting using ARIMA model. Journal of Advanced Management Science. 2016;4(2):117–121.
[61] Din MA. ARIMA by Box-Jenkins methodology for estimation and forecasting models in higher education. Athens: ATINER's Conference Paper Series; 2016. No. EMS2015-1846.
[62] Kim S, Ko W, Nam H, Kim C, Chung Y, Bang S. Statistical model for forecasting uranium prices to estimate the nuclear fuel cycle cost. Nuclear Engineering and Technology. 2017;49(5):1063–1070.
[63] Darekar A, Reddy AA. Cotton price forecasting in major producing states. Economic Affairs. 2017;62(3):373–378.
[64] [64] Darekar A, Reddy AA. Forecasting of common paddy prices in India. SSRN Electronic Journal. 2017;10(1):71–75.
[65] Mado I, Soeprijanto A, Suhartono S. Applying of double seasonal ARIMA model for electrical power demand forecasting at PT. PLN Gresik Indonesia. International Journal of Electrical and Computer Engineering. 2018;8(6):4892–4901.
[66] Samal KKR, Babu KS, Das SK, Acharya A. Time series-based air pollution forecasting using SARIMA and Prophet model. In: Proceedings of the 2019 International Conference on Information Technology and Computer Communications; 2019 Sep 20–22; Singapore, Singapore. New York (NY): Association for Computing Machinery; 2019. pp. 80–85.
[67] Cong J, Ren M, Xie S, Wang P. Predicting seasonal influenza based on SARIMA model, in mainland China from 2005 to 2018. International Journal of Environmental Research and Public Health. 2019;16(23):4760.
[68] Ismail MA, El-Metaal EMA. Forecasting residential natural gas consumption in Egypt. Journal of Humanities and Applied Social Sciences. 2020;2(4):297–308.
[69] Divisekara RW, Jayasinghe GJMSR, Kumari KWSN. Forecasting the red lentils commodity market price using SARIMA models. SN Business & Economics. 2021;1(20):1–20.
[70] Yadav A, Jha C, Sharan A. Optimizing LSTM for time series prediction in Indian stock market. Procedia Computer Science. 2020;167:2091–2100.
[71] Manigandan P, Alam MDS, Alharthi M, Khan U, Alagirisamy K, Pachiyappan D, Rehman A. Forecasting natural gas production and consumption in United States—evidence from SARIMA and SARIMAX models. Energies. 2021;14(19):6021.
[72] Ray S, Das SS, Mishra P, Khatib AMG Al. Time series SARIMA modelling and forecasting of monthly rainfall and temperature in the South Asian countries. Earth Systems and Environment. 2021;5:531–546.
[73] Deretić N, Stanimirović D, Awadh MA, Vujanović N, Djukić A. SARIMA modelling approach for forecasting of traffic accidents. Sustainability. 2022;14(8):4403.
[74] Perone G. Comparison of ARIMA, ETS, NNAR, TBATS and hybrid models to forecast the second wave of COVID-19 hospitalizations in Italy. The European Journal of Health Economics. 2022;23:917–940.
[75] Mado I, Rajagukguk A, Triwiyatno A, Fadllullah A. Short-term electricity load forecasting model based DSARIMA. International Journal of Electrical, Energy and Power System Engineering. 2022;5(1):6–11.
[76] Orang A, Berke O, Poljak Z, Greer AL, Rees EE, Ng V. Forecasting seasonal influenza activity in Canada—Comparing seasonal Auto-Regressive Integrated Moving Average and artificial neural network approaches for public health preparedness. Zoonoses Public Health. 2024;71(3):304–313.
[77] Wu DCW, Ji L, He K, Tso KFG. Forecasting tourist daily arrivals with a hybrid SARIMA–LSTM approach. Journal of Hospitality and Tourism Research. 2021;45(1):52–67.
[78] Jenkins GM, Quenouille MH. Analysis of multiple time-series. Biometrika. 1959;46(1-2):267.
[79] Funke M. Assessing the forecasting accuracy of monthly vector autoregressive models: The case of five OECD countries. International Journal of Forecasting. 1990;6(3):363–378.
[80] Dhrymes PJ, Thomakos DD. Structural VAR, MARMA and open economy models. International Journal of Forecasting. 1998;14(2):187–198.
[81] Hafer RW, Sheehan RG. The sensitivity of VAR forecasts to alternative lag structures. International Journal of Forecasting. 1989;5(3):399–408.
[82] Kuo CY. Does the vector error correction model perform better than others in forecasting stock price? An application of residual income valuation theory. Economic Modelling. 2016;52:772–789.
[83] Khan F, Saeed A, Ali S. Modelling and forecasting of new cases, deaths and recover cases of COVID-19 by using Vector Autoregressive model in Pakistan. Chaos, Solitons & Fractals. 2020;140:110189.
[84] Katris C. Unemployment and COVID-19 Impact in Greece: A vector autoregression (VAR) data analysis. Engineering Proceedings. 2021;5(1):41.
[85] Rajab K, Kamalov F, Cherukuri AK. Forecasting COVID-19: Vector autoregression-based model. Arabian Journal for Science and Engineering. 2022;47:6851–6860.
[86] Litterman RB. Forecasting with Bayesian vector autoregressions—five years of experience. Journal of Business & Economic Statistics. 1986;4(1):25–38.
[87] Artis MJ, Zhang W. BVAR forecasts for the G-7. International Journal of Forecasting. 1990;6(3):349–362.
[88] Ramos FFR. Forecasts of market shares from VAR and BVAR models: A comparison of their accuracy. International Journal of Forecasting. 2003;19(1):95–110.
[89] LeSage JP, Magura M. Using interindustry input-output relations as a Bayesian prior in employment forecasting models. International Journal of Forecasting. 1991;7(2):231–238.
[90] Lopreite M, Zhu Z. The effects of ageing population on health expenditure and economic growth in China: A Bayesian-VAR approach. Social Science & Medicine. 2020;265:113513.
[91] Kopytin IA, Pilnik NP, Stankevich IP. Modelling five variables BVAR for economic policies and growth in Azerbaijan, Kazakhstan and Russia: 2005–2020. International Journal of Energy Economics and Policy. 2021;11(5):510–518.
[92] Han F, Ng TH. ASEAN-5 macroeconomic forecasting using a GVAR model. Asian Development Bank Working Paper. 2011.
[93] Greenwood-Nimmo M, Nguyen VH, Shin Y. Probabilistic forecasting of output growth, inflation, and the balance of trade in a GVAR framework. Journal of Applied Econometrics. 2012;27(4):554–573.
[94] Cao Z, Li G, Song H. Modelling the interdependence of tourism demand: The global vector autoregressive approach. Annals of Tourism Research. 2017;67:1–13.
[95] Gunter U. Conditional forecasts of tourism exports and tourism export prices of the EU-15 within a global vector autoregression framework. Journal of Tourism Futures. 2018;4(2):121–38.
[96] Gunter U, Zekan B. Forecasting air passenger numbers with a GVAR model. Annals of Tourism Research. 2021;89:103252.
[97] Li H, Shi Y. Forecasting mortality with international linkages: A global vector-autoregression approach. Insurance: Mathematics and Economics. 2021;100:59–75.
[98] Salisu AA, Gupta R, Demirer R. Oil price uncertainty shocks and global equity markets: Evidence from a GVAR model. Journal of Risk and Financial Management. 2022;15(8):355.
[99] Cuaresma JC, Feldkircher M, Huber F. Forecasting with global vector autoregressive models: A Bayesian approach. Journal of Applied Econometrics. 2016;31(7):1371–1391.
[100] Assaf AG, Li G, Song H, Tsionas MG. Modeling and forecasting regional tourism demand using the Bayesian global vector autoregressive (BGVAR) model. Journal of Travel Research. 2018;58(3):383–397.
[101] Engle RF, Granger CWJ. Co-integration and error correction: Representation, estimation, and testing. Applied Econometrics. 2015;39(3):106–35.
[102] Shoesmith GL. Multiple cointegrating vectors, error correction, and forecasting with Litterman’s model. International Journal of Forecasting. 1995;11(4):557–567.
[103] Tegene A, Kuchler F. Evaluating forecasting models of farmland prices. International Journal of Forecasting. 1994;10(1):65–80.
[104] Ampountolas A. Forecasting hotel demand uncertainty using time series Bayesian VAR models. Tourism Economics. 2019;25(5):734–756.
[105] Cliff AD, Ord JK. Space-Time Modelling with an Application to Regional Forecasting. Transactions of the Institute of British Geographers. 1975;64:119–128.
[106] Pfeifer PE, Deutsch SJ. A three-stage iterative procedure for space-time modeling. Technometrics. 1980;22(1):35–47.
[107] Pfeifer PE, Deutsch SJ. Identification and interpretation of first order space-time ARMA models. Technometrics. 1980;22(3):397–408.
[108] Borovkova S, Lopuhaä H, Ruchjana B. Consistency and asymptotic normality of least squares estimators in generalized STAR models. Statistical Neerlandica. 2008;62(4):482–508.
[109] Ruchjana B, Borovkova S, Lopuhaä H. Least squares estimation of generalized space-time autoregressive (GSTAR) model and its properties. AIP Conference Proceedings. 2012;1450:61–64.
[110] Mukhaiyar U, Pasaribu US. A new procedure for generalized STAR modeling using IAcM approach. Journal of Mathematical and Fundamental Sciences. 2013;44(2):1721–1792.
[111] Zewdie M, Wubit G, Ayele A. G-STAR Model for Forecasting Space-Time Variation of Temperature in Northern Ethiopia. Turkish Journal of Forecasting. 2018;2(1):9–19.
[112] Akbar MS, Setiawan, Suhartono, Ruchjana BN, Prastyo DD, Muhaimin A, et al. A Generalized Space-Time Autoregressive Moving Average (GSTARMA) Model for Forecasting Air Pollutant in Surabaya. Journal of Physics: Conference Series. 2020;1490(1):012022.
[113] Nikolopoulos K, Buxton S, Khammash M, Stern P. Forecasting branded and generic pharmaceuticals. International Journal of Forecasting. 2016;32(2):344–357.
[114] Jan F, Shah I, Ali S. Short-Term electricity prices forecasting using functional time series analysis. Energies. 2022;15(9):3423.
[115] Taylor, James W. Triple seasonal methods for short-term electricity demand forecasting. European Journal of Operational Research, 2010;204(1):139–152.
[116] Naim I, Mahara T, Idrisi AR. Effective short-term forecasting for daily time series with complex seasonal patterns. Procedia Computer Science. 2018;132:1832–1841.
[117] Cleveland RB, Cleveland WS, McRae JE, Terpenning I. STL: A seasonal-trend decomposition procedure based on loess. Journal of Official Statistics. 1990;6(1):3–73.
[118] Theodosiou M. Forecasting monthly and quarterly time series using STL decomposition. International Journal of Forecasting. 2011;27(4):1178–1195.
[119] Li Y, Bao T, Gong J, Shu X, Zhang K. The prediction of dam displacement time series using STL, Extra-Trees, and stacked LSTM neural network. IEEE Access. 2020;8:94440–94452.
[120] Yin H, Jin D, Gu YH, Park CJ, Han SK, Yoo SJ. STL-ATTLSTM: Vegetable price forecasting using STL and attention mechanism-based LSTM. Agriculture. 2020;10(12):612.
[121] Ouyang Z, Ravier P, Jabloun M. STL decomposition of time series can benefit forecasting done by statistical methods but not by machine learning ones. Engineering Proceedings. 2021;5(1):42.
[122] Tian F, Wang D, Wu Q, Wei D. An empirical study on network conversion of stock time series based on STL method. Chaos. 2022;32(10):103111.
[123] Scotch CG, Murgulet D, Constantz J. Time-series temperature analyses indicate conduction and diffusion are dominant heat-transfer processes in fine sediment, low-flow streams. Science of the Total Environment. 2021;768:144367.
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