Document Type : Research Articles


Babol Noshirvani University of Technology, Babol, Iran.


Nowadays time series analysis is an important challenge in engineering problems. In this paper, we proposed the Comprehensive Learning Polynomial Autoregressive Model (CLPAR) predict linear and nonlinear time series. The presented model is based on the autoregressive (AR) model but developed in a polynomial aspect to make it more robust and accurate. This model predicts future values by learning the weights of the weighted sum of the polynomial combination of previous data. The learning process for the hyperparameters and properties of the model in the training phase is performed by the metaheuristic optimization method. Using this model, we can predict nonlinear time series as well as linear time series. The intended method was implemented on eight standard stationary and non-stationary large-scale real-world datasets. This method outperforms the state-of-the-art methods that use deep learning in seven time series and has better results compared to all other methods in six datasets. Experimental results show the advantage of the model accuracy over other compared methods on the various prediction tasks based on root mean square error (RMSE).


Main Subjects

[1] V. Ediger, S. Akar, ARIMA forecasting of primary energy
demand by fuel in Turkey, Energy Policy. 35 (2007) 1701–

[2] M. Khashei, F.M. Rafiei, M. Bijari, Hybrid fuzzy
auto-regressive integrated moving average (FARIMAH)
model for forecasting the foreign exchange markets, Int. J.
Comput. Intell. Syst. 6 (2013) 954–968.

[3] K. Kumar, V.K. Jain, Autoregressive integrated moving
averages (ARIMA) modelling of a traffic noise time series,
Appl. Acoust. 58 (1999) 283–294.

[4] F.-L. Chu, A fractionally integrated autoregressive moving
average approach to forecasting tourism demand, Tour.
Manag. 29 (2008) 79–88.

[5] H.-K. Yu, N.-Y. Kim, S.S. Kim, C. Chu, M.-K. Kee,
Forecasting the number of human immunodeficiency virus
infections in the Korean population using the
autoregressive integrated moving average model, Osong
Public Heal. Res. Perspect. 4 (2013) 358–362.

[6] G. Zheng, J.L. Starck, J.G. Campbell, F. Murtagh,
Multiscale transforms for filtering financial data streams, J.
Comput. Intell. Financ. 7 (1999).

[7] M.D. Chinn, M. LeBlanc, O. Coibion, The predictive
characteristics of energy futures: Recent evidence for crude oil, natural gas, gasoline and heating oil, Nat. Gas, Gasol.
Heat. Oil (October 2001). UCSC Econ. Work. Pap. (2001).

[8] C. Morana, A semiparametric approach to short-term oil
price forecasting, Energy Econ. 23 (2001) 325–338.

[9] W.K. Buchanan, P. Hodges, J. Theis, Which way the
natural gas price: an attempt to predict the direction of
natural gas spot price movements using trader positions,
Energy Econ. 23 (2001) 279–293.

[10] M.T. Hagan, S.M. Behr, The time series approach to short
term load forecasting, IEEE Trans. Power Syst. 2 (1987)

[11] G.E.P. Box, G.M. Jenkins, Time series analysis: forecasting
and control, Holden-Day, 1976.

[12] O. Renaud, J.-L. Starck, F. Murtagh, Wavelet-based
combined signal filtering and prediction, IEEE Trans. Syst.
Man, Cybern. Part B. 35 (2005) 1241–1251.

[13] G. Zhang, B.E. Patuwo, M.Y. Hu, Forecasting with artificial
neural networks: The state of the art, Int. J. Forecast. 14
(1998) 35–62.

[14] T.-S. Quah, B. Srinivasan, Improving returns on stock
investment through neural network selection, Expert Syst.
Appl. 17 (1999) 295–301.

[15] L.R. Rabiner, A tutorial on hidden Markov models and
selected applications in speech recognition, Proc. IEEE. 77
(1989) 257–286.

[16] J. Roman, A. Jameel, Backpropagation and recurrent neural
networks in financial analysis of multiple stock market
returns, in: Proc. HICSS-29 29th Hawaii Int. Conf. Syst.
Sci., 1996: pp. 454–460.

[17] S. A. Ghoreishi, and H. Khaloozadeh, "Application of
Covariance Matrix Adaptation-Evolution Strategy to
Optimal Portfolio," International Journal of Industrial
Electronics, Control and Optimization, Vol. 2, No. 2, pp.
81-90, 2019.

[18] A. Setare, O. Hesam, M. Homayun, Application of a fuzzy
method for predicting based on high-order time series, in:
2014 Iran. Conf. Intell. Syst., 2014: pp. 1–6.

[19] H.-K. Yu, Weighted fuzzy time series models for TAIEX
forecasting, Phys. A Stat. Mech. Its Appl. 349 (2005) 609–

[20] S.-M. Chen, Forecasting enrollments based on fuzzy time
series, Fuzzy Sets Syst. 81 (1996) 311–319.

[21] K. Huarng, Heuristic models of fuzzy time series for
forecasting, Fuzzy Sets Syst. 123 (2001) 369–386.

[22] J.L. Ticknor, A Bayesian regularized artificial neural
network for stock market forecasting, Expert Syst. Appl.
40 (2013) 5501–5506.

[23] L. Wang, Y. Zeng, T. Chen, Back propagation neural
network with adaptive differential evolution algorithm for
time series forecasting, Expert Syst. Appl. 42 (2015) 855–

[24] T.C. Jo, The effect of virtual term generation on the
neural-based approaches to time series prediction, in: 2003
4th Int. Conf. Control Autom. Proc., 2003: pp. 516–520.
[25] A.B. Geva, ScaleNet-multiscale neural-network architecture for time series prediction, IEEE Trans. Neural Networks. 9 (1998) 1471–1482.
[26] P. Liu, J. Liu, K. Wu, CNN-FCM: System modeling
promotes stability of deep learning in time series prediction,
Knowledge-Based Syst. 203 (2020) 106081.

[27] K. Wu, J. Liu, P. Liu, S. Yang, Time Series Prediction
Using Sparse Autoencoder and High-order Fuzzy
Cognitive Maps, IEEE Trans. Fuzzy Syst. (2019).

[28] J.G. Carvalho Jr, C.T. Costa Jr, Identification method for
fuzzy forecasting models of time series, Appl. Soft Comput.
50 (2017) 166–182.

[29] O.C. Yolcu, F. Alpaslan, Prediction of TAIEX based on
hybrid fuzzy time series model with single optimization
process, Appl. Soft Comput. 66 (2018) 18–33.

[30] J.-S. Jang, ANFIS: adaptive-network-based fuzzy inference
system, IEEE Trans. Syst. Man. Cybern. 23 (1993) 665–

[31] S. Yang, J. Liu, Time-series forecasting based on high-order
fuzzy cognitive maps and wavelet transform, IEEE Trans.
Fuzzy Syst. 26 (2018) 3391–3402.

[32] E. Bas, E. Egrioglu, C.H. Aladag, U. Yolcu,
Fuzzy-time-series network used to forecast linear and
nonlinear time series, Appl. Intell. 43 (2015) 343–355.

[33] H. Akaike, Fitting autoregressive models for prediction,
Ann. Inst. Stat. Math. 21 (1969) 243–247.

[34] S. Mirjalili, S.M. Mirjalili, A. Lewis, Grey Wolf Optimizer,
Adv. Eng. Softw. 69 (2014) 46–61.

[35] W. Lu, J. Yang, X. Liu, W. Pedrycz, The modeling and
prediction of time series based on synergy of high-order
fuzzy cognitive map and fuzzy c-means clustering,
Knowledge-Based Syst. 70 (2014) 242–255.

[36] S. Soltani, On the use of the wavelet decomposition for time
series prediction, Neurocomputing. 48 (2002) 267–277.

[37] H.S. Lopes, W.R. Weinert, EGIPSYS: an enhanced gene
expression programming approach for symbolic regression
problems, Int. J. Appl. Math. Comput. Sci. 14 (2004) 375–

[38] C. Ferreira, Gene expression programming: mathematical
modeling by an artificial intelligence, Springer, 2006.

[39] H. Cao, L. Kang, Y. Chen, J. Yu, Evolutionary modeling of
systems of ordinary differential equations with genetic
programming, Genet. Program. Evolvable Mach. 1 (2000)

[40] Y. Peng, C. Yuan, X. Qin, J. Huang, Y. Shi, An improved
gene expression programming approach for symbolic
regression problems, Neurocomputing. 137 (2014) 293–

[41] S.T.A. Niaki, S. Hoseinzade, Forecasting S&P 500 index
using artificial neural networks and design of experiments,
J. Ind. Eng. Int. 9 (2013) 1.

[42] R. Majhi, G. Panda, G. Sahoo, A. Panda, A. Choubey,
Prediction of S&P 500 and DJIA stock indices using
particle swarm optimization technique, in: 2008 IEEE Congr. Evol. Comput. (IEEE World Congr. Comput. Intell.,
2008: pp. 1276–1282.
[43] R. Tsaih, Y. Hsu, C.C. Lai, Forecasting S&P 500 stock
index futures with a hybrid AI system, Decis. Support Syst.
23 (1998) 161–174.

[44] M. Martens, Measuring and forecasting S&P 500
index-futures volatility using high-frequency data, J. Futur.
Mark. Futur. Options, Other Deriv. Prod. 22 (2002) 497–

[45] E. Hajizadeh, A. Seifi, M.H.F. Zarandi, I.B. Turksen, A
hybrid modeling approach for forecasting the volatility of
S&P 500 index return, Expert Syst. Appl. 39 (2012) 431–

[46] S.A. Hamid, Z. Iqbal, Using neural networks for forecasting
volatility of S&P 500 Index futures prices, J. Bus. Res. 57
(2004) 1116–1125.

[47] Yahoo, GSPC historical prices j S&P 500 stock, (n.d.).
EGSPC (accessed August 16, 2019).

[48] Monthly milk production: pounds per cow. Jan 62 - Dec 75,
e (accessed August 16, 2019).

[49] Monthly closings of the Dow-Jones industrial index, Aug.
1968 - Aug. 1981, (n.d.).
s=22v9&display=line (accessed August 16, 2019).

[50] Monthly critical radio frequencies in Washington, D.C.,
May 1934 - April 1954, (n.d.).

[51] Co2 (ppm) mauna loa, 1965-1980, (n.d.).!ds=22v1&display=line (accessed August
16, 2019).

[52] Monthly Lake Erie levels 1921 - 1970, (n.d.).
vels-1921-%0A1970#!ds=22pw&display=line (accessed
August 16, 2019).

[53] M.C. Mackey, L. Glass, Oscillation and chaos in
physiological control systems, Science (80-. ). 197 (1977)

[54] H.J. Song, C.Y. Miao, Z.Q. Shen, W. Roel, D.H. Maja, C.
Francky, Design of fuzzy cognitive maps using neural
networks for predicting chaotic time series, Neural
Networks. 23 (2010) 1264–1275.

[55] C.-F. Juang, Y.-W. Tsao, A self-evolving interval type-2
fuzzy neural network with online structure and parameter
learning, IEEE Trans. Fuzzy Syst. 16 (2008) 1411–1424.

[56] B.-T. Zhang, P. Ohm, H. Mühlenbein, Evolutionary
induction of sparse neural trees, Evol. Comput. 5 (1997)
[57] T. Hastie, R. Tibshirani, J. Friedman, Model assessment and
selection, in: Elem. Stat. Learn., Springer, 2009: pp. 219–

[58] Chicco, Davide, Matthijs J. Warrens, and Giuseppe Jurman.
"The coefficient of determination R-squared is more
informative than SMAPE, MAE, MAPE, MSE and RMSE
in regression analysis evaluation." PeerJ Computer Science
7 (2021): e623.