Uncertain time series is chronological sequence overtime where each period is described by an uncertain variable. In this paper, we investigate the smoothly clipped absolute deviation (SCAD) penalized estimation method to determine the unknown parameters in the uncertain autoregressive model, and the autoregressive model order can be simultaneously obtained for a pre-given thresholding parameter λ. Besides, an iterative algorithm based on local quadratic approximations for finding the penalized estimators is provided. Based on the fitted autoregressive model, the forecast value and the future value’s confidence interval are given. Besides, the sum of the squared error approach to select the optimal λ is discussed. Finally, some examples are used to validate the effectiveness of the proposed method by the comparative analysis.
Al-SmadiA. and Al-ZabenA., ARMA model order determination using edge detection: A new perspective, Circuits, Systems and Signal Processing24(6) (2005), 723–732.
ChenD. and YangX., Ridge estimation for uncertain autoregressive model with Imprecise observations, International Journal of Uncertainty, Fuzziness and Knowledge-based Systems29(1) (2021), 37–55.
4.
ChenD. and YangX., Maximum likelihood estimation for uncertain autoregressive model with application to carbon dioxide emissions, Journal of Intelligent & Fuzzy Systems40(1) (2021), 1391–1399.
5.
CochraneD. and OrcuttG., Applications of least squares regression to relationships containing autocorrelated errors, Journal of American Statistical Association44 (1949), 32–61.
6.
ReinselG.C., BasuS. and YapS.F., Maximum likelihood estimators in the multivariate autoregressive moving average model from a generalized least-squares viewpoint, Journal of Time Series Analysis13 (1992), 133–145.
7.
BoxG.E.P. and JenkinsG.M., Time series analysis: Forecasting and control, Journal of the American Statistical Association68(342) (1970), 199–201.
8.
LiangG., WilkesD.M. and CadzowJ.A., ARMA model order estimation based on the eigenvalues of the covariance matrix, IEEE Transactions on Signal Processing41(10) (1993), 3003–3009.
9.
SchwarzG., Estimating the dimension of a model, Annals of Statistics6 (1978), 461–464.
10.
WalkerG.T., On periodicity in series of related terms, Proceedings of the Royal Society of London Series A131 (1931), 518–532.
11.
YuleG.U., On a method of investigation periodicities in disturbed series with special reference to Wolfer’s sunspot numbers, Philosophical Transactions of the Royal Society of London226 (1927), 267–298.
12.
AkaikeH., A new look at the statistical model identification, IEEE Transactions on Automatic ControlAC-19(6) (1974), 716–723.
13.
AkaikeH., Statistical predictor identification, Annals of Institute of Statistical Mathematics, 1970, 203–217.
14.
McDonaldJ.B., Partially adaptive estimation of ARMA time series models, International Journal of Forecasting5(2) (1989), 217–230.
15.
DavisR.A., Gauss-Newton and M-estimation for ARMA processes with infinite variance, Stochastic Processes and their Applications63(1) (1976), 75–95.
16.
FanJ. and LiR., Variable selection via nonconcave penalized likelihood and its oracle properties, Journal of the American Statistical Association96(456) (2001), 1348–1360.
17.
FuchsJ.J., On estimating the order of an ARMA process, Automatica23(6) (1987), 779–782.
18.
PaarmannL.D. and KorenbergM.J., Estimation of the parameters of an ARMA signal model based on an orthogonal search, IEEE Transactions on Automatic Control37 (1992), 347–352.
19.
SadabadiM.S., ShafieeM. and KarrariM., Two-dimensional ARMA model order determination, ISA Transactions48 (2009), 247–253.
20.
YangX. and LiuB., Uncertain time series analysis with imprecise observations, Fuzzy Optimization and Decision Making18(3) (2019), 263–278.
21.
YangX. and NiY., Least squares estimation for uncertain moving average model, Communications in Statistics-Theory and Methods, DOI:10.1080/03610926.2020.1713373, 2020.
22.
YangX., ParkG. and HuY., Least absolute deviations estimation for uncertain autoregressive model, Soft Computing24(23) (2020), 18211–18217.
23.
LiuZ. and YangX., Cross validation for uncertain autoregressive model, Communications in Statistics-Simulation and Computation, DOI:10.1080/03610918.2020.1747077, 2020.
24.
ZhangZ.Q., YangX. and GaoJ.W., Uncertain autoregressive model via LASSO procedure, International Journal of Uncertainty, Fuzziness and Knowledge-based Systems28(6) (2020), 939–956.
