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Least absolute deviations estimation for uncertain autoregressive model

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Abstract

To predict future values based on imprecisely observed values, uncertain time series has been proposed, and the least-squares method has been presented to estimate the unknown parameters of uncertain autoregressive models. This paper considers the least absolute deviations estimation of uncertain autoregressive model, and a minimization problem is derived to calculate the unknown parameters in the uncertain autoregressive model. Finally, some numerical examples are given to illustrate the robustness of the least absolute deviations estimation compared with the least-squares estimation.

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Data availability

The data of Example 5.3 come from Earth System Research Laboratory (ftp://aftp.cmdl.noaa.gov/data/trace_gases/co2/flask/surface/co2_alt_surface-flask_1_ccgg_month.txt).

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Funding

This work was supported by the Program for Young Excellent Talents in UIBE (No. 18YQ06), and Ministry of Oceans and Fisheries (SMART-Navigation Project).

Author information

Authors and Affiliations

  1. School of Information Technology and Management, University of International Business and Economics, Beijing, 100029, China

    Xiangfeng Yang

  2. Korea Institute of Aids to Navigation, Sejong, 30100, Korea

    Gyei-Kark Park

  3. Navigation College, Shandong Jiaotong University, Weihai, 264209, China

    Yancai Hu

Authors
  1. Xiangfeng Yang
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  2. Gyei-Kark Park
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  3. Yancai Hu
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Corresponding author

Correspondence to Xiangfeng Yang.

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The authors declare that there is no conflict of interest regarding the publication of this paper.

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Appendix A: Absolute value of uncertain variable

Appendix A: Absolute value of uncertain variable

Let be an uncertainty space, whose measure satisfies normality, duality, subadditivity and product axioms. As a basic concept, the formal definition of an uncertain variable is given as follows. If \(\xi \) is an uncertain variable defined on an uncertainty space , and its uncertainty distribution is \(\varPhi (\cdot )\), then the expected of absolute value \(|\xi |\) is

Stipulation A.1

Let \(\xi \) be an uncertain variable with uncertainty distribution \(\varPhi \). Then the expected of absolute value \(|\xi |\) is

$$\begin{aligned} \mathrm{E}[|\xi |]=\int _0^{+\infty }(1-\varPhi (x)+\varPhi (-x))\mathrm{d}x. \end{aligned}$$

We can easily obtain the following formula via the change of variables and integration by parts,

$$\begin{aligned} \mathrm{E}[|\xi |]=\int _{-\infty }^{+\infty }|x|\mathrm{d}\varPhi (x). \end{aligned}$$
(18)

Furthermore, if \(\varPhi \) is regular, then we have

$$\begin{aligned} \mathrm{E}[|\xi |]=\int _{0}^{1}|\varPhi ^{-1}(\alpha )|\mathrm{d}\alpha \end{aligned}$$
(19)

form the change of variables.

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Yang, X., Park, GK. & Hu, Y. Least absolute deviations estimation for uncertain autoregressive model. Soft Comput 24, 18211–18217 (2020). https://doi.org/10.1007/s00500-020-05079-0

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