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Abstract

In recent years, there has been a great development in parameter estimation methods for uncertain differential equations (UDEs). However, the observations we can obtain in real life are limited, in which case the form of function in a UDE is unknown. When dealing with such UDEs, we may use observational data to make nonparametric estimates. There are many nonautonomous systems in real life, and nonautonomous UDEs can simulate some uncertain nonautonomous dynamical systems well. In this paper, a nonparametric estimation method based on the nonautonomous UDEs of the binary Legendre polynomial is proposed. Then, three numerical examples are given to verify the reliability of nonparametric estimation. As an application, a real data example of global average monthly temperatures is used to illustrate the effectiveness of our method.

Keywords

Uncertainty theory uncertain differential equations nonparametric estimation global average monthly temperature

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References

Liu

, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007.

Liu

, Some research problems in uncertainty theory, Journalof Uncertain Systems 3(1) (2009), 3–10.

Liu

, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems 2(1) (2008), 3–16.

Zhu

, Uncertain Optimal Control with application to a portfolioselection model, Cybernetics and Systems 41(7) (2010), 535–547.

Chen

and Liu

, Existence and uniqueness theorem for uncertaindifferential equations, Fuzzy Optimization and Decision Making 9(1) (2010), 69–81.

Yao

and Chen

, A numerical method for solving uncertaindifferential equations, Journal of Intelligent and FuzzySystems 25(3) (2013), 825–832.

Yao

, Extreme values and integral of solution of uncertaindifferential equation, Journal of Uncertainty Analysis andApplications 1 (2013), 1–21.

Yang

and Shen

, Runge-Kutta method for solving uncertaindifferential equations, Journal of Uncertainty Analysis andApplications 3(10) (2015), 1–12.

Yao

and Liu

, Parameter estimation in uncertain differentialequations, Fuzzy Optimization and Decision Making 19(2020), 1–12.

10.

Sheng

, Yao

and Chen

, Least squares estimation in uncertaindifferential equations, IEEE Transactions on Fuzzy Systems 28(10) (2020), 2651–2655.

11.

Yang

, Liu

and Park

, Parameter estimation of uncertaindifferential equation with application to financial market, Chaos, Solitons and Fractals 139(1) (2020), 110026.

12.

Liu

, Generalized moment estimation for uncertain differentialequations, Applied Mathematics and Computation 392(2021), 125724.

13.

Liu

and Liu

, Estimating unknown parameters in uncertaindifferential equation by maximum likelihood estimation, SoftComputing 26 (2022), 2773–2780.

14.

Liu

and Liu

, Residual analysis and parameter estimation ofuncertain differential equations, Fuzzy Optimization andDecision Making 21 (2022), 513–530.

15.

and Liu

, Uncertain hypothesis test with application touncertain regression analysis, Fuzzy Optimization and DecisionMaking 21 (2022), 157–174.

16.

and Liu

, Uncertain hypothesis test for uncertaindifferential equations, Fuzzy Optimization and Decision Making 22 (2022), 195–211.

17.

, Yan

and Zhu

, A numerical method for solving optimalcontrol problems via Legendre polynomials, EngineeringComputations 37(8) (2020), 2735–2759.

18.

, Zhu

and Gu

, Nonparametric estimation for uncertaindifferential equations, Fuzzy Optimization and Decision Making 22 (2023), 697–715.

19.

Lio

and Liu

, Residual and confidence interval for uncertainregression model with imprecise observations, Journal ofIntelligent and Fuzzy Systems 35 (2018), 2573–2583.

20.

Gao

, Li

, He

, et al. Absolute measurement of square flatswith Legendre polynomial fitting method, InternationalSymposium on Optoelectronic Technology and Application 2014: Laserand Optical Measurement Technology and Fiber OpticSensors.International Society for Optics and Photonics 9297 (2014), 90–96.

21.

Rudin

, Principles of Mathematical Analysis, 2nd edition, McGraw Hill, 1964.

22.

Lio

and Liu

, Uncertain maximum likelihood estimation withapplication to uncertain regression analysis, Soft Computing 24 (2020), 9351–9360.

23.

Lenssen

, Schmidt

, Hansen

, et al. Improvements in theGISTEMP uncertainty model, Journal of Geophysical Research:Atmospheres 124(12) (2019), 6307–6326.

Nonparametric estimation of nonautonomous uncertain differential equations with application to temperature models

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