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A robust nonparametric framework for reconstruction of stochastic differential equation models

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  • Rajabzadeh, Yalda
  • Rezaie, Amir Hossein
  • Amindavar, Hamidreza

Abstract

In this paper, we employ a nonparametric framework to robustly estimate the functional forms of drift and diffusion terms from discrete stationary time series. The proposed method significantly improves the accuracy of the parameter estimation. In this framework, drift and diffusion coefficients are modeled through orthogonal Legendre polynomials. We employ the least squares regression approach along with the Euler–Maruyama approximation method to learn coefficients of stochastic model. Next, a numerical discrete construction of mean squared prediction error (MSPE) is established to calculate the order of Legendre polynomials in drift and diffusion terms. We show numerically that the new method is robust against the variation in sample size and sampling rate. The performance of our method in comparison with the kernel-based regression (KBR) method is demonstrated through simulation and real data. In case of real dataset, we test our method for discriminating healthy electroencephalogram (EEG) signals from epilepsy ones. We also demonstrate the efficiency of the method through prediction in the financial data. In both simulation and real data, our algorithm outperforms the KBR method.

Suggested Citation

  • Rajabzadeh, Yalda & Rezaie, Amir Hossein & Amindavar, Hamidreza, 2016. "A robust nonparametric framework for reconstruction of stochastic differential equation models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 450(C), pages 294-304.
    • Handle: RePEc:eee:phsmap:v:450:y:2016:i:c:p:294-304
    DOI: 10.1016/j.physa.2016.01.016
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    References listed on IDEAS

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    1. Farahpour, F. & Eskandari, Z. & Bahraminasab, A. & Jafari, G.R. & Ghasemi, F. & Sahimi, Muhammad & Reza Rahimi Tabar, M., 2007. "A Langevin equation for the rates of currency exchange based on the Markov analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 385(2), pages 601-608.
    2. Lim, Gyuchang & Kim, SooYong & Scalas, Enrico & Kim, Kyungsik & Chang, Ki-Ho, 2008. "Analysis of price fluctuations in futures exchange markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(12), pages 2823-2830.
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    4. P. Rinn & H. Heißelmann & M. Wächter & J. Peinke, 2013. "Stochastic method for in-situ damage analysis," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 86(1), pages 1-5, January.
    5. C. Renner & J. Peinke & R. Friedrich, 2001. "Markov properties of high frequency exchange rate data," Papers cond-mat/0102494, arXiv.org, revised Apr 2001.
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    Cited by:

    1. Théo Michelot & Richard Glennie & Catriona Harris & Len Thomas, 2021. "Varying-Coefficient Stochastic Differential Equations with Applications in Ecology," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 26(3), pages 446-463, September.

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