A Novel Lag Window for Spectrum Estimation of the Ornstein-Uhlenbeck Process

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DOI: https://doi.org/10.28924/2291-8639-23-2025-223
Published: Sep 19, 2025
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Ali Sami Rashid, A Novel Lag Window for Spectrum Estimation of the Ornstein-Uhlenbeck Process, Int. J. Anal. Appl., 23 (2025), 223.

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Ali Sami Rashid

Abstract

In order to forecast and process the behavior of noisy data, spectral analysis is a crucial area of study in data analysis and interpretation. The goal of this study is to identify the optimal lag window to estimate the continuous-time Ornstein-Uhlenbeck (OU) process’s spectrum. The equivalent difference equation of the OU process was derived, and a consistent estimate of the spectral density function (SDF) was calculated using the most prominent lag window functions in the different parameter cases and time interval segmentation. A parameterized novel lag window (NLW) was proposed. The parameters can be changed to control the kurtosis and skewness of the NLW curve and reduce the influence of the tails of the estimated autocorrelation function on the consistent estimate of the SDF. The simulation results of comparing the SDF and the consistent estimate of the SDF with lag windows showed that the proposed NLW outperformed all other lag windows in estimating the spectrum of the OU process in all parameter cases and in all time-interval segmentation. The promising results of NLW can be used in signal processing and spectral analysis of phenomena subject to the influence of noise.

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References

  1. A.S. Rashid, M.J. Hawas Allami, A.K. Mutasher, Best Lag Window for Spectrum Estimation of Law Order Ma Process, Abstr. Appl. Anal. 2020 (2020), 9352453. https://doi.org/10.1155/2020/9352453.
  2. A. Seif, S.A.M. Loos, G. Tucci, É. Roldán, S. Goldt, The Impact of Memory on Learning Sequence-To-Sequence Tasks, Mach. Learn.: Sci. Technol. 5 (2024), 015053. https://doi.org/10.1088/2632-2153/ad2feb.
  3. B. Laala, S. Belaloui, K. Fang, A.M. Elsawah, Improving the Lag Window Estimators of the Spectrum and Memory for Long-Memory Stationary Gaussian Processes, Commun. Math. Stat. 13 (2023), 59–98. https://doi.org/10.1007/s40304-022-00304-8.
  4. C. Lu, C. Xu, Dynamic Properties for a Stochastic Seir Model with Ornstein–uhlenbeck Process, Math. Comput. Simul. 216 (2024), 288–300. https://doi.org/10.1016/j.matcom.2023.09.020.
  5. D. Bosq, H.T. Nguyen, A Course in Stochastic Processes, Springer, (2013).
  6. D. Yao, Application of Stochastic Process Models and Numerical Methods in Financial Product Pricing, J. Syst. Manag. Sci. 13 (2023), 483–496. https://doi.org/10.33168/jsms.2023.0531.
  7. F. Harris, On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform, Proc. IEEE 66 (1978), 51–83. https://doi.org/10.1109/proc.1978.10837.
  8. H. Hammood, A. H Ali, N. Jalil Aklo, The Effect of Sample Size on the Interpolation Algorithm of Frequency Estimation, Iraqi J. Electr. Electron. Eng. 21 (2024), 156–161. https://doi.org/10.37917/ijeee.21.1.15.
  9. J. Shang, W. Li, Dynamical Behaviors of a Stochastic Sirv Epidemic Model with the Ornstein–uhlenbeck Process, Adv. Contin. Discret. Model. 2024 (2024), 9. https://doi.org/10.1186/s13662-024-03807-6.
  10. J. Wang, N.K. Voulgarakis, Hierarchically Coupled Ornstein–uhlenbeck Processes for Transient Anomalous Diffusion, Physics 6 (2024), 645–658. https://doi.org/10.3390/physics6020042.
  11. K.G. Arbeev, O. Bagley, A.P. Yashkin, H. Duan, I. Akushevich, S.V. Ukraintseva, A.I. Yashin, Understanding Alzheimer’s Disease in the Context of Aging: Findings From Applications of Stochastic Process Models to the Health and Retirement Study, Mech. Ageing Dev. 211 (2023), 111791. https://doi.org/10.1016/j.mad.2023.111791.
  12. L. Stanković, D. Mandic, M. Daković, B. Scalzo, M. Brajović, E. Sejdić, A.G. Constantinides, Vertex-frequency Graph Signal Processing: a Comprehensive Review, Digit. Signal Process. 107 (2020), 102802. https://doi.org/10.1016/j.dsp.2020.102802.
  13. M.B. Priestley, Spectral Analysis and Time Series, Academic Press, 1981.
  14. M. Premkumar, S. Rajakumar, R. Subraja, Signal Processing Algorithms for Mean Square Error Analysis in Mimo Wireless Transceivers, Ingénierie Des Systèmes D Inf. 28 (2023), 1695–1700. https://doi.org/10.18280/isi.280628.
  15. C. Sushma, P. Nimmagadda, Design of Efficient Alterable Bandwidth Fir Filterbank for Hearing Aid System, e-Prime - Adv. Electr. Eng. Electron. Energy 7 (2024), 100478. https://doi.org/10.1016/j.prime.2024.100478.
  16. S.S.P. Shen, G.R. North, Statistics and Data Visualization in Climate Science with R and Python, Cambridge University Press, 2023. https://doi.org/10.1017/9781108903578.
  17. Y. Yan, W. Zhang, Y. Yin, W. Huo, An Ornstein–uhlenbeck Model with the Stochastic Volatility Process and Tempered Stable Process for VIX Option Pricing, Math. Probl. Eng. 2022 (2022), 4018292. https://doi.org/10.1155/2022/4018292.