Abstract
This paper addresses the problem of finding a direct operational method to disentangle the sum of two continuous Markovian stochastic processes, a more general case of the so-called measurement noise concept, given only a measured time series of the sum process. The presented method is based on a recently published approach for the analysis of multidimensional Langevin-type stochastic processes in the presence of strong correlated measurement noise (Lehle, J Stat Phys 152(6):1145–1169, 2013). The method extracts from noisy data the respective drift and diffusion coefficients corresponding to the Itô–Langevin equation describing each stochastic process. The method presented here imposes neither constraints nor parameters, but all coefficients are directly extracted from the multidimensional data. The method is introduced within the framework of existing reconstruction methods, and then applied to the sum of a two-dimensional stochastic process convoluted with an Ornstein–Uhlenbeck process.
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Acknowledgements
Authors gratefully acknowledge support from Fundação para a Ciência e a Tecnologia (FCT) through SFRH/BD/86934/2012 (TS), SFRH/BPD/65427/2009 (FR), UID/GEO/50019/2013-ILD-LA (FR), German Federal Ministry for Economic Affairs and Energy 0325577B (PGL), from FCT and German Academic Exchange Service DRI/DAAD/1208/2013 (TS, FR, P GL, MW) and for the internship fellowship through IPID4all from University of Oldenburg (TS). VVL thanks the Prometeo Project of SENESCYT (Ecuador) for financial support. This work is partially supported by FCT, UID/MAT/04561/2013 (VVL).
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Scholz, T., Raischel, F., Lind, P.G., Wächter, M., Lopes, V.V., Lehle, B. (2016). A Direct Method for the Langevin-Analysis of Multidimensional Stochastic Processes with Strong Correlated Measurement Noise. In: Rojas, I., Pomares, H. (eds) Time Series Analysis and Forecasting. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-28725-6_1
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