Property analysis of multivariate conditional linear random processes in the problems of mathematical modelling of signals
DOI:
https://doi.org/10.15587/2706-5448.2022.259906Keywords:
multivariate signal, conditional linear random process, stochastic integral, characteristic function, mathematical expectation, covariance functionAbstract
The object of research is the process of mathematical modelling of a multidimensional random signal, which in the structure of its generation is the sum of a large number of random impulses that occur at random times. Examples of stochastic signals of this type can be, in particular, electroencephalographic and cardiographic signals, photoplethysmography signals, resource consumption processes (electricity, gas, water consumption), radar signals, vibrations of bearings of electric machines and others.
A common mathematical model (especially in the multidimensional case) of this type of signal is a linear random process that allows the signal to be represented as the sum of a large number of stochastically independent random impulses that occur at Poisson moments. If the impulses are stochastically dependent (or the moments of time of their appearance are not Poisson), then the mathematical model is a conditional linear random process. The definition and analysis of the probabilistic properties of such processes for the multidimensional case have not been conducted.
The paper defines a multidimensional conditional linear random process, each component of which is represented as a stochastic integral of a random kernel driven by a process with independent increments. Expressions for the characteristic function and moment functions of the specified process are obtained. The approach used was to use the mathematical apparatus of conditional characteristic functions, as well as the known representation of an infinitely divisible characteristic function of a linear random process as a functional of a process with independent increments.
The obtained results provide a possibility for theoretical analysis of probabilistic properties of multichannel stochastic signals, the mathematical model of which is a multidimensional conditional linear random process. Justification of their properties of stationarity or cyclostationarity, which are the consequence of corresponding properties of the kernel and process with independent increments, can be carried out.
References
- Zvaritch, V., Mislovitch, M., Martchenko, B. (1994). White noise in information signal models. Applied Mathematics Letters, 7 (3), 93–95. doi: http://doi.org/10.1016/0893-9659(94)90120-1
- Babak, V. P., Babak, S. V., Zaporozhets, A. O., Myslovych, M. V., Zvaritch, V. M. (2020). Methods and Models for Information Data Analysis. Diagnostic Systems For Energy Equipments. Volume 281 of Studies in Systems, Decision and Control. Cham: Springer, 23–70. doi: http://doi.org/10.1007/978-3-030-44443-3_2
- Fryz, M., Scherbak, L., Karpinski, M., Mlynko, B. (2021). Characteristic Function of Conditional Linear Random Process. Proceedings of the 1st International Workshop on Information Technologies: Theoretical and Applied Problems. Ternopil, 129–135.
- Fryz, M. (2017). Conditional linear random process and random coefficient autoregressive model for EEG analysis. Proceedings of the 2017 IEEE First Ukraine Conference on Electrical and Computer Engineering. Kyiv, 305–309. doi: http://doi.org/10.1109/ukrcon.2017.8100498
- Pierre, P. A. (1971). Central Limit Theorems for Conditionally Linear Random Processes. SIAM Journal on Applied Mathematics, 20 (3), 449–461. doi: http://doi.org/10.1137/0120048
- Iwankiewicz, R. (1995). Dynamical mechanical systems under random impulses. Singapore: World Scientific Publishing Co. Pte. Ltd. doi: http://doi.org/10.1142/2767
- Demei, Y., Lan, L. (2016). Some results following from conditional characteristic functions. Communications in Statistics – Theory and Methods, 45 (12), 3706–3720. doi: http://doi.org/10.1080/03610926.2014.906614
- Bulinski, A. V. (2017). Conditional central limit theorem. Theory of Probability & Its Applications, 61 (4), 613–631. doi: http://doi.org/10.1137/s0040585x97t98837x
- Grzenda, W., Zieba, W. (2008). Conditional central limit theorem. International Mathematical forum, 3.31, 1521–1528.
- Napolitano, A. (2016). Cyclostationarity: New trends and applications. Signal Processing, 120, 385–408. doi: http://doi.org/10.1016/j.sigpro.2015.09.011
- Marchenko, B. G. (1973). Metod stokhasticheskikh integralnykh predstavlenii i ego prilozheniia v radiotekhnike. Kyiv: Naukova dumka, 192.
- Barndorff-Nielsen, O. E., Benth, F. E., Veraart, A. E. D. (2018). Ambit Stochastics. Springer Nature Switzerland AG. doi: http://doi.org/10.1007/978-3-319-94129-5
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Mykhailo Fryz, Bogdana Mlynko
This work is licensed under a Creative Commons Attribution 4.0 International License.
The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.
