Sign-error adaptive filtering algorithms involving Markovian parameters

Previous Article
Stabilization of hyperbolic equations with mixed boundary conditions
MCRF Home
This Issue
Next Article
Generalization on optimal multiple stopping with application to swing options with random exercise rights number

December 2015, 5(4): 781-806. doi: 10.3934/mcrf.2015.5.781

Sign-error adaptive filtering algorithms involving Markovian parameters

Araz Hashemi ^1, , George Yin ^2, and Le Yi Wang ^3,

1.	Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, United States
2.	Department of Mathematics, Wayne State University, Detroit, Michigan 48202
3.	Department of Electrical and Computer Engineering, Wayne State University, MI 48202

Received February 2015 Revised March 2015 Published October 2015

Abstract
Related Papers

Motivated by reduction of computational complexity, this work develops sign-error adaptive filtering algorithms for estimating randomly time-varying system parameters. Different from the existing work on sign-error algorithms, the parameters are time-varying and their dynamics are modeled by a discrete-time Markov chain. Another distinctive feature of the algorithms is the multi-time-scale framework for characterizing parameter variations and algorithm updating speeds. This is realized by considering the stepsize of the estimation algorithms and a scaling parameter that defines the transition rate of the Markov jump process. Depending on the relative time scales of these two processes, suitably scaled sequences of the estimates are shown to converge to either an ordinary differential equation, or a set of ordinary differential equations modulated by random switching, or a stochastic differential equation, or stochastic differential equations with random switching. Using weak convergence methods, convergence and rates of convergence of the algorithms are obtained for all these cases. Simulation results are provided for demonstration.

Keywords: tracking property., regime-switching model, convergence, mean squares error, stochastic approximation, Sign-error algorithm.

Mathematics Subject Classification: Primary: 93E10, 93E11, 93E24, 93E3.

Citation: Araz Hashemi, George Yin, Le Yi Wang. Sign-error adaptive filtering algorithms involving Markovian parameters. Mathematical Control & Related Fields, 2015, 5 (4) : 781-806. doi: 10.3934/mcrf.2015.5.781

References:

[1]	A. Benveniste, M. Metivier and P. Priouret, Adaptive Algorithms and Stochastic Approximations,, Springer-Verlag, (1990). doi: 10.1007/978-3-642-75894-2. Google Scholar
[2]	P. Billingsley, Convergence of Probability Measures,, J. Wiley, (1968). Google Scholar
[3]	H.-F. Chen and G. Yin, Asymptotic properties of sign algorithms for adaptive filtering,, IEEE Trans. Automat. Control, 48 (2003), 1545. doi: 10.1109/TAC.2003.816967. Google Scholar
[4]	E. Eweda, Convergence of the sign algorithm for adaptive filtering with correlated data,, IEEE Trans. Inform. Theory, 37 (1991), 1450. Google Scholar
[5]	J. Fang and H. Li, Adaptive distributed estimation of signal power from one-bit quantized data,, IEEE Transactions on Aerospace and Electronic Systems, 46 (2010), 1893. Google Scholar
[6]	A. Gersho, Adaptive filtering with binary reinforcement,, IEEE Trans. Inform. Theory, 30 (1984), 191. Google Scholar
[7]	L. Guo, Stability of recursive stochastic tracking algorithms,, SIAM Journal on Control and Optimization, 32 (1994), 1195. doi: 10.1137/S0363012992225606. Google Scholar
[8]	M. L. Honig and H. V. Poor, Adaptive interference suppression in wireless communication systems,, in Wireless Communications: Signal Processing Perspectives* (eds. H. V. Poor and G. W. Wornell)*, (1998). Google Scholar
[9]	V. Krishnamurthy, G. Yin and S. Singh, Adaptive step size algorithms for blind interference suppression in DS/CDMA systems,, IEEE Trans. Signal Processing, 49 (2001), 190. Google Scholar
[10]	H. J. Kushner and A. Shwartz, Weak convergence and asymptotic properties of adaptive filters with constant gains,, IEEE Trans. Inform. Theory, 30 (1984), 177. doi: 10.1109/TIT.1984.1056897. Google Scholar
[11]	H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications,, 2nd ed., (2003). Google Scholar
[12]	L. Y. Wang, G. Yin, J.-F. Zhang and Y. L. Zhao, System Identification with Quantized Observations: Theory and Applications,, Birkhäuser, (2010). doi: 10.1007/978-0-8176-4956-2. Google Scholar
[13]	B. Widrow and S. D. Stearns, Adaptive Signal Processing,, Prentice-Hall, (1985). Google Scholar
[14]	G. Yin, Adaptive filtering with averaging,, in Adaptive Control, (1995), 375. doi: 10.1007/978-1-4419-8568-2_18. Google Scholar
[15]	G. Yin and H.-F. Chen, On asymptotic properties of a constant-step-size sign-error algorithm for adaptive filtering,, Scientia Sinica, 45 (2002), 321. doi: 10.1007/BF02714090. Google Scholar
[16]	G. Yin, A. Hashemi and L. Y. Wang, Sign-regressor adaptive filtering algorithms for Markovian parameters,, Asian J. Control, 16 (2014), 95. doi: 10.1002/asjc.678. Google Scholar
[17]	G. Yin and V. Krishnamurthy, Least mean square algorithms with Markov regime switching limit,, IEEE Trans. Automat. Control, 50 (2005), 577. doi: 10.1109/TAC.2005.847060. Google Scholar
[18]	G. Yin and Q. Zhang, Discrete-time Markov Chains: Two-time-scale Methods and Applications,, Springer, (2005). Google Scholar
[19]	G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications,, Springer, (2010). doi: 10.1007/978-1-4419-1105-6. Google Scholar

show all references

References:

[1]	A. Benveniste, M. Metivier and P. Priouret, Adaptive Algorithms and Stochastic Approximations,, Springer-Verlag, (1990). doi: 10.1007/978-3-642-75894-2. Google Scholar
[2]	P. Billingsley, Convergence of Probability Measures,, J. Wiley, (1968). Google Scholar
[3]	H.-F. Chen and G. Yin, Asymptotic properties of sign algorithms for adaptive filtering,, IEEE Trans. Automat. Control, 48 (2003), 1545. doi: 10.1109/TAC.2003.816967. Google Scholar
[4]	E. Eweda, Convergence of the sign algorithm for adaptive filtering with correlated data,, IEEE Trans. Inform. Theory, 37 (1991), 1450. Google Scholar
[5]	J. Fang and H. Li, Adaptive distributed estimation of signal power from one-bit quantized data,, IEEE Transactions on Aerospace and Electronic Systems, 46 (2010), 1893. Google Scholar
[6]	A. Gersho, Adaptive filtering with binary reinforcement,, IEEE Trans. Inform. Theory, 30 (1984), 191. Google Scholar
[7]	L. Guo, Stability of recursive stochastic tracking algorithms,, SIAM Journal on Control and Optimization, 32 (1994), 1195. doi: 10.1137/S0363012992225606. Google Scholar
[8]	M. L. Honig and H. V. Poor, Adaptive interference suppression in wireless communication systems,, in Wireless Communications: Signal Processing Perspectives* (eds. H. V. Poor and G. W. Wornell)*, (1998). Google Scholar
[9]	V. Krishnamurthy, G. Yin and S. Singh, Adaptive step size algorithms for blind interference suppression in DS/CDMA systems,, IEEE Trans. Signal Processing, 49 (2001), 190. Google Scholar
[10]	H. J. Kushner and A. Shwartz, Weak convergence and asymptotic properties of adaptive filters with constant gains,, IEEE Trans. Inform. Theory, 30 (1984), 177. doi: 10.1109/TIT.1984.1056897. Google Scholar
[11]	H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications,, 2nd ed., (2003). Google Scholar
[12]	L. Y. Wang, G. Yin, J.-F. Zhang and Y. L. Zhao, System Identification with Quantized Observations: Theory and Applications,, Birkhäuser, (2010). doi: 10.1007/978-0-8176-4956-2. Google Scholar
[13]	B. Widrow and S. D. Stearns, Adaptive Signal Processing,, Prentice-Hall, (1985). Google Scholar
[14]	G. Yin, Adaptive filtering with averaging,, in Adaptive Control, (1995), 375. doi: 10.1007/978-1-4419-8568-2_18. Google Scholar
[15]	G. Yin and H.-F. Chen, On asymptotic properties of a constant-step-size sign-error algorithm for adaptive filtering,, Scientia Sinica, 45 (2002), 321. doi: 10.1007/BF02714090. Google Scholar
[16]	G. Yin, A. Hashemi and L. Y. Wang, Sign-regressor adaptive filtering algorithms for Markovian parameters,, Asian J. Control, 16 (2014), 95. doi: 10.1002/asjc.678. Google Scholar
[17]	G. Yin and V. Krishnamurthy, Least mean square algorithms with Markov regime switching limit,, IEEE Trans. Automat. Control, 50 (2005), 577. doi: 10.1109/TAC.2005.847060. Google Scholar
[18]	G. Yin and Q. Zhang, Discrete-time Markov Chains: Two-time-scale Methods and Applications,, Springer, (2005). Google Scholar
[19]	G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications,, Springer, (2010). doi: 10.1007/978-1-4419-1105-6. Google Scholar

[1]	Chao Xu, Yinghui Dong, Zhaolu Tian, Guojing Wang. Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2019072
[2]	Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019036
[3]	Ping Chen, Haixiang Yao. Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2018166
[4]	Yinghui Dong, Kam Chuen Yuen, Guojing Wang. Pricing credit derivatives under a correlated regime-switching hazard processes model. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1395-1415. doi: 10.3934/jimo.2016079
[5]	Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1965-1993. doi: 10.3934/jimo.2018132
[6]	Orazio Muscato, Wolfgang Wagner. A stochastic algorithm without time discretization error for the Wigner equation. Kinetic & Related Models, 2019, 12 (1) : 59-77. doi: 10.3934/krm.2019003
[7]	Hongfu Yang, Xiaoyue Li, George Yin. Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3743-3766. doi: 10.3934/dcdsb.2016119
[8]	Fuke Wu, George Yin, Zhuo Jin. Kolmogorov-type systems with regime-switching jump diffusion perturbations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2293-2319. doi: 10.3934/dcdsb.2016048
[9]	Jiaqin Wei. Time-inconsistent optimal control problems with regime-switching. Mathematical Control & Related Fields, 2017, 7 (4) : 585-622. doi: 10.3934/mcrf.2017022
[10]	Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control & Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237
[11]	Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100
[12]	Rua Murray. Approximation error for invariant density calculations. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 535-557. doi: 10.3934/dcds.1998.4.535
[13]	Liping Zhang, Soon-Yi Wu, Shu-Cherng Fang. Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems. Journal of Industrial & Management Optimization, 2010, 6 (2) : 333-346. doi: 10.3934/jimo.2010.6.333
[14]	Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124
[15]	Zhuo Jin, George Yin, Hailiang Yang. Numerical methods for dividend optimization using regime-switching jump-diffusion models. Mathematical Control & Related Fields, 2011, 1 (1) : 21-40. doi: 10.3934/mcrf.2011.1.21
[16]	Jiapeng Liu, Ruihua Liu, Dan Ren. Investment and consumption in regime-switching models with proportional transaction costs and log utility. Mathematical Control & Related Fields, 2017, 7 (3) : 465-491. doi: 10.3934/mcrf.2017017
[17]	Kun Fan, Yang Shen, Tak Kuen Siu, Rongming Wang. On a Markov chain approximation method for option pricing with regime switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 529-541. doi: 10.3934/jimo.2016.12.529
[18]	JaEun Ku. Maximum norm error estimates for Div least-squares method for Darcy flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1305-1318. doi: 10.3934/dcds.2010.26.1305
[19]	Lin Xu, Rongming Wang, Dingjun Yao. Optimal stochastic investment games under Markov regime switching market. Journal of Industrial & Management Optimization, 2014, 10 (3) : 795-815. doi: 10.3934/jimo.2014.10.795
[20]	Nguyen Huu Du, Nguyen Thanh Dieu, Tran Dinh Tuong. Dynamic behavior of a stochastic predator-prey system under regime switching. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3483-3498. doi: 10.3934/dcdsb.2017176

2018 Impact Factor: 1.292

American Institute of Mathematical Sciences