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

Sign-error adaptive filtering algorithms involving Markovian parameters

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

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.
Citation: Araz Hashemi, George Yin, Le Yi Wang. Sign-error adaptive filtering algorithms involving Markovian parameters. Mathematical Control and 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, Berlin, 1990. doi: 10.1007/978-3-642-75894-2.

[2]

P. Billingsley, Convergence of Probability Measures, J. Wiley, New York, 1968.

[3]

H.-F. Chen and G. Yin, Asymptotic properties of sign algorithms for adaptive filtering, IEEE Trans. Automat. Control, 48 (2003), 1545-1556. doi: 10.1109/TAC.2003.816967.

[4]

E. Eweda, Convergence of the sign algorithm for adaptive filtering with correlated data, IEEE Trans. Inform. Theory, 37 (1991), 1450-1457.

[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), 1893-1905.

[6]

A. Gersho, Adaptive filtering with binary reinforcement, IEEE Trans. Inform. Theory, 30 (1984), 191-199.

[7]

L. Guo, Stability of recursive stochastic tracking algorithms, SIAM Journal on Control and Optimization, 32 (1994), 1195-1225. doi: 10.1137/S0363012992225606.

[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), Prentice Hall, 1998.

[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), 190-201.

[10]

H. J. Kushner and A. Shwartz, Weak convergence and asymptotic properties of adaptive filters with constant gains, IEEE Trans. Inform. Theory, 30 (1984), 177-182. doi: 10.1109/TIT.1984.1056897.

[11]

H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, 2nd ed., Springer-Verlag, New York, NY, 2003.

[12]

L. Y. Wang, G. Yin, J.-F. Zhang and Y. L. Zhao, System Identification with Quantized Observations: Theory and Applications, Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4956-2.

[13]

B. Widrow and S. D. Stearns, Adaptive Signal Processing, Prentice-Hall, Englewood, Cliffs, NJ, 1985.

[14]

G. Yin, Adaptive filtering with averaging, in Adaptive Control, Filtering and Signal Processing (eds. K. Aström, G. Goodwin and P. R. Kumar), IMA Volumes in Mathematics and Its Applications, 74, Springer-Verlag, New York, 1995, 375-396. doi: 10.1007/978-1-4419-8568-2_18.

[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), 321-334. doi: 10.1007/BF02714090.

[16]

G. Yin, A. Hashemi and L. Y. Wang, Sign-regressor adaptive filtering algorithms for Markovian parameters, Asian J. Control, 16 (2014), 95-106. doi: 10.1002/asjc.678.

[17]

G. Yin and V. Krishnamurthy, Least mean square algorithms with Markov regime switching limit, IEEE Trans. Automat. Control, 50 (2005), 577-593. doi: 10.1109/TAC.2005.847060.

[18]

G. Yin and Q. Zhang, Discrete-time Markov Chains: Two-time-scale Methods and Applications, Springer, New York, NY, 2005.

[19]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.

show all references

References:
[1]

A. Benveniste, M. Metivier and P. Priouret, Adaptive Algorithms and Stochastic Approximations, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-75894-2.

[2]

P. Billingsley, Convergence of Probability Measures, J. Wiley, New York, 1968.

[3]

H.-F. Chen and G. Yin, Asymptotic properties of sign algorithms for adaptive filtering, IEEE Trans. Automat. Control, 48 (2003), 1545-1556. doi: 10.1109/TAC.2003.816967.

[4]

E. Eweda, Convergence of the sign algorithm for adaptive filtering with correlated data, IEEE Trans. Inform. Theory, 37 (1991), 1450-1457.

[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), 1893-1905.

[6]

A. Gersho, Adaptive filtering with binary reinforcement, IEEE Trans. Inform. Theory, 30 (1984), 191-199.

[7]

L. Guo, Stability of recursive stochastic tracking algorithms, SIAM Journal on Control and Optimization, 32 (1994), 1195-1225. doi: 10.1137/S0363012992225606.

[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), Prentice Hall, 1998.

[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), 190-201.

[10]

H. J. Kushner and A. Shwartz, Weak convergence and asymptotic properties of adaptive filters with constant gains, IEEE Trans. Inform. Theory, 30 (1984), 177-182. doi: 10.1109/TIT.1984.1056897.

[11]

H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, 2nd ed., Springer-Verlag, New York, NY, 2003.

[12]

L. Y. Wang, G. Yin, J.-F. Zhang and Y. L. Zhao, System Identification with Quantized Observations: Theory and Applications, Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4956-2.

[13]

B. Widrow and S. D. Stearns, Adaptive Signal Processing, Prentice-Hall, Englewood, Cliffs, NJ, 1985.

[14]

G. Yin, Adaptive filtering with averaging, in Adaptive Control, Filtering and Signal Processing (eds. K. Aström, G. Goodwin and P. R. Kumar), IMA Volumes in Mathematics and Its Applications, 74, Springer-Verlag, New York, 1995, 375-396. doi: 10.1007/978-1-4419-8568-2_18.

[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), 321-334. doi: 10.1007/BF02714090.

[16]

G. Yin, A. Hashemi and L. Y. Wang, Sign-regressor adaptive filtering algorithms for Markovian parameters, Asian J. Control, 16 (2014), 95-106. doi: 10.1002/asjc.678.

[17]

G. Yin and V. Krishnamurthy, Least mean square algorithms with Markov regime switching limit, IEEE Trans. Automat. Control, 50 (2005), 577-593. doi: 10.1109/TAC.2005.847060.

[18]

G. Yin and Q. Zhang, Discrete-time Markov Chains: Two-time-scale Methods and Applications, Springer, New York, NY, 2005.

[19]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.

[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 and Management Optimization, 2020, 16 (6) : 2603-2623. doi: 10.3934/jimo.2019072

[2]

Kehan Si, Zhenda Xu, Ka Fai Cedric Yiu, Xun Li. Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2415-2433. doi: 10.3934/jimo.2021074

[3]

Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control and Related Fields, 2020, 10 (1) : 189-215. doi: 10.3934/mcrf.2019036

[4]

Ping Chen, Haixiang Yao. Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching. Journal of Industrial and Management Optimization, 2020, 16 (2) : 531-551. doi: 10.3934/jimo.2018166

[5]

Ishak Alia, Mohamed Sofiane Alia. Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022048

[6]

Yinghui Dong, Kam Chuen Yuen, Guojing Wang. Pricing credit derivatives under a correlated regime-switching hazard processes model. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1395-1415. doi: 10.3934/jimo.2016079

[7]

Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1965-1993. doi: 10.3934/jimo.2018132

[8]

Meiqiao Ai, Zhimin Zhang, Wenguang Yu. Valuing equity-linked death benefits with a threshold expense structure under a regime-switching Lévy model. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022007

[9]

Orazio Muscato, Wolfgang Wagner. A stochastic algorithm without time discretization error for the Wigner equation. Kinetic and Related Models, 2019, 12 (1) : 59-77. doi: 10.3934/krm.2019003

[10]

Hongfu Yang, Xiaoyue Li, George Yin. Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3743-3766. doi: 10.3934/dcdsb.2016119

[11]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4887-4905. doi: 10.3934/dcdsb.2020317

[12]

Miaomiao Gao, Daqing Jiang, Tasawar Hayat, Ahmed Alsaedi, Bashir Ahmad. Dynamics of a stochastic HIV/AIDS model with treatment under regime switching. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3177-3211. doi: 10.3934/dcdsb.2021181

[13]

Rua Murray. Approximation error for invariant density calculations. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 535-557. doi: 10.3934/dcds.1998.4.535

[14]

Fuke Wu, George Yin, Zhuo Jin. Kolmogorov-type systems with regime-switching jump diffusion perturbations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2293-2319. doi: 10.3934/dcdsb.2016048

[15]

Christoforidou Amalia, Christian-Oliver Ewald. A lattice method for option evaluation with regime-switching asset correlation structure. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1729-1752. doi: 10.3934/jimo.2020042

[16]

Jiaqin Wei. Time-inconsistent optimal control problems with regime-switching. Mathematical Control and Related Fields, 2017, 7 (4) : 585-622. doi: 10.3934/mcrf.2017022

[17]

Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control and Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237

[18]

Engel John C Dela Vega, Robert J Elliott. Conditional coherent risk measures and regime-switching conic pricing. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 267-300. doi: 10.3934/puqr.2021014

[19]

Wensheng Yin, Jinde Cao, Yong Ren. Inverse optimal control of regime-switching jump diffusions. Mathematical Control and Related Fields, 2022, 12 (3) : 567-579. doi: 10.3934/mcrf.2021034

[20]

Jun Li, Fubao Xi. Exponential ergodicity for regime-switching diffusion processes in total variation norm. Discrete and Continuous Dynamical Systems - B, 2022, 27 (10) : 6125-6146. doi: 10.3934/dcdsb.2021309

2021 Impact Factor: 1.141

Metrics

  • PDF downloads (132)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]