Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T05:10:06.340Z Has data issue: false hasContentIssue false
  • English
  • Français

Averaging analysis of a point process adaptive algorithm

Part of: Stochastic analysis Stochastic processes Qualitative theory

Published online by Cambridge University Press:  14 July 2016

Victor Solo
Victor Solo*
Affiliation:
School of Electrical Engineering, University of New South Wales, Sydney, NSW 2052, Australia. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Motivated by a problem in neural encoding, we introduce an adaptive (or real-time) parameter estimation algorithm driven by a counting process. Despite the long history of adaptive algorithms, this kind of algorithm is relatively new. We develop a finite-time averaging analysis which is nonstandard partly because of the point process setting and partly because we have sought to avoid requiring mixing conditions. This is significant since mixing conditions often place restrictive history-dependent requirements on algorithm convergence.

Keywords

Averagingpoint processadaptive algorithm

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 34C29: Averaging method 60G55: Point processes
Type
Part 6. Stochastic processes
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 361 - 372
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Baῐnov, D. and Simeonov, R (1992). Integral Inequalities and Applications. Kluwer, Dordrecht.Google Scholar
[2] Benveniste, A., Metivier, M. and Priouret, P. (1990). Adaptive Algorithms and Stochastic Approximations. Springer, New York.CrossRefGoogle Scholar
[3] Blankenship, G. and Papanicoloau, G. C. (1978). Stability and control of stochastic systems with wide band noise disturbances. SIAM J. Appl. Math. 34, 437–476.CrossRefGoogle Scholar
[4] Brillinger, D. R. (1972). The spectral analysis of stationary interval functions. In Proc. 6th Berkeley Symp. Math. Statist. Prob. , Vol. 1, Theory of Statistics , eds Le Cam, L., Neyman, J. and Scott, E. L., University of California Press, Berkeley, CA, pp. 483–513.Google Scholar
[5] Brillinger, D. R. (1976). Estimation of the second-order intensities of a bivariate stationary point process. J. R. Statist. Soc. B 38, 60–66.Google Scholar
[6] Brown, E. et al. (2001). An analysis of neural receptive field dynamics by point process adaptive filtering. Proc. Nat. Acad. Sci. USA 98, 12261–12266.Google Scholar
[7] Ethier, S. and Kurtz, T. (1986). Markov Processes: Characterisation and Convergence. John Wiley, New York.Google Scholar
[8] Frackowiak, R. et al. (1997). Human Brain Function. Academic Press, Toronto.Google Scholar
[9] Freidlin, M. I. and Wentzell, A. D. (1984). Random Perturbations of Dynamical Systems. Springer, Heidelberg.Google Scholar
[10] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.Google Scholar
[11] Klebaner, F. (1998). Introduction to Stochastic Calculus with Applications. Imperial College Press, London.Google Scholar
[12] Krishnan, V. (1984). Nonlinear Filtering and Smoothing. John Wiley, New York.Google Scholar
[13] Kushner, H. (1984). Approximation and Weak Convergence Methods for Random Processes with Application to Stochastic System Theory. MIT Press, Cambridge, MA.Google Scholar
[14] Kushner, H. J. and Yin, G. (1997). Stochastic Approximation Algorithms and Applications. Springer, New York.Google Scholar
[15] Liptser, R. Sh. and Shiryayev, A. N. (1989). Theory of Martingales. Kluwer, Dordrecht.CrossRefGoogle Scholar
[16] O'Keefe, J. (1976). Place units in the hippocampus of the freely moving rat. Exper. Neurol. 51, 78–109.Google Scholar
[17] Rieke, F., Warland, D., De Ruyter Van Stevenink, R. and Bialek, W. (1997). Spikes: Exploring the Neural Code. MIT Press, Boston, MA.Google Scholar
[18] Robbins, H. and Munro, S. (1951). A stochastic approximation method. Ann. Math. Statist. 22, 400–407.CrossRefGoogle Scholar
[19] Sastry, S. and Bodson, M. (1989). Adaptive Control. Prentice-Hall, New York.Google Scholar
[20] Solo, V. (1996). Deterministic adaptive control with slowly-varying parameters: an averaging analysis. Internat. J. Control 64, 99–125.Google Scholar
[21] Solo, V. and Brown, E. (1999). Adaptive neural encoding. In Proc. 1st Joint IEEE BMES/EMBS Meeting (Atlanta, GA, October 1999).Google Scholar
[22] Solo, V. and Kong, X. (1995). Adaptive Signal Processing Algorithms. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[23] Volkonskii, V. A. and Rozanov, Y. A. (1959). Some limit theorems for random functions. Theory Prob. Appl. 4, 178–197.Google Scholar
[24] Widrow, B. and Stearns, S. (1985). Adaptive Signal Processing. Prentice-Hall, London.Google Scholar