Nonparametric output prediction for nonlinear fading memory systems

The authors construct a class of elementary nonparametric output predictors of an unknown discrete-time nonlinear fading memory system. Their algorithms predict asymptotically well for every bounded input sequence, every disturbance sequence in certain classes, and every linear or nonlinear system that is continuous and asymptotically time-invariant, causal, and with fading memory. The predictor is based on k/sub n/-nearest neighbor estimators from nonparametric statistics. It uses only previous input and noisy output data of the system without any knowledge of the structure of the unknown system, the bounds on the input, or the properties of noise. Under additional smoothness conditions the authors provide rates of convergence for the time-average errors of their scheme. Finally, they apply their results to the special case of stable linear time-invariant (LTI) systems.

[1]  Clas A. Jacobson.  Control Oriented System Identification , 1993 .

[2]  Adam Krzyżak,et al.  Identification of non-linear systems by recursive kernel regression estimates , 1993 .

[3]  L. Devroye.  Necessary and sufficient conditions for the pointwise convergence of nearest neighbor regression function estimates , 1982 .

[4]  Sanjeev R. Kulkarni,et al.  Rates of convergence of nearest neighbor estimation under arbitrary sampling , 1995, IEEE Trans. Inf. Theory.

[5]  Adam Krzyzak,et al.  On estimation of a class of nonlinear systems by the kernel regression estimate , 1990, IEEE Trans. Inf. Theory.

[6]  R. M. Dudley,et al.  Real Analysis and Probability , 1989 .

[7]  P. Mäkilä.  Robust identification and Galois sequences , 1991 .

[8]  L. Gyorfi.  The rate of convergence of k_n -NN regression estimates and classification rules (Corresp.) , 1981 .

[9]  W. Greblicki.  Nonparametric identification of Wiener systems by orthogonal series , 1994, IEEE Trans. Autom. Control..

[10]  Neri Merhav,et al.  Universal prediction of individual sequences , 1992, IEEE Trans. Inf. Theory.

[11]  John N. Tsitsiklis,et al.  Worst-Case Identification of Nonlinear Fading Memory Systems , 1992, 1992 American Control Conference.

[12]  J. Tsitsiklis,et al.  Optimal asymptotic identification under bounded disturbances , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[13]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[14]  C. J. Stone,et al.  Optimal Rates of Convergence for Nonparametric Estimators , 1980 .

[15]  J. Doyne Farmer,et al.  Exploiting Chaos to Predict the Future and Reduce Noise , 1989 .

[16]  C. J. Stone,et al.  Consistent Nonparametric Regression , 1977 .

[17]  Thomas M. Cover,et al.  Estimation by the nearest neighbor rule , 1968, IEEE Trans. Inf. Theory.

[18]  László Györfi,et al.  The Rate of Convergence of k ,-NN Regression Estimates and Classification Rules , 1978 .

[19]  Sanjeev R. Kulkarni,et al.  Data-dependent k/sub n/-NN estimators consistent for arbitrary processes , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[20]  Carl N. Nett,et al.  Control oriented system identification: a worst-case/deterministic approach in H/sub infinity / , 1991 .

[21]  K. Poolla,et al.  On the time complexity of worst-case system identification , 1994, IEEE Trans. Autom. Control..

[22]  Miroslaw Pawlak,et al.  Dynamic system identification with order statistics , 1994, IEEE Trans. Inf. Theory.

[23]  J. Tsitsiklis,et al.  The sample complexity of worst-case identification of FIR linear systems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.