IDENTIFICATION WITH FINITELY MANY DATA POINTS: THE LSCR APPROACH

Abstract This paper gives an overview of LSCR (Leave-out Sign-dominant Correlation Regions), a general technique for system identification. Under normal conditions, observations contain information corrupted by disturbances and measurement noise so that only an approximate description of the underlying system can at best be obtained from a finite data set. This is similar to describing an object seen through a frosted glass. Differently from standard identification methods that deliver single models, LSCR generates a model set. As information increases, the model set shrinks around the true system and, for any finite sample size, the set is guaranteed to contain the true system with a precise probability chosen by the user. LSCR only assumes a minimum amount of prior information on the noise.

[1]  M. Campi,et al.  Finite sample properties of system identification methods , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[2]  Marco C. Campi,et al.  Learning dynamical systems in a stationary environment , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[3]  M. Kenward,et al.  An Introduction to the Bootstrap , 2007 .

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

[5]  Alexander Goldenshluger,et al.  Nonparametric Estimation of Transfer Functions: Rates of Convergence and Adaptation , 1998, IEEE Trans. Inf. Theory.

[6]  Erik Weyer,et al.  Non-asymptotic confidence ellipsoids for the least-squares estimate , 2002, Autom..

[7]  Erik Weyer,et al.  Non-asymptotic quality assessment of generalised FIR models with periodic inputs , 2004, Autom..

[8]  Erik Weyer,et al.  Non-asymptotic confidence ellipsoids for the least squares estimate , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[9]  H. Hjalmarsson,et al.  An exact finite sample variance expression for a class of frequency function estimates , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[10]  Graham C. Goodwin,et al.  Finite sample properties of indirect nonparametric closed-loop identification , 2002, IEEE Trans. Autom. Control..

[11]  A. Vicino,et al.  Sequential approximation of feasible parameter sets for identification with set membership uncertainty , 1996, IEEE Trans. Autom. Control..

[12]  Andrea Garulli,et al.  Conditional central algorithms for worst case set-membership identification and filtering , 2000, IEEE Trans. Autom. Control..

[13]  Munther A. Dahleh,et al.  On system identification of complex systems from finite data , 2001, IEEE Trans. Autom. Control..

[14]  J. A. Hartigan.  Exact Confidence Intervals in Regression Problems with Independent Symmetric Errors , 1970 .

[15]  Ron Meir,et al.  Nonparametric Time Series Prediction Through Adaptive Model Selection , 2000, Machine Learning.

[16]  R. Bitmead,et al.  Empirical Estimation of Parameter Distributions in System Identification , 2003 .

[17]  Roy S. Smith,et al.  Continuous-time control model validation using finite experimental data , 1996, IEEE Trans. Autom. Control..

[18]  J. A. Ward,et al.  Sample complexity of worst-case H ∞ -identification , 1996 .

[19]  Marco Lovera,et al.  Bootstrap-based estimates of uncertainty in subspace identification methods , 2000, Autom..

[20]  J. Spall.  Uncertainty bounds for parameter identification with small sample sizes , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[21]  Antonio Vicino,et al.  Optimal estimation theory for dynamic systems with set membership uncertainty: An overview , 1991, Autom..

[22]  Dharmendra S. Modha,et al.  Memory-Universal Prediction of Stationary Random Processes , 1998, IEEE Trans. Inf. Theory.

[23]  Laura Giarré,et al.  Model quality evaluation in set membership identification , 1997, Autom..

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

[25]  Laura Giarré,et al.  Identification of approximated hammerstein models in a worst-case setting , 2002, IEEE Trans. Autom. Control..

[26]  E. Weyer,et al.  Parametric Identification of Nonlinear Systems: Guaranteed Confidence Regions , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[27]  Mario Milanese,et al.  H∞ identification and model quality evaluation , 1997, IEEE Trans. Autom. Control..

[28]  Bin Yu.  RATES OF CONVERGENCE FOR EMPIRICAL PROCESSES OF STATIONARY MIXING SEQUENCES , 1994 .

[29]  William P. Heath,et al.  Bias of indirect non-parametric transfer function estimates for plants in closed loop , 2001, Autom..

[30]  Håkan Hjalmarsson,et al.  Variance error quantifications that are exact for finite-model order , 2004, IEEE Transactions on Automatic Control.

[31]  M. Vidyasagar,et al.  Rates of uniform convergence of empirical means with mixing processes , 2002 .

[32]  John Odenckantz,et al.  Nonparametric Statistics for Stochastic Processes: Estimation and Prediction , 2000, Technometrics.

[33]  Mario Milanese,et al.  Hinfinity set membership identification: A survey , 2005, Autom..

[34]  S. Bittanti,et al.  Assessing the quality of identified models through the asymptotic theory - when is the result reliable? , 2004, Autom..

[35]  L Nyström,et al.  Statistical Analysis , 2008, Encyclopedia of Social Network Analysis and Mining.

[36]  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.

[37]  L. Gordon,et al.  Completely Separating Groups in Subsampling , 1974 .

[38]  Feng Ding,et al.  Performance bounds of forgetting factor least-squares algorithms for time-varying systems with finite measurement data , 2005, IEEE Transactions on Circuits and Systems I: Regular Papers.

[39]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[40]  Sergio Bittanti,et al.  The asymptotic model quality assessment for instrumental variable identification revisited , 2006, Syst. Control. Lett..

[41]  Dimitris N. Politis,et al.  Computer-intensive methods in statistical analysis , 1998, IEEE Signal Process. Mag..

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

[43]  Peter L. Bartlett,et al.  Prediction Algorithms: Complexity, Concentration and Convexity , 2003 .

[44]  Lennart Ljung,et al.  Using the bootstrap to estimate the variance in the case of undermodeling , 2002, IEEE Trans. Autom. Control..

[45]  Erik Weyer,et al.  Guaranteed non-asymptotic confidence regions in system identification , 2005, Autom..

[46]  Mathukumalli Vidyasagar,et al.  A Theory of Learning and Generalization , 1997 .

[47]  Erik Weyer.  Finite sample properties of system identification of ARX models under mixing conditions , 2000, Autom..

[48]  Vladimir N. Vapnik,et al.  The Nature of Statistical Learning Theory , 2000, Statistics for Engineering and Information Science.

[49]  J. A. Hartigan,et al.  Using Subsample Values as Typical Values , 1969 .

[50]  Iven M. Y. Mareels,et al.  Finite sample properties of linear model identification , 1999, IEEE Trans. Autom. Control..

[51]  Mathukumalli Vidyasagar,et al.  A learning theory approach to system identification and stochastic adaptive control , 2008 .

[52]  M. Chavance.  [Jackknife and bootstrap]. , 1992, Revue d'epidemiologie et de sante publique.

[53]  Dharmendra S. Modha,et al.  Minimum complexity regression estimation with weakly dependent observations , 1996, IEEE Trans. Inf. Theory.