On a class of interval predictor models with universal reliability

Abstract An Interval Predictor Model (IPM) is a rule by which some observable variables (system inputs) are mapped into an interval that is used to predict an inaccessible variable (system output). IPMs have been studied in Campi et al. (2009), where the problem of fitting an IPM on a set of observations has been considered. In the same paper, upper-bounds on the probability that a future system output will fall outside the predicted interval (misprediction) have also been derived in a stationary and independent framework. While these bounds have the notable property of being valid independently of the unknown mechanism that has generated the data, in general the actual probability distribution of the misprediction does depend on the data generation mechanism and, hence, these bounds may introduce conservatism when applied to a specific case. In this paper, we study the reliability of an important class of IPMs, called minimax layers, and show that this class exhibits the special property that the probability distribution of the misprediction is known exactly and is universal, i.e., is always the same irrespective of the data generation mechanism. This result carries important consequences on the use of minimax layers in practice.

[1]  H. Harter.  The Method of Least Squares and Some Alternatives: Part III , 1975 .

[2]  A. Haar,et al.  Die Minkowskische Geometrie und die Annäherung an stetige Funktionen , 1917 .

[3]  Joyce Snell,et al.  6. Alternative Methods of Regression , 1996 .

[4]  Luis G. Crespo,et al.  RANDOM PREDICTOR MODELS FOR RIGOROUS UNCERTAINTY QUANTIFICATION , 2015 .

[5]  M. C. CAMPI,et al.  Random Convex Programs with L1-Regularization: Sparsity and Generalization , 2013, SIAM J. Control. Optim..

[6]  Yin Zhang.  Primal-dual interior point approach for computingl1-solutions andl∞-solutions of overdetermined linear systems , 1993 .

[7]  S. Garatti,et al.  Modulating robustness in control design: Principles and algorithms , 2013, IEEE Control Systems.

[8]  Marco C. Campi,et al.  FAST - Fast Algorithm for the Scenario Technique , 2014, Oper. Res..

[9]  Marco C. Campi,et al.  Scenario Min-Max Optimization and the Risk of Empirical Costs , 2015, SIAM J. Optim..

[10]  Giuseppe Carlo Calafiore,et al.  Interval predictor models: Identification and reliability , 2009, Autom..

[11]  O. J. Karst,et al.  Linear Curve Fitting Using Least Deviations , 1958 .

[12]  Giuseppe Carlo Calafiore,et al.  The scenario approach to robust control design , 2006, IEEE Transactions on Automatic Control.

[13]  Eric Walter,et al.  Guaranteed recursive non‐linear state bounding using interval analysis , 2002 .

[14]  Giuseppe Carlo Calafiore,et al.  Learning noisy functions via interval models , 2010, Syst. Control. Lett..

[15]  Ronald D. Armstrong,et al.  Min‐Max Estimates for a Linear Multiple Regression Problem , 1979 .

[16]  Roberto Tempo,et al.  Randomized methods for design of uncertain systems: Sample complexity and sequential algorithms , 2013, Autom..

[17]  J. Gates,et al.  Strict Discrete Approximation in the L1 and L∞ Norms , 1991 .

[18]  Marco C. Campi,et al.  The Exact Feasibility of Randomized Solutions of Uncertain Convex Programs , 2008, SIAM J. Optim..

[19]  Marco C. Campi,et al.  A Sampling-and-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality , 2011, J. Optim. Theory Appl..

[20]  Eric Walter,et al.  Guaranteed non-linear estimation using constraint propagation on sets , 2001 .

[21]  Gautam Appa,et al.  On L1 and Chebyshev estimation , 1973, Math. Program..

[22]  Ian Barrodale,et al.  Algorithm 495: Solution of an Overdetermined System of Linear Equations in the Chebychev Norm [F4] , 1975, TOMS.

[23]  H. M. Wagner.  Linear Programming Techniques for Regression Analysis , 1959 .

[24]  J. Norton,et al.  Bounding Approaches to System Identification , 1996 .

[25]  E. Cheney.  Introduction to approximation theory , 1966 .

[26]  Balázs Csanád Csáji,et al.  Certified system identification: Towards distribution-free results , 2012 .

[27]  Luis G. Crespo,et al.  Interval Predictor Models With a Linear Parameter Dependency , 2016 .

[28]  Eric Walter,et al.  Identification of Parametric Models: from Experimental Data , 1997 .