Statistical Estimates for Generalized Splines

Magnus Egerstedt; Clyde Martin

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 9, page 553-562
  • ISSN: 1292-8119

Abstract

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In this paper it is shown that the generalized smoothing spline obtained by solving an optimal control problem for a linear control system converges to a deterministic curve even when the data points are perturbed by random noise. We furthermore show that such a spline acts as a filter for white noise. Examples are constructed that support the practical usefulness of the method as well as gives some hints as to the speed of convergence.

How to cite

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Egerstedt, Magnus, and Martin, Clyde. "Statistical Estimates for Generalized Splines." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 553-562. <http://eudml.org/doc/90710>.

@article{Egerstedt2010,
abstract = { In this paper it is shown that the generalized smoothing spline obtained by solving an optimal control problem for a linear control system converges to a deterministic curve even when the data points are perturbed by random noise. We furthermore show that such a spline acts as a filter for white noise. Examples are constructed that support the practical usefulness of the method as well as gives some hints as to the speed of convergence. },
author = {Egerstedt, Magnus, Martin, Clyde},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Optimal control; smoothing splines; linear systems; interpolation.; optimal control; interpolation},
language = {eng},
month = {3},
pages = {553-562},
publisher = {EDP Sciences},
title = {Statistical Estimates for Generalized Splines},
url = {http://eudml.org/doc/90710},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Egerstedt, Magnus
AU - Martin, Clyde
TI - Statistical Estimates for Generalized Splines
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 553
EP - 562
AB - In this paper it is shown that the generalized smoothing spline obtained by solving an optimal control problem for a linear control system converges to a deterministic curve even when the data points are perturbed by random noise. We furthermore show that such a spline acts as a filter for white noise. Examples are constructed that support the practical usefulness of the method as well as gives some hints as to the speed of convergence.
LA - eng
KW - Optimal control; smoothing splines; linear systems; interpolation.; optimal control; interpolation
UR - http://eudml.org/doc/90710
ER -

References

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  1. N. Agwu and C. Martin, Optimal Control of Dynamic Systems: Application to Spline Approximations. Appl. Math. Comput.97 (1998) 99-138.  Zbl0944.41006
  2. M. Camarinha, P. Crouch and F. Silva-Leite, Splines of Class Ck on Non-Euclidean Spaces. IMA J. Math. Control Inform.12 (1995) 399-410 Zbl0860.58013
  3. P. Crouch and J.W. Jackson, Dynamic Interpolation for Linear Systems, in Proc. of the 29th. IEEE Conference on Decision and Control. Hawaii (1990) 2312-2314  
  4. P. Crouch, G. Kun and F. Silva-Leite, Generalization of Spline Curves on the Sphere: A Numerical Comparison, in Proc. CONTROLO'98, 3rd Portuguese Conference on Automatic control. Coimbra, Portugal (1998).  
  5. P. Crouch and F. Silva-Leite, The Dynamical Interpolation Problem: On Riemannian Manifolds, Lie Groups and Symmetric Spaces. J. Dynam. Control Systems1 (1995) 177-202.  Zbl0946.58018
  6. M. Egerstedt and C. Martin, Optimal Trajectory Planning and Smoothing Splines. Automatica37 (2001).  Zbl0989.93038
  7. M. Egerstedt and C. Martin, Monotone Smoothing Splines, in Proc. of MTNS. Perpignan, France (2000).  Zbl1203.49022
  8. D. Nychka, Splines as Local Smoothers. Ann. Statist.23 (1995) 1175-1197.  Zbl0842.62025
  9. C. Martin, M. Egerstedt and S. Sun, Optimal Control, Statistics and Path Planning. Math. Comput. Modeling33 (2001) 237-253.  Zbl0976.65057
  10. R.C. Rodrigues, F. Silva-Leite and C. Sim oes, Generalized Splines and Optimal Control, in Proc. ECC'99. Karlsruhe, Germany (1999).  
  11. S. Sun, M. Egerstedt and C. Martin, Control Theoretic Smoothing Splines. IEEE Trans. Automat. Control45 (2000) 2271-2279.  Zbl0971.49022
  12. G. Wahba, Spline Models for Observational Data. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1990).  Zbl0813.62001
  13. E.J. Wegman and I.W. Wright, Splines in Statistics. J. Amer. Statist. Assoc.78 (1983).  Zbl0534.62017
  14. Z. Zhang, J. Tomlinson and C. Martin, Splines and Linear Control Theory. Acta Math. Appl.49 (1997) 1-34.  Zbl0892.41008

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