Statistical estimates for generalized splines

Magnus Egerstedt; Clyde Martin

ESAIM: Control, Optimisation and Calculus of Variations (2003)

  • 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 (2003): 553-562. <http://eudml.org/doc/245032>.

@article{Egerstedt2003,
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},
language = {eng},
pages = {553-562},
publisher = {EDP-Sciences},
title = {Statistical estimates for generalized splines},
url = {http://eudml.org/doc/245032},
volume = {9},
year = {2003},
}

TY - JOUR
AU - Egerstedt, Magnus
AU - Martin, Clyde
TI - Statistical estimates for generalized splines
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2003
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
UR - http://eudml.org/doc/245032
ER -

References

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  1. [1] N. Agwu and C. Martin, Optimal Control of Dynamic Systems: Application to Spline Approximations. Appl. Math. Comput. 97 (1998) 99-138. Zbl0944.41006MR1643083
  2. [2] M. Camarinha, P. Crouch and F. Silva–Leite, Splines of Class C k on Non-Euclidean Spaces. IMA J. Math. Control Inform. 12 (1995) 399-410 Zbl0860.58013
  3. [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. [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. [5] P. Crouch and F. Silva–Leite, The Dynamical Interpolation Problem: On Riemannian Manifolds, Lie Groups and Symmetric Spaces. J. Dynam. Control Systems 1 (1995) 177-202. Zbl0946.58018
  6. [6] M. Egerstedt and C. Martin, Optimal Trajectory Planning and Smoothing Splines. Automatica 37 (2001). Zbl0989.93038
  7. [7] M. Egerstedt and C. Martin, Monotone Smoothing Splines, in Proc. of MTNS. Perpignan, France (2000). 
  8. [8] D. Nychka, Splines as Local Smoothers. Ann. Statist. 23 (1995) 1175-1197. Zbl0842.62025MR1353501
  9. [9] C. Martin, M. Egerstedt and S. Sun, Optimal Control, Statistics and Path Planning. Math. Comput. Modeling 33 (2001) 237-253. Zbl0976.65057MR1812548
  10. [10] R.C. Rodrigues, F. Silva–Leite and C. Simões, Generalized Splines and Optimal Control, in Proc. ECC’99. Karlsruhe, Germany (1999). 
  11. [11] S. Sun, M. Egerstedt and C. Martin, Control Theoretic Smoothing Splines. IEEE Trans. Automat. Control 45 (2000) 2271-2279. Zbl0971.49022MR1807308
  12. [12] G. Wahba, Spline Models for Observational Data. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1990). Zbl0813.62001MR1045442
  13. [13] E.J. Wegman and I.W. Wright, Splines in Statistics. J. Amer. Statist. Assoc. 78 (1983). Zbl0534.62017MR711110
  14. [14] Z. Zhang, J. Tomlinson and C. Martin, Splines and Linear Control Theory. Acta Math. Appl. 49 (1997) 1-34. Zbl0892.41008MR1482878

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