Statistical estimates for generalized splines
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 553-562.

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.

DOI : 10.1051/cocv:2003026
Classification : 93-xx
Mots-clés : optimal control, smoothing splines, linear systems, interpolation 
@article{COCV_2003__9__553_0,
     author = {Egerstedt, Magnus and Martin, Clyde},
     title = {Statistical estimates for generalized splines},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {553--562},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2003},
     doi = {10.1051/cocv:2003026},
     mrnumber = {1998714},
     zbl = {1070.41003},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2003026/}
}
TY  - JOUR
AU  - Egerstedt, Magnus
AU  - Martin, Clyde
TI  - Statistical estimates for generalized splines
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2003
SP  - 553
EP  - 562
VL  - 9
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2003026/
DO  - 10.1051/cocv:2003026
LA  - en
ID  - COCV_2003__9__553_0
ER  - 
%0 Journal Article
%A Egerstedt, Magnus
%A Martin, Clyde
%T Statistical estimates for generalized splines
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2003
%P 553-562
%V 9
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2003026/
%R 10.1051/cocv:2003026
%G en
%F COCV_2003__9__553_0
Egerstedt, Magnus; Martin, Clyde. Statistical estimates for generalized splines. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 553-562. doi : 10.1051/cocv:2003026. http://www.numdam.org/articles/10.1051/cocv:2003026/

[1] N. Agwu and C. Martin, Optimal Control of Dynamic Systems: Application to Spline Approximations. Appl. Math. Comput. 97 (1998) 99-138. | MR | Zbl

[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 | Zbl

[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 Systems 1 (1995) 177-202. | Zbl

[6] M. Egerstedt and C. Martin, Optimal Trajectory Planning and Smoothing Splines. Automatica 37 (2001). | Zbl

[7] M. Egerstedt and C. Martin, Monotone Smoothing Splines, in Proc. of MTNS. Perpignan, France (2000).

[8] D. Nychka, Splines as Local Smoothers. Ann. Statist. 23 (1995) 1175-1197. | MR | Zbl

[9] C. Martin, M. Egerstedt and S. Sun, Optimal Control, Statistics and Path Planning. Math. Comput. Modeling 33 (2001) 237-253. | MR | Zbl

[10] R.C. Rodrigues, F. Silva-Leite and C. Simões, 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. Control 45 (2000) 2271-2279. | MR | Zbl

[12] G. Wahba, Spline Models for Observational Data. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1990). | MR | Zbl

[13] E.J. Wegman and I.W. Wright, Splines in Statistics. J. Amer. Statist. Assoc. 78 (1983). | MR | Zbl

[14] Z. Zhang, J. Tomlinson and C. Martin, Splines and Linear Control Theory. Acta Math. Appl. 49 (1997) 1-34. | MR | Zbl

Cité par Sources :