Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-18T10:24:26.539Z Has data issue: false hasContentIssue false
  • English
  • Français

Statistical Estimates for Generalized Splines

Published online by Cambridge University Press:  15 September 2003

Magnus Egerstedt  and
Clyde Martin
Magnus Egerstedt
Affiliation:
School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA; magnus@ece.gatech.edu.
Clyde Martin
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA; martin@math.ttu.edu.
Get access

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.

Keywords

Optimal controlsmoothing splineslinear systemsinterpolation.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 553 - 562
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agwu, N. and Martin, C., Optimal Control of Dynamic Systems: Application to Spline Approximations. Appl. Math. Comput. 97 (1998) 99-138.
Camarinha, M., Crouch, P. and Silva-Leite, F., Splines of Class C k on Non-Euclidean Spaces. IMA J. Math. Control Inform. 12 (1995) 399-410 CrossRef
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
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).
Crouch, P. and Silva-Leite, F., The Dynamical Interpolation Problem: On Riemannian Manifolds, Lie Groups and Symmetric Spaces. J. Dynam. Control Systems 1 (1995) 177-202. CrossRef
M. Egerstedt and C. Martin, Optimal Trajectory Planning and Smoothing Splines. Automatica 37 (2001).
M. Egerstedt and C. Martin, Monotone Smoothing Splines, in Proc. of MTNS. Perpignan, France (2000).
Nychka, D., Splines as Local Smoothers. Ann. Statist. 23 (1995) 1175-1197. CrossRef
Martin, C., Egerstedt, M. and Sun, S., Optimal Control, Statistics and Path Planning. Math. Comput. Modeling 33 (2001) 237-253. CrossRef
R.C. Rodrigues, F. Silva-Leite and C. Sim oes, Generalized Splines and Optimal Control, in Proc. ECC'99. Karlsruhe, Germany (1999).
Sun, S., Egerstedt, M. and Martin, C., Control Theoretic Smoothing Splines. IEEE Trans. Automat. Control 45 (2000) 2271-2279. CrossRef
G. Wahba, Spline Models for Observational Data. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1990).
E.J. Wegman and I.W. Wright, Splines in Statistics. J. Amer. Statist. Assoc. 78 (1983).
Zhang, Z., Tomlinson, J. and Martin, C., Splines and Linear Control Theory. Acta Math. Appl. 49 (1997) 1-34. CrossRef