Convergence of Minimum–Entropy Robust Estimators: Applications in DSP and Instrumentation
José Ismael de la Rosa Vargas
Universidad Autónoma de Zacatecas – Unidad Académica de Ingeniería Eléctrica, Lab. de Procesamiento Digital de Señales, Av. López Velarde No. 821, Col. Centro, C.P. 98068, Zacatecas, Zacatecas. ismaelrv@ieee.org
Artículo recibido en junio 21, 2004
Aceptado en noviembre 8, 2006
Resumen
En este trabajo de investigación nos proponemos continuar con la misma línea de investigaciones iniciadas por Pronzato y Thierry (Pronzato et al, 2000a), (Pronzato et al, 2000b), (Pronzato et al, 2001) las cuales se abordaron ya en los trabajos de De la Rosa y Fleury (De la Rosa et al, 2002), (De la Rosa et al, 2003) en un marco de trabajo perteneciente a la instrumentación, y en donde se establece un modelo estocástico para representar ciertas señales y para el cual se formulan ciertas hipótesis limitadas sobre la naturaleza del ruido o perturbaciones que afectan los sistemas bajo estudio. La utilización de estimadores robustos es importante, ya que los sistemas reales están expuestos a perturbaciones continuas que son de naturaleza desconocida, esto se ha experimentado en aplicaciones propias de la instrumentación médica, en procesos industriales, y en telecomunicaciones entre otros. Presentamos algunos resultados complementarios a los presentados por Pronzato y Thierry sobre la estimación robusta, tanto para modelos lineales como para modelos no lineales.
Palabras clave: Entropía, Simulación de tipo Monte–Carlo, Estimación No Paramétrica, Regresión, Estimación Robusta.
Abstract
In this paper we propose to continue in the same research line initiated by Pronzato and Thierry (Pronzato et al, 2000a), (Pronzato et al, 2000b), (Pronzato et al, 2001), recent works inspired in the minimum–entropy estimation have been published by De la Rosa and Fleury (De la Rosa et al, 2002), (De la Rosa et al, 2003) in the instrumentation framework. An statistical model has been established to represent some instrumental signals, similarly, some limited hypothesis over such a model have been made. In fact, we assume limited knowledge of the noise or external perturbations distribution that interact into the system. The use of robust estimators in such situations is very helpful, since the real systems are always exposed to continuous perturbations of unknown nature. Some applications where the last is true are: medical instrumentation, industrial processes, in telecommunications among others. Some results of new minimum–entropy estimators for linear and nonlinear models are presented, such results complement those presented by Pronzato and Thierry.
Keywords: Entropy, Monte Carlo Simulation, Nonparametric Estimation, Regression, Robust Estimation.
DESCARGAR ARTÍCULO EN FORMATO PDF
]]>Agradecimientos
Quiero agradecer de la manera más atenta, la disponibilidad, amabilidad y ayuda del Prof. Luc Pronzato del Laboratorio I3 S situado en la Universidad de Nice–Shopia Antipolis en Francia. Por otro lado, también agradezco el apoyo brindado por PROMEP para la realización de este trabajo que esta auspiciado por el convenio 103.5/03/1127.
Referencias
1. (Berlinet et al, 1994) Berlinet A., and Devroye L., "A comparison of kernel density estimates," Publications de l'Institut de Statistique de l'Université de Paris, Vol. 38 (3), 1994, pp. 3 – 59. [ Links ]
2. (De la Rosa et al, 2002) De la Rosa J. I., and Fleury G., "On the Kernel selection for Minimum–Entropy estimation," Proc. of the IEEE Instrumentation and Measurement Technology Conference, Anchorage, AK (USA), 21 – 23 May 2002, Vol. 2, pp. 1205 – 1210. [ Links ]
3. (De la Rosa et al, 2003) De la Rosa J. I., Fleury G., and Davoust M.–E., "Minimum–Entropy, pdf approximation and Kernel selection for measurement estimation," IEEE Transactions on Instrumentation and Measurement, Vol. 52, No. 4, August 2003, pp. 1009 – 1020. [ Links ]
4. (De la Rosa et al, 2006) De la Rosa J. I., and Fleury G., "Bootstrap methods for a measurement estimation problem," IEEE Transactions on Instrumentation and Measurement, Vol. 55, No. 3, June 2006, pp. 820 – 827. [ Links ]
5. (Devroye, 1992) Devroye L., "A note on the usefulness of superkernels in density estimation," The Annals of Statistics, Vol. 20, 1992, pp. 2037 – 2056. [ Links ]
6. URL: http://jeff.cs.mcgill.ca/ [ Links ]
7. (Devroye, 1989) Devroye L., "The double kernel method in density estimation," Annales de l'Institut Henri Poincaré, Vol. 25, 1989, pp. 533 – 580. [ Links ]
8. (Devroye et al, 1999) Devroye L., and Krzyzak A., "On the Hilbert kernel density estimate," Statistics and Probability Letters, Vol. 44, 1999, pp. 299 – 308. [ Links ]
9. (Devroye, 1997) Devroye L., "Universal smoothing factor selection in density estimation: theory and practice," Test, Vol. 6, 1997, pp. 223 – 320. [ Links ]
10. (Ö. Egecioglu et al, 2000) Egecioglu Ö., and Srinivasan A., "Efficent non–parametric density estimation on the sphere with applications in fluid mechanics," SIAM Journal on Scientific Computing, Vol. 22, 2000, No. 1, pp. 152 – 176. [ Links ]
11. (Fleury, 1999) Fleury G., "Optimal nonlinear modeling and reparametrization," IEEE International workshop on Intelligent Signal Proc., sept. 1999, pp. 72 – 76. [ Links ]
12. (Loader, 1999) Loader C. M., "Bandwidth selection: classical or plug–in?," The Annals of Statistics, Vol. 27, No. 3, 1999, pp. 415 – 438. [ Links ]
13. (Masry, 1983) Masry E., "Probability density estimation from sampled data," IEEE Trans. on Information Theory, Vol. IT–29, No.5, Sept. 1983, pp. 697 – 709. [ Links ]
14. (Pronzato et al, 2000a) Pronzato L., and Thierry E., "A minimum–entropy estimator for regression problems with unknown distribution of observation errors," MaxEnt 2000, Edited by A. Mohammad–Djafari, American Institute of Physics, July 8 – 13, 2000, pp. 169 – 180. [ Links ]
15. (Pronzato et al, 2000b) Pronzato L., and Thierry E., "A minimum–entropy estimator for regression problems with unknown distribution of observation errors," Tech. Rep. 00–08, Laboratoire I3 S, CNRS–Université de Nice–Shopia Antipolis, France, 2000. [ Links ]
16. (Pronzato et al, 2001) Pronzato L., and Thierry E., "Entropy minimization of parameter estimator with unknown distribution of observation errors," Proc. of the IEEE International Conference in Acoustics, Speech and Signal Processing, Vol. 6, March 2001, pp. 3993 – 3996. [ Links ]
17. (Terrell, 1990) Terrell G. P., "The maximal smoothing principle in density estimation,"Journal of the American Statistical Association, Vol. 85, 1990, pp. 470 – 477. [ Links ]
18. (Terrell et al, 1985) Terrell G. P., and Scott D. W., "Oversmoothed nonparametric density estimation," Journal of the American Statistical Association, Vol. 80, 1985, pp. 209 – 214. [ Links ] ]]>