Finding the principal points of a random variable

Emilio Carrizosa; E. Conde; A. Castaño; D. Romero–Morales

RAIRO - Operations Research (2010)

  • Volume: 35, Issue: 3, page 315-328
  • ISSN: 0399-0559

Abstract

top
The p-principal points of a random variable X with finite second moment are those p points in minimizing the expected squared distance from X to the closest point. Although the determination of principal points involves in general the resolution of a multiextremal optimization problem, existing procedures in the literature provide just a local optimum. In this paper we show that standard Global Optimization techniques can be applied.

How to cite

top

Carrizosa, Emilio, et al. "Finding the principal points of a random variable." RAIRO - Operations Research 35.3 (2010): 315-328. <http://eudml.org/doc/197828>.

@article{Carrizosa2010,
abstract = { The p-principal points of a random variable X with finite second moment are those p points in $\{\mathbb R\}$ minimizing the expected squared distance from X to the closest point. Although the determination of principal points involves in general the resolution of a multiextremal optimization problem, existing procedures in the literature provide just a local optimum. In this paper we show that standard Global Optimization techniques can be applied. },
author = {Carrizosa, Emilio, Conde, E., Castaño, A., Romero–Morales, D.},
journal = {RAIRO - Operations Research},
keywords = {Principal points; d.c. functions; branch and bound.; principal points; branch and bound},
language = {eng},
month = {3},
number = {3},
pages = {315-328},
publisher = {EDP Sciences},
title = {Finding the principal points of a random variable},
url = {http://eudml.org/doc/197828},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Carrizosa, Emilio
AU - Conde, E.
AU - Castaño, A.
AU - Romero–Morales, D.
TI - Finding the principal points of a random variable
JO - RAIRO - Operations Research
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 3
SP - 315
EP - 328
AB - The p-principal points of a random variable X with finite second moment are those p points in ${\mathbb R}$ minimizing the expected squared distance from X to the closest point. Although the determination of principal points involves in general the resolution of a multiextremal optimization problem, existing procedures in the literature provide just a local optimum. In this paper we show that standard Global Optimization techniques can be applied.
LA - eng
KW - Principal points; d.c. functions; branch and bound.; principal points; branch and bound
UR - http://eudml.org/doc/197828
ER -

References

top
  1. E. Carrizosa, E. Conde, A. Casta no, I. Espinosa, I. González and D. Romero-Morales, Puntos principales: Un problema de Optimización Global en Estadística, Presented at XXII Congreso Nacional de Estadística e Investigación Operativa. Sevilla (1995).  
  2. D.R. Cox, A use of complex probabilities in the theory of stochastic processes, in Proc. of the Cambridge Philosophical Society, Vol. 51 (1955) 313-319.  Zbl0066.37703
  3. B. Flury, Principal points. Biometrika77 (1990) 33-41.  Zbl0691.62053
  4. B. Flury and T. Tarpey, Representing a Large Collection of Curves: A Case for Principal Points. Amer. Statist.47 (1993) 304-306.  
  5. R. Fourer, D.M. Gay and B.W. Kernigham, AMPL, A modeling language for Mathematical Programming. The Scientific Press, San Francisco (1993).  
  6. E. Gelenbe and R.R. Muntz, Probabilistic Models of Computer Systems-Part I. Acta Inform.7 (1976) 35-60.  Zbl0343.60066
  7. R. Horst, An Algorithm for Nonconvex Programming Problems. Math. Programming10 (1976) 312-321.  Zbl0337.90062
  8. R. Horst and H. Tuy, Global Optimization. Deterministic Approaches. Springer-Verlag, Berlin (1993).  Zbl0704.90057
  9. S.P. Lloyd, Least Squares Quantization in PCM. IEEE Trans. Inform. Theory 28 (1982) 129-137.  Zbl0504.94015
  10. L. Li and B. Flury, Uniqueness of principal points for univariate distributions. Statist. Probab. Lett.25 (1995) 323-327.  Zbl0837.62017
  11. K. Pötzelberger and K. Felsenstein, An asymptotic result on principal points for univariate distribution. Optimization28 (1994) 397-406.  Zbl0813.62012
  12. S. Rowe, An Algorithm for Computing Principal Points with Respect to a Loss Function in the Unidimensional Case. Statist. Comput.6 (1997) 187-190.  
  13. T. Tarpey, Two principal points of symmetric, strongly unimodal distributions. Statist. Probab. Lett.20 (1994) 253-257.  Zbl0799.62019
  14. T. Tarpey, Principal points and self-consistent points of symmetric multivariate distributions. J. Multivariate Anal.53 (1995) 39-51.  Zbl0820.62047
  15. T. Tarpey, L. Li and B. Flury, Principal points and self-consistent points of elliptical distributions. Ann. Statist.23 (1995) 103-112.  Zbl0822.62042
  16. A. Zoppè, Principal points of univariate continuous distributions. Statist. Comput.5 (1995) 127-132.  
  17. A. Zoppè, On Uniqueness and Symmetry of self-consistent points of univariate continuous distribution. J. Classification14 (1997) 147-158.  Zbl0891.62005

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.