, Volume 75, Issue 7, pp 939–951 | Cite as

A conditional distribution approach to uniform sampling on spheres and balls in Lp spaces

  • Vladimír LackoEmail author
  • Radoslav Harman


Liang and Ng (Metrika 68:83–98, 2008) proposed a componentwise conditional distribution method for L p -uniform sampling on L p -norm n-spheres. On the basis of properties of a special family of L p -norm spherical distributions we suggest a wide class of algorithms for sampling uniformly distributed points on n-spheres and n-balls in L p spaces, generalizing the approach of Harman and Lacko (J Multivar Anal 101:2297–2304, 2010), and including the method of Liang and Ng as a special case. We also present results of a numerical study proving that the choice of the best algorithm from the class significantly depends on the value of p.


Lp-norm n-sphere n-ball Uniform distribution p-generalized normal distribution Monte Carlo simulation Probabilistic robustness analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables, 10th edn. No. 55 in Applied Mathematics Series, National Bureau of StandardsGoogle Scholar
  2. Billingsley P (1968) Convergence of Probability Measures. Wiley, New YorkzbMATHGoogle Scholar
  3. Box GEP, Muller ME (1958) A note on the generation of random normal deviates. Ann Math Stat 29: 610–611zbMATHCrossRefGoogle Scholar
  4. Calafiore G, Dabbene F, Tempo R (1998) Uniform sample generation in p balls for probabilistic robustness analysis. In: Proceedings of 37th IEEEGoogle Scholar
  5. Calafiore GC, Dabbene F, Tempo R (2000) Randomized algorithms for probabilistic robustness with real and complex structured uncertainty. IEEE T Automat Contr 45: 2218–2235MathSciNetzbMATHCrossRefGoogle Scholar
  6. Fang KT, Yang Z, Kotz S (2001) Generation of multivariate distributions by vertical density representation. Statistics 35: 281–293MathSciNetzbMATHCrossRefGoogle Scholar
  7. Fishman GS (1996) Monte Carlo: concepts, algorithms, and applications. Springer, New YorkzbMATHGoogle Scholar
  8. Goodman IR, Kotz S (1973) Multivariate θ-generalized normal distributions. J Multivar Anal 3: 204–219MathSciNetzbMATHCrossRefGoogle Scholar
  9. Gupta AK, Song D (1997) l p-norm spherical distribution. J Stat Plan Infer 60: 241–260MathSciNetzbMATHCrossRefGoogle Scholar
  10. Harman R, Lacko V (2010) On decompositional algorithms for uniform sampling from n-spheres and n-balls. J Multivar Anal 101: 2297–2304MathSciNetzbMATHCrossRefGoogle Scholar
  11. Hashorva E (2008) Conditional limiting distribution of beta-independent random vectors. J Multivar Anal 99: 1438–1459MathSciNetzbMATHCrossRefGoogle Scholar
  12. Jambunathan MV (1954) Some properties of beta and gamma distributions. Ann Math Stat 25: 401–405MathSciNetzbMATHCrossRefGoogle Scholar
  13. Liang J, Ng KW (2008) A method for generating uniformly scattered points on the l p-norm unit sphere and its applications. Metrika 68: 83–98MathSciNetCrossRefGoogle Scholar
  14. McLeish DL (2005) Monte Carlo simulation and finance. Wiley, HobokenzbMATHGoogle Scholar
  15. Muller ME (1959) A note on a method for generating points uniformly on n-dimensional spheres. Commun ACM 2: 19–20zbMATHCrossRefGoogle Scholar
  16. R Development Core Team (2008) R: A language and environment for statistical computing. R Found Stat Comput, ViennaGoogle Scholar
  17. von Jöhnk KMD (1964) Erzeugung von betaverteilten und gammaverteilten Zufallszahlen. Metrika 8: 5–15MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and InformaticsComenius UniversityBratislavaSlovak Republic

Personalised recommendations