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Variational Analysis of Poisson Processes

  • Ilya Molchanov
  • Sergei ZuyevEmail author
Chapter
Part of the Bocconi & Springer Series book series (BS, volume 7)

Abstract

The expected value of a functional F(η) of a Poisson process η can be considered as a function of its intensity measure μ. The paper surveys several results concerning differentiability properties of this functional on the space of signed measures with finite total variation. Then, necessary conditions for μ being a local minima of the considered functional are elaborated taking into account possible constraints on μ, most importantly the case of μ with given total mass a. These necessary conditions can be phrased by requiring that the gradient of the functional (being the expected first difference \(F(\eta +\delta _{x}) - F(\eta )\)) is constant on the support of μ. In many important cases, the gradient depends only on the local structure of μ in a neighbourhood of x and so it is possible to work out the asymptotics of the minimising measure with the total mass a growing to infinity. Examples include the optimal approximation of convex functions, clustering problem and optimal search. In non-asymptotic cases, it is in general possible to find the optimal measure using steepest descent algorithms which are based on the obtained explicit form of the gradient.

Keywords

Poisson Process Point Process Cluster Centre Optimal Measure Poisson Point Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The authors thank the organisers of the Oberwolfach Mini-Workshop: Stochastic Analysis for Poisson point processes (Feb. 2013) and the hospitality of the Oberwolfach Mathematical Institute for an excellent forum to exchange ideas partly presented in this paper.

References

  1. 1.
    Albeverio, S., Kondratiev, Y.G., Röckner, M.: Analysis and geometry on configuration spaces. J. Funct. Anal. 154 (2), 444–500 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Atkinson, A.C., Donev, A.N.: Optimum Experimental Designs. Clarendon Press, Oxford (1992)zbMATHGoogle Scholar
  3. 3.
    Ben-Tal, A., Zowe, J.: A unified theory of first and second order conditions for extremum problems in topological vector spaces. Optimality and stability in mathematical programming. Math. Program. Stud. 19, 39–76 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cominetti, R.: Metric regularity, tangent sets and second-order optimality conditions. Appl. Math. Optim. 21, 265–287 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. General Theory and Structure, vol. II, 2nd edn. Springer, New York (2008)Google Scholar
  6. 6.
    Dunford, N., Schwartz, J.T.: Linear Operators. General Theory, vol. 1. Wiley, New York (1988)Google Scholar
  7. 7.
    Graf, S., Luschgy, H.: Foundations of Quantization for Probability Distributions. Lecture Notes in Mathematics, vol. 1730. Springer, Berlin (2000)Google Scholar
  8. 8.
    Gruber, P.M.: Optimum quantization and its applications. Adv. Math. 186, 456–497 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hall, P.: On the coverage of k-dimensional space by k-dimensional spheres. Ann. Probab. 13, 991–1002 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ho, Y., Cao, X.: Perturbation Analysis of Discrete Event Dynamic Systems. Kluwer Academic Publishers, Boston (1991)CrossRefzbMATHGoogle Scholar
  11. 11.
    Last, G.: Perturbation analysis of Poisson processes. Bernoulli 20 (2), 486–513 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Last, G.: Stochastic analysis for Poisson processes. In: Peccati, G., Reitzner, M. (eds.) Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Ito Chaos Expansions and Stochastic Geometry. Bocconi & Springer Series, vol. 7, pp. 1–36. Springer, Cham (2016)Google Scholar
  13. 13.
    Margulis, G.A.: Probabilistic characteristics of graphs with large connectivity. Probl. Pereda. Inf. 10 (2), 101–108 (1974) (in Russian)MathSciNetzbMATHGoogle Scholar
  14. 14.
    McClure, D.E., Vitale, R.A.: Polygonal approximation of plane convex bodies. J. Math. Anal. Appl. 51, 326–358 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Molchanov, I.S.: Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester (1997)zbMATHGoogle Scholar
  16. 16.
    Molchanov, I., Tontchev, N.: Optimal approximation and quantisation. J. Math. Anal. Appl. 325, 1410–1429 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Molchanov, I., Tontchev, N.: Optimal Poisson quantisation. Stat. Probab. Lett. 77, 1123–1132 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Molchanov, I.S., Zuyev, S.A.: Variational analysis of functionals of a Poisson process. Tech. Rep. RR-3302, INRIA, Sophia-Antipolis ftp://ftp.inria.fr/INRIA/publication/RR/RR-3302.ps.gz (1997)
  19. 19.
    Molchanov, I., Zuyev, S.: Tangent sets in the space of measures: with applications to variational analysis. J. Math. Anal. Appl. 249, 539–552 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Molchanov, I., Zuyev, S.: Variational analysis of functionals of Poisson processes. Math. Oper. Res. 25 (3), 485–508 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Molchanov, I., Zuyev, S.: Variational calculus in space of measures and optimal design. In: Atkinson, A., Bogacka, B., Zhigljavsky, A. (eds.) Optimum Experimental Design: Prospects for the New Millennium, pp. 79–90. Kluwer, Dordrecht (2000)Google Scholar
  22. 22.
    Molchanov, I., Zuyev, S.: Steepest descent algorithms in space of measures. Stat. Comput. 12, 115–123 (2002)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Molchanov, I., Zuyev, S.: Optimisation in space of measures and optimal design. ESAIM: Probab. Stat. 8, 12–24 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Molchanov, I., Chiu, S., Zuyev, S.: Design of inhomogeneous materials with given structural properties. Phys. Rev. E 62, 4544–4552 (2000)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Molchanov, I., van Lieshaut, M., Zuyev, S.: Clustering methods based on variational analysis in the space of measures. Biometrika 88 (4), 1021–1033 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. Wiley, Chichester (2000)CrossRefzbMATHGoogle Scholar
  27. 27.
    Penrose, M.D., Yukich, J.E.: Weak laws in geometric probability. Ann. Appl. Probab. 13, 277–303 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Polak, E.: Optimization: Algorithms and Consistent Approximations. Springer, New York (1997)CrossRefzbMATHGoogle Scholar
  29. 29.
    Röckner, M.: Stochastic analysis on configuration spaces: basic ideas and recent results. In: New Directions in Dirichlet Forms. AMS/IP Studies in Advanced Mathematics, vol. 8, pp. 157–231. American Mathematical Society, Providence (1998)Google Scholar
  30. 30.
    Rubinstein, R., Croese, D.: Simulation and the Monte-Carlo Method. Wiley, New York (2008)Google Scholar
  31. 31.
    Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 56, 229–237 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Schneider, R.: Random approximations of convex sets. J. Microsc. 151, 211–227 (1988)CrossRefzbMATHGoogle Scholar
  33. 33.
    Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and its Applications, 2nd edn. Wiley, Chichester (1995)zbMATHGoogle Scholar
  34. 34.
    Zuyev, S.A.: Russo’s formula for the Poisson point processes and its applications. Diskret. Mate. 4 (3), 149–160 (1992) (In Russian). English translation: Discrete Math. Appl. 3, 63–73 (1993)Google Scholar
  35. 35.
    Zuyev, S.: Stopping sets: gamma-type results and hitting properties. Adv. Appl. Probab. 31, 355–366 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute of Mathematical Statistics and Actuarial ScienceUniversity of BernBernSwitzerland
  2. 2.Department of Mathematical SciencesChalmers University of Technology and University of GothenburgGothenburgSweden

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