1 / 56

A scale invariant probabilistic model based on Leibniz- like pyramids

A scale invariant probabilistic model based on Leibniz- like pyramids. Antonio Rodríguez 1,2. 1 Dpto. Matemática Aplicada y Estadística. Universidad Politécnica de Madrid 2 G rupo I nterdisciplinar de S istemas C omplejos. Outline. One -dimensional model .

damon
Download Presentation

A scale invariant probabilistic model based on Leibniz- like pyramids

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. A scaleinvariantprobabilisticmodelbasedon Leibniz-likepyramids Antonio Rodríguez1,2 1Dpto. Matemática Aplicada y Estadística. Universidad Politécnica de Madrid 2Grupo Interdisciplinar de Sistemas Complejos

  2. Outline • One-dimensional model. • Scaleinvarianttriangles. • q-entropy. • Two-dimensional model. • Scaleinvarianttetrahedrons. • Conditional and marginal distributions. • Generalizationtoarbitrarydimension. • Conclusions.

  3. q-gaussianity scaleinvariance extensivity

  4. scaleinvariance marginal probabilitydistribution variables jointprobabilitydistribution N-1 variables joint N N-1 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  5. x1 0 1 p 1-p One-dimensional model. Ndistinguisable 1d-binary independent variables N=1 1 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  6. x2 0 1 p2 1 p(1-p) 0 p(1-p) (1-p)2 One-dimensional model. Ndistinguisable 1d-binary independent variables N=2 x1 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  7. One-dimensional model. Ndistinguisable 1d-binary independent variables N=2 x1 x2 0 1 p2 p 1 p(1-p) 0 p(1-p) (1-p)2 1-p p 1-p A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  8. p2(1-p) p(1-p)2 p(1-p)2 (1-p)3 p2 p(1-p) p(1-p) (1-p)2 p3 p2(1-p) p2(1-p) p(1-p)2 One-dimensional model. N=3 x3=0 x3=1 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  9. p2(1-p) p(1-p)2 p(1-p)2 p2(1-p) p2 p(1-p) p(1-p) One-dimensional model. N=3 (1-p)3 1 p 1-p N=0 N=1 N=2 p3 p2(1-p) p(1-p)2 (1-p)2 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  10. + + + p2 p(1-p) One-dimensional model. Leibniz rule 1 p 1-p N=0 N=1 N=2 (1-p)2 p2(1-p) p(1-p)2 p3 (1-p)3 N=3 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  11. p2 p(1-p) One-dimensional model. CLT Binomialdistribution Gaussian 1 p 1-p 1 N=0 N=1 N=2 1 1 (1-p)2 1 1 2 p2(1-p) p(1-p)2 1 3 p3 1 (1-p)3 3 N=3 Pascal triangle A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  12. p2 p(1-p) Scaleinvarianttriangles 1 p 1-p N=0 N=1 N=2 (1-p)2 p2(1-p) p(1-p)2 p3 (1-p)3 N=3 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  13. Scaleinvarianttriangles Leibniz triangle N=0 N=1 N=2 N=3 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  14. Scaleinvarianttriangles Leibniz triangle N=0 N=1 N=2 N=3 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  15. Scaleinvarianttriangles Leibniz triangle N=0 N=1 N=2 N=3 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  16. Scaleinvarianttriangles N=0 N=1 N=2 N=3 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  17. Scaleinvarianttriangles A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  18. Scaleinvarianttriangles A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  19. Scaleinvarianttriangles A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  20. Scaleinvarianttriangles A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

  21. Scaleinvarianttriangles R. Hanel, S. Thurner and C. Tsallis. Eur. Phys. J. B 72, 263 (2009)

  22. q-gaussianity scaleinvariance ? ? extensivity for

  23. q-entropy

  24. q-gaussianity scaleinvariance ? ? extensivity for

  25. Outline • One-dimensional model. • Scaleinvarianttriangles. • q-entropy. • Two-dimensional model. • Scaleinvarianttetrahedrons. • Conditional and marginal distributions. • Generalization to arbitrary dimension. • Conclusions

  26. Two dimensional model Ndistinguisableindependent variables 2d-ternary N=1 (x1 , y1) (1 ,0) (0 , 1) (0 ,0) p q 1-p-q 1 A. Rodríguez and C. Tsallis, J. Math. Phys53, 023302 (2012)

  27. Two dimensional model Ndistinguisable 2d-ternary independent variables N=2 (x1 , y1) (1 ,0) (0 ,1) • (0 ,0) (x2 , y2) p p2 • p(1-p-q) (1 ,0) pq pq q2 (0 ,1) • q(1-p-q) q • (0 ,0) p(1-p-q) 1-p-q (1-p-q) 2 q(1-p-q) p q 1-p-q A. Rodríguez and C. Tsallis, J. Math. Phys53, 023302 (2012)

  28. 1 N=0 N=2 N=1 p2 p3 pq q2 p2q p2(1-p-q) N=3 p(1-p-q) (1-p-q) 2 q(1-p-q) pq2 p(1-p-q) 2 pq(1-p-q) q(1-p-q) 2 q3 q2(1-p-q) p (1-p-q) 3 q 1-p-q

  29. 1 Generalized Leibniz rule N=0 p + + N=1 1-p-q q + p2 pq p(1-p-q) N=2 + + + q2 q(1-p-q) (1-p-q) 2 p3 p2q p2(1-p-q) N=3 + + pq2 p(1-p-q) 2 pq(1-p-q) + + + + q(1-p-q) 2 q3 q2(1-p-q) (1-p-q) 3

  30. CLT Trinomialdistribution 2d-Gaussian 1 1 Pascal pyramid N=0 p 1 N=1 1-p-q q 1 1 p2 1 pq 2 2 p(1-p-q) N=2 2 1 1 q2 q(1-p-q) (1-p-q) 2 1 p3 p2q 3 3 p2(1-p-q) N=3 pq2 6 3 3 p(1-p-q) 2 pq(1-p-q) 1 3 3 q(1-p-q) 2 q3 q2(1-p-q) (1-p-q) 3 1

  31. 1 N=0 p N=1 1-p-q q p2 pq p(1-p-q) N=2 q2 q(1-p-q) (1-p-q) 2 p3 p2q p2(1-p-q) N=3 pq2 p(1-p-q) 2 pq(1-p-q) q(1-p-q) 2 q3 q2(1-p-q) (1-p-q) 3

  32. N=0 N=1 Leibniz-like pyramid N=2 N=3

  33. N=0 N=1 Leibniz-like pyramid N=2 N=3

  34. N=0 N=1 Leibniz pyramid N=2 N=3

  35. N=0 N=1 N=2 N=3

  36. Scaleinvariantpyramids

  37. Scaleinvariantpyramids ? 2D q-Gaussian

  38. Outline • One-dimensional model. • Scaleinvarianttriangles. • q-entropy. • Two-dimensional model. • Scaleinvarianttetrahedrons. • Conditional and marginal distributions. • Generalization to arbitrary dimension. • Conclusions

  39. Conditionaldistributions

  40. Conditionaldistributions

  41. N=3 Marginal distributions

  42. Marginal distributions • Thethreedirectionsyieldidenticalnonsymmetricscale-invariantdistributions.

  43. Marginal distributions

  44. Marginal distributions • Thedirectionyields a symmetric nonscale-invariantdistribution

  45. Jointdistribution

  46. q-gaussianity scaleinvariance ? extensivity

  47. q-entropy

  48. Outline • One-dimensional model. • Scaleinvarianttriangles. • q-entropy. • Two-dimensional model. • Scaleinvarianttetrahedrons. • Conditional and marginal distributions. • Generalization to arbitrary dimension. • Conclusions

  49. Scaleinvarianthyperpyramids Ndistinguisableindependent variables 3d-cuaternary

More Related