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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 .

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A scale invariant probabilistic model based on Leibniz- like pyramids

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  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

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