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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 scaleinvariantprobabilisticmodelbasedon Leibniz-likepyramids Antonio Rodríguez1,2 1Dpto. Matemática Aplicada y Estadística. Universidad Politécnica de Madrid 2Grupo Interdisciplinar de Sistemas Complejos
Outline • One-dimensional model. • Scaleinvarianttriangles. • q-entropy. • Two-dimensional model. • Scaleinvarianttetrahedrons. • Conditional and marginal distributions. • Generalizationtoarbitrarydimension. • Conclusions.
q-gaussianity scaleinvariance extensivity
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)
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)
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)
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)
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)
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)
+ + + 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)
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)
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)
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)
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)
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)
Scaleinvarianttriangles N=0 N=1 N=2 N=3 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
Scaleinvarianttriangles A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
Scaleinvarianttriangles A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
Scaleinvarianttriangles A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
Scaleinvarianttriangles A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
Scaleinvarianttriangles R. Hanel, S. Thurner and C. Tsallis. Eur. Phys. J. B 72, 263 (2009)
q-gaussianity scaleinvariance ? ? extensivity for
q-gaussianity scaleinvariance ? ? extensivity for
Outline • One-dimensional model. • Scaleinvarianttriangles. • q-entropy. • Two-dimensional model. • Scaleinvarianttetrahedrons. • Conditional and marginal distributions. • Generalization to arbitrary dimension. • Conclusions
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)
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)
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
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
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
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
N=0 N=1 Leibniz-like pyramid N=2 N=3
N=0 N=1 Leibniz-like pyramid N=2 N=3
N=0 N=1 Leibniz pyramid N=2 N=3
N=0 N=1 N=2 N=3
Scaleinvariantpyramids ? 2D q-Gaussian
Outline • One-dimensional model. • Scaleinvarianttriangles. • q-entropy. • Two-dimensional model. • Scaleinvarianttetrahedrons. • Conditional and marginal distributions. • Generalization to arbitrary dimension. • Conclusions
N=3 Marginal distributions
Marginal distributions • Thethreedirectionsyieldidenticalnonsymmetricscale-invariantdistributions.
Marginal distributions • Thedirectionyields a symmetric nonscale-invariantdistribution
q-gaussianity scaleinvariance ? extensivity
Outline • One-dimensional model. • Scaleinvarianttriangles. • q-entropy. • Two-dimensional model. • Scaleinvarianttetrahedrons. • Conditional and marginal distributions. • Generalization to arbitrary dimension. • Conclusions
Scaleinvarianthyperpyramids Ndistinguisableindependent variables 3d-cuaternary