a scale invariant probabilistic model based on leibniz-like pyramids antonio rodríguez 1,2 1 dpto....
TRANSCRIPT
A scale invariant probabilistic model
based on Leibniz-like pyramids
Antonio Rodríguez1,2
1Dpto. Matemática Aplicada y Estadística. Universidad Politécnica de Madrid2Grupo Interdisciplinar de Sistemas Complejos
Outline One-dimensional model.
Scale invariant triangles. q-entropy.
Two-dimensional model. Scale invariant tetrahedrons. Conditional and marginal distributions.
Generalization to arbitrary dimension.
Conclusions.
scale invariance
extensivity
q-gaussiani
ty
1 2 1 1 2 1, , , , , ,N N N N Np x x x dx p x x x
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
joint probability distributionN-1 variables
probability distributionvariables
marginal N-1
joint N
scale invariance
One-dimensional model.
x1
p 1-p
1 0
N=1
N distinguisable 1d-binary independent variables
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
1 2 NX X X X
1
One-dimensional model.
x1
p2 p (1-p)
p (1-p) (1-p)2
1 0
1
0
x2
N=2
N distinguisable 1d-binary independent variables
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
One-dimensional model.
x1
p2 p (1-p)
p (1-p) (1-p)2
1 0
1
0
x2
p
1-p
p 1-p
N=2
N distinguisable 1d-binary independent variables
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
p3 p2(1-p)
p2(1-p) p(1-p)2
N=3
One-dimensional model.
p2 p(1-p)
p(1-p) (1-p)2
p2(1-p) p(1-p)2
p(1-p)2 (1-p)3
x3=1
x3=0
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
p3
p2(1-p) p(1-p)2
p2(1-p)
N=3
One-dimensional model.
p2 p(1-p)
(1-p)2p(1-p)
(1-p)3
p2(1-p) p(1-p)2
p(1-p)2 1 p 1-p
N=0
N=1 N=2
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
,N nr
p3 p2(1-p) p(1-p)2N=3
One-dimensional model.
p2 p(1-p) (1-p)2
(1-p)3
1 p 1-p
N=0
N=1 N=2
,N nr
+
+
+
, , 1 1,N n N n N nr r r Leibniz rule
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
p3 p2(1-p) p(1-p)2N=3
One-dimensional model.
p2 p(1-p) (1-p)2
(1-p)3
1 p 1-p
N=0
N=1 N=2
,N nr,N n
N
np
111
2 11
3 31 1
Pascal triangle
CLT
N
2
1
2 (1- )
2 (1 )( )Np p
x Np
Np pp x e
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
Binomial distribution Gaussian
p3 p2(1-p) p(1-p)2N=3
Scale invariant triangles
p2 p(1-p) (1-p)2
(1-p)3
1 p 1-p
N=0
N=1 N=2
,N nr
0,0r
1,0r 1,1r
2,0r 2,1r 2,2r
3,0r3,1r 3,2r
3,3r
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
N=3
N=0
N=1 N=2
0,0r
1,0r 1,1r
2,0r 2,1r 2,2r
3,0r3,1r 3,2r
3,3r
1
1
121
12
1
4
1
4
1
3
1
31
6
1
2
1
2
Scale invariant triangles
Leibniz triangle
1
(1),
1
1N n
Nr
nN
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
N=3
N=0
N=1 N=2
1
1
121
12
1
4
1
4
1
3
1
31
6
1
2
1
2
(1),
1
1N nrN
Scale invariant triangles
Leibniz triangle
(1),N n
Np
n
N
2xe
1
3 1
41
4
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
N=3
N=0
N=1 N=2
1
1
121
12
1
4
1
4
1
3
1
31
6
1
2
(2),N nr
Scale invariant triangles
Leibniz triangle
1
2
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
1
21
2
1
1
53
10
3
101
51
5
1
101
10
(1),N nr
N=3
N=0
N=1 N=2
1
1
10
1
51
5
3
10
3
101
5
1
2
(2),N nr
1
10
2
7
3
142
7 5
28
5
283
28
3
28
Scale invariant triangles
1
2
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
11
2
1
2
(3),N nr
Scale invariant triangles
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
(1)2( 1), 1( )
, (1)2( 1), 1
N nN n
rr
r
; 1, 2,
( ),N nr ( )
,N n
Np
n
Scale invariant triangles1
2N
N
n
2 1
4(2 1)i jX X
0
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
( ),N nr ( )
,N n
Np
n
N
lim
21; ( 1)
1q
Scale invariant triangles
12 1
2
nx
N
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
2
lim lim( ) x
q qq x C e
( ),N nr ( )
,N n
Np
n
N
2
lim lim( ) x
q qq x C e
lim 0q lim 1/ 2q lim 3/ 4q
lim 8 / 9q
lim 1q 500N
Scale invariant triangles
A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)
( , )
( , );
B n N n
B
Scale invariant triangles
R. Hanel, S. Thurner and C. Tsallis. Eur. Phys. J. B 72, 263 (2009)
0
lim
21; ( 1)
1q
12 1
2
nx
N
lim
2 31; ( 0)
2 1q
/ 2
1/ 2( )
n Nx
n N n
(1)2( 1), 1
(1)2( 1), 1
N nr
r
; 1, 2, ( ),N nr
scale invariance
extensivity
q-gaussiani
ty
?
for 1q
?
q-entropy ( )
,01
( )1
qN
N nn
q
Nr
nS N k
q
ent 1q
scale invariance
extensivity
q-gaussiani
ty
?
for 1q
?
Outline One-dimensional model.
Scale invariant triangles. q-entropy.
Two-dimensional model. Scale invariant tetrahedrons. Conditional and marginal distributions.
Generalization to arbitrary dimension.
Conclusions
Two dimensional model
N=1
N distinguisable independent variables
1
2d-ternary
(x1 , y1)
p q
(1 , 0) (0 , 1)
1-p-q
(0 , 0)
1 1 2 2, , , ,N NX Y X Y X Y X Y
A. Rodríguez and C. Tsallis, J. Math. Phys 53, 023302 (2012)
Two dimensional model
N=2
N distinguisable 2d-ternary independent variables
(x1 , y1)
p2 pq
pq q2
(1 , 0) (0 , 1)
(1 , 0)
(0 , 1)
(x2 , y2)
(0 , 0)
(0 , 0)
p(1-p-q) q(1-p-q) (1-p-q) 2
p(1-p-q)
q(1-p-q)
p q 1-p-q
p
q 1-p-q
A. Rodríguez and C. Tsallis, J. Math. Phys 53, 023302 (2012)
N=2
N=0
N=1
, ,N n mr
1-p-qp q
p2
pq q2
p(1-p-q) (1-p-q) 2N=3
1
q3 (1-p-q) 3
p3
p2q pq2
p2(1-p-q)
p(1-p-q) 2
q2(1-p-q) q (1-p-q) 2
pq (1-p-q) q(1-p-q)
N=2
p2
1N=0
N=1
, ,N n mr
N=3
p
1-p-qq
p(1-p-q)
(1-p-q) 2q(1-p-q)q2
pq
q3
p2q pq2
p2(1-p-q)
q2(1-p-q) q (1-p-q) 2
pq (1-p-q)
p3
p(1-p-q) 2
(1-p-q) 3
++ +
+
+
+
+
+
++ ++
, , , 1, 1 , , 1 1, , 1N n m N n m N n m N n mr r r r
Generalized Leibniz rule
N=2
p2
1N=0
N=1
, ,N n mr
q2
q
q3
N=3
, , ,N n m
N
n mp
Pascal pyramidTrinomial distribution
1
1
1
CLT
N
111 2
2( , )
TX Xp x y e
2d-Gaussian
;,
N N n m
n m n m m
1-p-q1
p(1-p-q)
q(1-p-q) (1-p-q) 21
pq
2
p3
p2(1-p-q)
p(1-p-q) 2
p2q pq2
q (1-p-q) 2 q2(1-p-q)
363
3pq (1-p-q)
1 3
2
p
1
2
1
33
1(1-p-q) 3
1
N=2
p2
p
1N=0
N=1
, ,N n mr
1-p-q
p(1-p-q)
(1-p-q) 2q(1-p-q)q2
pq
q
q3 (1-p-q) 3
p3
p2q pq2
p2(1-p-q)
p(1-p-q) 2
q2(1-p-q) q (1-p-q) 2
pq (1-p-q) N=3
1,0,0r
2,1,0r
1,1,0r
0,0,0r
2,0,1r2,0,0r
1,0,1r
2,0,2r 2,1,1r 2,2,0r
3,0,2r
3,0,0r
3,2,0r3,0,1r
3,1,1r3,0,3r 3,1,2r 3,2,1r
3,1,0r
3,3,0r
N=2
N=0
N=1
, ,N n mr
N=3
1,0,0r
2,1,0r
1,1,0r
0,0,0r
2,0,1r2,0,0r
1,0,1r
2,0,2r 2,1,1r
3,0,2r
3,0,0r
3,2,0r3,0,1r
3,1,1r3,0,3r 3,1,2r 3,2,1r
3,1,0r
3,3,0r
1
(1), ,
2
,( 1)( 2)N n m
Nr
n mN N
1
2,2,0r
1
3 1
3
1
3
1
6
1
6
1
12
1
61
12
1
12
1
10
1
101
10
1
301
30 1
30
1
30
1
30
1
301
60
Leibniz-like pyramid
N=2
N=0
N=1
, ,N n mr
N=3
1
(1), ,
2
,( 1)( 2)N n m
Nr
n mN N
11
3 1
3
1
3
1
6
1
6
1
12
1
61
12
1
12
1
10
1
101
10
1
301
30 1
30
1
30
1
30
1
301
60
1
6
1
61
6
1
10
1
10
1
10
1
10
1
10
1
10
1
101
10
(1) (1), , , ,
2
, ( 1)( 2)N n m N n m
Np r
n m N N
Leibniz-like pyramid
0 m n N
(1), ,N n mp
n
m
30N
N=2
N=0
N=1
N=3
11
3 1
3
1
3
1
6
1
6
1
12
1
61
12
1
12
1
101
10
1
301
30 1
30
1
30
1
30
1
301
60
1
10
Leibniz pyramid
(1), ,N n mr
111
3 1
3
1
3
(2), ,N n mr
(2), ,N n mp
n
m
50N
N=2
N=0
N=1
N=3
11
3 1
3
1
3
1
7
1
7
2
21
1
72
21
2
21
1
141
14
1
281
28 1
30
1
28
1
28
1
28
1
14
1
42
(2), ,N n mr
1
1
3 1
3
1
3
1
(3), ,N n mr
(3), ,N n mp
n
m
50N
( ), ,
( , ) ( 2 , )
( , ) ( , 2 )N n m
B n m B n m N n m
B Br
Scale invariant pyramids(1)
3( 1), 1, 1( ), , (1)
3( 1), 1, 1
N n mN n m
rr
r
; 1, 2, ; 0
( ) ( ), , , ,,N n m N n m
Np r
n m
(3), ,N n mr
2 2 22
9(3 1)i j i jX X X Y
0
1
, 3N
N
n m
( ), ,
( , ) ( 2 , )
( , ) ( , 2 )N n m
B n m B n m N n m
B Br
Scale invariant pyramids
; 0
( ) ( ), , , ,,N n m N n m
Np r
n m
N
2D q-Gaussian?
Outline One-dimensional model.
Scale invariant triangles. q-entropy.
Two-dimensional model. Scale invariant tetrahedrons. Conditional and marginal distributions.
Generalization to arbitrary dimension.
Conclusions
Conditional distributions
, /3, /3n m N N
( ), ,N n mp
n
m
3
Nn
Conditional distributions
( )
, ,3
NN n
p
n
2
4
5
3
( ) ( ), | ,N n m N m np p
lim
2
1q
300N
Marginal distributions
N=3
0n
1n
2n
3n
0m
1m
2m
3m
0n m
1n m
2n m
3n m
( ) ( ), ,,
0
N n
N n mN nm
p p
( ) ( ), ,,
0
N m
N n mN mn
p p
( ) ( ), ,,
0
n m
N k n m kN n mk
p p
( )3,0,2p ( )
3,2,0p
( )3,0,1p
( )3,1,2p ( )
3,2,1p
( )3,1,0p
( )3,0,0p
( )3,1,1p
( )3,0,3p ( )
3,3,0p
Marginal distributions The three directions yield
identical nonsymmetric scale-invariant distributions.100N
Marginal distributions
2n m 0n m 2n m
3n m 1n m 1n m 3n m
( ) ( ), ,,
0
n m
N k n m kN n mk
p p
( )3,2,0p
( )3,1,0p
( )3,0,0p
( )3,0,1p
( )3,0,2p ( )
3,1,1p
( )3,1,2p ( )
3,3,0p ( )3,2,1p ( )
3,0,3p
Marginal distributions The direction yields a symmetric non scale-invariant distribution
Joint distribution
2 20 10
scale invariance
extensivity
q-gaussiani
ty
?
q-entropy
( ), ,0
1,
( )1
qN
N n mn m
q
Nr
n mS N k
q
ent 1q
Outline One-dimensional model.
Scale invariant triangles. q-entropy.
Two-dimensional model. Scale invariant tetrahedrons. Conditional and marginal distributions.
Generalization to arbitrary dimension.
Conclusions
Scale invariant hyperpyramids
N distinguisable independent variables3d-cuaternary
N=2N=0 N=1
N=3
1
(1), , ,
6
, ,( 1)( 2)( 3)N n m l
Nr
n m lN N N
Leibniz-like hyperpyramid
1
60 1
601
60
1
60
1
60
1
120
1
60 1
120
1
20
1
201
20
1
601
60 1
60
1
60
1
60
1
601
120
1
120
1
20
1
20 1
201
20
1
10
1
10
1
20
1
10 1
201
20
1
10
1
4 1
41
4
1
41
(1)4( 1), 1, 1, 1( )
, , , (1)4( 1), 1, 1, 1
N n m lN n m l
rr
r
( ) ( ) ( ) ( ) ( ), , , , 1, 1, , , 1, 1 , , 1, 1, , 1,N n m l N n m l N n m l N n m l N n m lr r r r r
(d + 1)-sided dice d-dimensional variable
d-dimensional (d+1)-ary variable taking values
Leibniz hyperpyramid
( ) 1 2d NX X X X ��������������������������������������������������������
:iX
��������������
1 2, , , ,0.de e e��������������������������������������������������������
1 2
1
(1), , , ,
1 2
!, , ,( 1)( 2) ( )lN n n n
d
Ndr
n n nN N N d
Outline One-dimensional model.
Scale invariant triangles. q-entropy.
Two-dimensional model. Scale invariant tetrahedrons. Conditional and marginal distributions.
Generalization to arbitrary dimension.
Conclusions
Conclusions and future work
We have generalized to an arbitrary dimension a one-dimensional discrete probabilistic model which, for one dimension, yields q-gaussians in the thermodynamic limit.
Our two-dimensional model, though containing one-dimensional conditional distributions yielding q-gaussians doesn’t seem to yield bidimensional q-gaussians as limiting probability distributions for .
The case of binary variables is special !!! The formulation of a probabilistic model yielding
multidimensional q-gaussians in the thermmodynamic limit is still an open question.
The relationship between scale invariance, q-gaussianity and extensivity is still an open question.
N
Thanks!