Finite samples, scaling and non-stationarity
Khurom Kiyani and Sandra Chapman
CFSA
Sunday, 16 November 2008
1
The question
What effects does the the partitioning of
data into smaller segments have on the
estimate of the scaling exponents?
Sunday, 16 November 2008
2
The question
What effects does the the partitioning of
data into smaller segments have on the
estimate of the scaling exponents?
• On the way to answering these questions we will explore:
1. apparent or pseudo-nonstationarity,
2. limit theorems,
3. and the role of extreme values from heavy-tailed statistics.
Sunday, 16 November 2008
3
The method
Stationary Brownian motion (Gaussian) N=106
#!!
!
!#!!
!%!!
!'!!
!+!!
!"!!!
!"#!!
!"%!!
!
"
#
$
,-./
%
&
'
*
()"!
Partition time series i = 1 · · · L parts
Calculate scaling exponents in these L parts
Sunday, 16 November 2008
4
Test statistic and its scaling
increments
yi (t, τ ) = xi (t + τ ) − xi (t)
)!!
'!!
%!!
τ
12-3
#!!
!
!#!!
!%!!
!'!!
!
"
#
$
%
&
-./0
'
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)
*
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&
+,"!
"
#
$
%
&
-./0
'
(
)
*
"!
&
+,"!
'
%
+1-2!3
#
!
!#
!%
!'
!
Sunday, 16 November 2008
5
Test statistic and its scaling
increments
yi (t, τ ) = xi (t + τ ) − xi (t)
)!!
'!!
%!!
τ
12-3
#!!
!
!#!!
!%!!
!'!!
!
"
#
$
%
&
-./0
'
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*
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&
+,"!
"
#
$
%
&
-./0
'
(
)
*
"!
&
+,"!
'
%
+1-2!3
#
second moment
N
!
1
2
2
Mi (τ ) =
(yi )j
N j=1
!
!#
!%
!'
!
Sunday, 16 November 2008
6
Test statistic and its scaling
increments
yi (t, τ ) = xi (t + τ ) − xi (t)
)!!
'!!
%!!
τ
12-3
#!!
!
!#!!
!%!!
!'!!
!
"
#
$
%
&
-./0
'
(
)
*
"!
&
+,"!
'
%
+1-2!3
#
second moment
N
!
1
2
2
Mi (τ ) =
(yi )j
N j=1
moment scaling
!
!#
2
Mi (τ )
!%
!'
!
"
#
$
%
&
-./0
'
(
)
*
"!
&
+,"!
2
log Mi (τ )
=
=
2
ζi (2)
Mi (1)τ
2
log Mi (1)
+ ζi (2) log τ
via ordinary least-squares regression
Sunday, 16 November 2008
7
Test statistic and its scaling
increments
yi (t, τ ) = xi (t + τ ) − xi (t)
)!!
'!!
%!!
τ
12-3
#!!
!
!#!!
!%!!
!'!!
!
"
#
$
%
We also use the iterative
conditioning technique
to handle
second moment
heavy-tailed statistics; see:
N
!
&
-./0
'
(
)
*
"!
&
+,"!
1
2
=
(yi )j
N j=1
K. Kiyani, S. C. Chapman and B. Hnat,
Phys. Rev. Emoment
74, 051122
(2006).
scaling
2
Mi (τ )
'
%
#
!
!#
2
Mi (τ )
!%
!'
!
"
#
$
%
&
-./0
'
(
)
*
"!
&
+,"!
2
log Mi (τ )
=
=
2
ζi (2)
Mi (1)τ
2
log Mi (1)
+ ζi (2) log τ
via ordinary least-squares regression
Sunday, 16 November 2008
8
The model time series
Lévy process
)!!
'!!
•
•
•
•
%!!
12-3
#!!
!
Infinite variance
Stationary increments
Heavy-tailed
Self-similar -- monofractal
!#!!
!%!!
!'!!
!
Stationary Levy process with !=1.4 (N=106)
"
#
$
%
&
-./0
'
(
)
*
"!
&
+,"!
"%!!!
"#!!!
Standard Brownian motion
"!!!!
+!!!
Finite variance
All moments finite
Stationary increments
Self-similar -- monofractal
'!!!
0
•
•
•
•
%!!!
#!!!
!
!#!!!
!%!!!
!
Sunday, 16 November 2008
"
#
$
,-./
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9
The model time series
#
#
p-model
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./&0$$'&#*'#1(2-2#%$*,'%3*4'#2&(4)
5&()'#6*!*,'%3*%'-27*!89:;
'(&!
•
•
•
•
"
!
.
!"
!#
Finite variance
Nonstationary increments
Heavy-tailed
Self-similar -- multifractal
!$
&#!
!%
"
#
*+,-
$
%
Nonstationary Brownian motion
• Finite variance
• Nonstationary increments
PDF broader with time
• Self-similar -- monofractal
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)
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-
1!23456-+*25-05(*50
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%!
$!
#!
"!
!-
Sunday, 16 November 2008
10
Pseudo-nonstationarity
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&!!
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%!
$!
#!
"!
!8
7
5
N=10
6
N=104
M2i (!=1)
5
4
3
2
1
0
Sunday, 16 November 2008
11
Pseudo-nonstationarity
&#!
-
0.2
&"!
%!
0.15
!i(2)
'()*+('(,-./*+0
&!!
1!23456-+*25-05(*50
718!9$:-;8&!$
$!
4
<!2> for N=10
0.1
6
<!2> for N=10
<!2> for N=105
#!
0.05
!2 for N=104
"!
5
!2 for N=10
0
!8
7
0.24
0.22
5
N=10
6
0.2
N=104
0.18
4
3
2
!i(2)
M2i (!=1)
5
0.16
0.14
0.12
0.1
1
0.08
0
0.06
0
Sunday, 16 November 2008
2
4
time
6
8
10
5
x 10
12
Pseudo-nonstationarity
&#!
-
0.2
&"!
0.15
%!
!i(2)
'()*+('(,-./*+0
&!!
1!23456-+*25-05(*50
718!9$:-;8&!$
$!
4
<!2> for N=10
0.1
6
<!2> for N=10
<!2> for N=105
#!
0.05
!2 for N=104
"!
5
!2 for N=10
0
!8
7
0.24
0.22
5
N=10
6
0.2
N=104
0.18
!i(2)
M2i (!=1)
5
4
3
0.16
0.14
0.12
2
0.1
1
0.08
0
0.06
0
2 points
Sunday, 16 November 2008
2
4
time
6
8
10
5
x 10
13
Quantifying scatter/variation of ζ(2) with N
!*
%
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Standard Brownian
+
$
6-78!8"9:59
(
)
#
*
"
&
!(
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"
#
$
,-./01(2341(5
%
&!
)
'(&!
Power law?
Sunday, 16 November 2008
14
Quantifying scatter/variation of ζ(2) with N
!*
%
'(&!
Standard Brownian
+
$
6-78!8"9:59
(
)
!!
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.
#
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&
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#
$
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Power law?
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.
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Answer: yes
Sunday, 16 November 2008
$
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%
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!"
&
!"
15
Quantifying scatter/variation of ζ(2) with N
−1
a.)
!*
%
'(&!
−1.5
(
−2
Var(ζ(2))
+
10
)
−3
−3.5
log
6-78!8"9:59
$
−2.5
#
Standard Brownian
p−model (p=0.6)
fBm H=0.8
Non−stationary Brownian
Cyclic−stationary Brownian
−ve unit gradient
−4
−4.5
*
−5
"
−5.5
3
&
!(
b.)
!
"
#
$
,-./01(2341(5
%
&!
)
'(&!
3.5
4
4.5
log10 N
5
5.5
5
5.5
1
0
log10 Var(ζ(2))
−1
−2
−3
Levy α=1.2
Levy α=1.4
Levy α=1.6
Levy α=1.8
LFSM H=0.44 α=1.4
LFSM H=0.9 α=1.6
−ve unit gradient
−4
−5
−6
−7
Sunday, 16 November 2008
3
3.5
4
4.5
log10 N
16
Quantifying error on ζ(2)
!*
%
'(&!
(
+
6-78!8"9:59
$
)
!!
!"
.
#
*
!'
!"
&
!(
!
"
#
$
,-./01(2341(5
!
V ar(ζ(2))
! 0.05
ζ(2)|L=1
3)45!5'6726
"
!#
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)
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!"
!%
!"
!&
!"
.
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$
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%
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!"
&
!"
V ar(ζ(2)) ! (0.05ζ(2)|L=1 )
2
Sunday, 16 November 2008
17
Quantifying error on ζ(2)
p-model N~105
Var(ζ(2))
−2
10
fBm, stationary & nonstationary
Brownian N~103;
−1.5
−2.5
−3
−3.5
log
finite variance
−1
a.)
Standard Brownian
p−model (p=0.6)
fBm H=0.8
Non−stationary Brownian
Cyclic−stationary Brownian
−ve unit gradient
−4
−4.5
−5
−5.5
3
Lévy α=1.8 & LFSM
(H=0.9,α=1.6) N~103;
Lévy α=1.6
N~104;
Lévy α=1.4, 1.2 & LFSM
(H=0.44,α=1.4) N~105;
Sunday, 16 November 2008
4
4.5
log10 N
5
5.5
5
5.5
1
0
−1
log10 Var(ζ(2))
infinite variance
b.)
3.5
−2
−3
Levy α=1.2
Levy α=1.4
Levy α=1.6
Levy α=1.8
LFSM H=0.44 α=1.4
LFSM H=0.9 α=1.6
−ve unit gradient
−4
−5
−6
−7
3
3.5
4
4.5
log10 N
18
Quantifying error on V ar(ζ(2))
b.)
a.)
0
−2
Var(ζ(2))
−3
10
−4
log
log10 Var(ζ(2))
−1
−5
−6
−2
−3
−4
3
3.5
4
4.5
5
−5
5.5
3
3.5
4
4.5
log10 N
log10 N
c.) −1.5
d.)
5
5.5
0
−2
−1
log10 Var(ζ(2))
log
10
Var(ζ(2))
−2.5
−3
−3.5
−4
−4.5
−5
−5.5
Sunday, 16 November 2008
−3
(Mean) Standard Brownian
linear fit ( y = −1.06*x + 1.39 )
3
3.5
4
4.5
log10 N
−2
5
5.5
−4
linear fit ( y = −1.4*x + 4.35 )
(Mean) Levy α=1.4
3
3.5
4
4.5
log N
5
5.5
6
10
19
Real-world data
0
log10 Var(ζ(2))
−1
−2
−3
−4
Levy α=1.4
ACE 2000 B2
WIND 1996 |V|
WIND 1996 Bz
Standard Brownian
−ve unit gradient
−5
2.5
3
3.5
4
4.5
log10 N
5
5.5
N~105 introduces an error of ~12%
Sunday, 16 November 2008
20
Limit theorems
‘All epistemological value of the theory of probability is based
on this: that large scale random phenomena in their collective
action create strict, non-random regularity.’
(Gnedenko and Kolmogorov, Limit Distributions for Sums of Independent
Random Variables)
Sunday, 16 November 2008
21
Limit theorems
Central Limit Theorem (De Moivre, Laplace, Lyapunov)
SN
N
!
1
yi
=√
N i=1
Sunday, 16 November 2008
lim SN → Gaussian
N →∞
22
Limit theorems
Central Limit Theorem (De Moivre, Laplace, Lyapunov)
SN
N
!
1
yi
=√
N i=1
lim SN → Gaussian
N →∞
Generalized Central Limit Theorem (Lévy)
SN =
Sunday, 16 November 2008
1
N 1/α
N
!
i=1
yi
lim SN → Lévy
N →∞
23
Limit theorems
Central Limit Theorem (De Moivre, Laplace, Lyapunov)
SN
N
!
1
yi
=√
N i=1
lim SN → Gaussian
N →∞
Generalized Central Limit Theorem (Lévy)
SN =
1
N 1/α
N
!
yi
i=1
and many others!
Sunday, 16 November 2008
lim SN → Lévy
N →∞
Stable processes
24
Limit theorems and self-similar processes
SN
N
!
1
yi
=√
N i=1
SN =
Sunday, 16 November 2008
1
N 1/α
N
!
i=1
yi
Coarse-graining/averaging
Scaling
study of limit theorems and
stable processes have a very
profound link to self-similar
processes
25
!(2)
H(!(2))
3
2.5
2.5
!(2)
!(2)
4
1.4
Statistics of ζ(2)
b ii.)
6
1.6
1.2
2
1
0
2
0
2
4
6
time
0.8
H(!(2))
b i.)
!(2)
7
x 10
1
0.4
0
1.5
Levy "=1.4 N=1000
Gumbel MLE
(!=1.15,#=0.25)
0.6
2
1.5
2
4
6
time
7
x 10
Levy "=1.4 N=10,000
Gumbel MLE
(!=1.32,#=0.13)
1
0.5
0.2
1.5
1
1.5
!(2)
7
2
4
6
time
1
1.1
!(2)
1.2
1.3
#!
'
.-#!
7
x 10
!
!"#
!"$
!"%
!"&
!*$+
2.5
!"'
!"(
2.5
!(2)
3
3.5
2
,
"
0
2
4
6
time
7
x 10
Levy "=1.4 N=10,000
Gumbel MLE
(!=1.32,#=0.13)
1
0
4
!"(
)
!&(
!
#%
!
'
/012
#!
%
'
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3!14526-*37!"(+-87#!!!
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d i.)
#
!)#*
$
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lim
!"#
!"#$
N →∞
!"%
!"%$
!*%+
!"&$
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d ii.)
1
0.5
!
0
1.5
"&'
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1.6
456,#!!!,7 ,8,",9!!
8:;2<=,>?6
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#
$
1.8
1.9
8
x 10
ACE 2000 B2 N " 5000
Normal MLE
(!=0.78, #=0.23)
0.5
%
1.7
time
1
#
!
!"&
0
0.5
1
!)#*
1.5
2
!(2)
√
#!
$
.-#!
N (ζ(2) − ζ̂(2)) ∼ N (0, σ 2 )
!"&
!"&$
!"'
via ordinary least-squares regression
1.5
2
"
1
0.5
!
1.5
"&'
"&0123
"&(
"&.
(
/,"!
!(2))
!"
Sunday, 16 November 2008
!
d ii.)
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!-
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!*$+
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(
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)
%
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#!
#
!
$
1.5
!&'
!,
#!
.-#!
!"%$
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Theorem:
&
2.5
-
#'
!!"$
2
#(
,*!*%++
!*$+
'
1.5
c ii.)
!"&
(
1
$
/012
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2
!&%
!(2)
-
!
)
!-
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"
1.5
!*%+
2
#!
'
1.5
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1.5
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(
#
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1
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%
0.5
0
,*!*$++
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"&#
0.2
!)#**
!
3!14526-*37!"(+-87#!!!
8491:6-;<=
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d i.)
3
1
Levy "=1.4 N=1000
Gumbel MLE
(!=1.15,#=0.25)
0.4
!&(
#%
H(!(2))
0
0.6
"
!
$
!-
0.9
!(2)
!(2)
0
0.8
c i.)
#'
&
2.5
!"%$
#(
!"
1
2
-
c ii.)
!"$
!!"$
1.5
!(2)
#
2
2
1
%
+)!)#**
H(!(2))
6
x 10
Standard Brownian N=10,000
Normal MLE
(!=0.99, "=0.04)
4
1.2
4
6
2.5
1.4
2
time
b ii.)
6
1.6
'
0
0
2
H(!(2))
b i.)
1
0.9
2
0.5
0
(
4
Standard Brownian N=1000
Normal MLE
(!=0.974, "=0.127)
0
4
!(2)
7
)
1
8
6
x 10
3.5
!"(
!)#*
4
time
H(!(2))
H(!(2))
2
3
,*!*%++
10
0
2
2.5
!(2)
!"&
2.5
0.5
2
!*$+
1
!(2)
3
1.1
12
1.5
-
c i.)
,*!*$++
a ii.)
1.5
!(2)
a i.)
3.5
1
!*%+
0
0
1.6
1.7
time
1.8
1.9
8
x 10
26
Statistics of ζ(2)
7
#%
!!"$
!
'
/012
&
#!
'
.-#!
3!14526-*37!"(+-87#!!!
8491:6-;<=
*!7!"#(>-"7!"!)+
$
!-
0.9
1
1.1
!(2)
1.2
1.3
4
time
0.8
6
7
x 10
Levy "=1.4 N=1000
Gumbel MLE
(!=1.15,#=0.25)
0.6
0.4
!
!"#
!"$
!"%
!"&
!*$+
!"'
!"(
,
2.5
!)#*
"
2
"
1.5
0
2
4
time
6
7
x 10
Levy "=1.4 N=10,000
Gumbel MLE
(!=1.32,#=0.13)
1
!&(
2.5
!(2)
3
3.5
4
Theorem:
0
1
1.5
2
"&'
"&0123
"&(
"&.
(
/,"!
!&'
!,
!"#$
!"%
!"%$
!*%+
456,#!!!,7 ,8,",9!!
8:;2<=,>?6
)!@!&''A,#@!&$"*
!
"
#
$
lim
N →∞
!"&$
!"'
1
0
1.6
1.7
time
1.8
1.9
8
x 10
1
ACE 2000 B2 N " 5000
Normal MLE
(!=0.78, #=0.23)
0.5
%
!"&
0.5
0
0.5
1
!)#*
√
$
1.5
2
#
!&%
2.5
!(2)
!"#
1.5
!&#
2
#!
.-#!
3!14526-*37!"(+-87#!9!!!
84:1;6-<=>
*!7!"#$)9-"7!"!%'+
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1.5
1
$
/012
d ii.)
"&#
0.2
1.5
!
)
!-
!")
#
0.5
1
#!
'
H(!(2))
2
!"#
!"!$
%
d i.)
3
!(2)
!(2)
0
!"#$
(
!"
0
!"%
!*%+
!
#
2
2
1
0
#'
%
+)!)#**
H(!(2))
7
!"$
What about explaining finite N?
4
1.2
6
x 10
Standard Brownian N=10,000
Normal MLE
(!=0.99, "=0.04)
2.5
1.4
4
6
b ii.)
6
1.6
2
time
0
2
H(!(2))
b i.)
1.5
!(2)
0
2
0.5
1
'
4
Standard Brownian N=1000
Normal MLE
(!=0.974, "=0.127)
0
0.9
8
6
x 10
1.5
1
(
!(2)
4
time
H(!(2))
H(!(2))
2
1
!"%$
#(
,*!*%++
10
0
2
)
-
c ii.)
!"&
2.5
0.5
!"(
!*$+
1
!(2)
3
1.1
12
-
c i.)
,*!*$++
a ii.)
1.5
!(2)
a i.)
3.5
1.5
2
!(2)
N (ζ(2) − ζ̂(2)) ∼ N (0, σ 2 )
via ordinary least-squares regression
Sunday, 16 November 2008
27
Statistics of ζ(2) -- finite N
N
!
1
2
2
Mi (τ ) =
(yi )j
N j=1
2
log Mi (τ )
Sunday, 16 November 2008
=
2
log Mi (1)
+ ζi (2) log τ
28
Statistics of ζ(2) -- finite N
N
!
1
2
2
Mi (τ ) =
(yi )j
N j=1
2
log Mi (τ )
=
2
log Mi (1)
+ ζi (2) log τ
1
= Tlog Z + √ !
N
! t
"−1 t
2
Z = Tlog Tlog
Tlog Mlog
2
Mlog
Sunday, 16 November 2008
29
Statistics of ζ(2) -- finite N
N
!
1
2
2
Mi (τ ) =
(yi )j
N j=1
2
log Mi (τ )
=
2
log Mi (1)
+ ζi (2) log τ
1
= Tlog Z + √ !
N
! t
"−1 t
2
Z = Tlog Tlog
Tlog Mlog
2
Mlog
ζi (2) =
k
!
j=1
Sunday, 16 November 2008
aj
"
#
2
log Mi (τj )
30
Statistics of ζ(2) -- finite N
Sum 1
Sum 2
N
!
1
2
2
Mi (τ ) =
(yi )j
N j=1
ζi (2) =
k
!
j=1
Sunday, 16 November 2008
aj
"
#
2
log Mi (τj )
31
Statistics of ζ(2) -- finite N
Sum 1
finite variance
N
!
1
2
2
Mi (τ ) =
(yi )j
N j=1
infinite variance
?
Sunday, 16 November 2008
32
Statistics of ζ(2) -- finite N
Sum 1
finite variance
N
!
1
2
2
Mi (τ ) =
(yi )j
N j=1
infinite variance
Gaussian
Sunday, 16 November 2008
33
Statistics of ζ(2) -- finite N
Sum 1
finite variance
N
!
1
2
2
Mi (τ ) =
(yi )j
N j=1
infinite variance
Frechet!
Extreme value distribution
for the maximum
Gaussian
P
Sunday, 16 November 2008
!
Mi2
"
=
Λ
1+α/2
2
2 (Mi )
#
exp −
Λ
α/2
2
α (Mi )
$
34
Statistics of ζ(2) -- finite N
ζi (2) =
k
!
aj
j=1
finite variance
Gumbel min-stable
x −ex
P (x) ∼ e e
Sunday, 16 November 2008
"
#
2
log Mi (τj )
infinite variance
Gumbel max-stable
−x −e−x
P (x) ∼ e
e
35
Statistics of ζ(2) -- finite N
Sum 2
ζi (2) =
k
!
j=1
finite variance
Gaussian
Convergence fast
Sunday, 16 November 2008
aj
"
#
2
log Mi (τj )
infinite variance
Gaussian
Convergence slow
36
2
4
8
6
time
H(!(2))
H(!(2))
0
2
7
x 10
1.5
H(!(2))
0.9
1
1.1
!(2)
b ii.)
!(2)
1.2
1
0
2
0
2
4
6
time
0.8
7
x 10
0.4
2
1.5
1
0
1.5
Levy "=1.4 N=1000
Gumbel MLE
(!=1.15,#=0.25)
0.6
1.3
2.5
4
2
1.2
3
2.5
1.4
7
Standard Brownian N=10,000
Normal MLE
(!=0.99, "=0.04)
0
2
6
1.6
6
x 10
6
!(2)
b i.)
1.5
!(2)
H(!(2))
Statistics of ζ(2)
1
4
2
0.5
0
2
time
4
Standard Brownian N=1000
Normal MLE
(!=0.974, "=0.127)
1
0
2
4
6
time
7
x 10
Levy "=1.4 N=10,000
Gumbel MLE
(!=1.32,#=0.13)
1
0.5
0.2
7
x 10
1.5
1.5
!(2)
7
#%
!
'
/012
#!
'
.-#!
3!14526-*37!"(+-87#!!!
8491:6-;<=
*!7!"#(>-"7!"!)+
$
!-
0.9
1
1.1
!(2)
1.2
1.3
2
4
time
6
7
x 10
0.6
0.4
!
!"#
!"$
!"%
!"&
!*$+
!"'
!"(
,
"
1.5
2
4
6
time
7
x 10
Levy "=1.4 N=10,000
Gumbel MLE
(!=1.32,#=0.13)
1
2.5
!(2)
3
3.5
0
4
!&(
!"(
!"$
#'
!
#%
!!"$
!
'
/012
&
#!
'
.-#!
3!14526-*37!"(+-87#!!!
8491:6-;<=
*!7!"#(>-"7!"!)+
!"&$
!"'
1
0
1.5
"&'
"&0123
"&(
"&.
(
/,"!
1.6
456,#!!!,7 ,8,",9!!
8:;2<=,>?6
)!@!&''A,#@!&$"*
"
#
!)#*
$
1.8
1.9
8
x 10
ACE 2000 B2 N " 5000
Normal MLE
(!=0.78, #=0.23)
0.5
%
1.7
time
1
#
!
!"&
0
0.5
1
1.5
2
!(2)
!"%
!"#
!"!$
#!
!
$
/012
#!
$
.-#!
)
3!14526-*37!"(+-87#!9!!!
84:1;6-<=>
*!7!"#$)9-"7!"!%'+
(
'
#
!"%$
!*%+
0.5
!&'
!,
!"%
!"#$
%
$
2.5
!"#$
!"%$
#(
,*!*%++
!*$+
'
2
-
c ii.)
!"&
(
1.5
!"#
!
!&%
!(2)
-
)
1
!*%+
2
$
1.5
!&#
1.5
#!
.-#!
3!14526-*37!"(+-87#!9!!!
84:1;6-<=>
*!7!"#$)9-"7!"!%'+
2
1.5
0
$
/012
d ii.)
"
2
0.2
1
!
)
!-
!")
#
0.5
0
!"!$
#!
'
"&#
2.5
1
Levy "=1.4 N=1000
Gumbel MLE
(!=1.15,#=0.25)
!"#
(
H(!(2))
0
!"%
!"#$
%
d i.)
3
!(2)
!(2)
0
0.8
,*!*$++
!
!"
1
c i.)
#'
&
2.5
!"%$
#(
#
2
2
2
-
c ii.)
!"$
!!"$
1.5
%
+)!)#**
H(!(2))
6
x 10
Standard Brownian N=10,000
Normal MLE
(!=0.99, "=0.04)
4
1.2
4
6
2.5
1.4
2
time
b ii.)
6
1.6
0
0
2
H(!(2))
b i.)
1
'
2
0.5
0
0.9
4
Standard Brownian N=1000
Normal MLE
(!=0.974, "=0.127)
1
(
1
!(2)
!"(
)
1
8
6
0
4
!(2)
4
time
H(!(2))
H(!(2))
2
3.5
,*!*%++
10
0
2
3
!"&
2.5
0.5
2.5
!(2)
!)#*
1
!(2)
3
12
2
-
c i.)
1.1
1.5
!*$+
a ii.)
1.5
,*!*$++
3.5
!(2)
a i.)
1
!*%+
0
%
Sunday, 16
! - November 2008
!
!"#
!"$
!"%
!"&
!"'
!"(
!")
!-
!"#
!"#$
!"%
!"%$
!"&
!"&$
!"'
37
Conclusions
• Finite N behaviour studied here.
• Apparent or pseudo nonstationarity is introduced due to small sample size -intuitively makes sense.
• Important note not considered in detail here: the errors determined here
depend on the statistical estimator used -- other estimators might be better;
conversely others might be worse.
• Distinguish scaling stationarity and parameter stationarity. If forecasting then
might be better to consider latter. But for investigating fundamental universal
feature (scaling) the former should suffice as long as one is sensible.
• BUT need to distinguish between more compact PDFs as opposed to the
more hazardous heavy or fat-tailed PDF -- some common results might not
apply; in particular convergence to limiting statistics is slow.
Sunday, 16 November 2008
38
Conclusions
Relevant papers are now on the ISSI group
directory:
‣K. Kiyani, S. C. Chapman and N. Watkins,
preprint (2008), submitted to Phys. Rev. E.
‣K. Kiyani, S. C. Chapman and B. Hnat, Phys.
Rev. E 74, 051122 (2006).
Sunday, 16 November 2008
39