Notes on Scaling- Sandra Chapman (MPAGS: Time Series Analysis)
Power Law Power Spectra and Scaling
Consider the autocorrelation function
"
R( ! ) =
# x ( t ) x ( t + ! )dt
!"
and its F.T. is the power spectral density G 2 ( f )
N !1
The discrete form of the ACF is R! = " xt xt+! .
t=0
We will discuss scaling in terms of increments (here ... denotes ensemble average
over t):
2
( x ( t + ! ) ! x ( t ))
~ ! 2H
H – Hurst exponent
Now
2
( x ( t + ! ) ! x ( t ))
= 2[ x 2 ( t ) ! R ( ! ) ]
where R ( ! ) = x ( t + ! )( x ( t ))
(stationary process)
so
and we insist that
x2 ( t + ! ) = x2 ( t )
R ( ! ) ~ ! 2H
Now relate this to the power spectrum.
Scaling argument
let
G2 ( f ) ~
1
f!
a 'power law' power spectrum
"
Now
R( ! ) =
# G ( f )e
2
2"if !
df
!"
Rewrite in new variables: f ! af and df = d "# af $% / a G 2 ( af ) =
"
then
R( ! ) =
# a G ( af ) e
"
2
!"
2#iaf ( ! / a )
d ( af
)
a
"
so
R( ! ) =
#a
G 2 ( af ) e2#iaf ( ! / a ) d ( af ) = a ! !1 R ( " / a )
"!1
!"
Thus
R ( " ) = a ! !1 R ( " / a ) or R ( a" ) = a ! !1 R ( " )
1
1
!
( af )
=
G2 ( f
a
!
)
Notes on Scaling- Sandra Chapman (MPAGS: Time Series Analysis)
so
R ( " ) ~ " ! !1
1
! R ( " ) ~ " !"1
f!
this is general, for stationary processes.
then
G2 ( f ) ~
then
2 H = ! ! 1 - Result, relates scaling exponent to power spectral exponent ! .
By evaluating the integral
"
# 1 !2"ift
df
% !e
$ f
IFT is:
!"
ft = x
substitute
giving
e!2!ix dx "
#!" x " t t
& +"
)+
(
!!1 ( # !2"ix dx +
+ = At !!1
= t (( % e
! +
x +++
(( $
' !"
*
"
Again, implies R ( " ) ~ " ! !1 so that 2 H = ! ! 1 .
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Notes on Scaling- Sandra Chapman (MPAGS: Time Series Analysis)
Notes on Fractal Dimension
Satellite image of the Himalayas- a natural fractal (courtesy USGS).
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Notes on Scaling- Sandra Chapman (MPAGS: Time Series Analysis)
Brownian
walk on a
grid in 2D
Consider P ( m, r )
- probability of finding m points in square side r
N
then
! P( m,r ) = 1
m=1
N
Defns: Mass dimension D given by: M ( r ) = ! mP ( m,r ) ~ r D
m=1
ln(N )
Box counting dimension: Dc = ! lim+
r"0 ln(r)
where N(r) boxes of side r are required to just cover the surface.
N
1
1
P ( m,r ) ~ D
r
m=1 m
However a practical definition is given by: N B ( r ) = !
N
These are all just examples of moments: M q ( r ) = ! mq P ( m,r ) .
m=1
For a fractal
!m$
P ( m,r ) ~ f ## D &&&
#" r %
then
N
!m$
q
m
P
m,r
~
' ( ) ' mq f ###" r D &&&%
m=1
m=1
N
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Notes on Scaling- Sandra Chapman (MPAGS: Time Series Analysis)
m! =
write
m
rD
N
M q (r) ~
" m! r
q qD
m ! =1
f ( m! ) ~ r qD
This will yield a straight line on a plot of log ( M q ) vz log ( r ) with exponent qD .
For a fractal, a plot of exponents as a function of q has slope D.
Relationship to Hurst exponent
For fBm (fractional Brownian motion) – shown here in 1D
x ( t + ! ) ! x(t)
2
~ ! 2H
on any t.
This has a power law power spectrum as above.
How many boxes are needed to cover the line?
For an interval ! , the line on average spans !x " x(t + # ) $ x(t)
2
1
2
and this will be
covered by !x " ~ " H #1 boxes.
Now consider the entire trace is divided in time by N intervals of length ! ie: ! ~
then the trace is covered by N
!x
boxes, and so
!
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1
,
N
Notes on Scaling- Sandra Chapman (MPAGS: Time Series Analysis)
N ( ! ) ~ N 2!H ~
so
D = 2! H
1
!
2!H
~
1
!D
In E Euclidean dimensions D = E + 1 ! H and for the zero sets, D = E ! H .
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