Scaling, structure functions and all

that…

S. C. Chapman

Notes for MPAGS MM1 Time series Analysis

•SCALING: Some generic concepts: universality, turbulence, fractals

and multifractals, stochastic models

•RESCALING PDFS AND STRUCTURE FUNCTIONS

•FINITE LENGTH TIMESERIES, UNCERTAINTIES, EXTREMES-’real

data’ examples

Scaling

Some ideas and examples

Scaling and universality-Branches

on a self-similar tree

Number N of branches of length L

Length L1/5x5 1/5 1

1

3

9=3x3 13 ,

5

log( ) log(3), log( ) log(5)

log(3)log( ) log( )

log(5)

so for any tree..

log( ) (a number) log( )

a

aN L

N a L a

N L

N L

Each branch grows 3 new branches, 1/5 as long as itself..

Segregation/coarsening- a

selfsimilar dynamics

Courtesy P. Sethna

Rules: each square changes to be like the majority of its neighbours

Coarsening, segregation, selfsimilarity

Solar corona over the solar cycle

SOHO-EIT image of the corona

at solar minimum and solar maximum

- Magnetic field structure

SOHO- LASCO image

of the outer corona

near solar maximum

The solar wind as a turbulence laboratory (will use as an example)

Solar wind at 1AU power spectra-

suggests inertial range of (anisotropic MHD) turbulence.

Multifractal scaling in velocity and magnetic field

components.. AND something else in B magnitude..

Goldstein and Roberts, POP 1999, See also Tu and Marsch, SSR, 1995

Quantifying scaling I

‘Fractal’ – self- affine scaling

Uncertainties, finite size effects

Link to SDE models (self- affine processes)

A regular fractal

Koch snowflake

line length (4 / 3)nl

A random fractal

16 particles- Brownian

random walk

1

2 2

11

2 2 2

1

2 2

12 2

consider a random walk in 2D

( )

2

so if n steps take time

or

n nn

n n nn

n n n n

n

n

n n

r t r r l

r r r r l l

r r r r l

r nl

t

r t r t

1 2

Consider a timeseries ( ) sampled with precision . We construct a timeseries

( , ) ( , ) ( ) ( ) so

( ) ( ) ( , ) and ( , ) is a random variable

then

( ) ( , ) ( , ) ...

x t differenced

x t y t x t x t

x t x t y t y t

x t y t y t

1

(1) (1) (1)

1 / 2

( ) ( ) ( )

1 / 2

( )

( , ) ( , ) ... ( , )

( , 2 ) .. ( , 2 ) .. ( , 2 )

( , 2 ) ... ( , 2 ) .. ( , 2 )

we seek a self affine scaling

' 2 , ' 2 , 2 , as 2

for arbitrary ,

n

k k N

k N

n n n n n n

k N

n n n

y t y t y t

y t y t y t

y t y t y t

y y y y

normalize such that

'( , ) ( , )

is a random variable, so we have the same PDF under transformation:

( ' ) ( )

the are not Gaussian iid. We need to find

consider CLT case..

y t y t

y

P y P y

y

Data Renormalization

Self –affine (‘fractal’) scaling in timeseries

Example-Brownian walk

Fluctuations:

Probability of wandering

different distances in a

given time (Gaussian)

( , ) ( ) ( )x t x t x t

RescaleThe height of the peaks is

power law- a single factor

rescales them

The same factor

rescales all the curves-=1/2

Self-similarity

Procedure to test for self affine scaling in a timeseries- Brownian

walk (simple fractal)

1) difference the timeseries ( ) on timescale to obtain ( , ) ( ) ( )

2) ( , ) are self- similar (fractal) - same function under single parameter rescaling

3) rescaling parameter comes from

x t y t x t x t

P y if

1the data eg ( ) ~ , here2

4) so moments of the PDF: ( , ) ~p p

ty t

example- ,B2 in the solar wind

slow sw shown, , B2

selfsimilar scaling up to ~few hrs

WIND 46/98s

Key Parameters ’95-’98

Approx 10^6 samples

Verified with ACE

Hnat, SCC et al GRL,2002, POP 2004

Diffusion- random walkBrownian random walk

is stochastic iid

dx

dt

2

diffusion equation

( , )( , )

( , ) is Gaussian

P y tD P y t

t

P y t

Renormalization-scaling system looks the same under

1' , ' and = ..........which implies ( ', ') ( , )

2

( , ) is Gaussian, the fixed point under RG

t yt y P y t P y t

P y t

Note: ( ) is distance

travelled in interval

a differenced variable

y t

t

Fokker- Planck models(see also fractional kinetics and Lévy flights)

Langevin equation

( ) ( )

stochastic iid

dxx x

dt

Fokker- Planck equation

( , )( ( ) ( , ) ( ) ( , ))

can solve for ( , )

P y tA y P y t B y P y t

t

P y t

Renormalization-scaling system looks the same under

1' , ' and ..........which implies ( ', ') ( , )

2

t yt y P y t P y t

Note: ( ) is distance

travelled in interval

a differenced variable

y t

t

A not so simple fractal timeseries- financial markets

• Mantegna and Stanley- Nature,1995

• S+P500 index

• ‘heavy tailed’ distributions

• Brownian walk in log(price) is the

basis of Black Scholes (FP model for

price dynamics)

• Non- Gaussian PDF, fractal scaling-

Fractional Kinetics or non- linear FP

The efficient market

dSdX dt

S

Efficient- arbitrageurs constantly trade to exploit differences in price

As a consequence any price differences are very short lived

The market is a ‘fair game’

Implies

Fluctuations are uncorrelated

Fluctuations aggregate many (N) trades, thus an equilibrium, large N

model implies Gaussian statistics (CLT)

Change in price S, dS in t-t+dt governed by:

Black-Scholes and all that..

2

Anticipate a Diffusion equation for -since

provided we have the self- similar scaling for

the stochastic variable

I

we can write an equation for price evolution

II ( , )

log( )dS

S dX dtS

dX d

dX

dS A S

t

t

( , )

can then write a Taylor expansion for any ( ) using I.

This leads to the B-S SDE for the price of options...

Riskless portfolio ( ) , ( ) is an option on stock

key phenomenology is th

dX B S t dt

f S

f S S f S S

scat a of ling

Nonlinear F-P model for self similar

fluctuations- asymptotic result

(alternative- fractional kinetics)

If the PDF of fluctuations ( ) ( ) on timescale is :

( , ) ( )

is then a solution of a eq

selfsimilar

Fokker- Pl uation:

, where transport coefficients ( ), ( )

w

anck

it

s

y x t x t

P y P y

P

PAP B P A A y B B y

1 1/ 2 1/h , we solve the Fokker- Planck for

This corresponds to a : ( ) ( ) ( )

and we can obtain , via the Fokk

Langevin e

er- Planck coefficients

see Hnat, SCC et al. Phys.

quation

Rev

sA y B y P

dxx x t

dt

. E (2003), Chapman et al, NPG (2005)

Fokker Planck fit to PDFs

Procedure:

1) Measure exponent

2) Solve FP for PDF functional form

3) Check this fits the observed PDF

Turbulence

a la Komogorov, intermittency

beyond power spectra...

(NB we will introduce intermittency in the context of

turbulence, but methods are quite general)

Turbulence

Dynamics are complex

Statistics are simple

Assume:

Isotropic

Stationary

Homogeneous

Example- strong multifractal

solar wind v,B

moments ( )

( ) quadratic in

m m mS x

m m

Intermittent turbulence-topology

3

1

1

Consider simple finite sized scaling system, scale lengths

/ , 1... with levels

from a smallest size = to the system size , patches on length

Non space filling, inter

scale

j

j j

N j j

l

l l l N N

l l L m l

( ) ( ) ( )

1 1

* * *j j-1 *N

j

1

= = where is 'active quantit

mi

y' per patch, lengthscale

active quant

ttent patches

ity per lengthsca

:

Fractal supp

le

ort:

Conservation:

q q q q q q

j j j j N

j

j j

m l m l m L

ll l L

*

j

0

( ) ( )

*

0

which fixes (1) or (1) 0

when these combine to give:

( )

j j

j

q q q

j jq q q q

j N N

m

l lm

L L

2

r

velocity difference across an eddy ( ) ( )

eddy time ( ) and energy transfer rate

have as the eddy t

Intermittency-

as a deviation from a space filling cascade (Kolmogorov turbulence)

r

r

d v v l r v l

d vT r

T

T

3

r

( )3 3

urnover time so that

If the flow is , independent for any p

~ - ( ) linear with

corr

non- intermittent

intermitten ectio

( ) scalin

cy n

g

-

r

r

p p

r

p pp p

r

d vrTd v r

r

d v r r p p p selfsimilar fractal

r

3 ( )3 3

steady st

dependence

~ - ( ) quadratic in

independent of ( ) soate

( ) must monotonically increase (and

(

( ) 1 for some

1) 0,

in situ single point obse

)

r

pp p

r

pp pp p

r

r

rL

Ld v r r p pr

r

p p p

( )vations take :

6 needed to measure

measure ( ) from

(2) ! predicted from phenomenolo y

~

g

p p

tpr t

p

d v t

Quantifying scaling II

Multifractal scaling and structure functions

Turbulence and scaling

r

Reproducible, predict

structures on many le

abl

ngth/timescales.

single spacecraft- time interval a proxy for space

to focus on any particular scale take a difference:

( ,

e in a sense.

)

statistica

r

y l

l

r x

( ) ( )

look at the statistics of ( , )

power spectra- compare power in Fourier modes on

different scales

l r x l

y l r

r

DNS of 2D compressible MHD turbulence

Merrifield, SCC et al, POP 2006,2007

Quantifying scaling

r

x(t3)x(t4)

x(t2)x(t1)

x(t0)

structures on many length/timescales.

look at (time-space) differences:

( , ) ( ) ( )

( , ) ( ) ( )

for all available of the timeseries ( )

Reproducible, predictable in a sense.

t

k k

y r l x r l x r

y t x t x t

t x

stati

t

stical

( )

( )

2

statistical scaling

structure functions ( ) | ( , ) |

or ( ) | ( , ) |

we want to measure the ( )

fractal (self- affi

est for i.e

( ) ~

(

ne)

multifractal ) ~ ...

would like | (

p p

p

p p

p

p

S r y

p

p

r l l

y t

p

p

y

S

p

B

, ) | | | ( , )

UT finite system/ data!

p pr l y P y l dy

Theory-data comparisons- examples

2 and 3D MHD simulations

Muller & Biskamp PRL 2000

Fluid experiments,

Anselmet et al, PSS, 2001

KO41

How large can we take p? See eg Dudok De Wit, PRE, 2004

A nice quiet fast interval of solar wind- CLUSTER high

cadence B field spanning IR and dissipation range

CLUSTER STAFF and FGM shown overlaid.

Kiyani, SCC et al PRL 2009

CLUSTER STAFF and FGM shown overlaid.

Kiyani, SCC et al PRL 2009,

Dissipation range- fractal

( )| ( ) ( ) | ~ , plot log( ) vz. log( ) to obtain ( )p p

p pS x t x t S p

Inertial range- multifractal

Quantifying scaling III

Uncertainties, extreme events, finite size effects

Will discuss structure functions but remarks relate to other measures of scaling

Finite sample effect- error on exponent ζ(2) as a function of sample size N

Time stationary

Brownian walk

Time stationary

P-model

Non- stationary

Brownian walk

Kiyani, SCC et al, PRE (2009)

Errors decrease in

power law with N!

p

( )

structure functioDefine (generalized variogram) for differenced timeseries:

( , ) ( ) ( )

( ) | ( , ) | if scaling

We would like to ca

n

Structure functions-estimating the ( ) from data

p p

p

S

y t x t x t

S y t

p

s

( )

lculate ( ) | ( , ) | | | ( , )

then

Conditioning- an estimate is:

| | | | ( , ) where [10 20] ( )

strictly ok if selfsimilar: y , ,

if

)

(

)

p

(

(

p p

p

p

s s s

A

p

p p

A

s

p

S y t y P y dy

y Pdy

y y P y dy A

y P P p p

S

) is quadratic in p (multifractal)- weaker estimate

Structure functions- sensitive to undersampling of largest events

(example - in slow sw)( )( , ) ( ) ( ) test for scaling - ( ) | ( , ) | m m

my t x t x t S y t

remove y >10()

2 sources of uncertainty in exponent

1) Fitting error of lines (error bar estimates)

2) Outliers- Shown: removed < 1% of the data

ACE 98-01 (4years)-106 samples.

Threshold 450 km/sec.

fractal or multifractal? 2

( ) ~

( ) ~ ...

fractal (self- affine)

multifractal

p p

p p p

cf Fogedby et al PRE ‘anomalous diffusion in a box’

Chapman et al, NPG, 2005,Kiyani et al PRE, 2006

1

22

( ) ~ , ,1 2 power law tails, self similar

for a finite length flight ( ) ~

so 2 is Gaussian distributed, Brownian walk

CP x x

x

x x t

( )( )Stru | ( , ) |

1expect ( ) ~

cture functio

,

ns: p p

pS x t

p p

sxPDF re xscaling , sP P

Outliers and a more precise test for fractality-

example-Lévy flight (‘fractal’)

1

22

( ) ~ , ,1 2 power law tails, self similar

for a finite length flight ( ) ~

so 2 is Gaussian distributed, Brownian walk

CP x x

x

x x t

( )( )Stru | ( , ) |

1expect ( ) ~

cture functio

,

ns: p p

pS x t

p p

sxPDF re xscaling , sP P

Chapman et al, NPG, 2005,Kiyani, SCC et al PRE (2006)

A more precise test for

fractality-

outliers and convergence:

example-Lèvy flight (‘fractal’)

Distinguishing self- affinity (fractality) and multifractality

P-model -multifractalLévy flight -fractal

Fractal signature ‘embedded’ in (multifractal) solar wind

inertial range turbulence-coincident with complex

coronal magnetic topology

Solar cycle variation WIND Inertial Range- |B|2

Kiyani et al, PRL 2007, Hnat et al, GRL 2007

2000 - Solar max

fractal

1996 - Solar min

multifractal

Left: B² fluctuation PDF solar max and solar min

Right: solar max, FP and Lévy fit

WIND 1996 min (◊), 2000 max (◦), ACE 2000 max (□)

Hnat, SCC et al, GRL, (2007)

Quantifying scaling IV

Extended self similarity

3 4

( )( )( )

ˆ =< and its remainder versus ,

ESS tests

Generalized or extended self simlarity- ESS plot

i.e. ( )

gives exponents when e.g. (3) 1 or (4) 1

s:

p

p

ppq

p q p

S S S

S S S G

v b

v v

End

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