Distributions of Extreme Bursts Above Thresholds in a Fractional Lévy toy Model

of Natural Complexity

Nick Watkins

Chapman Conference on Complexity and Extreme Events in

Geosciences, Hyderabad, India, 19th February 2010.

With: *Sam Rosenberg (now Cambridge), Raul Sanchez (Oak Ridge), Sandra Chapman (Warwick Physics), *Dan Credgington (now UCL), Mervyn Freeman (BAS),Christian Franzke (BAS),Bobby Gramacy (Cambridge Statslab),*Tim Graves (Cambridge Statslab)& *John Greenhough (now Edinburgh)

For more on fractional Levy models & their uses see: Watkins et al, Space Sci. Rev., 121, 271-284 (2005)

For bursts in fractional Levy models: Watkins et al, Phys. Rev. E 79, 041124 (2009) [DROPPED CTRW COMPARISON FROM TALK]

For bursts in multifractals: Watkins et al, Comment in Phys. Rev. Lett. , 103, 039501 (2009) [TIME PERMITTING]

Work is fruit of BAS Natural Complexity project-see Watkins and Freeman, Science, 2008

Mission was to apply complexity ideas and methods in:

• Magnetosphere [op. cit.; Freeman & Watkins, Science, 2002]

• Biosphere [Andy Edwards et al, Nature, 2007; Chapman et al, submitted]

• Atmosphere [Franzke, NPG, 2009]

• Cryosphere [Liz Edwards et al, GRL, 2009; Davidsen et al, PRE, 2010]

and hopefully to feed back to fundamental aspects of complexity

e.g. Chapman et al, Phys. Plasmas, 2009.

Context: I will talk about how interplay of 2 parameters: d [long range memory] & α[heavy tails] affects Prob(size, duration, of bursts above threshold) in a non-Gaussian, long range correlated, non-stationary walk(linear fractional stable motion, textbook model, extends Brownian walks) ...... complements talks by Lennartz, Bunde & Santhanam on effect of d on return times in long range correlated stationary Gaussiannoise.

My applications are to solar wind and ionosphere: “complex” both in

everyday sense ...

Solar wind

Magnetosphere

Ionosphere

c.f. Baker,

Sharma,

Weigel,

Eichner

inter alia

... & technical sense -

“burstiness” is just one symptom of

complexity

Magnetosphere

Space-based: Ultraviolet Imager on NASA Polar

Ground based: magnetometers and all-sky imager

Auroral index data

Solar wind

MagnetosphereIonospheric currents. Energy source = turbulent SW. To the eye looks more stationary on scale of 1 day than a few hours-c.f. DSt index studied by [Eichner (talk) ; Takalo et al, GRL, 1995; Takalo & Murula, 2001]

Ionosphere

12 magnetometer time series

AU

ALAE=AU-AL

1 day

“Fat tails”: one facet of burstiness .pdf (AE) at 15 min

“Noah effect”- original example -stable Lévy

motion applied to cotton prices

Mandelbrot [1963]

=1

Hnat et al, NPG [2004]

=2

“Econophysics” still inspires ...Bunde’s comment to Weigelon Tues reminded me thatHnat et al’s work GRL [2002] on solar wind data collapse wasdirectly inspired by Mantegna & Stanley [Nature, 1996] work on truncated Levy flights as amodel of log returns in S&P 500

Mantegna

& Stanley

[Nature, 1996]

M & S

book

Persistence is another face of bustiness

“Joseph effect”-e.g. fractional Brownian motion (fBm) [Mandelbrot & van Ness, 1968]. In fBm p.s.d exponent is -2(1+d)

d= -1/2

d=0

Tsurutani et al, GRL [1990] S(f) ~ f-1

S(f) ~ f-2

Can define a simple spatiotemporal measure for “bursts” above threshold

Commonly used in 2D SOC models-introduced into space physics by

Takalo, 1994;Consolini, 1997 on both data and sandpile models.

Can measure “bursts” e.g. solar wind

log s

log

P(T)

log

P()

logT

log

Poynting flux in solar wind plasma from NASA Wind Spacecraft at Earth-Sun L1 point Freeman et al [PRE, 2000]

log

P(s)

But how to model bursts ?

size

length

waiting time

Naive: Brownian, self-similar, walk

14

Standard dev. of difference pdf grows with time, pdf peak P(0) shrinks in synchrony

“the “normal” model of

natural fluctuations …”

Mandelbrot (1995)

[pun intended]

Exponents H, governing fall of the pdf peak P(0), and J, for growth of pdf width ,

are here both the same = ½

P(0)

~ -H

σ~ J

15

Brownian motion is prototype of

monoscaling

But not always what we see

P(0)

σ

P(0) & σ scale same way in top 3 lines (all auroral) but differently in bottom one (solar wind)

Hnat et al [GRL,2003-2004]; Watkins et al, Space Sci. Rev. [2005]

More echoes of “econophysics”

This difference between scaling of P(0) and scaling of was remarked on by Mantegna & Stanley in Nature, 1996 (and their book on Econophysics).

They had recently proposed a truncated Levy model for S&P 500, and Ghashgaie et al [1996]

had then suggested a turbulence-inspired Castaing model as an alternative.

In a response to Ghashgaie et al, M&S contrasted S&P 500 where standard deviation of (log) price differences grew approx. as +1/2

with wind tunnel data in which it grew approx. as +1/3

Mantegna & Stanley, 1996

S&PWind tunnel

M&S then noted that P(0) for S&P 500 fell faster than -1/2 while in turbulence it also fell but not with a clear power law

dependence.

Mantegna & Stanley, 1996

S&P Wind tunnel

What’s going on ?

In stable processes community well known that in the simplest stable, self similar

models, the self-similarity exponent H sums two contributions

H=H(d,1/α)=1/α+d

Here 1/α refers to heavy tails & d to long range memory

This is the same relationship H=L+[J-1/2]

discussed in Mandelbrot’s selecta volumesHere L=1/α refers to Noah effect

and J=d+1/2 to Joseph effect

http://www.math.yale.edu/~bbm3/webbooks.html

Example limiting cases:

1. Fractional Brownian: Gaussian so α=2 hence L=1/α=1/2, H=J so measuring H

measures J - this equivalence is why

Mandelbrot originally used “H” quite freely

and only later favoured reserving J for

“Joseph” exponent, as also measured by R/S method [again see his selecta]

2. Ordinary Levy: α<2, H=1/α, J=1/2,

so H≠J, whether you measure H or J

depends on whether you want to measure

self similarity or long range dependence.

S&P seems close to ordinary Levy with H=1/α =0.71, J=0.53Mantegna & Stanley, 1996

H J

In turbulence “H” not same as “J”. P(0) is not actually straight while “J” takes Kolmogorov 1/3 value. Data is in fact

strongly multifractal.

Mantegna & Stanley, 1996

Ambiguities led Mandelbrot & Wallis [1969] to study a “fractional hyperbolic” model (i.e. fBm with

power law jumps) which exhibited both Noah & Joseph effects.

Nowadays the stochastic stable processes community studies linear

fractional stable motion 1 1

1( ) ( ) ( ) ( )H H

H HR

X t C t s s dL s

1/d H

e.g. textbooks of Samorodnitsky & Taqqu and Janicki & Weron. Allows H to vary with both Noah parameter α, and Joseph parameter d-allows a subdiffusive H<1/2 to coexist with a superdiffusive α >2 , c.f. our space data application

Can now return to “burst” diagnostics[Kearney & Majumdar, 2005]gave simple argument for tails of pdfs of “burst sizes” in Brownian case.

If curve height scales as t 1/2 then burst sizes s scale as~ T 3/2 i.e. with exponent =3/2

They could then then exploit the identity of burst duration & first passage problem inBrownian case to give a duration scaling P() ~ -3/2 & use Jacobian to get P(s) ~ s and =-4/3. In fact in BM case they were able to solve pdf exactly.

We adapted Kearney-Majumdar argument to pdf tails in LFSM case. A well known consequence of fractal nature of fBm trace, that exponent is =2-H for length of burst, enabled us to predict =-2/(1+H) for size of bursts.

Same scalings and found by Carbone et al [PRE, 2004] for fBm only-they used running average threshold rather than our fixed one (see also Rypdal and Rypdal, PRE 2008, again for fBm).

Simulate numerically

with Stoev-Taqqu

algorithm.

Exponents obtained

using maximum

likelihood fit codes

of Clauset et al,

SIAM Review, 2009.

Only power law case

used so far.

fBm: 40 trials per exponent value

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

H

Burst length exponent, , vs. H for =2, &40 trials / exponent

<Simulation>

= 2-H

Agreement with averaged exponents not terrible, but not great either -we

would like to quantify how “good” and reasons for discrepancy.

fBm: one way to gauge agreement is box plots

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

H

Burst length exponent, , vs. H for =2, &40 trials / exponent

Boxes show median

(red line),upper and lower

quartiles, with outliers as

red crosses.

fBm: now checking predicted scaling of burst size

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

H

Burst size exponent, , vs. H for =2, &40 trials / exponent

<Simulation>

= 2/(1+H)

fBm: and again, more informative comparison via box plot

1

1.5

2

2.5

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

H

Burst size exponent, , vs. H for =2, &40 trials / exponent

LFSM, alpha =1.6 case, burst length

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

H

Burst length exponent, , vs. H for =1.6, &40 trials / exponent

<Simulation>

= 2-H

One might have guessed that fit would be poorer than fBm, but for LFSM

expressions for & show similar levels of agreement even for

as low as 1.6. Again, not perfect but “in the ballpark”.

LFSM alpha =1.6 case, burst length

1

1.5

2

2.5

3

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

H

Burst length exponent, , vs. H for =1.6, &40 trials / exponent

LFSM alpha =1.6 case, burst size

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

H

Burst size exponent, , vs. H for =1.6, &40 trials / exponent

<Simulation>

= 2/(1+H)

LFSM alpha =1.6 case, burst size

1

1.5

2

2.5

3

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

H

Burst size exponent, , vs. H for =1.6, &40 trials / exponent

LFSM alpha =1.2 case, burst length

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

H

Burst length exponent, , vs. H for =1.2, &40 trials / exponent

<Simulation>

= 2-H

By the very heavy tailed case of =1.2, there is clearly a problem.

LFSM alpha =1.2 case, burst length

1

1.5

2

2.5

3

3.5

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

H

Burst length exponent, , vs. H for =1.2, &40 trials / exponent

LFSM alpha =1.2 case, burst size

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

H

Burst size exponent, , vs. H for =1.2, &40 trials / exponent

<Simulation>

= 2/(1+H)

LFSM alpha =1.2 case, burst size

1

1.5

2

2.5

3

3.5

4

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

H

Burst size exponent, , vs. H for =1.2, &40 trials / exponent

Work in progress on two issues:

1. How big is the intrinsic scatter on maximum likelihood estimates of power law tails-c.f. recent work of Edwards, [Journal Animal Ecology, 2008]; Clauset et al [arXiv, 2007;SIAM Review, 2009] i.e. “how big a scatter would we expect anyway ?”

2. If form of burst size or duration pdfs were in fact not a power law asymptotically but a stretched exponential [c.f. the return intervals in Lennartz et al, EPL, 2008; Bogachev et al, EPJB, 2008], or a product of the two [Santhanam], how would our empirical scaling arguments then behave ? Hope to have preliminary results at EGU.

But what if self-similar additive model is thought not to be the best one for other a priori reasons ?

Could for example believe that physics of system is intrinsically a turbulent cascade-especially true of solar wind-then expect multifractality.

Meneveau & Sreenivasan’sp-model of cascade

Filtered p-model: burst sizes

Watkins et al., 2009

Noah

Conclusion:Need to model burstiness in complex systems

Monofractal Gaussian models sometimes clearly insufficient.

(Additive) linear fractional stable motion offers good controllable prototype for better models in some contexts-and a useful source of insight.

Has allowed us to make a start to be made on accounting for measured “burst distributions” of data. Now examining in parallel with cascade-based models

Thanks for your attention and the invitation ...

Magnetosphere

Contrast LFSM with CTRW

Watkins et al, Space Sci. Rev., 121, 271-284 (2005)

Watkins et al, Phys. Rev. E 79, 041124 (2009)

Watkins et al, Comment in Phys. Rev. Lett. , 103, 039501 (2009)

Filtered p-model: multifractality Watkins et al. [2009]

Some diagnostics measure self-similarity exponent H e.g. variable

bandwidth method [VBW]

VBW calculates average ranges and standard deviations as a function of

scale, delivering two exponents [e.g. Schmittbuhl et al, PRE, 1995].

Franzke et al,

in preparation.

Fractional BrownianOrdinary Levy

Others find long range dependence exponent J e.g. celebrated R/S

method ...Franzke et al,

in preparation.

Fractional Brownian

Ordinary Levy

In fBm case H=J so doesn’t matter, but in ordinary Levy case R/S returns not

H but J (=1/2) . Dangerous if intuition solely built on fBm/fGn.

Ordinary Levy

... and DFA (here DFA1)Franzke et al,

in preparation.

Fractional Brownian

Obviously this is a plus if what you want is the long range dependence exponent !

“Bursty” isn’t in many dictionaries...

Solar wind

Magnetosphere

... But is in lexicon of complexity, as both a

– common symptom :- needs explanation &

– common property :- seen in models e.g. avalanching sandpiles and turbulent cascades