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Invited talk at 2010 American Geophysical Union Chapman Conference in Hyderabad, India.
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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
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