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

1. 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.

2. 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)

3. 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]

4. 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.

5. 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 Gaussian
noise.

6. 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

7. ... & technical
sense -
“burstiness” is just
one symptom of
complexity
Magnetosphere
Space-based: Ultraviolet Imager on NASA Polar
Ground based: magnetometers and all-sky imager

8. 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
AL
AE=AU-AL
1 day

9. “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

10. “Econophysics” still inspires ...
Bunde’s comment to Weigel
on Tues reminded me that
Hnat et al’s work GRL [2002]
on solar wind data collapse was
directly inspired by Mantegna
& Stanley [Nature, 1996] work
on truncated Levy flights as a
model of log returns in
S&P 500
Mantegna
& Stanley
[Nature, 1996]
M & S
book

11. 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

12. 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.

13. 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

14. 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]

15. 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

16. 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]

17. 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.

18. 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&P
Wind tunnel

19. 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 ?

20. 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

21. This is the same relationship
H=L+[J-1/2]
discussed in Mandelbrot’s selecta volumes
Here L=1/α refers to Noah effect
and J=d+1/2 to Joseph effect
http://www.math.yale.edu/~bbm3/webboo
ks.html

22. 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]

23. 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

24. 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

25. Ambiguities led Mandelbrot & Wallis [1969] to
study a “fractional hyperbolic” model (i.e. fBm with
power law jumps) which exhibited both Noah &
Joseph effects.

26. Nowadays the stochastic stable
processes community studies linear
fractional stable motion
1 1
1
( ) ( ) ( ) ( )
H H
H H R
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

27. 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 in
Brownian 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.

28. 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).

29. 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 1
1
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.

30. 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.

31. 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 1
1
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)

32. 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


33. 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 1
1
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”.

34. 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


35. 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 1
1
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)

36. 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


37. 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 1
1
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.

38. 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


39. 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 1
1
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)

40. 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


41. 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.

42. 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.

43. Meneveau & Sreenivasan’s
p-model of cascade

44. Filtered p-model: burst sizes
Watkins et al., 2009
Noah

45. 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

46. Thanks for your attention and the
invitation ...
Magnetosphere

47. Contrast LFSM with CTRW

48. 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)

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

50. 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 Brownian
Ordinary Levy

51. 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.

52. 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 !

53. “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