Distributions of Extreme Bursts Above
Thresholds in a Fractional Lévy toy Model
of Natural Complexity
Nick Watkins
Chapman...
With: *Sam Rosenberg (now Cambridge),
Raul Sanchez (Oak Ridge),
Sandra Chapman (Warwick Physics),
*Dan Credgington (now UC...
For more on fractional Levy models & their uses
see: Watkins et al, Space Sci. Rev., 121, 271-284
(2005)
For bursts in fra...
Work is fruit of BAS Natural Complexity project-
see Watkins and Freeman, Science, 2008
Mission was to apply complexity id...
Context: I will talk about how interplay of 2
parameters: d [long range memory] & α
[heavy tails] affects Prob(size, durat...
My applications are to solar wind and
ionosphere: “complex” both in
everyday sense ...
Solar wind
Magnetosphere
Ionosphere...
... & technical
sense -
“burstiness” is just
one symptom of
complexity
Magnetosphere
Space-based: Ultraviolet Imager on NA...
Auroral index data
Solar wind
MagnetosphereIonospheric currents. Energy source = turbulent SW. To the
eye looks more stati...
“Fat tails”: one facet of burstiness .
pdf (AE) at 15 min
“Noah effect”- original
example -stable Lévy
motion applied to...
“Econophysics” still inspires ...
Bunde’s comment to Weigel
on Tues reminded me that
Hnat et al’s work GRL [2002]
on solar...
Persistence is another face of
bustiness
“Joseph effect”-e.g. fractional
Brownian motion (fBm)
[Mandelbrot & van Ness, 196...
Can define a simple spatiotemporal
measure for “bursts” above threshold
Commonly used in 2D SOC models-introduced into spa...
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 ...
Naive: Brownian, self-similar, walk
14
Standard dev.  of difference pdf
grows with time, pdf peak P(0)
shrinks in synchro...
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
...
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 (sola...
More echoes of “econophysics”
This difference between scaling of P(0) and scaling
of  was remarked on by Mantegna & Stanl...
In a response to Ghashgaie et al, M&S contrasted S&P 500 where
standard deviation of (log) price differences grew approx. ...
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 la...
In stable processes community well known
that in the simplest stable, self similar
models, the self-similarity exponent H ...
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...
Example limiting cases:
1. Fractional Brownian: Gaussian so α=2
hence L=1/α=1/2, H=J so measuring H
measures J - this equi...
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 simila...
In turbulence “H” not same as “J”. P(0)
is not actually straight while “J” takes
Kolmogorov 1/3 value. Data is in fact
str...
Ambiguities led Mandelbrot & Wallis [1969] to
study a “fractional hyperbolic” model (i.e. fBm with
power law jumps) which ...
Nowadays the stochastic stable
processes community studies linear
fractional stable motion
1 1
1
( ) ( ) ( ) ( )
H H
H H R...
Can now return to “burst” diagnostics
[Kearney & Majumdar, 2005]
gave simple argument for tails of
pdfs of “burst sizes” i...
We adapted Kearney-Majumdar argument
to pdf tails in LFSM case. A well known
consequence of fractal nature of fBm trace,
t...
Simulate numerically
with Stoev-Taqqu
algorithm.
Exponents obtained
using maximum
likelihood fit codes
of Clauset et al,
S...
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 0.8...
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....
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
B...
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

Bu...
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...
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...
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, ,...
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

Bur...
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 expo...
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...
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 expone...
Work in progress on two issues:
1. How big is the intrinsic scatter on maximum likelihood
estimates of power law tails-c.f...
But what if self-similar additive model
is thought not to be the best one for
other a priori reasons ?
Could for example b...
Meneveau & Sreenivasan’s
p-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.
(Addit...
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 i...
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 an...
Others find long range dependence
exponent J e.g. celebrated R/S
method ...Franzke et al,
in preparation.
Fractional Brown...
Ordinary Levy
... and DFA (here DFA1)
Franzke et al,
in preparation.
Fractional Brownian
Obviously this is a plus if what ...
“Bursty” isn’t in many
dictionaries...
Solar wind
Magnetosphere
... But is in lexicon of complexity, as both a
– common sy...
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Hyderabad 2010 Distributions of extreme bursts above thresholds in a fractional Levy toy model of natural complexity

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

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Hyderabad 2010 Distributions of extreme bursts above thresholds in a fractional Levy toy model of natural complexity

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 43. Meneveau & Sreenivasan’s p-model of cascade
  44. 44. Filtered p-model: burst sizes Watkins et al., 2009 Noah
  45. 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. 46. Thanks for your attention and the invitation ... Magnetosphere
  47. 47. Contrast LFSM with CTRW
  48. 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. 49. Filtered p-model: multifractality Watkins et al. [2009]
  50. 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. 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. 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. 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

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