This document discusses two topics in the history of complexity: "bunched black swans" and the "minds eye". It summarizes the work of Benoit Mandelbrot in developing self-similar models, such as fractional Brownian motion and fractional hyperbolic motion, to model long-range dependence and heavy-tailed distributions seen in natural phenomena like cotton prices and the Nile River. It discusses how Mandelbrot later developed multifractal models to address limitations with purely self-similar models, such as their inability to capture "volatility clustering". The document also discusses cognitive styles and individual differences in visual versus verbal thinking and their relevance to the history of mathematics, science, and future developments in information and communication

1. Two topics in history of
complexity: “Bunched black
swans” and the minds eye
Nick Watkins nww@bas.ac.uk
NERC British Antarctic Survey, Cambridge, UK
Visiting Fellow, Centre for the Analysis of Time Series, LSE
Associate Fellow, Department of Physics, University of Warwick
“Those who cannot remember the past are condemned to repeat it”-Santayana

2. Thank many people: Tim Graves (Cambridge), Dan
Credgington (Now UCL) , Sam Rosenberg (Now Barclays
Capital), Christian Franzke (BAS), Bogdan Hnat (Warwick),
Sandra Chapman (Warwick), Nicola Longden (BAS),
Mervyn Freeman (BAS), Bobby Gramacy (Chicago), Jean
Boulton, Ed Bullmore, Bill Faw, Brian Levine, ...
Watkins et al, Space Sci. Rev., 121, 271-284 (2005)
Watkins et al, Phys. Rev. E 79, 041124 (2009a)
Watkins et al, Phys. Rev. Lett. , 103, 039501 (2009b)
Franzke et al, Phil Trans Roy Soc, (2012)
Watkins et al, in press AGU Hyderabad Chapman Conference Proceedings (2012)

3. Summary
• Why have I been involved in complexity ?
• Self-similar models, burst distributions and
SOC [c.f. Watkins et al, PRE; PRL, 2009.]
• Multifractal models-why did Mandelbrot
come to embrace them ? And what route did
he take to get there ?
• What can we learn about complexity (and
science in general) from the history of &
frontiers of research into cognitive styles – a
link to Future ICT (in the broadest sense ...) ?

4. How did an Antarctic scientist get
interested in complexity ?
coupled solar
wind-ionosphereSolar wind
Magnetosphere
Ionosphere
Ultraviolet Imager –
NASA Polar
Instrumentation
Solar wind
Problem

5. log s
log
P(T)
log
P(τ)
logT
log τ
Poynting flux in solar wind plasma from NASA
Wind Spacecraft at Earth-Sun L1 point
Freeman, Watkins & Riley [PRE, 2000].
log
P(s) size
length
waiting time
“Fat tailed” burst pdfs seen in
solar wind data ...
Data
... and ionospheric in currents (not shown).

6. Our initial question (1997-98): …
18 July 2012 6
Does Bak et al’s SOC paradigm apply to
magnetospheric energy storage/release cycle ?
Lui et al, GRL, 2001

7. Bak et al’s aim was to unify fractals in space
with “1/f” noise in time directly, via a physical
mechanism:
Answering Kadanoff’s
question: [spacetime] ...”fractals: where’s the
physics” ? (often traduced, was plea, not a criticism)

8. Another way ?
Experience with SOC and complexity in space physics
[summarised in Freeman & Watkins, Science, 2002; Chapman
& Watkins, Space Science Reviews], and the difficulty of
uniquely attributing complex natural phenomena led us to
“back up” one step.
Got interested in applying the known models for non-Gaussian
and non iid random walks. Partly to try and see what physics
was embodied in any particular choice, partly for “calibration”
of the measurement tools. Link to risk and extremes. Such
models go beyond the CLT. They are not always general “laws”,
but they are mapping out a range of widely observed
“tendencies”. In learning about these we have become
interested in the history of Mandelbrot’s paradigmatic models
and their relatives.

9. Another way ?

10. Four giant leaps made by Mandelbrot between 1963
and 1974---”well known” and yet process is instructive
1. BBM observes heavy tailed fluctuations in 1963 in cotton
prices---proposes alpha-stable model , self-similarity idea
2. BBM hears about River Nile and “Hurst effect”. Initially (see
his Selecta) believes this will be explained by heavy tails,
but when he sees that fluctuations are ~ Gaussian
applies self-similarity [Comptes Rendus1965] in the form of a
long range dependent (LRD)
model, roots of fractional Brownian motion. BBM’s classic series of papers on fBm in
mathematical & hydrological literature with Van Ness and Wallis in 1968-1969.
3. BBM demonstrates a new self-similar model, fractional hyperbolic motion, in
1969 paper with Wallis on robustness of R/S. Combines 1 & 2 above (heavy tails & LRD)
4. BBM becomes dissatisfied with purely self-similar models, develops multifractal
cascade, initially in context of turbulence, JFM 1974. Later applications include finance.

11. Earthquakes & “Black Swans”
Light tailed
Gaussian
Heavy
tailed
power law
Plot number of
events (#) versus
magnitude (x). In
red “normal” case,
a magnitude 25
event essentially
never happens.
In the blue heavy
tailed case, it
becomes a “1 in
2000” event.
“Extreme events
… [are ] the norm”
-John Prescott
This matters because it applies in many
natural and man-made situations
e.g. Gutenberg-Richter law,
insurer’s “80-20” rule of thumb
#

12. “Noah effect”- e.g. Lévy
flights where a < 2
increases tail fatness
a=1
e.g. Hnat et al, NPG [2004]
a=2
Levy flight model
Ionospheric
Data
BBM observes heavy tailed fluctuations in 1963 in
cotton prices---proposes alpha-stable model ,
abstracts out self-similarity idea

13. Droughts & Bunching
18 July 2012 13
Mandelbrot’s climate example: Pharoah’s dream 7 years of plenty
(green boxes) and 7 years of drought (red boxes). Now shuffle ...

14. Droughts & Bunching
18 July 2012 14
Mandelbrot’s climate example: Pharoah’s dream 7 years of plenty
(green boxes) and 7 years of drought (red boxes). Now shuffle ...
Point is that frequency distribution is same (c.f. Previous slide) but
that the two series represent very different hazards. Don’t even
need to come from heavy tails, e.g. a long run of very hot days ...

15. “Joseph effect”- e.g. fractional
Brownian (fBm) walk: steepness
of log(psd) with log(f) increases
with memory parameter d
d=-1/2
d=0
S(f) ~ f-2(1+d)
Fractional Brownian
walk model
BBM hears about River Nile and “Hurst
effect”. Initially (see
his Selecta) believes this will be explained
by heavy tails,
but when he sees that fluctuations are ~
Gaussian
applies self-similarity [Comptes
Rendus1965] in the form of a
long range dependent (lrd)
model, roots of fractional Brownian
motion.
BBM’s classic series of papers on
fBm in mathematical &
hydrological literature with
Van Ness and Wallis in 1968-1969.

16. What if heavy tailed and LRD ?
• Mandelbrot looked at this in 1969 with Wallis, proposed a
version of fractional Brownian motion which substitutes
heavy tailed “hyperbolic” innovations for the Gaussian ones –
hence fractional hyperbolic motion
• In such a model you not only get “black swan” (heavy tail)
events, but they are “bunched” by the long range dependence
...

17. Financial Bunched Black Swans
“ “We were seeing things that were 25-standard deviation
moves, several days in a row,” said David Viniar,
Goldman’s CFO ... [describing catastrophic losses on their
flagship Global Alpha hedge fund]. “What we have to look
at more closely is the phenomenon of the crowded trade
overwhelming market
fundamentals”,
he said. “It makes
you reassess how
big the extreme
moves can be””.
--- FT, August 13th, 2007

18. Financial Bunched Black Swans
“ “We were seeing things that were 25-standard deviation
moves, several days in a row,” said David Viniar,
Goldman’s CFO ... [describing catastrophic losses on their
flagship Global Alpha hedge fund]. “What we have to look
at more closely is the phenomenon of the crowded trade
overwhelming market
fundamentals”,
he said. “It makes
you reassess how
big the extreme
moves can be””.
--- FT, August 13th, 2007

19. Financial Bunched Black Swans
“ “We were seeing things that were 25-standard deviation
moves, several days in a row,” said David Viniar,
Goldman’s CFO ... [describing catastrophic losses on their
flagship Global Alpha hedge fund]. “What we have to look
at more closely is the phenomenon of the crowded trade
overwhelming market
fundamentals”,
he said. “It makes
you reassess how
big the extreme
moves can be””.
--- FT, August 13th, 2007

20. 1 1
1
( ) ( ) ( ) ( )
H H
H H R
X t C t s s dL sα α
α α α
 − −−
 
 , , + +
 
= − − −∫
H = d+1/α: allows
H “subdiffusive” (i.e. < ½) while
α “superdiffusive” (i.e. <2).
R/S, DFA etc, see only d not alpha (e.g.
Franzke et al, Phil Trans Roy Soc, 2012)
Memory kernel:
Joseph
α-stable jump:
Noah
An H-selfsimilar, stable successor to
Mandelbrot’s model
3. BBM demonstrates a new self-similar model, fractional hyperbolic
motion, in 1969 paper op cit. Combines effects 1 & 2 (heavy tails &
LRD). Nowadays would use Linear Fractional Stable Motion:

21. Bursts in LFSM
• We have begun to study how bursts, defined as integrated
area above thresholds, scale for LFSM. [Watkins et al, PRE,
2009] Scaling depends both on alpha and d, via H.
• Links to work of Kearney and Majumdar [J Phys A, 2005] on
area defined by curve to its first return (for Brownian motion
started epsilon above a threshold) AND to Carbone and
Stanley , PRE and Physica A on bursts defined in fBm a la DFA.

22. Natural examples include ice cores (e.g. Davidsen and Griffin, PRE , 2009),
and returns of ionospheric AE index (above), see also Consolini et al, PRL,
1996. Man-made example from which name volatility is taken is finance.
Effect not seen in fractional Levy models c.f. Rypdal & Rypdal, JGR 2010
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10
4
-600
-400
-200
0
200
400
600
increments,r
First differences of AE index January-June 1979
-100 -80 -60 -40 -20 0 20 40 60 80 100
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
lag
acf
AE data: acf of returns
-100 -80 -60 -40 -20 0 20 40 60 80 100
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
lag
acf
AE data: acf of squared returns
First differenced AE data
ACF of diff. AE
ACF of (diff. AE) squared
Having introduced 3 models in 6 years, Why did BBM
remain dissatisfied ? Partly because his eyes told him ...
Effect that multifractals capture is “volatility clustering”

23. Meneveau
& Srinivasan
P-model
4. BBM becomes dissatisfied with purely self-similar models, develops
multifractal cascade, initially in context of turbulence, JFM 1974.
Later applications include finance in late 1990s by BBM, Ghashgaie et al.

24. Importance can be seen from simple thought experiment,
and the fact we don’t know the answer to it ...
If we could not only see and hear these talks, but also see the
pictures drawn (or not) in individual audience’s minds by them
---how much variety would we see ?
Many steps made over millions of years to allow human
beings to be convinced that we are communicating about
same thing. Language itself, cave painting [c.f. Herzog’s film
The Cave of Forgotten Dreams], maths, blackboards,
visualisation ….
Cognitive styles:

25. ...(& FutureICT !)
How can we (will we) make further leaps towards this collective
communication while discovering and appreciating why (it is
evolutionary advantage ?) that we don’t all see the same thing ?
Cognitive styles:

26. Memory & the prospective
brain:
Solar wind
Magnetosphere
Ionosphere
Endel Tulving distinguished in 1970s between episodic and semantic
memory. Episodic means re-experiencing the past
(“remembering”) while semantic is about
the facts we store about it (“knowing”).
More recently fMRI studies c. 2005 onwards
[Schacter group Nature; Tulving group
PNAS] are beginning to
link centres of the brain used in episodic
memory to the simulation of future events.
Tulving sees ep mem and future prognosis
as linked “mental time travel” capability,
which he sees as an element of “autonoetic”
consciousness. Has made provocative
suggestion that this is unique to H. Sapiens
(i.e. our “killer app” compared to Neanderthals).

27. William James [1890]
Solar wind
Magnetosphere
Ionosphere
“A person whose visual imagination is strong finds it
hard to understand how those who are without the
faculty can think at all. Some people undoubtedly
have no visual images at all worthy of the name,
and instead of seeing their breakfast-table, they tell
you that they remember it or know what was on it.
This knowing and remembering takes place
undoubtedly by means of verbal images, ...”
- The Principles of Psychology, Volume 2.

28. Dichotomy :
Solar wind
Magnetosphere
Ionosphere
I am fascinated by the possible ways in which Tulving’s episodic vs
semantic and James’ verbal/visual dichotomies may illuminate :
... (Tulving): the tension (metaphorical, and by many accounts sometimes
literal !) between the very visual thinking of BBM and the
non-visual, formal proofs of e.g. the Bourbaki, of which his uncle,
Szolem Mandelbrojt was a founder member ...
And various reports about mathematical and scientific thought in the 19th
and 20th centuries.

29. Dirac
• “Her fundamental laws do not govern the
world as it appears in our mental picture in
any very direct way, but instead they control a
substratum of which we cannot form a mental
picture without introducing irrelevancies."
--- Preface to The Principles of Quantum
Mechanics [1930]

30. Euclid
Solar wind
Magnetosphere
Ionosphere
“In Euclid, you find some drawings but it is known
that most of them were added after Euclid, in
later editions. Most of the drawings in the original
are abstract drawings. You make some reasoning
about some proportions and you draw some
segments, but they are not intended to be
geometrical segments, just representations of
some abstract notions.”
The Continuing Silence of Bourbaki—An Interview with Pierre Cartier [Bourbaki
1955-83], June 18, 1997
by Marjorie Senechal in The Mathematical Intelligencer, № 1 (1998) · pp.22–28

31. Lagrange & Russell
Solar wind
Magnetosphere
Ionosphere
Also Lagrange proudly stated, in his textbook
on mechanics, "You will not find any drawing
in my book!“
The analytical spirit was part of the French tradition
and part of the German tradition. And I suppose it
was also due to the influence of people like Russell,
who claimed that they could prove everything
formally—that so-called geometrical intuition
was not reliable in proof.
Senechal , op cit

32. 19th Century Scientists
Solar wind
Magnetosphere
Ionosphere
“The earliest results of my inquiry amazed me. I had begun by
questioning friends in the scientific world, as they were the most likely
class of men to give accurate answers concerning this faculty of
visualizing, to which novelists and poets continually allude, which has left
an abiding mark on the vocabularies of every language, and which
supplies the material out of which dreams and the well-known
hallucinations of sick people are built.”
“To my astonishment, I found that, the great majority of the men of
science to whom I first applied protested that mental imagery way
unknown to them, and they looked on me as fanciful and fantastic in
supposing that the words 'mental imagery' really expressed what I
believed everybody supposed them to mean. They had no more notion of
its true nature than a color-blind man, ... has of the nature of color.” -
Francis Galton quoted in James, op cit

33. No mind’s eye ?
Solar wind
Magnetosphere
Ionosphere
“They [...] naturally enough supposed that those who affirmed they
possessed [visual images] were romancing. To illustrate their mental
attitude it will be sufficient to quote a few lines from the letter of one of my
correspondents, who writes:
"These questions presuppose assent to some sort of a proposition
regarding the "mind's eye," and the "images" which it sees. . . . This
points to some initial fallacy. . . . It is only by a figure of speech that I can
describe my recollection of a scene as a "mental image" which I can "see"
with my "mind's eye. " . . . I do not see it . . . any more than a man sees
the thousand lines of Sophocles which under due pressure he is ready to
repeat. The memory possesses it,' etc”.”

34. Bourbaki
Solar wind
Magnetosphere
Ionosphere
“Again Bourbaki's abstractions and disdain for
visualization were part of a global fashion,
as illustrated by the abstract tendencies in the
music and the paintings of that period.“
Senechal op cit in The Mathematical Intelligencer, № 1 (1998) · pp.22–28
http://www.ega-math.narod.ru/Bbaki/Cartier.htm

35. Fashion ? Or selection ??
Solar wind
Magnetosphere
Ionosphere
Even British and continental European approaches differed
e.g. different attitudes to construction of models.
$64000 question ---was Galton’s sample real
And why would it not be same now ?

36. Another Dichotomy :
Solar wind
Magnetosphere
Ionosphere
left and right brain ...

37. The Master & His Emissary
Solar wind
Magnetosphere
Ionosphere
The coinage of Ian McGilchrist in a book of
the same name about the brain’s left and
right hemispheres.
See his Royal Society of Arts talk and the
10 minute animated summary

38. The Master & His Emissary
Solar wind
Magnetosphere
Ionosphere
“This book argues that the division of the brain into two
hemispheres is essential to human existence, making possible
incompatible versions of the world, with quite different priorities and
values.
Most scientists long ago abandoned the attempt to understand why
nature has so carefully segregated the hemispheres, or how to
make coherent the large, and expanding, body of evidence about
their differences. In fact to talk about the topic is to invite
dismissal. Yet no one who knows anything about the area would
dispute for an instant that there are significant differences: it's just
that no-one seems to know why. And we now know that every type
of function - including reason, emotion, language and imagery - is
subserved not by one hemisphere alone, but by both.
This book argues that the differences lie not, as has been
supposed, in the 'what' - which skills each hemisphere possesses -
but in the 'how', the way in which each uses them, and to what
end”.