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

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)

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 ...) ?

How did an Antarctic scientist get interested in complexity ?

coupled solar wind-ionosphere

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Ultraviolet Imager – NASA Polar

Instrumentation

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Problem

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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

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“Fat tailed” burst pdfs seen in solar wind data ...

Data

... and ionospheric in currents (not shown).

Our initial question (1997-98): …

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Does Bak et al’s SOC paradigm apply to magnetospheric energy storage/release cycle ?

Lui et al, GRL, 2001

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)

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.

Another way ?

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.

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

#

“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

Droughts & Bunching

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Mandelbrot’s climate example: Pharoah’s dream 7 years of plenty (green boxes) and 7 years of drought (red boxes). Now shuffle ...

Droughts & Bunching

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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 ...

“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.

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 ...

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

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

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

1 11( ) ( ) ( ) ( )H HH H R

X t C t s s dL sα αα 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:

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.

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

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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”

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.

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:

...(& 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:

Memory & the prospective brain:

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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).

William James [1890]

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“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.

Dichotomy :

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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.

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]

Euclid

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“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

Lagrange & Russell

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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

19th Century Scientists

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“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

No mind’s eye ?

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“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”.”

Bourbaki

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“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

Fashion ? Or selection ??

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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 ?

Another Dichotomy :

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left and right brain ...

The Master & His Emissary

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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

The Master & His Emissary

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“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”.