Bunched black swans at the British Antarctic Survey ...

Nick Watkins: NickWatkins@mykolab.com

NERC British Antarctic Survey, Cambridge, UKVisiting Fellow, Centre for the Analysis of Time Series, LSE

Associate Fellow, Department of Physics, University of Warwick

http://www.mendeley.com/profiles/nicholas-watkins1/

IMA Annual Conference: Mathematics of Planet Earth,

14th March 2013

Addendum

• My last talk as a BAS employee, this was an invited presentation at the Institute

of Mathematics and its Applications Annual Conference in London, on 14th March

(Pi Day) 2013.

The meeting’s theme was Mathematics for Planet Earth, and details

of the programme are at http://www.ima.org.uk/viewitem.cfm-cit_id=384212.html

Apart from changing the email address to one which is still in use, I have not

changed or corrected slides.

NWW

30/7/14

Summary

• THE WHAT: Will talk about long range correlated, wild, events (colloquially, in my

title, following Taleb, I have called them “bunched black swans”). Will discuss a

particular type of wild event, the “burst” above a threshold.

• THE WHY: Will intersperse examples of why scientists looking at earth’s

climate, space weather (and finance) care about bursts. BAS and beyond …

• THE HOW: Will talk a bit about how my colleagues & I have been modelling one

class of “routinely extreme” burst problem using an approach that builds on

Mandelbrot’s pioneering work in the 60s, and the more recent

fractional stable models (e.g. Samorodnitsky & Taqqu).

• MORE DETAIL: Short review in Watkins, GRL Frontiers, 2013, doi:

10.1002/grl.50103, intended to act as a portal setting Mandelbrot’s work

in context in the large and growing literature in this area.

Why BAS ?

Auroral physics

Two examples of many

Foraging

The sun and space weather

Solar wind

Magnetosphere

Example of diagnostic is magnetometers,

sensing ionospheric currents. Ultimate

energy source turbulent solar wind.

Ionosphere

Fat tails & the sun

Riley, 2012

“Fat” tailed distribution of

velocity of coronal mass

ejections. Lognormal ?

Truncated power law ?

How “fat” is “fat” ?

= How likely is next severe

“Carrington event” ?

Persistence & heat waves

• Rather than coming from a fat-tailed distribution of amplitudes, the severity of an event might be a result of a long duration.

• Runs of hot days above a fixed threshold, e.g. summer 1976 in UK, or summer 2003 in France.

• Direct link to weather

derivatives [e.g. book

by Steve Jewson et al]

Finance: persistence & fat tails ? “ “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 makesyou reassess how big the extreme moves can be””.

--- FT, August 13th, 2007

Several ways to conceptualise “wild” events

30 July 2014 9

“ Is it a bird, an extreme event, or a cliche ... ?”

– The Economist, March 2011

Colloquial, but resonant

with much experienceRigorous, but limitations.

Inspired by what we

see in many time series,

and by models of

intermittency

Black Swans

30 July 2014 10

• “Rarity …

• … extreme impact …

•… and retrospective

(though not prospective)

predictability”

Extremes

30 July 2014 11

1 2 3{ , , , }max Nx x x x

Bursts

30 July 2014 12

2

1

[ ( ') ] 't

ts x t L dt 2

1

't

tT dt

Inspired by sandpiles

and SOC.

Compared, for

example to EVT,

does not assume

threshold is in some

sense high.

Can be measured on

time series, or fields,

has links to research on

sojourns, level sets,

etc.

Also several ways to conceptualise clustering of severe events

30 July 2014 13

Bunched Black

Swans

Several ways to think about dependent“wild” events

• If considering extreme values, the largest (or smallest) of a set, and can use, or infer, a model, then EV theory [e.g. Gumbel; Coles] is mature … & is now developed to encompass clustering.

• Rather than EVT, however, today concerned with burst framework (“grouped grey swans”) .

• Why is burst a different concept, and what kind of “routinely extreme” randomness does it represent ?

• And why do I use the phrase “routinely extreme” 30 July 2014 14

Grey swans: “routinely extreme”• In The Black Swan, Taleb observes that “Extremistan does not

always imply Black Swans. Some events can [be]... somewhat predictable [ ...]. They are near–Black Swans ... I call this special case of “gray” swans Mandelbrotian randomness”.

30 July 2014 15

600 700 800 900 1000 1100 1200 13009

10

11

12

13

14

15Annual minimum level of Nile: 622-1284

Annual m

inim

um

:

Time in years

As John Prescott put it (and was widely mocked) :

“Extreme events will be the norm”

Grey swans: “routinely extreme”• In The Black Swan, Taleb observes that “Extremistan does not

always imply Black Swans. Some events can [be]... somewhat predictable [ ...]. They are near–Black Swans ... I call this special case of “gray” swans Mandelbrotian randomness”.

30 July 2014 16

600 700 800 900 1000 1100 1200 13009

10

11

12

13

14

15Annual minimum level of Nile: 622-1284

Annual m

inim

um

:

Time in years

Or as JP might have put it:

“Extremely large &/or long-lived

events will be commonplace”

Gaussian (normal): “mild” variation

30 July 2014 17

21exp{ (1/ 2)[( ) / ]) }

2(p

If we want to model

positive rvs an exponential

gives light tails, for example.

30 July 201418

Light tail

Heavy tail

Tails may be not just fat however, but (rigorously) heavy …

Pareto distribution is one such

( ) ( )x

F x p d df

( ) 1 ( ) 0,F x F x x Tail of

df

0( ) ,

1( ) 0I

x

F y dx xF y

Integrated tail of df

Example, tail of Pareto distribution makes the integral

in the Cramer-Lundberg condition diverge:

0 0( ) (1 )x xe x dxF e x dx

Essential distinction of small & large

claim distributions [Embrechts et al book]

… can arise from large jumps

30 July 2014 19

Mandelbrot, J. Business, 1963

30 July 2014 20

100 200 300 400 500 600 700 800 900 1000-5

0

5

10

15

20

25

30

35

40

am

plit

ude

time

stable

Gaussian

Mandelbrot: “wild” symptoms …

30 July 2014 21

Heavier than

Normal tails

Wildly fluctuating

second moment

… proposed an α-stable cure ?

30 July 2014 22

Tails of stable df

Data

Auroral physics: fat tails

pdf of AE

Magnetometer time series

AU

AL

AE=AU-AL

Hnat et al, NPG [2004]

S&P 500

Mantegna & Stanley

White noise: fast fluctuations

30 July 2014 24

( ) ( ) ( ) ( )t t dt

Delta correlated

White noise

A new puzzle: the Nile …

Hurst (1880-1978) studied this dataset from point of view of design of “large and long term over-year storage … ‘century storage’”30 July 2014 25

600 700 800 900 1000 1100 1200 13009

10

11

12

13

14

15Annual minimum level of Nile: 622-1284

Annual m

inim

um

:

Time in years

Ideal reservoir

• Average influx over

years - we need to ensure annual released

volume equals mean influx:

Accumulated deviation

of the influx from the

mean:

30 July 2014 26

1

1( )

t

t

1

) { ( )( },t

u

uX t

Range:

Standard deviation:

Plot against interval (loglog).

In white Gaussian case Feller:

(Rescaled) range

30 July 2014 27

R

S

0 0max ) min , )( (,t tx xtR t

1S

1/2/ ~R S

Hurst found anomalous growth of R/S,

exponent J about 0.7

Hurst’s effect

30 July 2014 28

/ ~ JR S

Hurst, Nature, 1957

Bold ine, mean

value over

datasets.

Circles, trials from

Hurst’s own model.

Joseph effect

30 July 2014 29

Joseph effect

30 July 2014 30

Joseph formalised

30 July 2014 31

2 1( ) dk ck

0 1/ 2d

( )k

k

k

( )k

k

k

Beran, “Statistics for long

Memory processes”, 1994

Fast decaying autocorrelation

e.g. white noise, delta correlated;

or AR(1), exponentially correlated

Short range dependence (SRD) Long range dependence (LRD)

Modelling heavy tails and LRD

30 July 2014 32

Mandelbrot and Wallis, 1969

Fractional hyperbolic noise

Linear Fractional Stable Motion: combines heavy tails & long range

dependence

• C.f. Mandelbrot’s fractional Brownian motion but integrates -stable “Levy” (not Gaussian) noise (e.g. Samarodnitsky & Taqqu).30 July 2014 , 33

1 11( ) ( ) ( ) ( )

H H

H HR

X t C t s s dL s

Memory kernel: d measures LRD α-stable jump:

heavy tails

1d H

LFSM in Markovian (d=0), but heavy tailed, limit = stable motion

30 July 2014 34

1( ) ( )H HR

X t C L s

Memory kernel =1, here α-stable jump: heavy tails

1H

LFSM in long range dependent, light tailed limit, fractional

Brownian motion.

30 July 2014 35

1/21/2

2,2 2

1( ) ( ) ( ) ( )H

HR

H

HX t C t s s dL s

Memory kernel: Joseph effect Gaussian jump

2

1d H

Bursts

30 July 2014 36

2

1

) '( 't

ts x t dt

2

1

't

tT dt

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, Watkins & Riley [PRE, 2000].

log

P(s) size

length

waiting time

some bursts we made earlier

Data

Simulations of light-tailed bursts• Power laws for pdf of burst size s, and duration T

predicted to have exponents =-2/(1+H) & =2-H

respectively [Watkins et al, Phys. Rev. E, 2009 ].

• Good agreement in Gaussian (fBm) limit: Confirmed findings of Carbone et al [PRE, 2004] & Rypdal and Rypdal [PRE, 2008].

30 July 2014 38

=-2/(1+H) =2-H

gamma

H H

beta

Simulations of heavy-tailed bursts• Watkins et al, PRE, 2009 found expressions

also reasonable down to ~ 1.6, but to fail completely by =1.

Nonstationarity

of LFSM is one

conceptual isssue,

also technical ones,

work in progress.30 July 2014 39

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

Other approaches: multifractalsMay wish for an even more volatile time series--

- a multifractal then more useful. Dependence sometimes seen not in amplitude but in the unsigned magnitudes, counterpart of the “volatility bunching” seen in finance.

30 July 2014 40

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

-600

-400

-200

0

200

400

600

incre

ments

, 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

Recap• Some natural severe events may be rare extremes, tractable with EVT

[e.g. Coles, 2001], adapted to allow correlation if present.

• Others may belong to a different class [studied by Mandelbrot] “Bursty” time series may show comparatively frequent high amplitude events, and/or long range correlations between successive values. The frequent large values due to the first of these effects can give rise to an burst composed of successive wild events. Conversely, long range dependence, even in a light-tailed Gaussian model, can integrate ``mild” events into a extreme burst

• I showed a standard statistical time series model, linear fractional stable motion (LFSM), which allows these two effects to be varied independently. Presented results from a preliminary study of such bursts [Watkins et al, PRE, 2009].

• Other options for burst scaling modelling including multiplicative cascades (such as multifractals).

41

Thank many people including Tim Graves (Cambridge), Dan Credgington (now Cambridge) , Sam Rosenberg (Now Barclays Capital), Christian Franzke (BAS), Bogdan Hnat (Warwick), Sandra Chapman (Warwick), Nicola Longden (BAS), Mervyn Freeman (BAS), Bobby Gramacy (Chicago), Dave Stainforth and Lenny Smith (LSE), and my very patient GRL editor: Paul Williams

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, AGU Hyderabad Chapman Conference Proceedings (2012)

Watkins, GRL Frontiers, 2013, doi: 10.1002/grl.50103

Spares

30 July 2014 43

Hurst’s model(s)

30 July 2014 44

Obituary

Compound extremes ...

30 July 2014 45

Compound extremes ...

30 July 2014 46

Compound extremes ...

30 July 2014 47