A Random Walk Through Search Research

A Random Walk Through Search
Research
Nick Watkins
NickWatkins@mykolab.com

Thanks
• Rainer Klages for inviting me to participate in the ASG
• ONR for funding my participation in the ASG
• PKS for my visiting position in 2013-2014, hosted in
Holger Kantz’s group, and visits to this ASG in Dresden
• Ralf Metzler for hosting me at Potsdam during ASG
Discussions with many people including the above, Andy
Edwards, Mervyn Freeman, Sandra Chapman, Yu-Xi Chau,
Eli Barkai, Igor Sokolov, Eugene Murphy, Richard Philips,
Aleksei Chechkin, Gandhi Viswanathan, Gene Stanley,
Marcos Da Luz, and the 2015 ASG team.

Overview I
• Brownian paradigm: keeps Central Limit
Theorem, adapts walk models for movement
& search research
• Continuous Time Random Walk & compound
Brownian models as exemplar.
• Non-Brownian paradigms: arise from going to
ECLT and/or modifying time dependence
• Levy Flights/Walks, Fractional Time Process as
resulting exemplars
• Levy Foraging (or Environmental) Hypothesis
as resulting conjectures

Overview II
• My current interest: Consequences of
different ways of introducing time
dependence-either Markovian or ergodic
• Disclaimer: First parts strongly influenced by
my own experience in 2005-8 [Edwards,
Philips, Watkins et al, Nature, 2007].
Hopefully, however, its semi-historical nature
sets a stage for those who will go on to bring
you up to date [Zaburdaev, Bartumeus and
others].

Movement modelling
• In widespread use, for very diverse reasons …
• Home ranges of foraging animals [e.g. Randon-
Furling et al, PRL, 2009; Claussen et al, 2015]
• Tracking shoppers in malls [e.g. Path Intelligence].
THEORY PRACTICE

Search Research
• Chupeau diagram
iRobot’s Firstlook
Chupeau et al, 2015
THEORY
PRACTICE

Why (and which) stochastic model?
• Movement patterns of biological organisms,
including humans, very rich and complex.
• Fortunately, while reasonable to assume that
animal movements deterministic on small
spatiotemporal scales, on larger scales
foraging paths exhibit random patterns so
stochastic approaches to modelling exploited.
• But “randomness” comes in many flavours,
how to choose right one for given case?

Random walks & BM
• Pearson quote
The connection of
animal motion and
random walks was
made more than a
century ago by
pioneering
statistician and
biometrician Karl
Pearson.
[Pearson, Biometric
ser., 3, 54 (1906)].

• Pearson also coined the term random
walk when posing it as a problem in a
famous letter to Nature, which was
answered by Lord Rayleigh who
showed the Gaussian nature of the
solution and linked it to a problem
he had solved in physics.
• As the theory of Brownian random
walks and the Wiener process rapidly
developed in statistical physics and
mathematics, this concept came to
dominate the stochastic modeling of
movement in biology.

CTRW as archetype
1
( ) i
n
i
X t 

 
1
i
n
n
i
t 

 
/
( )x
 
 
Notation as Fulger et al, PRE (2008)
-8 -6 -4 -2 0 2 4 6 8
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
4

Histogram of jump sizes
0 2 4 6 8 10 12 14
10
0
10
1
10
2
10
3
10
4
10
5
10
6

Semilog histogram of waiting times
( , ) ( ) ( )      

2D CTRW
0 1 2 3 4 5 6 7 8 9 10
x 10
5
-1000
-500
0
500
1000
1500
2000
2500
3000
Time
x,ycomponents
Time series
x
y
-1000 -500 0 500 1000 1500
-500
0
500
1000
1500
2000
2500
3000
Spatial pattern
x
y

2D CTRW
0 1 2 3 4 5 6 7 8 9 10
x 10
5
-1000
-500
0
500
1000
1500
2000
2500
3000
Time
x,ycomponents
Time series
x
y
Another choice would
be single random vector
length R with uniform
random angle θ
– which choice is adopted
becomes more important for
non-Brownian cases
-1000 -500 0 500 1000 1500
-500
0
500
1000
1500
2000
2500
3000
Spatial pattern
x
y

CTRW = renewal reward process
Here jumps at {J}
become
rewards {W} and
waiting times
become holding
times {S}
Much scope for interplay of maths and physics

Compound Brownian ?
• As measurement techniques in experimental biology advanced,
was found that ordinary Brownian motion too simple to generally
represent foraging paths.
• Variations on simple random walk paradigm, notably composite
correlated random walks and complex intermittent search
strategies combining scanning and relocation modes developed and
applied to explain biological movement data with much success [e.g
Benhamou, 2014; Benichou et al, 2011; Campos et al, 2014].
• All generate “normal diffusion” in long time limit, mean squared
displacement of ensemble of foragers grows linearly in time.

Symptoms of complex transport: 1
Possibility of very long jumps
(“flights”) compared to the <jump>

Anomalous diffusion ?
• However, at least one observed feature of animal foraging (&
human movement) motivates something other than
Brownian, or even compound Brownian, picture.
• Many individuals across a broad range of species observed to
make small local random movements interspersed with rare
long-distance jumps. Particularly true of affluent modern
humans …
• In consequence Brownian theoretical framework for diffusion
in both biology and the physical sciences has been challenged
over the past three decades by a new emerging synthesis.
• This is drawn both from data and from the theory of those
stochastic processes which generate non-Brownian
“anomalous” diffusion, where the mean squared
displacement grows nonlinearly in the long time limit.

ECLT
• Brownian motion
paradigm for random
walks was closely linked
to central limit theorem
(CLT), but known for
much of the last century
that relaxing just one of
assumptions-finite
variance-gives a new
class of random walk.
• This class obeys extended
central limit theorem-has
a very different
probability distribution
for the size of a step-
asymptotic power law
tail.
(1 )
( ) 
   
 0<α< 2

Levy flights
• The resulting stochastic movement model is
called a Lévy flight. Intriguingly, like the
biological observations, a “Lévy flyer” has
apparent clusters linked by long jumps.
-0.5 0 0.5 1 1.5 2 2.5
x 10
4
-8000
-6000
-4000
-2000
0
2000
4000
Spatial pattern
x
y
α=1.5,
=1
0 2 4 6 8 10 12
x 10
5
-1
-0.5
0
0.5
1
1.5
2
2.5
x 10
4
Time
x,ycomponents
Time series
x
y

Symptoms of complex transport: 2
... longer waiting times

A Random Walk Through Search Research

Fractional Time Process
• Another type of
anomalous
diffusion-what
you get if you
keep Markovian
structure and
factorising
probability but
allow long tails in
waiting time
between jumps.
Example here is Mittag-Leffler, has advantage that one limit is exponential

0 1 2 3 4 5 6 7 8 9 10
x 10
5
-1000
-500
0
500
1000
1500
2000
2500
3000
Time
x,ycomponents
Time series
x
y
FTP Time series
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10
6
-3000
-2500
-2000
-1500
-1000
-500
0
500
1000
Time
x,ycomponents
Time series
x
y
α=2, β= 0.9
Brownian example for
comparison

Can Combine LF & FTP
0 0.5 1 1.5 2 2.5
x 10
8
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
x 10
4
Time
x,ycomponents
Time series
x
y
α=1.5, β=0.7
-14000 -12000 -10000 -8000 -6000 -4000 -2000 0 2000 4000
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
x 10
4
Spatial pattern
x
y
time series
space
Called “Fully Fractional CTRW”

Dollar bills [Brockmann, 2006]
Data fitted to an FFCTRW with α, β both about 0.6
See also Bartumeus, Giuggioli et al’s work on Shearwaters, 2010

08/09/2015
From supp. Info. of [Brockmann et al, 2006] comes very useful schematic-
NB they defined (,) opposite way to Fulger and others.
Actually not fBm,
but rather in fact the
fractional time
process (FTP).
Don’t believe
everything you
read in Nature ;-)
Can Combine LF & FTP

Another way: couple space to time
Gives a finite velocity by introducing a jump duration τ’ & coupling
the jump size to it – idea known as Lévy walk
[Shlesinger & Klafter, PRL (1985)].
( , ') ( '| ) ( )
(| | ') ( )
      
    
 
 
Lévy walk
( , ) ( ) ( )      Uncoupled CTRW
In above τ’ means flight duration in Levy walk, and
τ waiting time in CTRW.

Viswanathan et al
1996 [V96]
“In the days of sail the bird often
accompanied ships for days, not merely
following it, but wheeling in wide circles
around it without ever being observed to
land on the water.“ - Wikipedia

Time series data is number of consecutive hours that
bird is not wet.
13:00 14:00 15:00 16:00 17:00
Dry for full hour
Wet at least once Wet at least once
Data:

Levy Environmental Hypothesis I
V96
•

Levy Environmental Hypothesis II
V96
•

Lévy Foraging Hypothesis I
Schlesinger & Klafter ended their comparison of Lévy
walks & Lévy flights in a landmark 1985 NATO ASI
volume with this comment: “ It has been suggested
[B. Ninham, priv. comm.] that certain animals such as
ants perform Levy walks when searching for food in a
new area. The above analysis may imply that starving
Levy walk ants possess a slight evolutionary
advantage over ants performing other walks, such as
even the [self avoiding walk]. Flying ants can be
considered by the reader.” – in “On Growth & Form”,
Stanley & Ostrowsky (eds. ) Editions Nijhoff,
1986.

Lévy Foraging Hypothesis I
Schlesinger & Klafter ended their comparison of Lévy
walks & Lévy flights in a landmark 1985 NATO ASI
volume with this comment: “ It has been suggested
[B. Ninham, priv. comm.] that certain animals such as
ants perform Levy walks when searching for food in a
new area. The above analysis may imply that starving
Levy walk ants posess a slight evolutionary
advantage over ants performing other walks, such as
even the [self avoiding walk]. Flying ants can be
considered by the reader.” – in “On Growth & Form”,
Stanley & Ostrowsky (eds. ) Editions Nijhoff,
1986.

Lévy Foraging Hypothesis II
V96

Lévy Foraging Hypothesis II
V96
Viswanathan et al 99

Edwards, Philips, Watkins et al, 2007
• Time series including take off and landing

Edwards et al, op cit.
• Waiting times
after correction:
Short-tailed, unlike a Lévy walk ...but like the “ studies
[which] found [birds spent]... as long as 18 hours a
day in flight”: Poncet, Nat. Geographic, March 1989.

Re-examination of LW/LFH …
• Travis, 2007; Buchanan, 2008; Barabasi,
2010;Viswanathan et al 2012. ...

… only increased activity in the field
Pyke, 2014
Two particular legacies of time:
Renewed attention to statistical
inference and other issues around
measurement
Consideration of a wider range
of models and paradigms

How would cleverest random
walker play battleships
How does even young child play
battleships ?
Gedanken experiment

Models for “1/f”
Ionosphere
Magnetosphere
Selecta H
Selecta N

Choices for time dependence
• CTRW is Markovian in structure- “memory” is
modelled by using heavy tailed times between
jumps. [Mandelbrot, 65-67] knew that FTP-
like process was non-ergodic because its
periodogram grows with observation time !
• Alternative is an ergodic, non-Markovian
process where memory is embodied in the
kernel [Mandelbrot and co-authors, 65-68]

Ionosphere
Magnetosphere
• Abrupt state changes
• Fat distributions of switching times: “Levy” (E[t^2] = ∞) case.
“What you see is what you wait for”

The conditional spectrum:
Magnetosphere
Mandelbrot 1967 reviewed in N2, Selecta, 1999

The conditional spectrum:
Magnetosphere
Mandelbrot 1967 reviewed in N2, Selecta, 1999
• “Numerical … 1/f … spectrum … need not … estimate …
Wiener-Khinchine spectrum”. Instead “depends on
conditioning length T”. Unlike stationary LRD model,
singularity is an artefact.

Ionosphere
Magnetosphere
1 1
2 2
, 22fBm: ( ) ( ) ( )~ ( )
 
 
  
 
 
  
H
H R
H
X t t s s dL s
Infinite range
memory kernel
Gaussian
fBm

Ionosphere
Magnetosphere
[…], if infinite dependence is necessary it does not mean
that IBM's details of ten years ago influence IBM today,
because there's no mechanism within IBM for this
dependence. However, IBM is not alone. The River Nile is
[not] alone. They're just one-dimensional corners of
immensely big systems. The behaviour of IBM stock ten
years ago does not influence its stock today through IBM, but
IBM the enormous corporation has changed the environment
very strongly. The way its price varied, went up or went up
and fluctuated, had discontinuities, had effects upon all kinds
of other quantities, and they in turn affect us. –
Mandelbrot, interviewed in 1998 by B. Sapoval for Web of
Stories
What does fBm mean ?

Ionosphere
Magnetosphere
[…], if infinite dependence is necessary it does not mean
that IBM's details of ten years ago influence IBM today,
because there's no mechanism within IBM for this
dependence. However, IBM is not alone. The River Nile is
[not] alone. They're just one-dimensional corners of
immensely big systems. The behaviour of IBM stock ten
years ago does not influence its stock today through IBM, but
IBM the enormous corporation has changed the environment
very strongly. The way its price varied, went up or went up
and fluctuated, had discontinuities, had effects upon all kinds
of other quantities, and they in turn affect us. –
Mandelbrot, interviewed in 1998 by B. Sapoval for Web of
Stories
What does fBm mean ?
• Resolution of apparent paradox is that world as a whole is
Markovian, the memory is a consequence of looking at a
piece of it. Generalises the Mori-Zwanzig idea.

Conclusions
• Random walks and biology co-evolved from
earliest days
• Still a productive dialogue to which ASG will
contribute
• Also links to deep and current issues in
fundamental statistical mechanics such as
weak ergodicity breaking

Theme
Hurst
effect
Will today distinguish
three things often taken
as same
• Observed growth of
range in time series:
“Hurst effect”

Theme
1/f
Hurst
effect
Will today distinguish
three things often taken
as same
• Observed growth of
range in time series:
“Hurst effect”
• Observation of a
singularity at zero in
Fourier spectra: “1/f”

Theme
(S)LRD
1/f
Hurst
effect
Will today distinguish three
things often taken as same
• Observed growth of range
in time series: “Hurst
effect”
• Observation of a singularity
at zero in Fourier spectra:
“1/f”
• The long range dependence
seen in stationary 1/f case:
(S)LRD.
• Using 1/f as a diagnostic of
LRD assumes stationarity

Fact: Anomalous growth of range
Hurst
Effect
Hurst, Nature, 1957
“I heard about the … Nile … in '64, ... the variance doesn't draw l
time span as you take bigger and bigger integration intervals;
it goes like time to a certain power different from one. … Hurst …
was getting results that were incomprehensible”. – Mandelbrot, 1

Formula: Long Range Dependence
(S)LRD
Hurst
Effect
• Mandelbrot, van Ness, and
Wallis, 1965-69
• First [history in Graves et al,
arXiv, 2014a] demonstration
that Hurst effect could be
explained by stationary long
range dependent process
• Model, fractional Gaussian
noise [see also Kolmogorov’s
“Wiener Spiral”], had singular
spectral density at lowest
frequencies.
'( ) ~ 
S f f

The 1/f “paradox”
If spectral density '( )
then i) it is singular as
and ii) if we define an autocorrelation
function via ( ) ( ) ( )
and use Wiener-Khinchine theorem to
get from Fourier transform of
~
0

  


  

S f f
f
x t x t
S
falls off as power law, and
'( )
then
summed lags "blow up"
its
( )

   
f

Fractional motions and noises
0 100 200 300 400 500 600 700 800 900 1000
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
2
Fractional Brownian motion, H=0.7
1 1
2 2
, 22fBm: ( ) ( ) ( )~ ( )
 
 
  
 
 
  
H
H R
H
X t t s s dL s
Build a nonstationary, self
similar walk … (used wfbm in
Matlab)
fractional motion
2 1  H

Fractional motions and noises
0 100 200 300 400 500 600 700 800 900 1000
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
2
Fractional Brownian motion, H=0.7
1 1
2 2
, 22fBm: ( ) ( ) ( )~ ( )
 
 
  
 
 
  
H
H R
H
X t t s s dL s
Build a nonstationary, self
similar walk … (used wfbm in
Matlab)
0 100 200 300 400 500 600 700 800 900 1000
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Fractional Gaussian noise
fractional motion
Then differentiate to give a
stationary LRD noise
fractional noise
2 1  H 2 1  H

1/f without (S)LRD
(S)LRD
1/f
Hurst
effect
• Before (S)LRD models,
Mandelbrot [1963-67]
had proposed other 1/f
models which were not
stationary LRD in same
sense as fGn.
• Solved 1/f paradox by a
different route. Still
little known in the
geosciences [but see
Klemes, WRR, 1974].

Formula versus fact
“Like the ear, the eye is very
sensitive to features that the
spectrum does not reflect. Seen
side by side, different 1/f noises,
Gaussian [i.e. fGn], dustborne [i.e.
fractional renewal] and multifractal,
obviously differ from one another”-
Mandelbrot, Selecta N, 1999.
“Nothing can be more fatal to
progress than a too confident
reliance on mathematical symbols;
for the student is only too apt to …
consider the formula and not the
fact as the physical reality”.
Thomson (Kelvin) & Tait, 1890
edition.

Open (L) v closed (R)Brownian
walks
-2.5 -2 -1.5 -1 -0.5 0 0.5
-0.5
0
0.5
1
1.5
2
-1 -0.5 0 0.5
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
α=2

Open (L) v closed (R) Lévy flights
-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
-1 -0.5 0 0.5 1 1.5 2 2.5 3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
α=1.6

Boyer et al (2008)
• Boyer et al, arXiv, 2008 pointed out that jump
sizes of foraging model in Viswanathan et al,
1999 not same as those of simple Levy walk
used in Viswanathan et al, 1996. Worked out
distribution explicitly:

A Random Walk Through Search Research

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