Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
Quantifying spatial correlation in the turbulent solar
wind flow using mutual information: simultaneous
in-situ spacecraft observations from WIND and
ACE.
R. T. Wicks, S. C. Chapman, R. O. Dendy
Center for Fusion, Space and Astrophysics
University of Warwick
Coventry
UK
r.wicks@warwick.ac.uk
Thanks to the WIND and ACE magnetometer and SWE teams
Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
1
Background
Introduction
Histograms
2
Mutual Information: A Test Model
Vicsek Model Rules
Behaviour
Order Parameters
Results
Limited Data
3
Application: The Solar Wind
Solar Wind MI
Correlation vs. MI
4
Conclusions
Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
Introduction
Histograms
Introduction
Developed by Shannon in 1949
Mutual information (MI) is a measure of shared entropy between
two signals (X and Y )
I (X ; Y ) = H(X ) + H(Y ) − H(X , Y )
H(X ) = −
P
i
P(Xi )log2 (P(Xi ))
Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
Introduction
Histograms
Histograms
50
0.04
40
0.03
30
P(θ)
Raw
data
X
Probabilities P(X ), P(Θ) and P(X , Θ) estimated by frequency of
occurrence
20
0.02
0.01
10
0
−3 −2 −1
0
θ
1
2
0
−3 −2 −1
3
0.06
0
θ
1
2
3
50
40
0.04
X
P(X)
X
position
histogram
Θ value
histogram
30
20
0.02
10
0
0
10
20
X
30
40
50
−3 −2 −1
0
θ
1
2
3
Joint
probability
distribution
P(X , Θ)
Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
Vicsek Model Rules
Behaviour
Order Parameters
Results
Limited Data
Vicsek Model Rules
A model for co-orienting self-propelled particles
Particles align within radius R
x in+1 = x in + v in δt
v in = v0 cos θni x̂ + sin θni ŷ
i
θn+1
= hθnNR i + δθni
δθni
∈ [−η, η]
Vicsek et al 1995.
Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
Vicsek Model Rules
Behaviour
Order Parameters
Results
Limited Data
Behaviour
Low noise → ordered motion
High noise → disorder
Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
Vicsek Model Rules
Behaviour
Order Parameters
Results
Limited Data
Order Parameters
Order parameter:
φ =
1
Nv0
N
X
vi
i=1
Susceptibility:
χ =
1
hφ2 i − hφi2
N
Wicks et al PRE 2007 (in press)
Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
Vicsek Model Rules
Behaviour
Order Parameters
Results
Limited Data
Vicsek Model MI
MI is calculated between position (X , Y ) and angle of motion Θ
using a histogram method
I (X , Θ) =
X
I (Y , Θ) =
X
P(Xi , Θj )
P(Xi )P(Θj )
P(Yi , Θj )
P(Yi )P(Θj )
P(Xi , Θj ) log2
i,j
P(Yi , Θj ) log2
i,j
I
=
I (X , Θ) + I (Y , Θ)
2
Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
Vicsek Model Rules
Behaviour
Order Parameters
Results
Limited Data
Vicsek Model MI
MI and χ are calculated on snapshots of the system
ηc defined by peak χ
Error on MI much
smaller than on χ
around peak
Matsuda et al Int. J. Theor.
Phys. 1996
Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
Vicsek Model Rules
Behaviour
Order Parameters
Results
Limited Data
Limited Data
Real world measurements often measure local or reduced
dimensional data from a many degree of freedom system
Take timeseries data from
only 10 particles from
3000
Divide into 10 smaller
sections
Calculate MI
χ=
MI
φ
1
N
hφ2 i − hφi2
Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
Vicsek Model Rules
Behaviour
Order Parameters
Results
Limited Data
Limited Data
Error on χ is very large
Peak approximately
correct for MI
χ cannot identify ηc
Wicks et al PRE 2007 (in press)
Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
Solar Wind MI
Correlation vs. MI
Solar Wind MI
ACE and WIND data
MI calculated with lag
Min MI 0.0001
Max MI 1.42
MI gives indication of
typical correlation length
within solar wind
Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
Solar Wind MI
Correlation vs. MI
Mean Correlation Coefficient
Correlation vs. MI
0.8
0.6
0.4
0.2
0
−0.2
−180
0.8
|B|
B
−120
−60
0
60
120
180
x
By
Mean MI
Bz
0.6
0.4
0.2
−180
−120
−60
0
60
Lag (minutes)
MI and correlation
agree on time lag and
ordering of the data.
120
180
MI peaks sharper
than correlation.
Linear Correlation
identifies the
convecting structures.
Distinct signature in
MI.
Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
We used a simple model with dynamics and a phase transition
to compare susceptibility and MI.
Mutual information is able to identify the phase transition
accurately.
When knowledge of the system is limited, MI performs better
than susceptibility at identifying the phase transition.
MI demonstrated on ‘real world’ dataset of multiple spacecraft
measurements of the solar wind.
Both linear correlation and MI detect convecting structures.
Result from MI and linear correlation are distinct. Reflects
presence of both linearly convecting structures and evolving
turbulence in the flow.
Wicks et al PRE 2007 (in press)
Background
Mutual Information: A Test Model
Application: The Solar Wind
Conclusions
Correlation in θ
Linear Correlation
1
0.75
0.5
Linear correlation of
Vicsek model
0.25
0
0
1
2
3
4
Correlation in x
Noise η
0.9
0.8
No indication of phase
transition
0.7
0.6
0.5
0
1
2
Noise η
3
4