Old and New Methods in the Age of Big Data

Zoltán Somogyvári1,2

1Department of TheoryWigner Research Center for Physics

of the Hungarian Academy of Sciences2National Institute for Clinical Neuroscience

How do we find structures behind

the data?

Transformation of time series into

connections

How to find connection between data series?

The traditional method: Correlation(more precisely, the linear correlation coefficient)

Days

Exc

hang

e ra

te

USD vs GBP

2*EUR vs GBP

How to find connection between data series?

The traditional method: Correlation(more precisely, the linear correlation coefficient)

US

D v

s G

BP

2*EUR vs GBP

R=0.6

What does the correlation tells us? Problem 1: it is possible, that there is a clear connection between the two time series, but the correlation is 0 because of the non-linear form of connection.

Causality or common cause?

Is the temporal delay shows us the direction of the causality?

Cross correlation function:Correlation between delayed signals

Unfortunately not, because it assumes, that we observe the twoSignals with the same delays.

USD vs GBPleads

EUR vs GBPleads

Delay

Is there any way to infer causality?

Granger-causalityThe original idea came from Norbert Winer

x → y, if the inclusion of past x values improves the prediction quality on y

?Clive GrangerPublication 1969

Nobel price in Economic Sciences 2003

Granger-causality

Presumtions:– Stationary processes– Zero-mean– Uncorrelated

Gaussian noise– We have data of

every important valiable

?Linear autoregression:

Application – rat hippocampus

● Data:– Local Field Potential

→ Microelectrode-array● 256 channels● 20 kHz freq

● Information stream in the hippocamus

– options:→ State-dependent differences

● ie. sleep-awake→ Event-related

● Spike-related information transfer

Problems with the Granger-causality

Uses linear models (there are nonlinear extensions)

Assumes weak interactions (separability)

Unreliable results in case of circular causality

Has problems in deterministic (non-stochastic) cases

Cross Convergence Map:A new framework for causality analysis

A new approach, promising

● Detection of circular causality● Causality in nonlinear system● Deterministic (chaotical) system

Science 338, 496 (2012)

The model system: The logistic map

A one dimensional, discreet-time dynamical system implementingstretching an folding transformations.

xn+1=rxxn(1-xn)

The model system: The logistic map

It can exhibit different dynamical behavior, from stable fixpoint, throughperiodic oscillations to chaos, depending on the parameter r.

Taken's time delay embedingtheorem

First coordinate: the data itselfSecond coordinate: the data delayed by tauThird coordinate: the data delayed by 2 tau…....

The trajectory reconstructed in the state space is topologically equivalentWith the trajectory of the system's original trajectory in its real space.

Our model system: Two coupled logistic maps

xn+1=xn(rx(1-xn)+byxyn) yn+1=yn(ry(1-yn)+bxyxn)

rx=ry=3.8 so both maps are in the chaotic regime

Phase-space reconstruction based on delayed maps

Reconstructed state-space from thefirst data series in 3 embedding dimension

Reconstructed state-space from thesecond data series in 3 embedding dimension

Both dataset formed a 2D manifold in the 3D embedding space

xn+1=xn(rx(1-xn)+byxyn) yn+1=yn(ry(1-yn)+bxyxn)

Reconstructed state-space from thefirst data series in 3 embedding dimension

Reconstructed state-space from thesecond data series in 3 embedding dimension

Both dataset formed a 2D manifold in the 3D embedding space

In case of causal connections, the the reconstructed manifoldsholud be topologically equivalent according to the Takens' theorem.

But, how to test it?

Reconstructed state-space from thefirst data series in 3 embedding dimension

Reconstructed state-space from thesecond data series in 3 embedding dimension

Both dataset formed a 2D manifold in the 3D embedding space

Choose a point

Sugihara's method: Convergent Cross mapping

Reconstructed state-space from thefirst data series in 3 embedding dimension

Reconstructed state-space from thesecond data series in 3 embedding dimension

Both dataset formed a 2D manifold in the 3D embedding space

Find its neighborhood

Sugihara's method: Convergent Cross mapping

Reconstructed state-space from thefirst data series in 3 embedding dimension

Reconstructed state-space from thesecond data series in 3 embedding dimension

Lets do it for many points! If the neighbors in the first space are neighborsin the the second space as well, then the second variable is causal to thefirst one.

Find the same time points in the other state space

Sugihara's method: Convergent Cross mapping

In case of circular causality the mapping should work in both directions

Reconstructed state-space from thefirst data series in 3 embedding dimension

Reconstructed state-space from thesecond data series in 3 embedding dimension

Let us do it into the other direction!

Sugihara's method: Convergent Cross mapping

Reconstructed state-space from thefirst data series in 3 embedding dimension

Reconstructed state-space from thesecond data series in 3 embedding dimension

Let us do it into the other direction!

The chosen point

Sugihara's method: Convergent Cross mapping

Reconstructed state-space from thefirst data series in 3 embedding dimension

Reconstructed state-space from thesecond data series in 3 embedding dimension

Let us do it into the other direction!

The neighborhood

Sugihara's method: Convergent Cross mapping

Reconstructed state-space from thefirst data series in 3 embedding dimension

Reconstructed state-space from thesecond data series in 3 embedding dimension

The mapping worked well into both directions!This is the sign of circular causality.

Mapping

Sugihara's method: Convergent Cross mapping

Cross mapping in case of unidirectional interactions

How can be the topological equivalence is an asymmetric relation?

Reconstructed state-space from thefirst data series in 3 embedding dimension

Reconstructed state-space from thesecond data series in 3 embedding dimension

While the first dataset formed a 2D manifold, the second dataset resultedan only 1D manifold in the 3D embedding space!

yn+1=ryyn(1-yn)xn+1=xn(rx(1-xn)+byxyn)

Cross mapping in case of unidirectional interactions

How can be the topological equivalence is an asymmetric relation?

Reconstructed state-space from thefirst data series in 3 embedding dimension

Reconstructed state-space from thesecond data series in 3 embedding dimension

While the first dataset formed a 2D manifold, the second dataset resultedan only 1D manifold in the 3D embedding space!

Mapping works well in this direction

Cross mapping in case of unidirectional interactions

How can be the topological equivalence is an asymmetric relation?

Reconstructed state-space from thefirst data series in 3 embedding dimension

Reconstructed state-space from thesecond data series in 3 embedding dimension

The mapping worked well from x to y but failed from y to x, showing,that y is causal to x but x is not causal to y.

But spread out in the other direction!

MRI with implanted electrodes

4*8 channels in the grid plus 2*8 channelsIn two strip electrodes, 1024 Hz sampling

EEG signal of an epileptic seizure recorded on 48 channels

Ele

ctric

Po

ten

tial [

mV

]

Time [ms]

The initiation of the seizureE

lect

ric

Pot

entia

l [m

V]

Time [ms]

Connection dynamics during seizure

CausalityRight→LeftLeft→Right

This seizure appeared only on in the right hippocampus.

It is clear, that the right hippocampus has large effect to the Left hippocampus, while there is only mild effect in the backward direction.

Rig

ht F

OLe

ft F

O

LFP

Time [s]

Rig

ht F

OLe

ft F

O

LFP

Causality

Right→LeftLeft→Right

The seizure was more pronounced in the left hippocampus,

Although,

The right hippocampusdrove the left during the first period of the seizure, thena circular connection structure emerged.

Connection dynamics in seizure

May help in surgical preparation

Time [s]