Nonlinear estimators and time-embedding

Nonlinear estimatorsand time embedding

Raul Vicente [email protected]

FIASFrankfurt, 08-08-2007

OUTLINE

Introduction to nonlinear systems

2

Phase space methods

Exponents and dimensions

Interdependence measures

Take-home messages

OUTLINE


3

Definition

Why nonlinear methods?

Linear techniques

Phase space methods



Take-home messages

INTRODUCTIONDefinition

A nonlinear system is one whose behavior can‘t be expressed asa sum of the behaviors of its parts. In technical terms, the behaviorof nonlinear systems is not subject to the principle of superposition.

4

„Nonlinear“ is a very popular word in (neuro)science but what does it really mean?

The brain as a whole is a nonlinear „device“

Ex: our perception can be more than the sum of responsesto individual stimulus Surface completion



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Individual neurons are also nonlinear

Excitable cells with all or none responses Double the input does not

mean double the output

Nonlinear frequency response



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Individual neurons are also nonlinear

Double the input does not mean double the output

The brain as a whole is a nonlinear „device“

Ex: our perception can be more than the sum of responsesto individual stimulus

INTRODUCTIONWhy nonlinear methods?

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„The study of non-linear physics is like the study of non-elephant biology“Unknown

Neuronal activity is highly nonlinear

Nonlinear features will be present in the recorded neurophysiological data

From neuronal action potentials (spikes) to integrated activity (EEG, MEG, fMRI)

Linear techniques might fail to capture key information

Nonlinear indices: measure complexity of EEG, monitoring depth of anaesthesia, studies of epilepsy, detection of interdependence, etc...

INTRODUCTIONLinear techniques

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Linear systems always need irregular inputs to produce bounded irregular signals

Linear SystemLinear System

Most simple system which produces nonperiodic (interesting) signals is a linear stochastic process

...,Sn-1, Sn, Sn+1,...

Measurement of state Sn at time n of such a process p(s) probability dist.

Information about p(s) can be inferred from the time series:

1

1 N

nn

s sN

2

2

1

1

1

N

nn

s sN


7

Linear methods interpret all regular structure in a data set, such as a dominantfrequency, as linear correlations (time or frequency domain)

2

2 2

1 n nn n

s s sc s s s s

Autocorrelation at lag

Sn

Sn

-

Sn

Sn

-

Sn

Sn

-0c 0c 0c

Periodic signalStochastic processChaotic system

Periodic autocorrelationDecaying autocorrelationExponential decay ?


8

Cross-correlation function: measures the linear correlation between two variablesX and Y as a function of their delay time ()

1

1( ) ( ) ( )

N

XYk

C x k y kN

Cross-correlation at lag

( ) 0XYC ( ) 0XYC ( ) 0XYC

tendency to have similar values with the same signtendency to have similar values with opposite signsuggest lack of linear interdependence

EEG time series recorded from the two hemispheres in a rat

X´=X4

Y´=Y4

rxy = 0.63 rxy = 0.25

that maximizes this function

estimator delay between signals


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Cross-correlation function: measures the linear correlation between two variablesX and Y as a function of their delay time ()

1

1( ) ( ) ( )

N

XYk

C x k y kN

Cross-correlation at lag

( ) 0XYC ( ) 0XYC ( ) 0XYC

tendency to have similar values with the same signtendency to have similar values with opposite signsuggest lack of linear interdependence

that maximizes this function

estimator delay between signals

Cross-correlogram histogram is also used to reveal the temporal coherence in the firing of neurons

MT neurons in visual cortex of a macaque monkey


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Coherence: measures the linear correlation between two signals as a function ofthe frequency

2( )

( )( ) ( )

XYXY

XX YY

S fK f

S f S f Coherence at frequency f

0( ) 0XYK f

0( ) 1XYK f activities of the signals in this frequency are linearly independentmaximum linear correlation for this frequency

In forming an estimate of coherence, it is always essential to simulate ensemble averaging. EEGand MEG signals are subdivided in epochs or forevent-related data spectra are averaged over trials

( ( ))XYFFT C


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Prediction: we have a sequence of measuraments sn, n = 1,...,N and we want to predict the outcome of the following measurement, sN+1

Linear prediction1

1

m

n j n m jj

s a s

minimising the error

2

1 1

N

n nn m

s s

In-sample Out-of-sample

ja


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Causality: (Nobert Wiener in 1956) for two simultaneously measured signals, if one can predict the first signal better by incorporating the past information from the second signal than using only information from the first one, then the second signal can be called causal to the first one

Predicting the future of X improves when incorporating the information about the past of Y → Y is causal to XX1

Y1

Time

X2

In neurophysiology, a question of great interest is whether there exists a causal relation between twobrain regions. Inferring causality from the time delay in the cross-correlation is not always straighforward


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Causality: (Nobert Wiener in 1956) for two simultaneously measured signals, if one can predict the first signal better by incorporating the past information from the second signal than using only information from the first one, then the second signal can be called causal to the first one

X1

Y1

Time

X2

Granger just applied this definition in the context of linear stochastic models. If X is influencing Y, then adding the past values of the first variable to the regression of the second one will improve its prediction error.

Univariate fitting Bivariate fitting

Prediction performance: is assessed by the variances of the prediction errors

Granger causality

OUTLINE


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The concept of phase space

Attractor reconstruction

Time embedding

Application: nonlinear predictor

Phase space methods



Take-home messages

Phase spaceThe concept

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Phase space: is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space

Ex: the Fitzhugh-Nagumo model is a two-dimensional simplification of the Hodgkin-Huxley model of spike generation

In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space. Ex: dim(HH) = 4

- membrane potential- recovery variable

(V, W)V(t)

W(t)

For deterministic systems (no noise), the system state at time t consists of all information needed to uniquely determine the future system states for times > t

Phase spaceAttractor reconstruction

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Attractor: a set of points in phase space such that for "many" choices of initial point the system will evolve towards them. It is a set to which the system evolves after a long enough time

Attractor setA pointA curve

A manifold

BehaviorConstantPeriodic

Possibly chaotic

Van der Pol limit cycle attractor

Lorenz attractor

Strange attractors produce chaotic behavior.Nonlinear systems: irregular dynamics without invoking noise!

Phase spaceAttractor reconstruction

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In general (and especially in biological systems) it is impossible to access all relevant variables of a system. Ex: usually in electrophysiology we just measure membrane voltage.

[ ( ), ( ), ( 2 )]tX x t x t T x t T

These vectors constructed from a single variable play a role similar to [x(t),y(t),z(t)]

How from a single measured quantity can onereconstruct the original attractor?

?

http://www.scholarpedia.org/wiki/images/d/db/Lor3.png

http://www.scholarpedia.org/wiki/images/c/ce/Lor1.png

http://www.scholarpedia.org/wiki/images/8/8c/Lor2.png

Phase spaceTime embedding

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In general (and especially in biological systems) it is impossible to access all relevant variables of a system. Ex: usually in electrophysiology we just measure membrane voltage.

[ ( ), ( ), ( 2 )]tX x t x t T x t T

( )( , ( ), ( ), ( ));

( )( , ( ), ( ), ( ));

( )( , ( ), ( ), ( ));

dx tf t x t y t z t

dtdy t

g t x t y t z tdt

dz th t x t y t z t

dt

3 2( ) ( ) ( )( , ( ))

d x t d x t dx ta b p t x t

dt dt dt

2( ) ( )[ ( ), , ]

dx t d x tx t

dt dt2

( ) ( ) ( ) 2 ( ) ( 2 )[ ( ), , ]

x t x t T x t x t T x t Tx t

T T

Time delay embedding

[ ( ), ( ), ( 2 ),..., ( )]tX x t x t T x t T x t mT

m > 2DF

T ~ first zero autocorrelation

Phase spaceApplication: nonlinear predictor

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- A signal does not change is easy to predict: take the last observation as a forecast for the next one

Depending on the type of signals the power of predictability and the best strategy changes:

- A periodic system is also easy one observed for a full cycle

- For independent random numbers the best prediction is the mean value

- Interesting signals are not periodic but contain some kind of structure which can be exploited to obtain better predictions

If the source of predictability are linear correlations in time: next observations will be given approximately by a linear combination of preceding observations

1 01

m

n j n m jj

s a a s

What if I know that my series is nonlinear?

Phase spaceApplication: nonlinear predictor

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For nonlinear deterministic systems all future states are unambiguosly determinedby specifying its present state. Nonlinear correlations can be exploited with new techniques

Lorenz method of analogues:Prediction for the future state XN+1

Look for recorded states close to the one we want to predictPredict the average of the next states of the past neighbors

Current state

Neighbors

[ ( ), ( ), ( 2 )]tX x t x t T x t T

Next values of

( )

1

( )n N

N n n nX U XN

X XU X

Better prediction for short time scales than linear predictors

Predicted state

OUTLINE


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Sensibility to initial conditions: Lyapunov exp.

Self-similarity: correlation dimension

Phase space methods



Take-home messages

Exponents and dimensionsSensibility to initial conditions

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The most striking feature of chaos is the long-term unpredictability of the futuredespite a deterministic time evolution.

The cause is the inherent instability of the solutions, reflected in their sensitivedependence on initial conditions.

Amplification of errors since nearby trajectories separate exponentially fast. How fast it is measured by the Lyapunov exponent .

( ) (0)exp( )t t

type of motion maximal Lyap. exp.

stable fixed point < 0

limit cycle = 0

chaos 0 < < ∞

noise = ∞

The inverse of the maximal Lyap. Exp.defines the time beyond which predictability is impossible.

Exponents and dimensionsSensibility to initial conditions

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The most striking feature of chaos is the long-term unpredictability of the futuredespite a deterministic time evolution.

The cause is the inherent instability of the solutions, reflected in their sensitivedependence on initial conditions.

( ) (0)exp( )t t

Maximal Lyapunov exponent from time series:

- Delay embedding- Compute the average diverging rate:

0

0 001 ( )

1 1( ) ln

( )n n

N

n n n nn X U Xn

S n X XN U X

- The slope of S(n) is an estimate of the maximal LE

Exponents and dimensionsCorrelation dimension

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Strange attractors with fractal dimension are typical of chaotic systems. Non integerdimensions are assigned to geometrical objects which exhibit self-similarity and structure on all length scales.

Box-counting dimension:

( ) , 0.FDN r r r

For time series:

- Delay embedding- Compute the correlation sum: 1 1

2( ) ( )

( 1)

N N

i ji j i

C r rN N

X X

0

ln ( )lim

lnr

C rD

r

Kaplan-Yorke conjecturerelates D to Lyapunov spectra

Exponents and dimensionsApplications

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Nonlinear statistics such exponents, dimensions, prediction errors, etc., can becomputed to characterize non-trivial differences in signals (EEG) between different stages (brain states: sleep/rest, eyes open/closed).

Word of caution: such quantities are used to compare data from similar situations!

Ex: ECG series taken during exercise are more noisy due to sweat of patient skin.

The different noise levels at rest and exercising can affect the former nonlinear estimators and erroneusly conclude a higher complexity of the heart duringexercise just because of sweat on the skin.

OUTLINE


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Synchronization

Phase space methods



Take-home messages

Interdependence measuresSynchronization

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Synchronization is the dynamical process by which two or more oscillators adjust their rhythms due to their weak interaction.

A universal phenomenon found everywhere:

Mechanical systems (pendula, London’s bridge, …)

Electrical generators (power grids, Josephson junctions, …)

Life sciences (biological clocks, firing neurons, pacemaker cells, …)

Chemical reactions (Belousov-Zhabotinsky)

...

Synchronization refers to the way in which coupled elements, due to their dynamics, communicate and exhibit collective behavior.

In large populations of oscillators synchronization can be understood as a self-organization process.

Without a master o leader the individuals spontaneously tend to oscillate in synchrony.

Neural synchronization is one of the most promising mechanism to underlay the flexible formation of cell assemblies and thus bind the information processed at different areas.

Interdependence measuresSynchronization hallmarks

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Before coupling: f

F=F2-F1=0f2f1

After coupling: F

in-phase

Higher order: F2/F1=q/p

anti-phase

Phase shift is fixed:

Phase can be extracted from data by several techniques:

• Hilbert transform• Wavelet transform• Poincare map

__________________________Frequency locking___________________________

____________________________Phase locking______________________________

Interdependence measuresSynchronization solutions

Information theory: mutual information, entropies,... 26

Different types of synchronization capture different relationships betweenthe signals x1(t) and x2(t) of two interacting systems:

Classical: adjustment of rhythms in periodic oscillators.

Identical: coincidence of outputs due to their coupling, x1(t)=x2(t).

Generalized: captures a more general relationship like x1(t)=F(x2(t)).

Phase: expresses the regime where the phase difference between two irregular oscillators is bounded but their amplitudes are uncorrelated.

Lag: accounts for relation between two systems when compared atdifferent times such as x1(t)=x2(t-).

Noise-induced: synchronization induced by a common noise source.

OUTLINE


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Phase space methods



Take-home messages

Take-home messagesLinear vs nonlinear

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- Linear techniques are much well understood and rigorous.

- Linear and nonlinear estimates may assess different characteristics of the signals.

- Complementary approaches to the analysis of temporal series.

Even though most of systems in Nature are nonlinear do not underestimate linear methods

Take-home messagesPhase space methods

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- Attractor reconstruction is a powerful technique to recover the topological structureof an attractor given a scalar time series.

- Useful complexity quantifiers of the signal can be computed after the reconstruction.

- Word of caution in their use.

Take-home messagesDo it yourself... with a little help

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- TISEAN is a very complete software package for nonlinear time series analysis

- „Nonlinear time series analysis“ by Holger Kantz and Thomas Schreiber, Cambridge University Press.

Technology

Nonlinear estimators and time-embedding