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Maximum Likelihood Estimation for Multiepoch EEG Analysis

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### Maximum Likelihood Estimation for Multiepoch EEG Analysis

J.Kříž

Department of physics,

University of Hradec Králové

Quantum Circle

January 30, 2007

EEG = Electroencephalographymeasures electric potentials on the scalp (generated by neuronal activity in the brain)

Multiepoch EEG: Evoked potentials

= responses to the external stimulus (auditory, visual, etc.) sensory and cognitive processing in the brain

Common properties of EEGand radar signal processing :

- spatial–temporal character
- data of the form X = S + W
- low signal to noise ratio (SNR)

MOTIVATION

Is MLE suitable topic for QC seminar?

YES !!!

QC seminar: On various aspects of the quantum theory, for students in the first place

MLE: Hradil, Řeháček, Fiurášek, Ježek, Maximum Likelihood Methods in Quantum Mechanics, in Quantum State Estimation, Lecture Notes in Physics (ed. M.G.A. Paris, J. Rehacek), 59-112, Springer, 2004.

Basic concept of MLE

- originally developed by R.A. Fisher in 1920’s
- assume pdf fof random vector ydepending on a parameter set w, i.e. f(y|w)
- it determines the probability of observing the data vector y (in dependence on the parameters w)
- however, we are faced with an inverse problem: we have given data vector and we do not know parameters
- define likelihood functionl by reversing the roles of data and parameter vectors, i.e. l(w|y) = f(y|w).
- MLE maximizes l over all parameters w
- that is, given the observed data (and a model of interest), find the pdf, that is most likely to produce the given data.

MLE for EEG evoked response analysis

Baryshnikov, B.V., Van Veen, B.D. and Wakai R.T.,

IEEE Trans. Biomed. Eng. 51 ( 2004), p. 1981 – 1993.

Assumptions: response is the same across all epochs

noise is independent from trial to trial,

it is temporally white, but spatially coloured

it is normally distributed with zero mean

Experiment: pattern reversal evoked potentials

63 – channel EEG device

100 epochs

sampling rate of 1 kHz

MLE for EEG evoked response analysis

Experiment:

MLE for EEG evoked response analysis: Model

N …spatial channels , T… time samples per epoch

J … number of epochs ( N=63, T=666, J=100)

- data for j-th epoch: Xj= S + Wj... N x T matrix
- The estimate of repeated signal Scan be expressed in the form
- S=HqCT
- C … known TxL matrix of temporal basis vectors, i.e.
- rows of S are linear combinations of columns of C
- known frequency band of interest is used to construct C
- H … unknown NxPmatrix of spatial basis vectors, i.e.
- columns of S are linear combinations of columns of H
- … unknown PxL matrix of coefficients

Model is purely linear, both spatial and temporal nonlocal

MLE for EEG evoked response analysis: Model

Full dataset of Jepochs: X=[ X1 X2 ... XJ ] ... NxJTmatrix

Noise over J epochs: W=[ W1W2 ... WJ] ... N x JTmatrix

X = [ S S ... S ] + W,

[ S S ... S ] =HqDT, whereDT = [ CT CT... CT ]

Noise covariance „supermatrix“ is modeled as the Kronecker product of spatial and temporal covariance matrices, i.e.:

every element of N x N„spatial matrix“ is JTx JT „temporal matrix“

RT= WTW… JT x JT temporal cov. matrix, (RT=1 in our model)

R = WWT … N x N spatial cov. matrix (unknown in our model)

MLE for EEG evoked response analysis: Model

Temporal basis matrix C

Processes of interests in EEG are usually in the frequency band 1-20 Hz.

Temporal basis vectors can be chosen as (discretized):

sin(2pft), cos(2pft)

to cover the frequency band of interest.

The number of basis vectors L is given by frequency band.

In the case L=T we may choose C=1(we take all frequencies)

Under all above assumptions, the pdfcan be written as

Maximum-likelihood parameter estimation

Thus, we are looking for unknown matricesR, qand Hto maximize the likelihood function for our data X.

It was done by Baryshnikov et al.

However, the parameter P (rank of matrix H) remains free.

The question of suitable choice of P is discussed.

Comparison of MLE with independent methods

- Filtering and averaging
- 1. Filter data (4th order Butterworth filter with passband 1-20 Hz)
- 2. Average data over all epochs
- - local in both temoral and spatial dimension
- Principal component method (PCA)
- Project the data to the subspace given by eigenvectors corresponding to the largest eigenvalues of data covariance matrix.
- PCA in EEG evoked potentials analysis requires signal-free data for noise whitening (pre-stimulus based whitening).

Comparison of MLE with independent methods

- Theoretically
- for C=1, P=N, MLE gives exactly the mean over epochs
- for C≠1, P=N, MLE gives the mean over „filtered“ epochs
- for C=1, P ≠N, matrix Hcontains eigenvectors corresponding to P largest eigenvalues of

Link to PCA

columns of H … eigenvectors corr. to Plargest eigenvalues

increasing P… MLE tends to „filtering and averaging“

low values of P are interesting

Comparison of MLE with filtering/averaging method

Green … nonfiltered mean over epochs

Blue … filtered (1-20 Hz) mean over epochs

Red … MLE (P = 5, frequency band 1-20 Hz)

stimulus onset at 200 ms

Dependence of MLE on P

Differences betweem two matrices are calculated in the norm

Questeion of suitable value of P

Problem: we do not know the signal of interest S, we cannot determine for which P is the MLE closest to S.

Solution: simulated EEG data: take some signal of interest S and add a (coloured) noise to it.

Questeion of suitable value of P: simulated EEG data

Green … real signal of interest

Blue … filtered (1-20 Hz) mean over epochs

Red … MLE (P = 5, frequency band 1-20 Hz)

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