MATH347

MATH347 L16: A first realistic application - Electroencephalogram analysis

New concepts:
- Data organization into vectors, matrices
- Practical interpretation of fundamental matrix subspaces
- Working with data in Octave/Matlab

Electroencephalogram data

Figure 1. Brain neurons transmit electrical signals recorded by electrodes placed on the skull

>>	load eeg; data=EEG.data';

>>	[m,n]=size(data); pdata=data./max(data)+meshgrid(0:n-1,0:m-1);

>>	hold on; for j=1:n plot(pdata(:,j)); end; hold off;

Epilepsy prediction

Figure 2. Electrode placement, normal EEG, epileptic seizure EEG (PN12-3, t=800, t=950)

>>	edata=edfread("PN12-3.edf"); plot(cell2mat(edata(800,1).EEGFp1)); plot(cell2mat(edata(950,1).EEGFp1));

Linear algebra approaches

Separate long time recording of normal activity into pieces, form $𝑨 \in ℝ^{m_{S} \times n_{S}}$
>>

nElec=1; mS=256; nS=20; A=reshape(pdata(1:mS*nS,nElec),mS,nS);
Ask: when is a recording $𝒃$ indicative of something abnormal?

Possible answer: when $𝒃$ has much larger component in $N (𝑨^{T})$ than in $C (𝑨)$
Approach: project $𝒃$ onto $C (𝑨)$ to obtain $𝒄 = 𝑷_{𝑸} 𝒃$ , find fraction of size of $𝒃$ within $C (𝑨)$ by computing ratio $r = || 𝒄 || / || 𝒃 ||$ .
>>

[Q,R]=qr(A); Qt=Q';
>>

b=pdata(mS*nS+1:mS*nS+256,1); c=Q*(Qt*b); norm(c)/norm(b)
Establish a threshold $r$ that indicates onset of epileptic seizure
Algorithm can be made into a phone/watch app together with continual sensors to obtain a wearable device to alert/treat onset of epileptic seizures
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