Functional Brain Signal Processing: EEG & fMRI

Lesson 8

Kaushik Majumdar

Indian Statistical Institute Bangalore Center

kmajumdar@isibang.ac.in

M.Tech. (CS), Semester III, Course B50

Artificial Neural Network (ANN)

What does a single node in an ANN do?

x1

x2

x3

x4

x5

w12

w22

w32w42

w52

y2 5

21

exp i ii

b w x

5

21

i ii

w x b

More Nodes

x1

x2

x3

x4

x5

x6

y1

y2

y3

y4

output

6

1

1

1 expj

j ij iji

y

b w x

6

1 1 11

i ii

b w x

6

2 2 21

i ii

b w x

6

3 3 31

i ii

b w x

6

4 4 41

i ii

b w x

Hidden layer

Input layer

Output layer

1 if inside, 0 if outside the closed region

Number of Hidden Layers

There must be two hidden layers to identify the following annulus.

A neural network is basically a function approximator, which can approximate continuous functions by piecewise linear functions (interpolation). Neural networks are also known as universal approximator.

Separation or Classification

A separation or classification is nothing but approximating the surface separating the (mixed) data. In other words it approximates a continuous function generating the separating surface.

A classifier will have to approximate the function whose graph is this curve.

Classification by ANN

Most classification tasks are accomplished by separating the data with curve(s) consisting only a single line. Therefore for most classification tasks ANNs with a single hidden layer is sufficient.

However number of nodes in the hidden layer is to be determined by trial and error for optimal classification.

Universal Approximation

For any continuous mapping there must exist a three-layer neural network (having an input or ‘fanout’ layer with n processing elements, a hidden layer with 2n + 1 processing elements, and an output layer with m processing elements) that implements

exactly. Hecht-Nielsen, 1988.

:[0,1]n n mf

f

Backpropagation Neural Network

By far the most widely used type of neural network. It is simple yet powerful neural network even for

complex models having hundred of thousands of parameters.

Its conceptual simplicity and high success rate makes it a mainstay in adaptive pattern recognition.

Offers means to calculate input to hidden layer weights.

Duda et al., Chapter 6, p. 283 & 289

Regularization

It is a deep issue concerning complexity of the network. Number of input and output nodes is fixed. But number of hidden nodes and connection weights are not. These are free parameters. If there are too few of them the training set cannot be adequately learned. If there are too many of them, generalization of the network will be poor

Regularization (cont.)

(apart from enhanced computational complexity). That is, its performance on the test data set will fall down (while on training data set its performance may remain very high).

Training seizure pattern Testing seizure pattern

Backpropagation Architecture

Hecht-Nielsen, 1988

x1 x2 x3 x4

y1 y2

GeneralThree layer

Backpropagation Architecture (cont.)

Hecht-Nielsen, 1988

Backpropagation Algorithm

22

1

1 1( ) ( )

2 2

c

k kk

J t z

w t z has to be minimized, where t and z are target and network output vectors respectively. c is # output nodes.

J

w

wwhere is the learning rate.

( 1) ( ) ( )m m m w w w m stands for the m’th iteration.

Epileptic EEG Signal

Subasi and Ercelebi, Comp. Meth. Progr. Biomed., 78: 87 – 99, 2005

DB4 Wavelet

DB wavelets do not have closed form representation (cannot be expressed by an elegant mathematical formula, like Morlet wavelet).

http://en.wikipedia.org/wiki/Daubechies_wavelet

DB4 Wavelet Generation: Cascade Algorithm

11

0

( ) ( ) 2 (2 )N

k k

n

t h n t n

( ) ( ) 2 (2 )k

n

t g n t n

g(n), h(n) are impulse response functions. Ψ(t) is the wavelet. DB4 will contain only 4 taps or coefficients.

1 3(0)

4 2

3 3(1)

4 2

h

h

3 3(2)

4 2

1 3(3)

4 2

h

h

(0) (3) (2) (1)

(1) (2) (3) (0)

g h g h

g h g h

http://www.bearcave.com/misl/misl_tech/wavelets/daubechies/index.html

EEG Data

Electrode placement was according to 10 – 20 system.

4 signals selected as F7 – C3, F8 – C4, T5 – O1 and T6 – O2.

Sample frequency 200 Hz. Band-pass filtered in 1 – 70 Hz range upon

acquisition. EEG was segmented at 1000 time point window

(5s).

Feature Extraction by DB4 Wavelets

EEG signals decomposed by high-pass (called ‘detail signal’) and low-pass (called ‘approximation’) FIR filtering

Assignment

Preprocess depth EEG signals (to be given) by wavelet transforms (DB4 wavelet is seen to be more efficient than other wavelets, see Subasi & Ercelebi, 2005 and Vardhan & Majumdar, 2011). This will extract features from the signals.

Use a three layer (that is, with only one hidden layer) perceptron neural network to

Assignment (cont.)

classify the features to separate out the seizure portion from non-seizure portion in the signals.

References

A. Subasi and E. Ercelebi, Classification of EEG signals using neural networks and logistic regression, Comp. Meth. Progrm. Biomedicine, 78: 87 – 99, 2005.

I. Kaplan, Daubechies D4 wavelet transform, http://www.bearcave.com/misl/misl_tech/wavelets/daubechies/index.html

References (cont.)

R. Hecht-Nielsen, Theory of the backpropagation neural network, INNS 1988, p. I-593 – I-605. Freely available at http://s112088960.onlinehome.us/annProjects/Research%20Paper%20Library/backPropTheory.pdf

I. Daubechies, Ten lectures on wavelets, SIAM, 1992. p. 115, 132, 194, 242.

THANK YOU

This lecture is available at http://www.isibang.ac.in/~kaushik