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Artificial Neural Networks
(ANNs)

Background and Motivation
• Computers and the Brain: A Contrast









Arithmetic:
Vision:

1 brain = 1/10 pocket calculator
1 brain = 1000 super computers

Memory of arbitrary details: computer wins
Memory of real-world (related) facts: brain wins
A computer must be programmed explicitly
The brain can learn by experiencing the world

What are Artificial Neural Networks?
A system loosely modeled on the human brain that simulates the
multiple layers of simple processing elements called neurons.
There are various classes of ANN models
depending on:
q Problem type

Prediction, Classification , Clustering
q Design / Structure of the model
q Model building algorithm

A bit of biology . . .
Most important functional unit in human brain – a class of cells called –
NEURONS

Dendrites
Cell Body
Axon
Synapse

Actual Neuron

Schematic

• Dendrites – Receive information • Cell Body – Process information
• Axon – Carries processed information to other neurons
• Synapse – Junction between Axon end and Dendrites of other Neurons

Artificial Neurons
• simulated on hardware or by software
• input connections - the receivers
• node, unit, Processing Unit or Processing Element
simulates neuron body
• output connection and transfer function - the
transmitter (axon)
• activation function employs a threshold or bias
• connection weights act as synaptic junctions
Learning (for a particular model) occurs via changes in
value of the various weights and choice of functions.

An Artificial Neuron: Structure
Dendrites

X1
X2

w1
w2

..
.
Xp

wp

Cell Body

Axon

Direction of flow of Information

I

f

V = f(I)

I = w1X1 + w2X2
+ w3X3 +… + wpXp

• Receives Inputs X 1 X2 … Xp from other neurons or environment
• Inputs fed-in through connections with ‘weights’
• Total Input = Weighted sum of inputs from all sources
• Transfer function and Activation function convert the input to output
• Output goes to other neurons or environment

Behaviour of an artificial neural network
to any particular input depends upon
• structure of each node (threshold /
activation / transfer function)
• structure of the network (architecture)
• weights on each of the connections
.... these must be learned !

Activity
in
an
ANN
Node
Mathematical Model of a Node
Adder _ Function
a0
w0
Incoming
activation

wi
ai



Θ

Outgoing
activation

wn

an

Activation_ Function

Activity in an ANN Node
Adder _ Function
n

a0

I = ∑ xi wi

w0
Incoming
activation

wi
ai



Θ

i =0

Outgoing
activation

wn

an

Activation_ Function

Activity in an ANN Node
Adder _ Function
a0
w0
Incoming
activation

wi
ai



Θ

Outgoing
activation

wn

Activation_ Function
an

 1 if I > t,
f (I ) = 
0 if I <= t

Transfer Functions
There are various choices for Transfer / Activation functions

1

1

1

0.5

-1

Tanh
f(x) =
(ex – e-x) / (ex + e-x)

0

Logistic
f(x) = ex / (1 + ex)

0

Threshold
0 if x< 0
f(x) =
1 if x >= 1

Artificial Neural Network Architecture
The most common neural network
architectures has three layers:
- 1st Layer: input layer
-2nd Layer: hidden layer
-3rd Layer: output layer

More complex ANN Architectures
Number of hidden layers can be

None

One

More

Feed-forward Artificial Neural Network Architecture
A collection of neurons form a ‘Layer’
X1

X2

X3

X4

Input Layer

Direction of information flow

- Each neuron gets ONLY
one input, directly from outside

Hidden Layer
- Connects Input and Output
layers

Output Layer
- Output of each neuron
directly goes to outside

y1

y2

Feed-forward Artificial Neural Network Architecture
A few restrictions
X1

X2

X3

X4

• Within a layer neurons are NOT
connected to each other.
• Neuron in one layer is connected
to neurons ONLY in the NEXT
layer. (Feed-forward)
• Jumping of layer is NOT
allowed

y1

y2

An ANN model
What do we mean by ‘A particular Model’ ?

Input: X1 X2 X3
For an ANN :

Output: Y

Model: Y = f(X1 X2 X3)

Algebraic form of f(.) is too complicated to write down.

However it is characterized by
• # Input Neurons
• # Hidden Layers
• # Neurons in each Hidden Layer
• # Output Neurons
• The adder, activation and transfer functions
• WEIGHTS for all the connections
‘Fitting’ an ANN model = Specifying values for all those parameters

ANN Model – an Example
Input: X1 X2 X3

Output: Y
X2

X1
0.6

-0.1

X3

-0.2

0.1

0.7

0.5

0.1

Model: Y = f(X1 X2 X3)
Parameters

Example

# Input Neurons

3

# Hidden Layers

1

# Hidden Layer Size

2

# Output Neurons

1

-0.2
Weights

Y

Decided by the structure
of the problem

# Input Nrns = # of X’s
# Output Nrns = # of Y’s

Learnt

Free parameters

Prediction using a particular ANN Model
Input: X1 X2 X3
X1 =1

Output: Y
X2=-1

0.6

-0.1

X3 =2

-0.2

0.1

0.7

0.5
0.2
f (0.2) = 0.55
0.55

0.9
f (0.9) = 0.71
0.71

0.1

Model: Y = f(X1 X2 X3)
0.2 = 0.5 * 1 –0.1*(-1) –0.2 * 2

f(x) = ex / (1 + ex)
f(0.2) = e0.2 / (1 + e0.2) = 0.55

Predicted Y = 0.478

-0.2
Suppose Actual Y = 2

-0.087
f (-0.087) = 0.478
0.478

Then
Prediction Error = (2-0.478) =1.522

Real Estate Appraisal by Neural Networks

Building ANN Model
Input: X1 X2 X3

Output: Y Model: Y = f(X1 X2 X3)

# Input Neurons = # Inputs = 3
# Hidden Layer = ???
# Neurons in Hidden Layer = ???

# Output Neurons = # Outputs = 1
Try 1
Try 2

Architecture is now defined …

No fixed strategy.
By trial and error
How to get the weights ???

Given the Architecture There are 8 weights to decide.
W = (W1, W 2, …, W 8)
Training Data: (Yi , X1i, X2i, …, Xpi ) i= 1,2,…,n
Given a particular choice of W, we will get predicted Y’s

( V1,V2,…,Vn)
They are function of W.
Choose W such that over all prediction error E is minimized

E = Σ (Yi – Vi) 2

Training the Model

Back Propagation

Feed forward

• Start with a random set of weights.

E = Σ (Yi – Vi) 2

• Feed forward the first observation through the net
X1
Network
V1
; Error = (Y1 – V1)
• Adjust the weights so that this error is reduced
( network fits the first observation well )
• Feed forward the second observation.
Adjust weights to fit the second observation well
• Keep repeating till you reach the last observation
• This finishes one CYCLE through the data
• Randomize the order of the input (observations)
•Perform many such training cycles till the
overall prediction error E is small.

Back Propagation
Each weight ‘Shares the Blame’ for prediction
error and tries to ‘adjust’ with other weights.
Back Propagation algorithm decides how to
distribute the blame among all weights and
adjust the weights accordingly.
Small portion of blame leads to small
adjustment.
Large portion of the blame leads to large
adjustment.

E = Σ (Yi – Vi) 2

Weight adjustment during Back Propagation
Weight adjustment formula in Back Propagation
Vi – the prediction for ith observation –
is a function of the network weights vector W = ( W1, W 2,….)
Hence, E, the total prediction error is also a function of W

E( W ) = Σ [ Yi – Vi( W ) ] 2
Gradient Descent Method :
For every individual weight W i, updation formula looks like

Wnew = Wold + α * ( ∂E / ∂W) | Wold
α = Learning Parameter (between 0 and 1)

Another slight variation is also used sometimes

W(t+1) = W (t) + α * ( ∂E / ∂W) | W(t) + β * (W(t) - W(t-1) )
β = Momentum (between 0 and 1)

Geometric interpretation of the Weight adjustment
Consider a very simple network with 2 inputs and 1 output. No hidden layer.
There are only two weights whose values needs to be specified.

E( w1, w2 ) = Σ [ Yi – Vi(w1, w2 ) ] 2
• A pair ( w1, w2 ) is a point on 2-D plane.

w1

w2

• For any such point we can get a value of E.
• Plot E vs ( w1, w2 ) - a 3-D surface - ‘Error Surface’
• Aim is to identify that pair for which E is minimum
• That means – identify the pair for which the height of
the error surface is minimum.

Gradient Descent Algorithm
• Start with a random point ( w1, w2 )

• Move to a ‘better’ point ( w’1, w’2 ) where the height of error surface is lower.
• Keep moving till you reach ( w*1, w*2 ), where the error is minimum.

Crawling the Error Surface
14.0

Error Surface

12.0

Local Minima

10.0

8.0
Error
6.0

4.0

w*

w0

Weight Space

-2.000

-1.000

0.000

1.000

2.000

3.000

4.000

5.000

6.000
6.000

5.000

4.000

3.000

2.000

1.000

0.000

-1.000

W2

-2.000

-3.000

0.0

-3.000

2.0

Global Minima

W1

Training Algorithm
Decide the Network architecture
(# Hidden layers, #Neurons in each Hidden Layer)
Decide the Learning parameter and Momentum
Initialize the Network with random weights
Do till Convergence criterion is not met
For I = 1 to # Training Data points
Feed forward the I-th observation thru the Net
Compute the prediction error on I-th observation
Back propagate the error and adjust weights

E = Σ (Yi – Vi)

2

Next I
Check for Convergence
End Do

Convergence Criterion
When to stop training the Network ?
Ideally – when we reach the global minima of the error surface
How do we know we have reached there ?

We don’t …

Suggestions:
1. Stop if the decrease in total prediction error (since last cycle) is small.
2. Stop if the overall changes in the weights (since last cycle) are small.
Drawback:
Error keeps on decreasing. We get a very good fit to training data.
BUT … The network thus obtained have poor generalizing power on unseen data
The phenomenon is also known as - Over fitting of the Training data
The network is said to Memorize the training data.
- so that when an X in training set is given,
the network faithfully produces the corresponding Y.
-However for X’s which the network didn’t see before, it predicts poorly.

Convergence Criterion
Modified Suggestion:
Partition the training data into Training set and Validation set
Use:

Training set - build the model
Validation set - test the performance of the model on unseen data
Typically as we have more and more training cycles
Error on Training set keeps on decreasing.
Error on Validation set first decreases and then increases.

Error

Validation

Training

Cycle

Stop training when the error on
Validation set starts increasing

Choice of Training Parameters
Learning Parameter and Momentum
- needs to be supplied by user from outside. Should be between 0 and 1
What should be the optimal values of these training parameters ?
- No clear consensus on any fixed strategy.
- However, effects of wrongly specifying them are well studied.
Learning Parameter
Too big – Large leaps in weight space – risk of missing global minima.
Too small –
- Takes long time to converge to global minima
- Once stuck in local minima, difficult to get out of it.
Suggestion
Trial and Error – Try various choices of Learning Parameter and Momentum
See which choice leads to minimum prediction error

Summary
q

Artificial Neural network (ANN)
– A class of models inspired by biological Neurons

q

Used for various modeling problems – Prediction, Classification, Clustering, ..

q

One particular subclass of ANN’s – Feed forward Back propagation networks
v
v
v
v

Organized in layers – Input, hidden, Output
Each layer is a collection of a number of artificial Neurons
Neurons in one layer in connected to neurons in next layer
Connections have weights

q

Fitting an ANN model is to find the values of these weights.

q

Given a training data set – weights are found by Feed forward Back
propagation algorithm, which is a form of Gradient Descent Method – a
popular technique for function minimization.

q

Network architecture as well as the training parameters are decided upon by
trial and error. Try various choices and pick the one that gives lowest
prediction error.

Self-Organizing Maps

Overview
• A Self-Organizing Map (SOM) is a way to
represent higher dimensional data in an usually 2D or 3-D manner, such that similar data is grouped
together.
• It runs unsupervised and performs the grouping on
its own.
• Once the SOM converges, it can classify new data.
• SOMs run in two phases:
– Training phase: the map is built, the network organizes
using a competitive process, it is trained using a large
numbers of inputs.
– Mapping phase: new vectors are quickly given a
location on the converged map, easily classifying or
categorizing the new data.

Example

• Example: Data on poverty levels in different countries.
– SOM does not show poverty levels, rather it shows how similar the
poverty levels of different countries are to each other. (Similar
color = similar poverty level).

Forming the Map

The Basic Process
1)
2)
3)
4)
5)
6)
7)

8)
9)

Each pixel is a node / unit.
Initialize each node’s weights with random values.
Take the next vector from training data and present it to the
SOM.
Every node is examined to find the Best Matching Unit /
Node (BMU).
Adjust BMU’s weights to make it come closer to the input
vector. The rate of adjustment is determined by the ‘learning
rate’. The learning rate decreases with each iteration.
The radius of the neighborhood around the BMU is
calculated. The size of the neighborhood decreases with each
iteration.
Each node in the BMU’s neighborhood has its weights
adjusted to become more like the BMU. Nodes closest to the
BMU are altered more than the nodes further away in the
neighborhood.
Repeat from step 3 to 7 for each vector in the training data.
Repeat from step 3 to 8 for enough iterations for convergence.

Calculating the Best Matching Unit
• Calculating the BMU is done according to the
Euclidean distance between the node’s weights
(W1, W2, … , Wn) and the input vector’s values
(V1, V2, … , Vn).
– This gives a good measurement of how similar the two
sets of data are to each other.
– The node closest to the input vector is chosen as the
BMU (or ‘winning’ node).

Determining the BMU
Neighborhood


Size of the neighborhood: uses an exponential decay function that shrinks
on each iteration until eventually the neighborhood is just the BMU itself.



Effect of location within the neighborhood: The neighborhood is defined by a
gaussian curve so that nodes that are closer are influenced more than farther
nodes.

Modifying the Nodes’ Weights
• The new weight for a node is the old weight, plus a fraction (L) of
the difference between the old weight and the input vector…
adjusted (by theta) depending on the distance from the BMU.

• The learning rate, L, is also an exponential decay function.
– This ensures that the SOM will converge.

• The lambda represents a constant, and t is the time step

SOM Application: Text Clustering

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