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Fundamentals of Neural Networks
What is Neural Net?
- A neural tries to (ANN) is net is an artificial representation of the human brain that simulate its learning process. An artificial neural network often called a "Neural Network" or simply Neural Net (NN).
- Traditionally, the word neural network is referred to a network of biological neurons in the nervous system that process and transmit information.
- Artificial neural network is an interconnected group of artificial neurons that uses a mathematical model or computational model for information processing based on a connectionist approach to computation.
- The artificial neural networks are made of interconnecting artificial neurons which may share some properties of biological neural networks.
- Artificial Neural network is a network of simple processing elements (neurons) which can exhibit complex global behavior, determined by the connections between the processing elements and element parameters.
SC - Neural Network – Introduction
1. Introduction Neural Computers mimic certain processing capabilities of the human brain. - Neural Computing is an information processing paradigm, inspired by biological system, composed of a large number of highly interconnected processing elements (neurons) working in unison to solve specific problems. - Artificial Neural Networks (ANNs), like people, learn by example. - An ANN is configured for a specific application, such as pattern recognition or data classification, through a learning process. - Learning in biological systems involves adjustments to the synaptic connections that exist between the neurons. This is true of ANNs as well.
1. 2 Research History SC^ -^ Neural Network^ –^ Introduction The history is relevant because for nearly two decades the future of Neural network remained uncertain. McCulloch and Pitts ( 1943 ) are generally recognized as the designers of the first neural network. They combined many simple processing units together that could lead to an overall increase in computational power. They suggested many ideas like : a neuron has a threshold level and once that level is reached the neuron fires. It is still the fundamental way in which ANNs operate. The McCulloch and Pitts's network had a fixed set of weights. Hebb ( 1949 ) developed the first learning rule, that is if two neurons are active at the same time then the strength between them should be increased. In the 1950 and 60 's, many researchers (Block, Minsky, Papert, and Rosenblatt worked on perceptron. The neural network model could be proved to converge to the correct weights, that will solve the problem. The weight adjustment (learning algorithm) used in the perceptron was found more powerful than the learning rules used by Hebb. The perceptron caused great excitement. It was thought to produce programs that could think. Minsky & Papert ( 1969 ) showed that perceptron could not learn those functions which are not linearly separable. The neural networks research declined throughout the 1970 and until mid 80 's because the perceptron could not learn certain important functions. Neural network regained importance in 1985 - 86. The researchers, Parker and LeCun discovered a learning algorithm for multi-layer networks called back propagation that could solve problems that were not linearly separable.
SC - Neural Network – Introduction 1.3 Biological Neuron Model The human brain consists of a large number, more than a billion of neural cells that process information. Each cell works like a simple processor. The massive interaction processing only makes the brain's Fig. Structure of Neuron between all cells and their parallel abilities possible. Dendrites are branching fibers that extend from the cell body or soma. Soma or cell body of a neuron contains the nucleus and other structures, support chemical processing and production of neurotransmitters. Axon is a singular fiber carries information away from the soma to the synaptic sites of other neurons (dendrites and somas), muscles, or glands. Axon hillock is the site of summation for incoming information. At any moment, the collective influence of all neurons that conduct impulses to a given neuron will determine whether or not an action potential will be initiated at the axon hillock and propagated along the axon. Myelin Sheath consists of fat-containing cells that insulate the axon from electrical activity. This insulation acts to increase the rate of transmission of signals. A gap exists between each myelin sheath cell along the axon. Since fat inhibits the propagation of electricity, the signals jump from one gap to the next. Nodes of Ranvier are the gaps (about 1 m) between myelin sheath cells long axons are Since fat serves as a good insulator, the myelin sheaths speed the rate of transmission of an electrical impulse along the axon. Synapse is the point of connection between two neurons or a neuron and a muscle or a gland. Electrochemical communication between neurons takes place at these junctions. Terminal Buttons of a neuron are the small knobs at the end of an axon that release chemicals called neurotransmitters.
SC - Neural Network – Introduction 1.4 Artificial Neuron Model An artificial neuron is a mathematical function conceived as a simple model of a real (biological) neuron.
- The^ McCulloch-Pitts^ Neuron This is a simplified model of real neurons, known as a Threshold Logic Unit. Input 1 Input (^2) Input (^) n ■ (^) A set of input connections brings in activations from other neurons. ■ A processing unit sums the inputs, and then applies a non-linear activation function (i.e. squashing / transfer / threshold function). ■ An output line transmits the result to other neurons. In other words , - The input to a neuron arrives in the form of signals. - The signals build up in the cell. - Finally the cell discharges (cell fires) through the output. - The cell can start building up signals again.
Output
SC - Neural Network – Introduction 1.5 Notations Recaps : Scalar, Vectors, Matrices and Functions
- Scalar^ : The^ number^ xi^ can be added up^ to^ give a scalar^ number. **s = x 1 + x 2 + x 3 +.... + xn = xi
- Vectors**^ :^ An^ ordered sets of related^ numbers. Row^ Vectors^ (1 x^ n ) X = ( x 1 , x 2 , x 3 ,.. ., xn ) , Y = ( y 1 , y 2 , y 3 ,.. ., yn ) Add : Two vectors of same length added to give another vector. Z = X + Y = (x 1 + y 1 , x 2 + y 2 ,.... , xn + yn) Multiply: Two vectors of same length multiplied to give a scalar.
p = X. Y = x 1 y 1 + x 2 y 2 +.... + xnyn = xi yi
i= n i= n
SC - Neural Network – Introduction
6. Functions The Function y= f(x) describes a relationship, an input-output mapping, from x to y. ■ Threshold or Sign function : sgn(x) defined as Sign(x) 1 if x 0 sgn (x) = 0 if x **0 O/P
- 4 - 3 - 2 - 1 or Sign function : sigmoid(x) 0 1** defined 2 3 4 I/P ■ (^) Threshold as a smoothed (differentiable) form of the threshold function 1 sigmoid (x) = 1 + e - x Sign(x) **O/P
- 4 - 3 - 2 - 1 0 1 2 3 4 I/P 1 . . . . 0 . 0 1 . .**
SC - Neural Network – Artificial Neuron Model
2. Model of Artificial Neuron A very simplified model of real neurons is known as a Threshold Logic Unit (TLU). The model is said to have : - A set of synapses (connections) brings in activations from other neurons. - A processing unit sums the inputs, and then applies a non-linear activation function (i.e. squashing / transfer / threshold function). - An output line transmits the result to other neurons. 2.1 McCulloch-Pitts (M-P) Neuron Equation McCulloch-Pitts neuron is a simplified model of real biological neuron. Input (^1) Input (^2) Input (^) n Simplified Model of Real Neuron (Threshold Logic Unit) The equation for the output of a McCulloch-Pitts neuron as a function of 1 to n inputs is written as
Output = sgn ( Input i - )
where is the neuron’s activation threshold.
If Input i then Output = 1
If Input i then Output = 0
In this McCulloch-Pitts neuron model, the missing features are :
- Non-binary input and output,
- Non-linear summation,
- Smooth thresholding,
- Stochastic, and
- Temporal information processing.
Output
i= n
i= n
i= n
SC - Neural Network – Artificial Neuron Model ■ (^) Threshold for a Neuron In practice, neurons generally do not fire (produce an output) unless their total input goes above a threshold value. The total input for each neuron is the sum of the weighted inputs to the neuron minus its threshold value. This is then passed through the sigmoid function. The equation for the transition in a neuron is : a = 1 /( 1 + exp(- x)) where x = ai wi - Q a is the activation for the neuron ai is the activation for neuron i wi is the weight Q is the threshold subtracted ■ (^) Activation Function An activation function f performs a mathematical operation on the signal output. The most common activation functions are:
- Linear Function,
- Piecewise Linear Function,
- Tangent hyperbolic function
- Threshold Function,
- Sigmoidal (S shaped) function, The activation functions are chosen depending upon the type of problem to be solved by the network.
i
SC - Neural Network – Artificial Neuron Model
2. Activation Functions f - Types Over the years, researches tried several functions to convert the input into an outputs. The most commonly used functions are described below. - I/P Horizontal axis shows sum of inputs. - O/P Vertical axis shows the value the function produces ie output. - All functions f are designed to produce values between 0 and 1. - Threshold^ Function A threshold (hard-limiter) activation function is either a binary type or a bipolar type as shown below. binary threshold O/p I/P Output of a binary threshold function produces : 1 if the weighted sum of the inputs is +ve, 0 if the weighted sum of the inputs is - ve. 1 if I 0 Y = f (I) = 0 if I 0 bipolar threshold O/p I/P Output of a bipolar threshold function produces : 1 if the weighted sum of the inputs is +ve, - 1 if the weighted sum of the inputs is - ve. 1 if I **0 Y = f (I) =
- 1 if I** 0 Neuron with hard limiter activation function is called McCulloch-Pitts model. **1 1
- 1**
SC - Neural Network – Artificial Neuron Model Sigmoidal Function (S-shape function) The nonlinear curved S-shape function is called the sigmoid function. This is most common type of activation used to construct the neural networks. It is mathematically well behaved, differentiable and strictly increasing function. Sigmoidal function A^ sigmoidal^ transfer^ function^ can^ be written in the form: 1 Y = f (I) = , 0 f(I) 1 1 + e - ^ I = 1/(1 + exp(- I)) , 0 f(I) 1 This is explained as 0 for large - ve input values, 1 for large +ve values, with a smooth transition between the two. is slope parameter also called shape parameter; symbol the is also used to represented this parameter. The sigmoidal
By varying
function is achieved using exponential equation. different shapes of the function can be obtained which adjusts the abruptness of the function as it changes between the two asymptotic values. **1 O/P
I/P**
- 4 - 2 0 1 2
SC - Neural Network – Artificial Neuron Model
- Example^ : The neuron shown consists of four inputs with the weights. I = XT. W = 1 2 5^8 = 14 **- 1
= (1 x 1) + (2 x 1) + (5 x - 1) + (8 x 2) = 14** With a binary activation function the outputs of the neuron is: y (threshold) = 1; Synaptic Weights Fig Neuron Structure of Example The output I of the network, prior to the activation function stage, is + +
x 1 =
x 2 =
xn = 8
Activation Function Summing Junction = 0 Threshold
y
X 3 =
I
SC - Neural Network – Architecture 3. 1 Single Layer Feed-forward Network The Single Layer Feed-forward Network consists of a single layer of weights , where the inputs are directly connected to the outputs, via a series of weights. The synaptic links carrying weights connect every input to every output , but not other way. This way it is considered a network of feed- forward type. The sum of the products of the weights and the inputs is calculated in each neuron node, and if the value is above some threshold (typically 0 ) the neuron fires and takes the activated value (typically 1 ); otherwise it takes the deactivated value (typically - 1 ). Single layer Neurons Fig. Single Layer Feed-forward Network w 12 wn w1m w2m wn wnm w 11 w 21 w 22 y 1 y 2 ym
x 1
x 2
xn
input xi (^) weights wij output yj
SC - Neural Network – Architecture 3. 2 Multi Layer Feed-forward Network The name suggests, it consists of multiple layers. The architecture of this class of network, besides having the input and the output layers, also have one or more intermediary layers called hidden layers. The computational units of the hidden layer are known as hidden neurons_._ Fig. Multilayer feed-forward network in (ℓ – m – n) configuration****.
- The hidden layer does intermediate computation before directing the input to output layer.
- The input layer neurons are linked to the hidden layer neurons; the weights on these links are referred to as input-hidden layer weights.
- The hidden layer neurons and the corresponding weights are referred to as output-hidden layer weights.
- A multi-layer feed-forward network with ℓ input neurons, m 1 neurons in the first hidden layers, m 2 neurons in the second hidden layers, and n output neurons in the output layers is written as ( ℓ - m 1 - m 2 – n ). The Fig. above illustrates a multilayer feed-forward network with a configuration (ℓ - m – n). v 21 w1m v1m v2m vn Vℓm w 12 w 11
x 1
x 2
y 3
y 1
y 2
yn
y 1
ym
Hidden Layer neurons yj Output Layer neurons zk
xℓ
Input Layer neurons xi Input hidden layer weights vij v 11 Output hidden layer weights wjk w 11
