![$$delta_{inj}:=:displaystylesumlimits_{k=1}^m delta_{k}:w_{jk}$$
Error term can be calculated as follows −
$$delta_{j}:=:delta_{inj}f^{'}(Q_{inj})$$
On this basis, update the weight and bias as follows −
$$Delta w_{ij}:=:alphadelta_{j}x_{i}$$
$$Delta b_{0j}:=:alphadelta_{j}$$
Phase 3
Step 9 − Each output unit (ykk = 1 to m) updates the weight and bias as follows −
$$v_{jk}(new):=:v_{jk}(old):+:Delta v_{jk}$$
$$b_{0k}(new):=:b_{0k}(old):+:Delta b_{0k}$$
Step 10 − Each output unit (zjj = 1 to p) updates the weight and bias as follows −
$$w_{ij}(new):=:w_{ij}(old):+:Delta w_{ij}$$
$$b_{0j}(new):=:b_{0j}(old):+:Delta b_{0j}$$
Step 11 − Check for the stopping condition, which may be either the number of epochs
reached or the target output matches the actual output.
Generalized Delta Learning Rule
Delta rule works only for the output layer. On the other hand, generalized delta rule, also
called as back-propagation rule, is a way of creating the desired values of the hidden layer.
Mathematical Formulation
For the activation function $y_{k}:=:f(y_{ink})$ the derivation of net input on Hidden layer
as well as on output layer can be given by
$$y_{ink}:=:displaystylesumlimits_i:z_{i}w_{jk}$$
And $::y_{inj}:=:sum_i x_{i}v_{ij}$
Now the error which has to be minimized is
$$E:=:frac{1}{2}displaystylesumlimits_{k}:[t_{k}:-:y_{k}]^2$$
By using the chain rule, we have](https://image.slidesharecdn.com/neuralnetworksareparallelcomputingdevices-221119112528-611c1b8e/85/Neural-networks-are-parallel-computing-devices-docx-pdf-21-320.jpg)
![$$frac{partial E}{partial w_{jk}}:=:frac{partial }{partial
w_{jk}}(frac{1}{2}displaystylesumlimits_{k}:[t_{k}:-:y_{k}]^2)$$
$$=:frac{partial }{partial w_{jk}}lgroupfrac{1}{2}[t_{k}:-:t(y_{ink})]^2rgroup$$
$$=:-[t_{k}:-:y_{k}]frac{partial }{partial w_{jk}}f(y_{ink})$$
$$=:-[t_{k}:-:y_{k}]f(y_{ink})frac{partial }{partial w_{jk}}(y_{ink})$$
$$=:-[t_{k}:-:y_{k}]f^{'}(y_{ink})z_{j}$$
Now let us say $delta_{k}:=:-[t_{k}:-:y_{k}]f^{'}(y_{ink})$
The weights on connections to the hidden unit zj can be given by −
$$frac{partial E}{partial v_{ij}}:=:- displaystylesumlimits_{k} delta_{k}frac{partial
}{partial v_{ij}}:(y_{ink})$$
Putting the value of $y_{ink}$ we will get the following
$$delta_{j}:=:-displaystylesumlimits_{k}delta_{k}w_{jk}f^{'}(z_{inj})$$
Weight updating can be done as follows −
For the output unit −
$$Delta w_{jk}:=:-alphafrac{partial E}{partial w_{jk}}$$
$$=:alpha:delta_{k}:z_{j}$$
For the hidden unit −
$$Delta v_{ij}:=:-alphafrac{partial E}{partial v_{ij}}$$
$$=:alpha:delta_{j}:x_{i}$$
hese kinds of neural networks work on the basis of pattern association, which means they can
store different patterns and at the time of giving an output they can produce one of the stored
patterns by matching them with the given input pattern. These types of memories are also
called Content-Addressable Memory CAM
. Associative memory makes a parallel search with the stored patterns as data files.
Following are the two types of associative memories we can observe −
● Auto Associative Memory
● Hetero Associative memory
Auto Associative Memory](https://image.slidesharecdn.com/neuralnetworksareparallelcomputingdevices-221119112528-611c1b8e/85/Neural-networks-are-parallel-computing-devices-docx-pdf-22-320.jpg)
![wij=∑p=1P[2si(p)−1][2sj(p)−1]fori≠j
Case 2 − Bipolar input patterns
For a set of binary patterns sp
, p = 1 to P
Here, sp
= s1p, s2p,..., sip,..., snp
Weight Matrix is given by
wij=∑p=1P[si(p)][sj(p)]fori≠j
Testing Algorithm
Step 1 − Initialize the weights, which are obtained from training algorithm by using Hebbian
principle.
Step 2 − Perform steps 3-9, if the activations of the network is not consolidated.
Step 3 − For each input vector X, perform steps 4-8.
Step 4 − Make initial activation of the network equal to the external input vector X as follows
−
yi=xifori=1ton
Step 5 − For each unit Yi, perform steps 6-9.
Step 6 − Calculate the net input of the network as follows −
yini=xi+∑jyjwji
Step 7 − Apply the activation as follows over the net input to calculate the output −
yi=⎧⎩⎨1yi0ifyini>θiifyini=θiifyini<θi
Here θi
is the threshold.
Step 8 − Broadcast this output yi to all other units.
Step 9 − Test the network for conjunction.](https://image.slidesharecdn.com/neuralnetworksareparallelcomputingdevices-221119112528-611c1b8e/85/Neural-networks-are-parallel-computing-devices-docx-pdf-27-320.jpg)
Neural networks are parallel computing devices.docx.pdf
- 1. Neural networks are parallel computing devices, which is basically an attempt to make a computer model of the brain. The main objective is to develop a system to perform various computational tasks faster than the traditional systems. These tasks include pattern recognition and classification, approximation, optimization, and data clustering. What is Artificial Neural Network? Artificial Neural Network (ANN) is an efficient computing system whose central theme is borrowed from the analogy of biological neural networks. ANNs are also named as “artificial neural systems,” or “parallel distributed processing systems,” or “connectionist systems.” ANN acquires a large collection of units that are interconnected in some pattern to allow communication between the units. These units, also referred to as nodes or neurons, are simple processors which operate in parallel. Every neuron is connected with other neuron through a connection link. Each connection link is associated with a weight that has information about the input signal. This is the most useful information for neurons to solve a particular problem because the weight usually excites or inhibits the signal that is being communicated. Each neuron has an internal state, which is called an activation signal. Output signals, which are produced after combining the input signals and activation rule, may be sent to other units. A Brief History of ANN The history of ANN can be divided into the following three eras − ANN during 1940s to 1960s Some key developments of this era are as follows − ● 1943 − It has been assumed that the concept of neural network started with the work of physiologist, Warren McCulloch, and mathematician, Walter Pitts, when in 1943 they modeled a simple neural network using electrical circuits in order to describe how neurons in the brain might work. ● 1949 − Donald Hebb’s book, The Organization of Behavior, put forth the fact that repeated activation of one neuron by another increases its strength each time they are used. ● 1956 − An associative memory network was introduced by Taylor. ● 1958 − A learning method for McCulloch and Pitts neuron model named Perceptron was invented by Rosenblatt. ● 1960 − Bernard Widrow and Marcian Hoff developed models called "ADALINE" and “MADALINE.” ANN during 1960s to 1980s Some key developments of this era are as follows −
- 2. ● 1961 − Rosenblatt made an unsuccessful attempt but proposed the “backpropagation” scheme for multilayer networks. ● 1964 − Taylor constructed a winner-take-all circuit with inhibitions among output units. ● 1969 − Multilayer perceptron (MLP) was invented by Minsky and Papert. ● 1971 − Kohonen developed Associative memories. ● 1976 − Stephen Grossberg and Gail Carpenter developed Adaptive resonance theory. ANN from 1980s till Present Some key developments of this era are as follows − ● 1982 − The major development was Hopfield’s Energy approach. ● 1985 − Boltzmann machine was developed by Ackley, Hinton, and Sejnowski. ● 1986 − Rumelhart, Hinton, and Williams introduced Generalised Delta Rule. ● 1988 − Kosko developed Binary Associative Memory (BAM) and also gave the concept of Fuzzy Logic in ANN. The historical review shows that significant progress has been made in this field. Neural network based chips are emerging and applications to complex problems are being developed. Surely, today is a period of transition for neural network technology. Biological Neuron A nerve cell (neuron) is a special biological cell that processes information. According to an estimation, there are huge number of neurons, approximately 1011 with numerous interconnections, approximately 1015 . Schematic Diagram
- 3. Working of a Biological Neuron As shown in the above diagram, a typical neuron consists of the following four parts with the help of which we can explain its working − ● Dendrites − They are tree-like branches, responsible for receiving the information from other neurons it is connected to. In other sense, we can say that they are like the ears of neuron. ● Soma − It is the cell body of the neuron and is responsible for processing of information, they have received from dendrites. ● Axon − It is just like a cable through which neurons send the information. ● Synapses − It is the connection between the axon and other neuron dendrites. ANN versus BNN Before taking a look at the differences between Artificial Neural Network (ANN) and Biological Neural Network (BNN), let us take a look at the similarities based on the terminology between these two. Biological Neural Network (BNN) Artificial Neural Network (ANN) Soma Node Dendrites Input Synapse Weights or Interconnections Axon Output The following table shows the comparison between ANN and BNN based on some criteria mentioned.
- 4. Criteria BNN ANN Processing Massively parallel, slow but superior than ANN Massively parallel, fast but inferior than BNN Size 1011 neurons and 1015 interconnections 102 to 104 nodes (mainly depends on the type of application and network designer) Learning They can tolerate ambiguity Very precise, structured and formatted data is required to tolerate ambiguity Fault tolerance Performance degrades with even partial damage It is capable of robust performance, hence has the potential to be fault tolerant Storage capacity Stores the information in the synapse Stores the information in continuous memory locations Model of Artificial Neural Network The following diagram represents the general model of ANN followed by its processing. For the above general model of artificial neural network, the net input can be calculated as follows − $$y_{in}:=:x_{1}.w_{1}:+:x_{2}.w_{2}:+:x_{3}.w_{3}:dotso: x_{m}.w_{m}$$ i.e., Net input $y_{in}:=:sum_i^m:x_{i}.w_{i}$ The output can be calculated by applying the activation function over the net input. $$Y:=:F(y_{in}) $$ Output = function (net input calculated)
- 5. Network Topology A network topology is the arrangement of a network along with its nodes and connecting lines. According to the topology, ANN can be classified as the following kinds − Feedforward Network It is a non-recurrent network having processing units/nodes in layers and all the nodes in a layer are connected with the nodes of the previous layers. The connection has different weights upon them. There is no feedback loop means the signal can only flow in one direction, from input to output. It may be divided into the following two types − ● Single layer feedforward network − The concept is of feedforward ANN having only one weighted layer. In other words, we can say the input layer is fully connected to the output layer. ● Multilayer feedforward network − The concept is of feedforward ANN having more than one weighted layer. As this network has one or more layers between the input and the output layer, it is called hidden layers. Feedback Network
- 6. As the name suggests, a feedback network has feedback paths, which means the signal can flow in both directions using loops. This makes it a non-linear dynamic system, which changes continuously until it reaches a state of equilibrium. It may be divided into the following types − ● Recurrent networks − They are feedback networks with closed loops. Following are the two types of recurrent networks. ● Fully recurrent network − It is the simplest neural network architecture because all nodes are connected to all other nodes and each node works as both input and output. ● Jordan network − It is a closed loop network in which the output will go to the input again as feedback as shown in the following diagram. Adjustments of Weights or Learning Learning, in artificial neural network, is the method of modifying the weights of connections between the neurons of a specified network. Learning in ANN can be classified into three categories namely supervised learning, unsupervised learning, and reinforcement learning. Supervised Learning
- 7. As the name suggests, this type of learning is done under the supervision of a teacher. This learning process is dependent. During the training of ANN under supervised learning, the input vector is presented to the network, which will give an output vector. This output vector is compared with the desired output vector. An error signal is generated, if there is a difference between the actual output and the desired output vector. On the basis of this error signal, the weights are adjusted until the actual output is matched with the desired output. Unsupervised Learning As the name suggests, this type of learning is done without the supervision of a teacher. This learning process is independent. During the training of ANN under unsupervised learning, the input vectors of similar type are combined to form clusters. When a new input pattern is applied, then the neural network gives an output response indicating the class to which the input pattern belongs. There is no feedback from the environment as to what should be the desired output and if it is correct or incorrect. Hence, in this type of learning, the network itself must discover the patterns and features from the input data, and the relation for the input data over the output. Reinforcement Learning As the name suggests, this type of learning is used to reinforce or strengthen the network over some critic information. This learning process is similar to supervised learning, however we might have very less information.
- 8. During the training of network under reinforcement learning, the network receives some feedback from the environment. This makes it somewhat similar to supervised learning. However, the feedback obtained here is evaluative not instructive, which means there is no teacher as in supervised learning. After receiving the feedback, the network performs adjustments of the weights to get better critic information in future. Activation Functions It may be defined as the extra force or effort applied over the input to obtain an exact output. In ANN, we can also apply activation functions over the input to get the exact output. Followings are some activation functions of interest − Linear Activation Function It is also called the identity function as it performs no input editing. It can be defined as − $$F(x):=:x$$ Sigmoid Activation Function It is of two type as follows − ● Binary sigmoidal function − This activation function performs input editing between 0 and 1. It is positive in nature. It is always bounded, which means its output cannot be less than 0 and more than 1. It is also strictly increasing in nature, which means more the input higher would be the output. It can be defined as $$F(x):=:sigm(x):=:frac{1}{1:+:exp(-x)}$$ ● Bipolar sigmoidal function − This activation function performs input editing between -1 and 1. It can be positive or negative in nature. It is always bounded, which means its output cannot be less than -1 and more than 1. It is also strictly increasing in nature like sigmoid function. It can be defined as
- 9. $$F(x):=:sigm(x):=:frac{2}{1:+:exp(-x)}:-:1:=:frac{1:-:exp(x)}{1:+:exp(x )}$$ As stated earlier, ANN is completely inspired by the way biological nervous system, i.e. the human brain works. The most impressive characteristic of the human brain is to learn, hence the same feature is acquired by ANN. What Is Learning in ANN? Basically, learning means to do and adapt the change in itself as and when there is a change in environment. ANN is a complex system or more precisely we can say that it is a complex adaptive system, which can change its internal structure based on the information passing through it. Why Is It important? Being a complex adaptive system, learning in ANN implies that a processing unit is capable of changing its input/output behavior due to the change in environment. The importance of learning in ANN increases because of the fixed activation function as well as the input/output vector, when a particular network is constructed. Now to change the input/output behavior, we need to adjust the weights. Classification It may be defined as the process of learning to distinguish the data of samples into different classes by finding common features between the samples of the same classes. For example, to perform training of ANN, we have some training samples with unique features, and to perform its testing we have some testing samples with other unique features. Classification is an example of supervised learning. Neural Network Learning Rules We know that, during ANN learning, to change the input/output behavior, we need to adjust the weights. Hence, a method is required with the help of which the weights can be modified. These methods are called Learning rules, which are simply algorithms or equations. Following are some learning rules for the neural network − Hebbian Learning Rule This rule, one of the oldest and simplest, was introduced by Donald Hebb in his book The Organization of Behavior in 1949. It is a kind of feed-forward, unsupervised learning. Basic Concept − This rule is based on a proposal given by Hebb, who wrote − “When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased.” From the above postulate, we can conclude that the connections between two neurons might be strengthened if the neurons fire at the same time and might weaken if they fire at different times. Mathematical Formulation − According to Hebbian learning rule, following is the formula to increase the weight of connection at every time step.
- 10. $$Delta w_{ji}(t):=:alpha x_{i}(t).y_{j}(t)$$ Here, $Delta w_{ji}(t)$ = increment by which the weight of connection increases at time step t $alpha$ = the positive and constant learning rate $x_{i}(t)$ = the input value from pre-synaptic neuron at time step t $y_{i}(t)$ = the output of pre-synaptic neuron at same time step t Perceptron Learning Rule This rule is an error correcting the supervised learning algorithm of single layer feedforward networks with linear activation function, introduced by Rosenblatt. Basic Concept − As being supervised in nature, to calculate the error, there would be a comparison between the desired/target output and the actual output. If there is any difference found, then a change must be made to the weights of connection. Mathematical Formulation − To explain its mathematical formulation, suppose we have ‘n’ number of finite input vectors, x(n), along with its desired/target output vector t(n), where n = 1 to N. Now the output ‘y’ can be calculated, as explained earlier on the basis of the net input, and activation function being applied over that net input can be expressed as follows − $$y:=:f(y_{in}):=:begin{cases}1, & y_{in}:>:theta 0, & y_{in}:leqslant:thetaend{cases}$$ Where θ is threshold. The updating of weight can be done in the following two cases − Case I − when t ≠ y, then $$w(new):=:w(old):+;tx$$ Case II − when t = y, then No change in weight Delta Learning Rule (Widrow-Hoff Rule) It is introduced by Bernard Widrow and Marcian Hoff, also called Least Mean Square (LMS) method, to minimize the error over all training patterns. It is kind of supervised learning algorithm with having continuous activation function. Basic Concept − The base of this rule is gradient-descent approach, which continues forever. Delta rule updates the synaptic weights so as to minimize the net input to the output unit and the target value. Mathematical Formulation − To update the synaptic weights, delta rule is given by $$Delta w_{i}:=:alpha:.x_{i}.e_{j}$$ Here $Delta w_{i}$ = weight change for ith pattern; $alpha$ = the positive and constant learning rate;
- 11. $x_{i}$ = the input value from pre-synaptic neuron; $e_{j}$ = $(t:-:y_{in})$, the difference between the desired/target output and the actual output $y_{in}$ The above delta rule is for a single output unit only. The updating of weight can be done in the following two cases − Case-I − when t ≠ y, then $$w(new):=:w(old):+:Delta w$$ Case-II − when t = y, then No change in weight As the name suggests, supervised learning takes place under the supervision of a teacher. This learning process is dependent. During the training of ANN under supervised learning, the input vector is presented to the network, which will produce an output vector. This output vector is compared with the desired/target output vector. An error signal is generated if there is a difference between the actual output and the desired/target output vector. On the basis of this error signal, the weights would be adjusted until the actual output is matched with the desired output. Perceptron Developed by Frank Rosenblatt by using McCulloch and Pitts model, perceptron is the basic operational unit of artificial neural networks. It employs supervised learning rule and is able to classify the data into two classes. Operational characteristics of the perceptron: It consists of a single neuron with an arbitrary number of inputs along with adjustable weights, but the output of the neuron is 1 or 0 depending upon the threshold. It also consists of a bias whose weight is always 1. Following figure gives a schematic representation of the perceptron.
- 12. Perceptron thus has the following three basic elements − ● Links − It would have a set of connection links, which carries a weight including a bias always having weight 1. ● Adder − It adds the input after they are multiplied with their respective weights. ● Activation function − It limits the output of neuron. The most basic activation function is a Heaviside step function that has two possible outputs. This function returns 1, if the input is positive, and 0 for any negative input. Training Algorithm Perceptron network can be trained for single output unit as well as multiple output units. Training Algorithm for Single Output Unit Step 1 − Initialize the following to start the training − ● Weights ● Bias ● Learning rate $alpha$ For easy calculation and simplicity, weights and bias must be set equal to 0 and the learning rate must be set equal to 1. Step 2 − Continue step 3-8 when the stopping condition is not true. Step 3 − Continue step 4-6 for every training vector x. Step 4 − Activate each input unit as follows − $$x_{i}:=:s_{i}:(i:=:1:to:n)$$ Step 5 − Now obtain the net input with the following relation − $$y_{in}:=:b:+:displaystylesumlimits_{i}^n x_{i}.:w_{i}$$ Here ‘b’ is bias and ‘n’ is the total number of input neurons. Step 6 − Apply the following activation function to obtain the final output. $$f(y_{in}):=:begin{cases}1 & if:y_{in}:>:theta0 & if : -theta:leqslant:y_{in}:leqslant:theta-1 & if:y_{in}:<:-theta end{cases}$$ Step 7 − Adjust the weight and bias as follows − Case 1 − if y ≠ t then, $$w_{i}(new):=:w_{i}(old):+:alpha:tx_{i}$$ $$b(new):=:b(old):+:alpha t$$
- 13. Case 2 − if y = t then, $$w_{i}(new):=:w_{i}(old)$$ $$b(new):=:b(old)$$ Here ‘y’ is the actual output and ‘t’ is the desired/target output. Step 8 − Test for the stopping condition, which would happen when there is no change in weight. Training Algorithm for Multiple Output Units The following diagram is the architecture of perceptron for multiple output classes. Step 1 − Initialize the following to start the training − ● Weights ● Bias ● Learning rate $alpha$ For easy calculation and simplicity, weights and bias must be set equal to 0 and the learning rate must be set equal to 1. Step 2 − Continue step 3-8 when the stopping condition is not true. Step 3 − Continue step 4-6 for every training vector x. Step 4 − Activate each input unit as follows −
- 14. $$x_{i}:=:s_{i}:(i:=:1:to:n)$$ Step 5 − Obtain the net input with the following relation − $$y_{in}:=:b:+:displaystylesumlimits_{i}^n x_{i}:w_{ij}$$ Here ‘b’ is bias and ‘n’ is the total number of input neurons. Step 6 − Apply the following activation function to obtain the final output for each output unit j = 1 to m − $$f(y_{in}):=:begin{cases}1 & if:y_{inj}:>:theta0 & if : -theta:leqslant:y_{inj}:leqslant:theta-1 & if:y_{inj}:<:-theta end{cases}$$ Step 7 − Adjust the weight and bias for x = 1 to n and j = 1 to m as follows − Case 1 − if yj ≠ tj then, $$w_{ij}(new):=:w_{ij}(old):+:alpha:t_{j}x_{i}$$ $$b_{j}(new):=:b_{j}(old):+:alpha t_{j}$$ Case 2 − if yj = tj then, $$w_{ij}(new):=:w_{ij}(old)$$ $$b_{j}(new):=:b_{j}(old)$$ Here ‘y’ is the actual output and ‘t’ is the desired/target output. Step 8 − Test for the stopping condition, which will happen when there is no change in weight. Adaptive Linear Neuron (Adaline) Adaline which stands for Adaptive Linear Neuron, is a network having a single linear unit. It was developed by Widrow and Hoff in 1960. Some important points about Adaline are as follows − ● It uses bipolar activation function. ● It uses delta rule for training to minimize the Mean-Squared Error (MSE) between the actual output and the desired/target output. ● The weights and the bias are adjustable. Architecture The basic structure of Adaline is similar to perceptron having an extra feedback loop with the help of which the actual output is compared with the desired/target output. After comparison on the basis of training algorithm, the weights and bias will be updated.
- 15. Training Algorithm Step 1 − Initialize the following to start the training − ● Weights ● Bias ● Learning rate $alpha$ For easy calculation and simplicity, weights and bias must be set equal to 0 and the learning rate must be set equal to 1. Step 2 − Continue step 3-8 when the stopping condition is not true. Step 3 − Continue step 4-6 for every bipolar training pair s:t. Step 4 − Activate each input unit as follows − $$x_{i}:=:s_{i}:(i:=:1:to:n)$$ Step 5 − Obtain the net input with the following relation − $$y_{in}:=:b:+:displaystylesumlimits_{i}^n x_{i}:w_{i}$$ Here ‘b’ is bias and ‘n’ is the total number of input neurons. Step 6 − Apply the following activation function to obtain the final output − $$f(y_{in}):=:begin{cases}1 & if:y_{in}:geqslant:0 -1 & if:y_{in}:<:0 end{cases}$$ Step 7 − Adjust the weight and bias as follows −
- 16. Case 1 − if y ≠ t then, $$w_{i}(new):=:w_{i}(old):+: alpha(t:-:y_{in})x_{i}$$ $$b(new):=:b(old):+: alpha(t:-:y_{in})$$ Case 2 − if y = t then, $$w_{i}(new):=:w_{i}(old)$$ $$b(new):=:b(old)$$ Here ‘y’ is the actual output and ‘t’ is the desired/target output. $(t:-;y_{in})$ is the computed error. Step 8 − Test for the stopping condition, which will happen when there is no change in weight or the highest weight change occurred during training is smaller than the specified tolerance. Multiple Adaptive Linear Neuron (Madaline) Madaline which stands for Multiple Adaptive Linear Neuron, is a network which consists of many Adalines in parallel. It will have a single output unit. Some important points about Madaline are as follows − ● It is just like a multilayer perceptron, where Adaline will act as a hidden unit between the input and the Madaline layer. ● The weights and the bias between the input and Adaline layers, as in we see in the Adaline architecture, are adjustable. ● The Adaline and Madaline layers have fixed weights and bias of 1. ● Training can be done with the help of Delta rule. Architecture The architecture of Madaline consists of “n” neurons of the input layer, “m” neurons of the Adaline layer, and 1 neuron of the Madaline layer. The Adaline layer can be considered as the hidden layer as it is between the input layer and the output layer, i.e. the Madaline layer.
- 17. Training Algorithm By now we know that only the weights and bias between the input and the Adaline layer are to be adjusted, and the weights and bias between the Adaline and the Madaline layer are fixed. Step 1 − Initialize the following to start the training − ● Weights ● Bias ● Learning rate $alpha$ For easy calculation and simplicity, weights and bias must be set equal to 0 and the learning rate must be set equal to 1. Step 2 − Continue step 3-8 when the stopping condition is not true. Step 3 − Continue step 4-6 for every bipolar training pair s:t. Step 4 − Activate each input unit as follows − $$x_{i}:=:s_{i}:(i:=:1:to:n)$$ Step 5 − Obtain the net input at each hidden layer, i.e. the Adaline layer with the following relation − $$Q_{inj}:=:b_{j}:+:displaystylesumlimits_{i}^n x_{i}:w_{ij}:::j:=:1:to:m$$ Here ‘b’ is bias and ‘n’ is the total number of input neurons. Step 6 − Apply the following activation function to obtain the final output at the Adaline and the Madaline layer −
- 18. $$f(x):=:begin{cases}1 & if:x:geqslant:0 -1 & if:x:<:0 end{cases}$$ Output at the hidden (Adaline) unit $$Q_{j}:=:f(Q_{inj})$$ Final output of the network $$y:=:f(y_{in})$$ i.e. $::y_{inj}:=:b_{0}:+:sum_{j = 1}^m:Q_{j}:v_{j}$ Step 7 − Calculate the error and adjust the weights as follows − Case 1 − if y ≠ t and t = 1 then, $$w_{ij}(new):=:w_{ij}(old):+: alpha(1:-:Q_{inj})x_{i}$$ $$b_{j}(new):=:b_{j}(old):+: alpha(1:-:Q_{inj})$$ In this case, the weights would be updated on Qj where the net input is close to 0 because t = 1. Case 2 − if y ≠ t and t = -1 then, $$w_{ik}(new):=:w_{ik}(old):+: alpha(-1:-:Q_{ink})x_{i}$$ $$b_{k}(new):=:b_{k}(old):+: alpha(-1:-:Q_{ink})$$ In this case, the weights would be updated on Qk where the net input is positive because t = -1. Here ‘y’ is the actual output and ‘t’ is the desired/target output. Case 3 − if y = t then There would be no change in weights. Step 8 − Test for the stopping condition, which will happen when there is no change in weight or the highest weight change occurred during training is smaller than the specified tolerance. Back Propagation Neural Networks Back Propagation Neural (BPN) is a multilayer neural network consisting of the input layer, at least one hidden layer and output layer. As its name suggests, back propagating will take place in this network. The error which is calculated at the output layer, by comparing the target output and the actual output, will be propagated back towards the input layer.
- 19. Architecture As shown in the diagram, the architecture of BPN has three interconnected layers having weights on them. The hidden layer as well as the output layer also has bias, whose weight is always 1, on them. As is clear from the diagram, the working of BPN is in two phases. One phase sends the signal from the input layer to the output layer, and the other phase back propagates the error from the output layer to the input layer. Training Algorithm For training, BPN will use binary sigmoid activation function. The training of BPN will have the following three phases. ● Phase 1 − Feed Forward Phase ● Phase 2 − Back Propagation of error ● Phase 3 − Updating of weights All these steps will be concluded in the algorithm as follows Step 1 − Initialize the following to start the training − ● Weights ● Learning rate $alpha$ For easy calculation and simplicity, take some small random values.
- 20. Step 2 − Continue step 3-11 when the stopping condition is not true. Step 3 − Continue step 4-10 for every training pair. Phase 1 Step 4 − Each input unit receives input signal xi and sends it to the hidden unit for all i = 1 to n Step 5 − Calculate the net input at the hidden unit using the following relation − $$Q_{inj}:=:b_{0j}:+:sum_{i=1}^n x_{i}v_{ij}::::j:=:1:to:p$$ Here b0j is the bias on hidden unit, vij is the weight on j unit of the hidden layer coming from i unit of the input layer. Now calculate the net output by applying the following activation function $$Q_{j}:=:f(Q_{inj})$$ Send these output signals of the hidden layer units to the output layer units. Step 6 − Calculate the net input at the output layer unit using the following relation − $$y_{ink}:=:b_{0k}:+:sum_{j = 1}^p:Q_{j}:w_{jk}::k:=:1:to:m$$ Here b0k is the bias on output unit, wjk is the weight on k unit of the output layer coming from j unit of the hidden layer. Calculate the net output by applying the following activation function $$y_{k}:=:f(y_{ink})$$ Phase 2 Step 7 − Compute the error correcting term, in correspondence with the target pattern received at each output unit, as follows − $$delta_{k}:=:(t_{k}:-:y_{k})f^{'}(y_{ink})$$ On this basis, update the weight and bias as follows − $$Delta v_{jk}:=:alpha delta_{k}:Q_{ij}$$ $$Delta b_{0k}:=:alpha delta_{k}$$ Then, send $delta_{k}$ back to the hidden layer. Step 8 − Now each hidden unit will be the sum of its delta inputs from the output units.
- 21. $$delta_{inj}:=:displaystylesumlimits_{k=1}^m delta_{k}:w_{jk}$$ Error term can be calculated as follows − $$delta_{j}:=:delta_{inj}f^{'}(Q_{inj})$$ On this basis, update the weight and bias as follows − $$Delta w_{ij}:=:alphadelta_{j}x_{i}$$ $$Delta b_{0j}:=:alphadelta_{j}$$ Phase 3 Step 9 − Each output unit (ykk = 1 to m) updates the weight and bias as follows − $$v_{jk}(new):=:v_{jk}(old):+:Delta v_{jk}$$ $$b_{0k}(new):=:b_{0k}(old):+:Delta b_{0k}$$ Step 10 − Each output unit (zjj = 1 to p) updates the weight and bias as follows − $$w_{ij}(new):=:w_{ij}(old):+:Delta w_{ij}$$ $$b_{0j}(new):=:b_{0j}(old):+:Delta b_{0j}$$ Step 11 − Check for the stopping condition, which may be either the number of epochs reached or the target output matches the actual output. Generalized Delta Learning Rule Delta rule works only for the output layer. On the other hand, generalized delta rule, also called as back-propagation rule, is a way of creating the desired values of the hidden layer. Mathematical Formulation For the activation function $y_{k}:=:f(y_{ink})$ the derivation of net input on Hidden layer as well as on output layer can be given by $$y_{ink}:=:displaystylesumlimits_i:z_{i}w_{jk}$$ And $::y_{inj}:=:sum_i x_{i}v_{ij}$ Now the error which has to be minimized is $$E:=:frac{1}{2}displaystylesumlimits_{k}:[t_{k}:-:y_{k}]^2$$ By using the chain rule, we have
- 22. $$frac{partial E}{partial w_{jk}}:=:frac{partial }{partial w_{jk}}(frac{1}{2}displaystylesumlimits_{k}:[t_{k}:-:y_{k}]^2)$$ $$=:frac{partial }{partial w_{jk}}lgroupfrac{1}{2}[t_{k}:-:t(y_{ink})]^2rgroup$$ $$=:-[t_{k}:-:y_{k}]frac{partial }{partial w_{jk}}f(y_{ink})$$ $$=:-[t_{k}:-:y_{k}]f(y_{ink})frac{partial }{partial w_{jk}}(y_{ink})$$ $$=:-[t_{k}:-:y_{k}]f^{'}(y_{ink})z_{j}$$ Now let us say $delta_{k}:=:-[t_{k}:-:y_{k}]f^{'}(y_{ink})$ The weights on connections to the hidden unit zj can be given by − $$frac{partial E}{partial v_{ij}}:=:- displaystylesumlimits_{k} delta_{k}frac{partial }{partial v_{ij}}:(y_{ink})$$ Putting the value of $y_{ink}$ we will get the following $$delta_{j}:=:-displaystylesumlimits_{k}delta_{k}w_{jk}f^{'}(z_{inj})$$ Weight updating can be done as follows − For the output unit − $$Delta w_{jk}:=:-alphafrac{partial E}{partial w_{jk}}$$ $$=:alpha:delta_{k}:z_{j}$$ For the hidden unit − $$Delta v_{ij}:=:-alphafrac{partial E}{partial v_{ij}}$$ $$=:alpha:delta_{j}:x_{i}$$ hese kinds of neural networks work on the basis of pattern association, which means they can store different patterns and at the time of giving an output they can produce one of the stored patterns by matching them with the given input pattern. These types of memories are also called Content-Addressable Memory CAM . Associative memory makes a parallel search with the stored patterns as data files. Following are the two types of associative memories we can observe − ● Auto Associative Memory ● Hetero Associative memory Auto Associative Memory
- 23. This is a single layer neural network in which the input training vector and the output target vectors are the same. The weights are determined so that the network stores a set of patterns. Architecture As shown in the following figure, the architecture of Auto Associative memory network has ‘n’ number of input training vectors and similar ‘n’ number of output target vectors. Training Algorithm For training, this network is using the Hebb or Delta learning rule. Step 1 − Initialize all the weights to zero as wij = 0 i=1ton,j=1ton Step 2 − Perform steps 3-4 for each input vector. Step 3 − Activate each input unit as follows − xi=si(i=1ton) Step 4 − Activate each output unit as follows − yj=sj(j=1ton) Step 5 − Adjust the weights as follows − wij(new)=wij(old)+xiyj Testing Algorithm Step 1 − Set the weights obtained during training for Hebb’s rule. Step 2 − Perform steps 3-5 for each input vector.
- 24. Step 3 − Set the activation of the input units equal to that of the input vector. Step 4 − Calculate the net input to each output unit j = 1 to n yinj=∑i=1nxiwij Step 5 − Apply the following activation function to calculate the output yj=f(yinj)={+1−1ifyinj>0ifyinj⩽0 Hetero Associative memory Similar to Auto Associative Memory network, this is also a single layer neural network. However, in this network the input training vector and the output target vectors are not the same. The weights are determined so that the network stores a set of patterns. Hetero associative network is static in nature, hence, there would be no non-linear and delay operations. Architecture As shown in the following figure, the architecture of Hetero Associative Memory network has ‘n’ number of input training vectors and ‘m’ number of output target vectors. Training Algorithm For training, this network is using the Hebb or Delta learning rule. Step 1 − Initialize all the weights to zero as wij = 0 i=1ton,j=1tom Step 2 − Perform steps 3-4 for each input vector. Step 3 − Activate each input unit as follows − xi=si(i=1ton)
- 25. Step 4 − Activate each output unit as follows − yj=sj(j=1tom) Step 5 − Adjust the weights as follows − wij(new)=wij(old)+xiyj Testing Algorithm Step 1 − Set the weights obtained during training for Hebb’s rule. Step 2 − Perform steps 3-5 for each input vector. Step 3 − Set the activation of the input units equal to that of the input vector. Step 4 − Calculate the net input to each output unit j = 1 to m; yinj=∑i=1nxiwij Step 5 − Apply the following activation function to calculate the output yj=f(yinj)=⎧⎩⎨⎪⎪+10−1ifyinj>0ifyinj=0ifyinj<0 Hopfield neural network was invented by Dr. John J. Hopfield in 1982. It consists of a single layer which contains one or more fully connected recurrent neurons. The Hopfield network is commonly used for auto-association and optimization tasks. Discrete Hopfield Network A Hopfield network which operates in a discrete line fashion or in other words, it can be said the input and output patterns are discrete vector, which can be either binary 0,1 or bipolar +1,−1 in nature. The network has symmetrical weights with no self-connections i.e., wij = wji and wii = 0. Architecture Following are some important points to keep in mind about discrete Hopfield network − ● This model consists of neurons with one inverting and one non-inverting output. ● The output of each neuron should be the input of other neurons but not the input of self. ● Weight/connection strength is represented by wij.
- 26. ● Connections can be excitatory as well as inhibitory. It would be excitatory, if the output of the neuron is same as the input, otherwise inhibitory. ● Weights should be symmetrical, i.e. wij = wji The output from Y1 going to Y2, Yi and Yn have the weights w12, w1i and w1n respectively. Similarly, other arcs have the weights on them. Training Algorithm During training of discrete Hopfield network, weights will be updated. As we know that we can have the binary input vectors as well as bipolar input vectors. Hence, in both the cases, weight updates can be done with the following relation Case 1 − Binary input patterns For a set of binary patterns sp , p = 1 to P Here, sp = s1p, s2p,..., sip,..., snp Weight Matrix is given by
- 27. wij=∑p=1P[2si(p)−1][2sj(p)−1]fori≠j Case 2 − Bipolar input patterns For a set of binary patterns sp , p = 1 to P Here, sp = s1p, s2p,..., sip,..., snp Weight Matrix is given by wij=∑p=1P[si(p)][sj(p)]fori≠j Testing Algorithm Step 1 − Initialize the weights, which are obtained from training algorithm by using Hebbian principle. Step 2 − Perform steps 3-9, if the activations of the network is not consolidated. Step 3 − For each input vector X, perform steps 4-8. Step 4 − Make initial activation of the network equal to the external input vector X as follows − yi=xifori=1ton Step 5 − For each unit Yi, perform steps 6-9. Step 6 − Calculate the net input of the network as follows − yini=xi+∑jyjwji Step 7 − Apply the activation as follows over the net input to calculate the output − yi=⎧⎩⎨1yi0ifyini>θiifyini=θiifyini<θi Here θi is the threshold. Step 8 − Broadcast this output yi to all other units. Step 9 − Test the network for conjunction.
- 28. Energy Function Evaluation An energy function is defined as a function that is bonded and non-increasing function of the state of the system. Energy function Ef , also called Lyapunov function determines the stability of discrete Hopfield network, and is characterized as follows − Ef=−12∑i=1n∑j=1nyiyjwij−∑i=1nxiyi+∑i=1nθiyi Condition − In a stable network, whenever the state of node changes, the above energy function will decrease. Suppose when node i has changed state from y(k)i to y(k+1)i then the Energy change ΔEf is given by the following relation ΔEf=Ef(y(k+1)i)−Ef(y(k)i) =−(∑j=1nwijy(k)i+xi−θi)(y(k+1)i−y(k)i) =−(neti)Δyi Here Δyi=y(k+1)i−y(k)i The change in energy depends on the fact that only one unit can update its activation at a time.