Category: artificial Neural Network

Applications of Neural Networks Before studying the fields where ANN has been used extensively, we need to understand why ANN would be the preferred choice of application. Why Artificial Neural Networks? We need to understand the answer to the above question with an example of a human being. As a child, we used to learn the things with the help of our elders, which includes our parents or teachers. Then later by self-learning or practice we keep learning throughout our life. Scientists and researchers are also making the machine intelligent, just like a human being, and ANN plays a very important role in the same due to the following reasons − With the help of neural networks, we can find the solution of such problems for which algorithmic method is expensive or does not exist. Neural networks can learn by example, hence we do not need to program it at much extent. Neural networks have the accuracy and significantly fast speed than conventional speed. Areas of Application Followings are some of the areas, where ANN is being used. It suggests that ANN has an interdisciplinary approach in its development and applications. Speech Recognition Speech occupies a prominent role in human-human interaction. Therefore, it is natural for people to expect speech interfaces with computers. In the present era, for communication with machines, humans still need sophisticated languages which are difficult to learn and use. To ease this communication barrier, a simple solution could be, communication in a spoken language that is possible for the machine to understand. Great progress has been made in this field, however, still such kinds of systems are facing the problem of limited vocabulary or grammar along with the issue of retraining of the system for different speakers in different conditions. ANN is playing a major role in this area. Following ANNs have been used for speech recognition − Multilayer networks Multilayer networks with recurrent connections Kohonen self-organizing feature map The most useful network for this is Kohonen Self-Organizing feature map, which has its input as short segments of the speech waveform. It will map the same kind of phonemes as the output array, called feature extraction technique. After extracting the features, with the help of some acoustic models as back-end processing, it will recognize the utterance. Character Recognition It is an interesting problem which falls under the general area of Pattern Recognition. Many neural networks have been developed for automatic recognition of handwritten characters, either letters or digits. Following are some ANNs which have been used for character recognition − Multilayer neural networks such as Backpropagation neural networks. Neocognitron Though back-propagation neural networks have several hidden layers, the pattern of connection from one layer to the next is localized. Similarly, neocognitron also has several hidden layers and its training is done layer by layer for such kind of applications. Signature Verification Application Signatures are one of the most useful ways to authorize and authenticate a person in legal transactions. Signature verification technique is a non-vision based technique. For this application, the first approach is to extract the feature or rather the geometrical feature set representing the signature. With these feature sets, we have to train the neural networks using an efficient neural network algorithm. This trained neural network will classify the signature as being genuine or forged under the verification stage. Human Face Recognition It is one of the biometric methods to identify the given face. It is a typical task because of the characterization of “non-face” images. However, if a neural network is well trained, then it can be divided into two classes namely images having faces and images that do not have faces. First, all the input images must be preprocessed. Then, the dimensionality of that image must be reduced. And, at last it must be classified using neural network training algorithm. Following neural networks are used for training purposes with preprocessed image − Fully-connected multilayer feed-forward neural network trained with the help of back-propagation algorithm. For dimensionality reduction, Principal Component Analysis (PCA) is used. Learning working make money

Discuss Artificial Neural Network Neural networks are parallel computing devices, which are 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. This tutorial covers the basic concept and terminologies involved in Artificial Neural Network. Sections of this tutorial also explain the architecture as well as the training algorithm of various networks used in ANN. Learning working make money

Optimization Using Hopfield Network Optimization is an action of making something such as design, situation, resource, and system as effective as possible. Using a resemblance between the cost function and energy function, we can use highly interconnected neurons to solve optimization problems. Such a kind of neural network is Hopfield network, that consists of a single layer containing one or more fully connected recurrent neurons. This can be used for optimization. Points to remember while using Hopfield network for optimization − The energy function must be minimum of the network. It will find satisfactory solution rather than select one out of the stored patterns. The quality of the solution found by Hopfield network depends significantly on the initial state of the network. Travelling Salesman Problem Finding the shortest route travelled by the salesman is one of the computational problems, which can be optimized by using Hopfield neural network. Basic Concept of TSP Travelling Salesman Problem (TSP) is a classical optimization problem in which a salesman has to travel n cities, which are connected with each other, keeping the cost as well as the distance travelled minimum. For example, the salesman has to travel a set of 4 cities A, B, C, D and the goal is to find the shortest circular tour, A-B-C–D, so as to minimize the cost, which also includes the cost of travelling from the last city D to the first city A. Matrix Representation Actually each tour of n-city TSP can be expressed as n &times n matrix whose ith row describes the ith city’s location. This matrix, M, for 4 cities A, B, C, D can be expressed as follows − $$M = begin{bmatrix}A: & 1 & 0 & 0 & 0 \B: & 0 & 1 & 0 & 0 \C: & 0 & 0 & 1 & 0 \D: & 0 & 0 & 0 & 1 end{bmatrix}$$ Solution by Hopfield Network While considering the solution of this TSP by Hopfield network, every node in the network corresponds to one element in the matrix. Energy Function Calculation To be the optimized solution, the energy function must be minimum. On the basis of the following constraints, we can calculate the energy function as follows − Constraint-I First constraint, on the basis of which we will calculate energy function, is that one element must be equal to 1 in each row of matrix M and other elements in each row must equal to 0 because each city can occur in only one position in the TSP tour. This constraint can mathematically be written as follows − $$displaystylesumlimits_{j=1}^n M_{x,j}:=:1:for : x:in :lbrace1,…,nrbrace$$ Now the energy function to be minimized, based on the above constraint, will contain a term proportional to − $$displaystylesumlimits_{x=1}^n left(begin{array}{c}1:-:displaystylesumlimits_{j=1}^n M_{x,j}end{array}right)^2$$ Constraint-II As we know, in TSP one city can occur in any position in the tour hence in each column of matrix M, one element must equal to 1 and other elements must be equal to 0. This constraint can mathematically be written as follows − $$displaystylesumlimits_{x=1}^n M_{x,j}:=:1:for : j:in :lbrace1,…,nrbrace$$ Now the energy function to be minimized, based on the above constraint, will contain a term proportional to − $$displaystylesumlimits_{j=1}^n left(begin{array}{c}1:-:displaystylesumlimits_{x=1}^n M_{x,j}end{array}right)^2$$ Cost Function Calculation Let’s suppose a square matrix of (n × n) denoted by C denotes the cost matrix of TSP for n cities where n > 0. Following are some parameters while calculating the cost function − Cx, y − The element of cost matrix denotes the cost of travelling from city x to y. Adjacency of the elements of A and B can be shown by the following relation − $$M_{x,i}:=:1:: and:: M_{y,ipm 1}:=:1$$ As we know, in Matrix the output value of each node can be either 0 or 1, hence for every pair of cities A, B we can add the following terms to the energy function − $$displaystylesumlimits_{i=1}^n C_{x,y}M_{x,i}(M_{y,i+1}:+:M_{y,i-1})$$ On the basis of the above cost function and constraint value, the final energy function E can be given as follows − $$E:=:frac{1}{2}displaystylesumlimits_{i=1}^ndisplaystylesumlimits_{x}displaystylesumlimits_{yneq x}C_{x,y}M_{x,i}(M_{y,i+1}:+:M_{y,i-1}):+$$ $$:begin{bmatrix}gamma_{1} displaystylesumlimits_{x} left(begin{array}{c}1:-:displaystylesumlimits_{i} M_{x,i}end{array}right)^2:+: gamma_{2} displaystylesumlimits_{i} left(begin{array}{c}1:-:displaystylesumlimits_{x} M_{x,i}end{array}right)^2 end{bmatrix}$$ Here, γ1 and γ2 are two weighing constants. Learning working make money

Artificial Neural Network – Useful Resources The following resources contain additional information on Artificial Neural Network. Please use them to get more in-depth knowledge on this. Useful Video Courses Best Seller 88 Lectures 9.5 hours 16 Lectures 54 mins 12 Lectures 1 hours 21 Lectures 2 hours Most Popular 19 Lectures 58 mins 18 Lectures 1.5 hours Learning working make money

Artificial Neural Network – Quick Guide Artificial Neural Network – Basic Concepts 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 − 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 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. 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) Artificial Neural Network – Building Blocks Processing of ANN depends upon the following three building blocks − Network Topology Adjustments of Weights or Learning Activation Functions In this chapter, we will discuss in detail about these three building blocks of ANN 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

Artificial Neural Network – Genetic Algorithm Nature has always been a great source of inspiration to all mankind. Genetic Algorithms (GAs) are search-based algorithms based on the concepts of natural selection and genetics. GAs are a subset of a much larger branch of computation known as Evolutionary Computation. GAs was developed by John Holland and his students and colleagues at the University of Michigan, most notably David E. Goldberg and has since been tried on various optimization problems with a high degree of success. In GAs, we have a pool or a population of possible solutions to the given problem. These solutions then undergo recombination and mutation (like in natural genetics), producing new children, and the process is repeated over various generations. Each individual (or candidate solution) is assigned a fitness value (based on its objective function value) and the fitter individuals are given a higher chance to mate and yield more “fitter” individuals. This is in line with the Darwinian Theory of “Survival of the Fittest”. In this way, we keep “evolving” better individuals or solutions over generations, till we reach a stopping criterion. Genetic Algorithms are sufficiently randomized in nature, however they perform much better than random local search (in which we just try various random solutions, keeping track of the best so far), as they exploit historical information as well. Advantages of GAs GAs have various advantages which have made them immensely popular. These include − Does not require any derivative information (which may not be available for many real-world problems). Is faster and more efficient as compared to the traditional methods. Has very good parallel capabilities. Optimizes both continuous and discrete functions as well as multi-objective problems. Provides a list of “good” solutions and not just a single solution. Always gets an answer to the problem, which gets better over the time. Useful when the search space is very large and there are large number of parameters involved. Limitations of GAs Like any technique, GAs also suffers from a few limitations. These include − GAs are not suited for all problems, especially problems which are simple and for which derivative information is available. Fitness value is calculated repeatedly, which might be computationally expensive for some problems. Being stochastic, there are no guarantees on the optimality or the quality of the solution. If not implemented properly, GA may not converge to the optimal solution. GA – Motivation Genetic Algorithms have the ability to deliver a “good-enough” solution “fast-enough”. This makes Gas attractive for use in solving optimization problems. The reasons why GAs are needed are as follows − Solving Difficult Problems In computer science, there is a large set of problems, which are NP-Hard. What this essentially means is that, even the most powerful computing systems take a very long time (even years!) to solve that problem. In such a scenario, GAs prove to be an efficient tool to provide usable near-optimal solutions in a short amount of time. Failure of Gradient Based Methods Traditional calculus based methods work by starting at a random point and by moving in the direction of the gradient, till we reach the top of the hill. This technique is efficient and works very well for single-peaked objective functions like the cost function in linear regression. However, in most real-world situations, we have a very complex problem called as landscapes, made of many peaks and many valleys, which causes such methods to fail, as they suffer from an inherent tendency of getting stuck at the local optima as shown in the following figure. Getting a Good Solution Fast Some difficult problems like the Travelling Salesman Problem (TSP), have real-world applications like path finding and VLSI Design. Now imagine that you are using your GPS Navigation system, and it takes a few minutes (or even a few hours) to compute the “optimal” path from the source to destination. Delay in such real-world applications is not acceptable and therefore a “good-enough” solution, which is delivered “fast” is what is required. How to Use GA for Optimization Problems? We already know that optimization is an action of making something such as design, situation, resource, and system as effective as possible. Optimization process is shown in the following diagram. Stages of GA Mechanism for Optimization Process Followings are the stages of GA mechanism when used for optimization of problems. Generate the initial population randomly. Select the initial solution with the best fitness values. Recombine the selected solutions using mutation and crossover operators. Insert offspring into the population. Now if the stop condition is met, then return the solution with their best fitness value. Else, go to step 2. Learning working make money

Learning Vector Quantization Learning Vector Quantization (LVQ), different from Vector quantization (VQ) and Kohonen Self-Organizing Maps (KSOM), basically is a competitive network which uses supervised learning. We may define it as a process of classifying the patterns where each output unit represents a class. As it uses supervised learning, the network will be given a set of training patterns with known classification along with an initial distribution of the output class. After completing the training process, LVQ will classify an input vector by assigning it to the same class as that of the output unit. Architecture Following figure shows the architecture of LVQ which is quite similar to the architecture of KSOM. As we can see, there are “n” number of input units and “m” number of output units. The layers are fully interconnected with having weights on them. Parameters Used Following are the parameters used in LVQ training process as well as in the flowchart x = training vector (x1,…,xi,…,xn) T = class for training vector x wj = weight vector for jth output unit Cj = class associated with the jth output unit Training Algorithm Step 1 − Initialize reference vectors, which can be done as follows − Step 1(a) − From the given set of training vectors, take the first “m” (number of clusters) training vectors and use them as weight vectors. The remaining vectors can be used for training. Step 1(b) − Assign the initial weight and classification randomly. Step 1(c) − Apply K-means clustering method. Step 2 − Initialize reference vector $alpha$ Step 3 − Continue with steps 4-9, if the condition for stopping this algorithm is not met. Step 4 − Follow steps 5-6 for every training input vector x. Step 5 − Calculate Square of Euclidean Distance for j = 1 to m and i = 1 to n $$D(j):=:displaystylesumlimits_{i=1}^ndisplaystylesumlimits_{j=1}^m (x_{i}:-:w_{ij})^2$$ Step 6 − Obtain the winning unit J where D(j) is minimum. Step 7 − Calculate the new weight of the winning unit by the following relation − if T = Cj then $w_{j}(new):=:w_{j}(old):+:alpha[x:-:w_{j}(old)]$ if T ≠ Cj then $w_{j}(new):=:w_{j}(old):-:alpha[x:-:w_{j}(old)]$ Step 8 − Reduce the learning rate $alpha$. Step 9 − Test for the stopping condition. It may be as follows − Maximum number of epochs reached. Learning rate reduced to a negligible value. Flowchart Variants Three other variants namely LVQ2, LVQ2.1 and LVQ3 have been developed by Kohonen. Complexity in all these three variants, due to the concept that the winner as well as the runner-up unit will learn, is more than in LVQ. LVQ2 As discussed, the concept of other variants of LVQ above, the condition of LVQ2 is formed by window. This window will be based on the following parameters − x − the current input vector yc − the reference vector closest to x yr − the other reference vector, which is next closest to x dc − the distance from x to yc dr − the distance from x to yr The input vector x falls in the window, if $$frac{d_{c}}{d_{r}}:>:1:-:theta::and::frac{d_{r}}{d_{c}}:>:1:+:theta$$ Here, $theta$ is the number of training samples. Updating can be done with the following formula − $y_{c}(t:+:1):=:y_{c}(t):+:alpha(t)[x(t):-:y_{c}(t)]$ (belongs to different class) $y_{r}(t:+:1):=:y_{r}(t):+:alpha(t)[x(t):-:y_{r}(t)]$ (belongs to same class) Here $alpha$ is the learning rate. LVQ2.1 In LVQ2.1, we will take the two closest vectors namely yc1 and yc2 and the condition for window is as follows − $$Minbegin{bmatrix}frac{d_{c1}}{d_{c2}},frac{d_{c2}}{d_{c1}}end{bmatrix}:>:(1:-:theta)$$ $$Maxbegin{bmatrix}frac{d_{c1}}{d_{c2}},frac{d_{c2}}{d_{c1}}end{bmatrix}: Updating can be done with the following formula − $y_{c1}(t:+:1):=:y_{c1}(t):+:alpha(t)[x(t):-:y_{c1}(t)]$ (belongs to different class) $y_{c2}(t:+:1):=:y_{c2}(t):+:alpha(t)[x(t):-:y_{c2}(t)]$ (belongs to same class) Here, $alpha$ is the learning rate. LVQ3 In LVQ3, we will take the two closest vectors namely yc1 and yc2 and the condition for window is as follows − $$Minbegin{bmatrix}frac{d_{c1}}{d_{c2}},frac{d_{c2}}{d_{c1}}end{bmatrix}:>:(1:-:theta)(1:+:theta)$$ Here $thetaapprox 0.2$ Updating can be done with the following formula − $y_{c1}(t:+:1):=:y_{c1}(t):+:beta(t)[x(t):-:y_{c1}(t)]$ (belongs to different class) $y_{c2}(t:+:1):=:y_{c2}(t):+:beta(t)[x(t):-:y_{c2}(t)]$ (belongs to same class) Here $beta$ is the multiple of the learning rate $alpha$ and $beta:=:m alpha(t)$ for every 0.1 < m < 0.5 Learning working make money

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 input pattern belongs. In this, there would be no feedback from the environment as to what should be the desired output and whether it is correct or incorrect. Hence, in this type of learning the network itself must discover the patterns, features from the input data and the relation for the input data over the output. Winner-Takes-All Networks These kinds of networks are based on the competitive learning rule and will use the strategy where it chooses the neuron with the greatest total inputs as a winner. The connections between the output neurons show the competition between them and one of them would be ‘ON’ which means it would be the winner and others would be ‘OFF’. Following are some of the networks based on this simple concept using unsupervised learning. Hamming Network In most of the neural networks using unsupervised learning, it is essential to compute the distance and perform comparisons. This kind of network is Hamming network, where for every given input vectors, it would be clustered into different groups. Following are some important features of Hamming Networks − Lippmann started working on Hamming networks in 1987. It is a single layer network. The inputs can be either binary {0, 1} of bipolar {-1, 1}. The weights of the net are calculated by the exemplar vectors. It is a fixed weight network which means the weights would remain the same even during training. Max Net This is also a fixed weight network, which serves as a subnet for selecting the node having the highest input. All the nodes are fully interconnected and there exists symmetrical weights in all these weighted interconnections. Architecture It uses the mechanism which is an iterative process and each node receives inhibitory inputs from all other nodes through connections. The single node whose value is maximum would be active or winner and the activations of all other nodes would be inactive. Max Net uses identity activation function with $$f(x):=:begin{cases}x & if:x > 0\0 & if:x leq 0end{cases}$$ The task of this net is accomplished by the self-excitation weight of +1 and mutual inhibition magnitude, which is set like [0 < ɛ < $frac{1}{m}$] where “m” is the total number of the nodes. Competitive Learning in ANN It is concerned with unsupervised training in which the output nodes try to compete with each other to represent the input pattern. To understand this learning rule we will have to understand competitive net which is explained as follows − Basic Concept of Competitive Network This network is just like a single layer feed-forward network having feedback connection between the outputs. The connections between the outputs are inhibitory type, which is shown by dotted lines, which means the competitors never support themselves. Basic Concept of Competitive Learning Rule As said earlier, there would be competition among the output nodes so the main concept is – during training, the output unit that has the highest activation to a given input pattern, will be declared the winner. This rule is also called Winner-takes-all because only the winning neuron is updated and the rest of the neurons are left unchanged. Mathematical Formulation Following are the three important factors for mathematical formulation of this learning rule − Condition to be a winner Suppose if a neuron yk wants to be the winner, then there would be the following condition $$y_{k}:=:begin{cases}1 & if:v_{k} > v_{j}:for:all::j,:j:neq:k\0 & otherwiseend{cases}$$ It means that if any neuron, say, yk wants to win, then its induced local field (the output of the summation unit), say vk, must be the largest among all the other neurons in the network. Condition of the sum total of weight Another constraint over the competitive learning rule is the sum total of weights to a particular output neuron is going to be 1. For example, if we consider neuron k then $$displaystylesumlimits_{k} w_{kj}:=:1::::for:all::k$$ Change of weight for the winner If a neuron does not respond to the input pattern, then no learning takes place in that neuron. However, if a particular neuron wins, then the corresponding weights are adjusted as follows − $$Delta w_{kj}:=:begin{cases}-alpha(x_{j}:-:w_{kj}), & if:neuron:k:wins\0 & if:neuron:k:lossesend{cases}$$ Here $alpha$ is the learning rate. This clearly shows that we are favoring the winning neuron by adjusting its weight and if a neuron is lost, then we need not bother to re-adjust its weight. K-means Clustering Algorithm K-means is one of the most popular clustering algorithm in which we use the concept of partition procedure. We start with an initial partition and repeatedly move patterns from one cluster to another, until we get a satisfactory result. Algorithm Step 1 − Select k points as the initial centroids. Initialize k prototypes (w1,…,wk), for example we can identifying them with randomly chosen input vectors − $$W_{j}:=:i_{p},::: where:j:in lbrace1,….,krbrace:and:p:in lbrace1,….,nrbrace$$ Each cluster Cj is associated with prototype wj. Step 2 − Repeat step 3-5 until E no longer decreases, or the cluster membership no longer changes. Step 3 − For each input vector ip where p ∈ {1,…,n}, put ip in the cluster Cj* with the nearest prototype wj* having the following relation $$|i_{p}:-:w_{j*}|:leq:|i_{p}:-:w_{j}|,:j:in lbrace1,….,krbrace$$ Step 4 − For each cluster Cj, where j ∈ { 1,…,k}, update the prototype wj to be the centroid of all samples currently in Cj , so that $$w_{j}:=:sum_{i_{p}in C_{j}}frac{i_{p}}{|C_{j}|}$$ Step 5 − Compute the total quantization error as follows − $$E:=:sum_{j=1}^ksum_{i_{p}in w_{j}}|i_{p}:-:w_{j}|^2$$ Neocognitron It is a multilayer feedforward network, which was developed by Fukushima in 1980s. This model is based on supervised learning and is used for visual pattern recognition, mainly hand-written characters. It is basically an extension of Cognitron network, which was also developed by Fukushima

Adaptive Resonance Theory This network was developed by Stephen Grossberg and Gail Carpenter in 1987. It is based on competition and uses unsupervised learning model. Adaptive Resonance Theory (ART) networks, as the name suggests, is always open to new learning (adaptive) without losing the old patterns (resonance). Basically, ART network is a vector classifier which accepts an input vector and classifies it into one of the categories depending upon which of the stored pattern it resembles the most. Operating Principal The main operation of ART classification can be divided into the following phases − Recognition phase − The input vector is compared with the classification presented at every node in the output layer. The output of the neuron becomes “1” if it best matches with the classification applied, otherwise it becomes “0”. Comparison phase − In this phase, a comparison of the input vector to the comparison layer vector is done. The condition for reset is that the degree of similarity would be less than vigilance parameter. Search phase − In this phase, the network will search for reset as well as the match done in the above phases. Hence, if there would be no reset and the match is quite good, then the classification is over. Otherwise, the process would be repeated and the other stored pattern must be sent to find the correct match. ART1 It is a type of ART, which is designed to cluster binary vectors. We can understand about this with the architecture of it. Architecture of ART1 It consists of the following two units − Computational Unit − It is made up of the following − Input unit (F1 layer) − It further has the following two portions − F1(a) layer (Input portion) − In ART1, there would be no processing in this portion rather than having the input vectors only. It is connected to F1(b) layer (interface portion). F1(b) layer (Interface portion) − This portion combines the signal from the input portion with that of F2 layer. F1(b) layer is connected to F2 layer through bottom up weights bij and F2 layer is connected to F1(b) layer through top down weights tji. Cluster Unit (F2 layer) − This is a competitive layer. The unit having the largest net input is selected to learn the input pattern. The activation of all other cluster unit are set to 0. Reset Mechanism − The work of this mechanism is based upon the similarity between the top-down weight and the input vector. Now, if the degree of this similarity is less than the vigilance parameter, then the cluster is not allowed to learn the pattern and a rest would happen. Supplement Unit − Actually the issue with Reset mechanism is that the layer F2 must have to be inhibited under certain conditions and must also be available when some learning happens. That is why two supplemental units namely, G1 and G2 is added along with reset unit, R. They are called gain control units. These units receive and send signals to the other units present in the network. ‘+’ indicates an excitatory signal, while ‘−’ indicates an inhibitory signal. Parameters Used Following parameters are used − n − Number of components in the input vector m − Maximum number of clusters that can be formed bij − Weight from F1(b) to F2 layer, i.e. bottom-up weights tji − Weight from F2 to F1(b) layer, i.e. top-down weights ρ − Vigilance parameter ||x|| − Norm of vector x Algorithm Step 1 − Initialize the learning rate, the vigilance parameter, and the weights as follows − $$alpha:>:1::and::0: $$0: Step 2 − Continue step 3-9, when the stopping condition is not true. Step 3 − Continue step 4-6 for every training input. Step 4 − Set activations of all F1(a) and F1 units as follows F2 = 0 and F1(a) = input vectors Step 5 − Input signal from F1(a) to F1(b) layer must be sent like $$s_{i}:=:x_{i}$$ Step 6 − For every inhibited F2 node $y_{j}:=:sum_i b_{ij}x_{i}$ the condition is yj ≠ -1 Step 7 − Perform step 8-10, when the reset is true. Step 8 − Find J for yJ ≥ yj for all nodes j Step 9 − Again calculate the activation on F1(b) as follows $$x_{i}:=:sitJi$$ Step 10 − Now, after calculating the norm of vector x and vector s, we need to check the reset condition as follows − If ||x||/ ||s|| < vigilance parameter ρ,⁡then⁡inhibit ⁡node J and go to step 7 Else If ||x||/ ||s|| ≥ vigilance parameter ρ, then proceed further. Step 11 − Weight updating for node J can be done as follows − $$b_{ij}(new):=:frac{alpha x_{i}}{alpha:-:1:+:||x||}$$ $$t_{ij}(new):=:x_{i}$$ Step 12 − The stopping condition for algorithm must be checked and it may be as follows − Do not have any change in weight. Reset is not performed for units. Maximum number of epochs reached. Learning working make money

Artificial Neural Network – Hopfield Networks 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. 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 s(p), p = 1 to P Here, s(p) = s1(p), s2(p),…, si(p),…, sn(p) Weight Matrix is given by $$w_{ij}:=:sum_{p=1}^P[2s_{i}(p)-:1][2s_{j}(p)-:1]:::::for:i:neq:j$$ Case 2 − Bipolar input patterns For a set of binary patterns s(p), p = 1 to P Here, s(p) = s1(p), s2(p),…, si(p),…, sn(p) Weight Matrix is given by $$w_{ij}:=:sum_{p=1}^P[s_{i}(p)][s_{j}(p)]:::::for:i:neq:j$$ 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 − $$y_{i}:=:x_{i}:::for:i:=:1:to:n$$ Step 5 − For each unit Yi, perform steps 6-9. Step 6 − Calculate the net input of the network as follows − $$y_{ini}:=:x_{i}:+:displaystylesumlimits_{j}y_{j}w_{ji}$$ Step 7 − Apply the activation as follows over the net input to calculate the output − $$y_{i}:=begin{cases}1 & if:y_{ini}:>:theta_{i}\y_{i} & if:y_{ini}:=:theta_{i}\0 & if:y_{ini}: Here $theta_{i}$ is the threshold. Step 8 − Broadcast this output yi to all other units. Step 9 − Test the network for conjunction. Energy Function Evaluation An energy function is defined as a function that is bonded and non-increasing function of the state of the system. Energy function Ef⁡, ⁡also called Lyapunov function determines the stability of discrete Hopfield network, and is characterized as follows − $$E_{f}:=:-frac{1}{2}displaystylesumlimits_{i=1}^ndisplaystylesumlimits_{j=1}^n y_{i}y_{j}w_{ij}:-:displaystylesumlimits_{i=1}^n x_{i}y_{i}:+:displaystylesumlimits_{i=1}^n theta_{i}y_{i}$$ Condition − In a stable network, whenever the state of node changes, the above energy function will decrease. Suppose when node i has changed state from $y_i^{(k)}$ to $y_i^{(k:+:1)}$ ⁡then the Energy change $Delta E_{f}$ is given by the following relation $$Delta E_{f}:=:E_{f}(y_i^{(k+1)}):-:E_{f}(y_i^{(k)})$$ $$=:-left(begin{array}{c}displaystylesumlimits_{j=1}^n w_{ij}y_i^{(k)}:+:x_{i}:-:theta_{i}end{array}right)(y_i^{(k+1)}:-:y_i^{(k)})$$ $$=:-:(net_{i})Delta y_{i}$$ Here $Delta y_{i}:=:y_i^{(k:+:1)}:-:y_i^{(k)}$ The change in energy depends on the fact that only one unit can update its activation at a time. Continuous Hopfield Network In comparison with Discrete Hopfield network, continuous network has time as a continuous variable. It is also used in auto association and optimization problems such as travelling salesman problem. Model − The model or architecture can be build up by adding electrical components such as amplifiers which can map the input voltage to the output voltage over a sigmoid activation function. Energy Function Evaluation $$E_f = frac{1}{2}displaystylesumlimits_{i=1}^nsum_{substack{j = 1\ j ne i}}^n y_i y_j w_{ij} – displaystylesumlimits_{i=1}^n x_i y_i + frac{1}{lambda} displaystylesumlimits_{i=1}^n sum_{substack{j = 1\ j ne i}}^n w_{ij} g_{ri} int_{0}^{y_i} a^{-1}(y) dy$$ Here λ is gain parameter and gri input conductance. Learning working make money