ARTIFICIAL NEURAL NETWORKS FOR DATA MINING


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1 ARTIFICIAL NEURAL NETWORKS FOR DATA MINING Amrender Kumar I.A.S.R.I., Library Avenue, Pusa, New Delhi Introduction Neural networks, more accurately called Artificial Neural Networks (ANNs), are computational models that consist of a number of simple processing units that communicate by sending signals to one another over a large number of weighted connections. They were originally developed from the inspiration of human brains. In human brains, a biological neuron collects signals from other neurons through a host of fine structures called dendrites. The neuron sends out spikes of electrical activity through a long, thin stand known as an axon, which splits into thousands of branches. At the end of each branch, a structure called a synapse converts the activity from the axon into electrical effects that inhibit or excite activity in the connected neurons. When a neuron receives excitatory input that is sufficiently large compared with its inhibitory input, it sends a spike of electrical activity down its axon. Learning occurs by changing the effectiveness of the synapses so that the influence of one neuron on another changes. Like human brains, neural networks also consist of processing units (artificial neurons) and connections (weights) between them. The processing units transport incoming information on their outgoing connections to other units. The "electrical" information is simulated with specific values stored in those weights that make these networks have the capacity to learn, memorize, and create relationships amongst data. A very important feature of these networks is their adaptive nature where "learning by example" replaces "programming" in solving problems. This feature makes such computational models very appealing in application domains where one has little or incomplete understanding of the problem to be solved but where training data is readily available. These networks are neural in the sense that they may have been inspired by neuroscience but not necessarily because they are faithful models of biological neural or cognitive phenomena. ANNs have powerful pattern classification and pattern recognition capabilities through learning and generalize from experience. ANNs are nonlinear data driven self adaptive approach as opposed to the traditional model based methods. They are powerful tools for modelling, especially when the underlying data relationship is unknown. ANNs can identify and learn correlated patterns between input data sets and corresponding target values. After training, ANNs can be used to predict the outcome of new independent input data. ANNs imitate the learning process of the human brain and can process problems involving nonlinear and complex data even if the data are imprecise and noisy. These techniques are being successfully applied across an extraordinary range of problem domains, in areas as diverse as finance, medicine, engineering, geology, physics, biology and agriculture. There are many different types of neural networks. Some of the most traditional applications include classification, noise reduction and prediction. 2. Review Genesis of ANN modeling and its applications appear to be a recent development. However, this field was established before the advent of computers. It started with the modeling the functions of a human brain by McCulloch and Pitts in 1943, proposed a model of computing element called McCulloch Pitts neuron, which performs weighted sum of the inputs to the element followed by a threshold logic operation. Combinations of these computing elements were used to
2 realize several logical computations. The main drawback of this model of computation is that the weights are fixed and hence the model could not learn from examples. Hebb (1949) proposed a learning scheme for adusting a connection weight based on pre and post synaptic values of the variables. Hebb s law became a fundamental learning rule in neuron network literature. Rosenblatt (1958) proposed the perceptron models, which have weights adustable by the perceptron learning law. Widrows and Hoff (1960) proposed an ADALINE (Adaptive Linear Element) model for computing elements and LMS (Least Mean Square) learning algorithm to adust the weights of an ADALINE model. Hopfield (1982) gave energy analysis of feed back neural networks. The analysis has shown the existence of stable equilibrium states in a feed back network, provided the network has symmetrical weights. Rumelhart et al. (1986) showed that it is possible to adust the weights of a multilayer feed forward neural network in a systematic way to learn the implicit mapping in a set of input output patterns pairs. The learning law is called generalized delta rule or error back propagation. Cheng and Titterington (1994) made a detailed study of ANN models visavis traditional statistical models. They have shown that some statistical procedures including regression, principal component analysis, density function and statistical image analysis can be given neural network expressions. Warner and Misra (1996) reviewed the relevant literature on neural networks, explained the learning algorithm and made a comparison between regression and neural network models in terms of notations, terminologies and implementation. Kaastra and Boyd (1996) developed neural network model for forecasting financial and economic time series. Dewolf and Francl (1997, 2000) demonstrated the applicability of neural network technology for plant diseases forecasting. Zhang et al. (1998) provided the general summary of the work in ANN forecasting, providing the guidelines for neural network modeling, general paradigm of the ANNs especially those used for forecasting. They have reviewed the relative performance of ANNs with the traditional statistical methods, wherein in most of the studies ANNs were found to be better than the latter. Sanzogni and Kerr (2001) developed models for predicting milk production from farm inputs using standard feed forward ANN. Chakraborty et al. (2004) utilized the ANN technique for predicted severity of anthracnose diseases in legume crop. Gaudart et al. (2004) compared the performance of MLP and that of linear regression for epidemiological data with regard to quality of prediction and robustness to deviation from underlying assumptions of normality, homoscedasticity and independence of errors and it was found that MLP performed better than linear regression. More general books on neural networks, to cite a few, Hassoun (1995), Patterson (1996), Schalkoff (1997), Yegnanarayana (1999), Anderson (2003) etc. are available. Software on neural networks has also been made, to cite a few, Statistica, Matlab etc. Commercial Software: Statistica Neural Network, TNs2Server,DataEngine, Know Man Basic Suite, Partek, Saxon, ECANSE  Environment for Computer Aided Neural Software Engineering, Neuroshell, Neurogen, Matlab:Neural Network Toolbar, Taran, FCM(Fuzzy Control manager) etc. Freeware Software: NetII, Spider Nets Neural Network Library, NeuDC, Binary Hopfeild Net with free Java source, Neural shell, PlaNet, Valentino Computational Neuroscience Work bench, Neural Simulation language versionnsl, etc. 3. Characteristics of neural networks The following are the basic characteristics of neural network: Exhibit mapping capabilities, that is, they can map input patterns to their associated output patterns. Learn by examples. Thus, NN architectures can be trained with known examples of a 158
3 problem before they are tested for their inference capability on unknown instances of the problem. They can, therefore, identify new obects previously untrained. Possess the capability to generalize. Thus, they can predict new outcomes from past trends. Robust systems and are fault tolerant. They can, therefore, recall full patterns from incomplete, partial or noisy patterns. 4. Basics of artificial neural networks The terminology of artificial neural networks has developed from a biological model of the brain. A neural network consists of a set of connected cells: The neurons. The neurons receive impulses from either input cells or other neurons and perform some kind of transformation of the input and transmit the outcome to other neurons or to output cells. The neural networks are built from layers of neurons connected so that one layer receives input from the preceding layer of neurons and passes the output on to the subsequent layer. A neuron is a real function of the input vector (y 1,, y k ). The output is obtained as k f (x ) = f ( w y ) where f is a function, typically the sigmoid (logistic or tangent i 1 i i hyperbolic) function. A graphical presentation of neuron is given in figure 1. Mathematically a MultiLayer Perceptron network is a function consisting of compositions of weighted sums of the functions corresponding to the neurons. Fig. 1: A single neuron 5. Neural networks architectures An ANNs is defined as a data processing system consisting of a large number of simple highly inter connected processing elements (artificial neurons) in an architecture inspired by the structure of the cerebral cortex of the brain. There are several types of architecture of ANNs. However, the two most widely used ANNs are discussed below: a. Feed forward networks In a feed forward network, information flows in one direction along connecting pathways, from the input layer via the hidden layers to the final output layer. There is no feedback (loops) i.e., the output of any layer does not affect that same or preceding layer. A graphical presentation of feed forward network is given in figure
4 Fig. 2: A multilayer feed forward neural network b. Recurrent networks These networks differ from feed forward network architectures in the sense that there is at least one feedback loop. Thus, in these networks, for example, there could exist one layer with feedback connections as shown in figure below. There could also be neurons with selffeedback links, i.e. the output of a neuron is fed back into itself as input. A graphical presentation of feed forward network is given in figure 3. Input layer Hidden layer Output layer Fig. 3: Recurrent neural network 6. Types of neural networks There are wide variety of neural networks and their architectures. Types of neural networks range from simple Boolean networks (perceptions) to complex selforganizing networks (Kohonen networks). There are also many other types of networks like Hopefield networks, Pulse networks, RadialBasis Function networks, Boltzmann machine. The most important class of neural networks for real world problems solving includes Multilayer Perceptron Radial Basis Function Networks Kohonen Self Organizing Feature Maps 6.1 Multilayer Perceptron The most popular form of neural network architecture is the multilayer perceptron (MLP). A multilayer perceptron: has any number of inputs. has one or more hidden layers with any number of units. 160
5 uses linear combination functions in the input layers. uses generally sigmoid activation functions in the hidden layers. has any number of outputs with any activation function. has connections between the input layer and the first hidden layer, between the hidden layers, and between the last hidden layer and the output layer. Given enough data, enough hidden units, and enough training time, an MLP with ust one hidden layer can learn to approximate virtually any function to any degree of accuracy. (A statistical analogy is approximating a function with nth order polynomials.) For this reason MLPs are known as universal approximators and can be used when you have little prior knowledge of the relationship between inputs and targets. Although one hidden layer is always sufficient provided you have enough data, there are situations where a network with two or more hidden layers may require fewer hidden units and weights than a network with one hidden layer, so using extra hidden layers sometimes can improve generalization. 6.2 Radial Basis Function Networks Radial basis functions (RBF) networks are also feedforward, but have only one hidden layer. A RBF network: has any number of inputs. typically has only one hidden layer with any number of units. uses radial combination functions in the hidden layer, based on the squared Euclidean distance between the input vector and the weight vector. typically uses exponential or softmax activation functions in the hidden layer, in which case the network is a Gaussian RBF network. has any number of outputs with any activation function. has connections between the input layer and the hidden layer, and between the hidden layer and the output layer. MLPs are said to be distributedprocessing networks because the effect of a hidden unit can be distributed over the entire input space. On the other hand, Gaussian RBF networks are said to be localprocessing networks because the effect of a hidden unit is usually concentrated in a local area centered at the weight vector. 6.3 Kohonen Neural Network Self Organizing Feature Map (SOFM, or Kohonen) networks are used quite differently to the other networks. Whereas all the other networks are designed for supervised learning tasks, SOFM networks are designed primarily for unsupervised learning (Patterson, 1996). At first glance this may seem strange. Without outputs, what can the network learn? The answer is that the SOFM network attempts to learn the structure of the data. One possible use is therefore in exploratory data analysis. A second possible use is in novelty detection. SOFM networks can learn to recognize clusters in the training data, and respond to it. If new data, unlike previous cases, is encountered, the network fails to recognize it and this indicates novelty. A SOFM network has only two layers: the input layer, and an output layer of radial units (also known as the topological map layer). Schematic representation of Kohonen network is given in Fig
6 Fig. 4: A Kohonen Neural Network Applications 7. Learning of ANNs The most significant property of a neural network is that it can learn from environment, and can improve its performance through learning. Learning is a process by which the free parameters of a neural network i.e. synaptic weights and thresholds are adapted through a continuous process of stimulation by the environment in which the network is embedded. The network becomes more knowledgeable about environment after each iteration of learning process. There are three types of learning paradigms namely, supervised learning, reinforced learning and selforganized or unsupervised learning. 7.1 Supervised learning In this, every input pattern that is used to train the network is associated with an output pattern, which is the target or the desired pattern. A teacher is assumed to be present during the learning process, when a comparison is made between the network s computed output and the correct expected output, to determine the error. The error can then be used to change network parameters, which result in an improvement in performance. Learning law describes the weight vector for the i th processing unit at time instant (t+1) in terms of the weight vector at time instant (t) as follows: wi ( t 1) wi ( t) wi ( t), where w i (t) is the change in the weight vector. The network adapts as follows: change the weight by an amount proportional to the difference between the desired output and the actual output. As an equation: Δ W i = η * (DY).I i where η is the learning rate, D is the desired output, Y is the actual output, and I i is the i th input. This is called the Perceptron Learning Rule. The weights in an ANN, similar to coefficients in a regression model, are adusted to solve the problem presented to ANN. Learning or training is term used to describe process of finding values of these weights. Supervised learning which 162
7 incorporates an external teacher, so that each output unit is told what its desired response to input signals ought to be. During the learning process global information may be required. An important issue concerning supervised learning is the problem of error convergence, i.e. the minimization of error between the desired and computed unit values. The aim is to determine a set of weights which minimizes the error. 7.2 Unsupervised learning With unsupervised learning, there is no feedback from the environment to indicate if the outputs of the network are correct. The network must discover features, regulations, correlations, or categories in the input data automatically. In fact, for most varieties of unsupervised learning, the targets are the same as inputs. In other words, unsupervised learning usually performs the same task as an autoassociative network, compressing information from the inputs. 7.3 Reinforced learning In supervised learning there is a target output value for each input value. However, in many situations, there is less detailed information available. In extreme situations, there is only a single bit of information after a long sequence of inputs telling whether the output is right or wrong. Reinforcement learning is one method developed to deal with such situations. Reinforcement learning is a kind of learning in that some feedback from the environment is given. However the feedback signal is only evaluative, not instructive. Reinforcement learning is often called learning with a critic as opposed to learning with a teacher. 8. Development of an ANN model The various steps in developing a neural network model are: 8.1 Variable selection The input variables important for modeling/ forecasting variable(s) under study are selected by suitable variable selection procedures. 8.2 Formation of training, testing and validation sets The data set is divided into three distinct sets called training, testing and validation sets. The training set is the largest set and is used by neural network to learn patterns present in the data. The testing set is used to evaluate the generalization ability of a supposedly trained network. A final check on the performance of the trained network is made using validation set. 8.3 Neural network structure Neural network architecture defines its structure including number of hidden layers, number of hidden nodes and number of output nodes etc. (a) Number of hidden layers: The hidden layer(s) provide the network with its ability to generalize. In theory, a neural network with one hidden layer with a sufficient number of hidden neurons is capable of approximating any continuous function. In practice, neural network with one and occasionally two hidden layers are widely used and have to perform very well. (b) Number of hidden nodes: There is no magic formula for selecting the optimum number of hidden neurons. However, some thumb rules are available for calculating number of hidden neurons. A rough approximation can be obtained by the geometric pyramid rule 163
8 (c) (d) proposed by Masters (1993). For a three layer network with n input and m output neurons, the hidden layer would have sqrt(n*m) neurons. Number of output nodes: Neural networks with multiple outputs, especially if these outputs are widely spaced, will produce inferior results as compared to a network with a single output. Activation function: Activation functions are mathematical formulae that determine the output of a processing node. Most units in neural network transform their net inputs by using a scalartoscalar function called an activation function, yielding a value called the unit's activation. Except possibly for output units, the activation value is fed to one or more other units. Activation functions with a bounded range are often called squashing functions. Appropriate differentiable function will be used as activation function. Some of the most commonly used activation functions are :  The sigmoid (logistic) function f ( x) ( 1 exp( x)) 1  The hyperbolic tangent (tanh) function f ( x) (exp( x) exp( x))/ (exp( x) exp( x))  The sine or cosine function f ( x) sin( x) or f ( x) cos( x) Activation functions for the hidden units are needed to introduce nonlinearity into the networks. The reason is that a composition of linear functions is again a linear function. However, it is the nonlinearity (i.e. the capability to represent nonlinear functions) that makes multilayer networks so powerful. Almost any nonlinear function does the ob, although for backpropagation learning it must be differentiable and it helps if the function is bounded. Therefore, the sigmoid functions are the most common choices. There are some heuristic rules for selection of the activation function. For example, Klimasauskas (1991) suggests logistic activation functions for classification problems which involve learning about average behaviour, and to use the hyperbolic tangent functions if the problem involves learning about deviations from the average such as the forecasting problem. 8.4 Model building Multilayer feed forward neural network or multi layer perceptron (MLP), is very popular and is used more than other neural network type for a wide variety of tasks. Multilayer feed forward neural network learned by back propagation algorithm is based on supervised procedure, i.e., the network constructs a model based on examples of data with known output. It has to build the model up solely from the examples presented, which are together assumed to implicitly contain the information necessary to establish the relation. An MLP is a powerful system, often capable of modeling complex, relationships between variables. It allows prediction of an output obect for a given input obect. The architecture of MLP is a layered feedforward neural network in which the nonlinear elements (neurons) are arranged in successive layers, and the information flow unidirectionally from input layer to output layer through hidden layer(s). An MLP with ust one hidden layer can learn to approximate virtually any function to any degree of accuracy. For this reason MLPs are known as universal approximates and can be used when we have litter prior knowledge of the relationship between input and targets. One hidden layer is always 164
9 sufficient provided we have enough data. Schematic representation of neural network is given in Fig. 5 Outputs Inputs Fig. 5: Schematic representation of neural network Each interconnection in an ANN has a strength that is expressed by a number referred to as weight. This is accomplished by adusting the weights of given interconnection according to some learning algorithm. Learning methods in neural networks can be broadly classified into three basic types (i) supervised learning (ii) unsupervised learning and (iii) reinforced learning. In MLP, the supervised learning will be used for adusting the weights. The graphic representation of this learning is given in Fig. 6 Input vector ANN model Output vector Target vector = Differences Adust weights Fig. 6 A learning cycle in the ANN model 8.5 Neural network training Training a neural network to learn patterns in the data involves iteratively presenting it with examples of the correct known answers. The obective of training is to find the set of weights between the neurons that determine the global minimum of error function. This involves decision regarding the number of iteration i.e., when to stop training a neural network and the selection of learning rate (a constant of proportionality which determines the size of the weight adustments made at each iteration) and momentum values (how past weight changes affect current weight changes). Backpropagation is the most commonly used method for training multilayered feedforward networks. It can be applied to any feedforward network with differentiable activation functions. For most networks, the learning process is based on a suitable error function, which is then minimized with respect to the weights and bias. If a network has differential activation functions, then the activations of the output units become differentiable functions of input 165
10 variables, the weights and bias. If we also define a differentiable error function of the network outputs such as the sum of square error function, then the error function itself is a differentiable function of the weights. Therefore, we can evaluate the derivative of the error with respect to weights, and these derivatives can then be used to find the weights that minimize the error function by either using optimization method. The algorithm for evaluating the derivative of the error function is known as backpropagation, because it propagates the errors backward through the network. Multilayer feed forward neural network or multilayered perceptron (MLP), is very popular and is used more than other neural network type for a wide variety of tasks. MLP learned by backpropagation algorithm is based on supervised procedure, i.e. the network constructs a model based on examples of data with known output. The Backpropagation Learning Algorithm is based on an error correction learning rule and specifically on the minimization of the mean squared error that is a measure of the difference between the actual and the desired output. As all multilayer feedforward networks, the multilayer perceptrons are constructed of at least three layers (one input layer, one or more hidden layers and one output layer), each layer consisting of elementary processing units (artificial neurons), which incorporate a nonlinear activation function, commonly the logistic sigmoid function. The algorithm calculates the difference between the actual response and the desired output of each neuron of the output layer of the network. Assuming that y(n) is the actual output of the th neuron of the output layer at the iteration n and d(n) is the corresponding desired output, the error signal e(n) is defined as: e (n) d (n) y (n) The instantaneous value of the error for the neuron is defined as (n) / 2 and correspondingly, the instantaneous total error E(n) is obtained by summing the neural error (n) / 2 over all neurons in the output layer. Thus, 1 2 E (n) e (n) 2 In the above formula, runs over all the neurons of the output layer. If we define N to be the total number of training patterns that consist the training set applied to the neural network during the training process, then the average squared error Eav is obtained by summing E(n) over all the training patterns and then normalizing with respect to the size N of the training set. Thus, N 1 E E(n ) av 2 n 1 It is obvious, that the instantaneous error E(n), as well as the average squared error Eav, is a function of all the free parameters of the network. The obective of the learning process is to modify these free parameters of the network in such a way that Eav is minimized. To perform this minimization, a simple training algorithm is utilized. The training algorithm updates the synaptic weights on a patternbypattern basis until one epoch, that is, one complete presentation of the entire training set is completed. The correction (modification) w i (n) that is applied on the synaptic weight w i (indicating the synaptic strength of the synapse originating from neuron i and directing to neuron ), after the application of the n th training pattern is proportional to the E(n) partial derivative. Specifically, the correction applied is given by: w (n) i 166 e 2 e 2
11 E(n) w i w i (n) In the above formula (this is also known as delta rule), η is the learningrate parameter of the backpropagation algorithm. The use of the minus sign in above equation accounts for the gradientdescent in weightspace, reflecting the seek of a direction for weight change that reduces the value of E(n). The exact value of the learning rate η is of great importance for the convergence of the algorithm since it modulates the changes in the synaptic weights, from iteration to iteration. The smaller the value of η, the smoother the traectory in the weight space and the slower the convergence of the algorithm. On the other hand, if the value of η is too large, the resulting large changes in the synaptic weights may result the network to exhibit unstable (oscillatory) behaviour. Therefore, the momentum term was introduce for generational of the above equation, Thus E(n) w i w i (n 1) w i (n) In this equation α is the is a positive number called the momentum constant is called the Generalized Delta Rule and it includes the Delta Rule as a special case (α =0). The weight update can be obtained as w (n) w i i (n 1) The weight adustment w i is made only after the entire training set has been presented to the network (Konstantinos, A.; 2000). With respect to the convergence rate the backpropagation algorithm is relatively slow. This is related to the stochastic nature of the algorithm that provides an instantaneous estimation of the gradient of the error surface in weight space. In the case that the error surface is fairly flat along a weight dimension, the derivative of the error surface with respect to that weight is small in magnitude, therefore the synaptic adustment applied to the weight is small and consequently many iterations of the algorithms may be required to produce a significant reduction in the error performance of the network. 9. Evaluation criteria The most common error function minimized in neural networks is the sum of squared errors. Other error functions offered by different software include least absolute deviations, least fourth powers, asymmetric least squares and percentage differences. 10. Conclusions ANNs has an ability to learn by example makes them very flexible and powerful which make them quite suitable for a variety of problem areas. Hence, to best utilize ANNs for different problems, it is essential to understand the potential as well as limitations of neural networks. For some tasks, neural networks will never replace conventional methods, but for a growing list of applications, the neural architecture will provide either an alternative or a complement to these existing techniques. ANNs have a huge potential for prediction and classification when they are integrated with Artificial Intelligence, Fuzzy Logic and related subects. (n)y i (n) 167
12 PRACTICAL ON ARTIFICAL NEURAL METWORKS MODELS FOR PREDICTION USING SAS MINER The snapshots for Opening of proect, importing the file from the desire directory and linking of Models (ANNs ) to data file in SAS miner are given below. 168
13 The output of the ANNs models 169
14 References Anderson, J. A. (2003). An Introduction to neural networks. Prentice Hall. Chakraborty, S., Ghosh. R, Ghosh, M., Fernandes, C.D. and Charchar, M.J. (2004). Weatherbased prediction of anthracnose severity using artificial neural network models. Plant Pathology, 53, Cheng, B. and Titterington, D. M. (1994). Neural networks: A review from a statistical perspective. Statistical Science, 9, Dewolf, E.D., and Francl, L.J., (1997). Neural network that distinguish in period of wheat tan spot in an outdoor environment. Phytopathalogy, 87(1) pp Dewolf, E.D. and Francl, L.J. (2000) Neural network classification of tan spot and stagonespore blotch infection period in wheat field environment. Phytopathalogy, 20(2), Gaudart, J. Giusiano, B. and Huiart, L. (2004). Comparison of the performance of multilayer perceptron and linear regression for epidemiological data. Comput. Statist. & Data Anal., 44, Hassoun, M. H. (1995). Fundamentals of Artificial Neural Networks. Cambridge: MIT Press. Hebb,D.O. (1949) The organization of behaviour: A Neuropsychological Theory, Wiley, New York Hopfield, J.J. (1982). Neural network and physical system with emergent collective computational capabilities. In proceeding of the National Academy of Science (USA) 79, Kaastra, I. and Boyd, M.(1996): Designing a neural network for forecasting financial and economic time series. Neurocomputing, 10(3), pp (1996) Klimasauskas, C.C. (1991). Applying neural networks. Part 3: Training a neural network, PCAI, May/ June, Konstantinos, A. (2000). Application of Back Propagation Learning Algorithms on Multilayer Perceptrons, Proect Report, Department of Computing, University of Bradford, England. Mcculloch, W.S. and Pitts, W. (1943) A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophy., 5, Patterson, D. (1996). Artificial Neural Networks. Singapore: Prentice Hall. Rosenblatt, F. (1958). The perceptron: A probabilistic model for information storage ang organization in the brain. Psychological review, 65, Rumelhart, D.E., Hinton, G.E and Williams, R.J. (1986). Learning internal representation by error propagation, in Parallel distributed processing: Exploration in microstructure of cognition, Vol. (1) ( D.E. Rumelhart, J.L. McClelland and the PDP research gropus, edn.) Cambridge, MA: MIT Press, Saanzogni, Louis and Kerr, Don (2001) Milk production estimate using feed forward artificial neural networks. Computer and Electronics in Agriculture, 32, Schalkoff, R. J. (1997). Artificial neural networks. The Mc GrawHall Warner, B. and Misra, M. (1996). Understanding neural networks as statistical tools. American Statistician, 50, Widrow, B. and Hoff, M.E. (1960). Adapative switching circuit. IREWESCON convention record, 4, Yegnanarayana, B. (1999). Artificial Neural Networks. Prentice Hall Zhang, G., Patuwo, B. E. and Hu, M. Y. (1998). Forecasting with artificial neural networks: The state of the art. International Journal of Forecasting,14,
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