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Conclusion
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At this point we can conclude the discussion on single neurons. While this is not a complete treatment of all aspects of single ANs, it introduced those concepts required for the rest of the chapters. In the next chapter we explain learning rules for networks of neurons, expanding on the different types of learning rules available.
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2.6. ASSIGNMENTS
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Assignments
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1. Explain why the threshold 9 is necessary. What is the effect of 0, and what will the consequences be of not having a threshold 2. Explain what the effects of weight changes are on the separating hyperplane. 3. Which of the following Boolean functions can be realized with a single neuron which implements a SU Justify your answer.
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(a) x
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(c) x1 + x2
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where x1x2 denotes x1 AND x2; x1 + x2 denotes x1 OR x2; x1 denotes NOT xi, 4. Is it possible to use a single PU to learn problems which are not linearly separable 5. Why is the error per pattern squared 6. Can the function |tp op| be used instead of (tp o p ) 2 7. Is the following statement true or false: A single neuron can be used to approximate the function f ( z ) = z2 Justify your answer. 8. What are the advantages of using the hyperbolic tangent activation function instead of the sigmoid activation function
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Single neurons have limitations in the type of functions they can learn. A single neuron (implementing a SU) can be used to realize linearly separable functions only. As soon as functions that are not linearly separable need to be learned, a layered network of neurons is required. Training these layered networks is more complex than training a single neuron, and training can be supervised, unsupervised or through reinforcement. This chapter deals with supervised training. Supervised learning requires a training set which consists of input vectors and a target vector associated with each input vector. The NN learner uses the target vector to determine how well it has learned, and to guide adjustments to weight values to reduce its overall error. This chapter considers different NN types that learn under supervision. These network types include standard multilayer NNs, functional link NNs, simple recurrent NNs, time-delay NNs and product unit NNs. We first describe these different architectures in Section 3.1. Different learning rules for supervised training are then discussed in Section 3.2. The chapter ends with a discussion on ensemble NNs in Section 3.4.
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Neural Network Types
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Various multilayer NN types have been developed. Feedforward NNs such as the standard multilayer NN, functional link NN and product unit NN receive external signals and simply propagate these signals through all the layers to obtain the result (output) of the NN. There are no feedback connections to previous layers. Recurrent NNs, on the other hand, have such feedback connections to model the temporal characteristics of the problem being learned. Time-delay NNs, on the other hand,
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CHAPTER 3. SUPERVISED LEARNING NEURAL NETWORKS
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Feedforward Neural Networks
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Figure 3.1 illustrates a standard feedforward neural network (FFNN), consisting of three layers: an output layer, a hidden layer and an output layer. While this figure illustrates only one hidden layer, a FFNN can have more than one hidden layer. However, it has been proved that FFNNs with monotonically increasing differentiable functions can approximate any continuous function with one hidden layer, provided that the hidden layer has enough hidden neurons [Hornik 1989]. A FFNN can also have direct (linear) connections between the input layer and the output layer.
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Figure 3.1: Feedforward neural network The output of a FFNN for any given input pattern p is calculated with a single forward pass through the network. For each output unit ok we have (assuming no direct connections between the input and output layers),
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o k,p = fok(netok,p)
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