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Fundamentals of Neural Networks by Laurene Fausett - Ebook download as PDF File .pdf), Text File .txt) or read book online. systematic study of the artificial neural network. Four years later The interest in neural networks comes from the networks' ability to mimic Chapter 2 − Fundamentals of NN Fausett, L., Fundamentals of Neural Networks, Prentice- Hall. Aug 10, Fundamentals of Neural Networks: Soft Computing Course Lecture 7 – 14, notes , slides Learning methods in neural networks: unsupervised Laurene V. Fausett, (), Prentice Hall, Chapter, page 6.
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Of course. During training. This may or may not be appropriate for a particular problem. The requirement for a positive response from the output unit is that the net input it receives. The weights on these input signals correspond to the importance the person places on each factor.
In this case. If one thinks in terms of a threshold. The threshold would be different for different people. Before discussing the particular nets which is to say. Depending on the number of input units in the network. Since we want one of two responses. Since it is the relative values of the weights. For a particular output unit. A "yes" response is represented by an output signal of 1. The analysis also extends easily to nets.
The value of the function is 1 if the net input is positive and. It is convenient. Notice in the following examples that there are many different lines that will serve to separate the input points that have different target values. The region where y is positive is separated from the region where it is negative by the line These two regions are often called decision regions for the net.
The first question we consider is. Any point that is not on the decision boundary can be used to determine which side of the boundary is positive and which is negative. There are four different bipolar input patterns we can use to train a net with two input units. Several of these functions are familiar from elementary logic.
For this very simple net. An example of weights that would give the decision boundary illustrated in the figure. One possible de- cision boundary for this function is shown in Figure 2.
INPUT It is easy to see that no single straight line can separate the points for which a positive response is desired from those for which a negative response is desired. Not all simple two-input. The input points to be classified positive can be separated from the input points to be clas- sified negative by a straight line. Note that if a bias weight were not included in these examples. In many cases- including Examples 2.
We will return to these examples to illustrate each of the learning rules in this chapter.. The equations of the decision boundaries are not unique.
Fundamentals of Neural Networks by Laurene Fausett
Binary representation is also not as good as bipolar if we want the net to generalize i. Since we are considering a single-layer net. Using bipolar input. The Hebb rule is also used for training other specific nets that are discussed later.
The remainder of this chapter focuses on three methods of training single- layer neural nets that are useful for pattern classification: Both are iterative techniques that are guaranteed to converge under suitable circumstances.
Hebb proposed that learning occurs by modification of the synapse strengths weights in a manner such that if two interconnected neurons are both "on" at the same time.
Many early neural network models used binary rep- resentation. The original statement only talks about neurons firing at the same time and does not say anything about reinforcing neurons that do not fire at the same time. The Hebb rule. We shall refer to a single-layer feedforward neural net trained using the extended Hebb rule as a Hebb net.
We shall discuss some of the issues relating to the choice of binary versus bipolar representation further as they apply to particular neural nets. The form of the data may change the problem from one that can be solved by a simple neural net to one that cannot. If data are represented in bipolar form. There are several methods of implementing the Hebb rule for learning. Set activation for output unit: Step 1. Step 3. Step 4. Adjust the bias: Note that the bias is adjusted exactly like a weight from a "unit" whose output signal is always 1.
Step 2. Set activations for input units: For each input training vector and target output pair. The weight update can also be expressed in vector form as This is often written in terms of the weight change. The foregoing algorithm requires only one pass through the training set. Only one iteration through the training vectors is required. The weight updates for the first input are as follows: The new weights are the sum of the previous weights and the' weight change.
Logic functions Example 2. Presenting the second. The algorithm is the same as that just given. The next example shows that the A N D function can be solved if we modify its representation to express the inputs as well as the targets in bipolar form. The choice of training patterns c a n play a significant role in determining which problems can be solved using the Hebb rule. Bipolar representation of the inputs and targets allows modification of a weight when the input unit and the target value are both "on" at the same time and when they are both "off" at the same time.
Pre- senting the second input vector and target results in the following situation: Presenting the last point. The graph in Figure 2. Presenting the third input vector and target yields the following: To convert from the two-dimensional pattern to an input vector.
For computer simulations. Character recognition Example 2.
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Pattern 1 then becomes 1. Pattern 1 Pattern To treat this example as a pattern classification problem with one output class. The patterns can be represented as. That is easy to do by assigning each the value 1 and each ". There are two types of changes that can be made to one of the input patterns that will generate a new input pattern for which it is reasonable to expect a response.
The second type of change is called "missing data. The net input for any input pattern is the dot product of the input pattern with the weight vector. Consider the following input and target output pairs: Adding the weight change to the weights representing the first pattern gives the final weights: For the second training pattern.
Now consider. The correct response for the second pattern is "off. The bias weight is 0. The first type of change is usually referred to as "mistakes in the data. For the first training vector. It is easy to see that these weights do not produce the correct output for the first pattern. We obtain: Weight change for first input pattern: The weights and bias are found by taking the sum of the weight changes that occur at each stage of the algorithm.
The figure also shows that a nonzero bias will be necessary. The weight change is simply the input pattern augmented by the fourth component. This plane corresponds to a weight vector of 1 1 1 and a bias of In this example. For each training input. The activation function for each associator unit was the binary step function with an arbitrary. If an error did not occur. Not too surpris- ingly. The net did not distinguish between an error in which the calculated output was zero and the target.
Under suitable assumptions. In either of these cases. If an error occurred for a particular training input pattern. Then the net would determine whether an error occurred for this pattern by comparing the calculated output with the target value. One particular simple perceptron [Block. The perceptron learning rule is a more powerful learning rule than the Hebb rule. A number of different types of perceptrons are described in Rosenblatt and in Minsky and Papert Although some perceptrons were self- organizing.
We will consider a proof of this theorem in Section 2. The goal of the net is to classify each input pattern as belonging. As the proof of the perceptron learning rule convergence theorem given in Section 2. The perceptron learning rule convergence theorem states that if weights exist to allow the net to respond correctly to all training patterns.
Since only the weights from the associator units to the output unit could be ad- justed. The net is trained to perform this classification by the iterative technique described earlier and given in the algorithm that follows. The threshold does not play the same role as in the step function illustrated in Section 2.
This means that as more training patterns produce the correct response. For each training pair s: Initialize weights and bias. For simplicity. Test stopping If no weights changed in Step 2. Update weights and bias if an error occurred for this pattern. Note that only weights connecting active input units 0 are updated. The algorithm is not particularly sen- sitive to the initial values of the weights or the value of the learning rate. While stopping condition is false.
Step 0. This is in contrast to the training of the Adaline units described in Section 2. Compute response of output unit: Set activations of input units: The role of the threshold is discussed following the presentation of the algorithm. The training data are as given in Example 2. Note that instead of one separating line. The form of the activation function for the output unit response unit is such that there is an "undecided" band of fixed width deter- mined by separating the region of positive response from that of negative re- sponse.
An adjustable bias is included. Presenting the first input. For the third input.
Presenting the second input yields the following: To complete the first epoch of training. The second epoch of training yields the following weight updates for the first input: I is not correct.
For the second input in the second epoch. In the second epoch. To complete the second epoch of training. The results for the seventh epoch are: The eighth epoch yields and the ninth Finally.
Since the proof of the perceptron learning rule convergence theorem Section 2. The target value is still bipolar. This variation provides the most direct comparison with Widrow-Hoff learning an net.
I Figure 2. This is a portion of the parity problem for three inputs. If there are two or three zeros in the input. The fourth component of the input vector is the input to the bias weight and is therefore always 1. The weight change vector is left blank if no error has occurred for a particular pattern. It seems intuitively obvious that a procedure that could continue to learn to improve its weights even after the classifications are all correct would be better than a learning rule in which weight updates cease a s soon a s all training patterns are classified correctly.
I if there is one zero input. Other simple examples Example 2. Since all the Aw's are 0 in epoch 2. Epoch 2: Epoch 3: Epoch 4: Epoch For each input vector x to be classified. Apply training algorithm to set the weights. Set activations of input units. Input from Font 2 A.. K Figure Input from Font 3 A Our net would have 63 input units and 2 output units. The net is as shown in Figure 2.
We could. In this type of application. The first output unit would correspond to "A or not-A". There are three examples of A and 18 examples of not-A in Figure 2. The architecture of such a net is shown in Figure 2. Update biases and weights. There are seven categories to which each input vector may belong. Initialize weights and biases 0 or small random values. A bipolar representation has better computational characteristics than does a binary representation. Each of the input. For ease of reading.
Step 6. The training patterns are illustrated in Figure 2. Test for stopping condition: If no weight changes occurred in Step 2. The input patterns may be converted to bipolar vectors as described in Example 2. For each bipolar training pair s: If then Else. The training input patterns and target responses must be converted to an ap- propriate form for the neural net to process.
The performance of the net shown in Figure Compute activation of each output unit. Set activation of each input unit. Step 5. Font Input from y.. Input from Font The pixels where the input pattern differs from the training pattern are indicated by for a pixel that is "on" now but was "off" in the training pattern. Each of these provides a slightly different perspective and insights into the essential aspects of the rule.
The fact that the weight vector is perpendicular to the plane separating the input patterns at each step of the learning processes [Hertz.
The perceptron learning rule convergence theorem is: The proof of the theorem is simplified by the observation that the training set can be considered to consist of two parts: The perceptron learning rule is as follows: We must show that this sequence is finite. If x 0 is the first training vector for which an error has occurred. As mentioned. If another error occurs.
The existence of a solution of the original problem. If the response of the net is incorrect for a given training input. Note that this will involve reversing the sign of all components including the input component corresponding to the bias for any input vectors for which the target was originally. In either case. Let the starting weights be denoted by the first new weights by w l. We now consider the sequence of input training vectors for which a weight change occurs.
We now sketch the proof of this remarkable convergence theorem. The Cauchy-Schwartz inequality states that for any vectors a and b. This shows that the squared length of the weight vector grows faster than where k is the number of time the weights have changed.
Combining the inequalities and shows that the number of times that the weights may change is bounded. Spe- cifically. The original restriction that the coefficients of the patterns be binary is un-.
The foregoing proof shows that many variations in the perceptron learning rule are possible. Several of these variations are explicitly mentioned in Chapter 11 of Minsky and Papert The also has a bias. The weights on the connections from the input units to the are adjustable. The actual target values do not matter.
All that is required is that there be a finite maximum norm of the training vectors or at least a finite upper bound to the norm. Variations on the learning step include setting the learning rate a to any nonnegative constant Minsky starts by setting it specifically to l.
Others usually indicate small random values. Training may take a long time a large number of steps if there are training vectors that are very small in norm. The learning rule minimizes the mean squared error between the activation and the target value.
This allows the net to continue learning on all training patterns. The argument of the proof is unchanged if a nonzero value of 0 is used although changing the value of 0 may change a problem from solvable to unsolvable or vice versa.
Minsky sets the initial weights equal to an arbitrary training pattern. Note that there is no requirement that there can be only finitely many training vectors. The rule Section 2. Note also that since the procedure will converge from an arbitrary starting set of weights.
In Section 2. Adalines are combined so that the output from some of them be- comes input for others of them. Initialize weights. Small random values are usually used. Such a multilayer net. Set learning rate a.
See comments following algorithm. An application algorithm is given in Section 2. If the net input to the Adaline is greater than or equal to 0. A single Adaline is shown in Figure 2. If the largest weight change that occurred in Step 2 is smaller than a specified tolerance. Compute net input to output unit: If too large a value is chosen. Setting the learning rate to a suitable value requires some care. According to Hecht-Nielsen If the target values are bivalent binary or bipolar.
The proof of the convergence of the Adaline training process is essentially contained in the derivation of the delta rule. The choice of learning rate and methods of modifying it are considered further in Chapter 6. Update bias and weights. For a single neuron. Set activations of the input units to x. Initialize weights from Adaline training algorithm given in Section 2. For each bipolar input vector x. The function is defined by the fol- lowing four training patterns: As indicated in the derivation of the delta rule Section 2.
The following procedure shows the step function for bipolar targets. Good approximations to these values can be found using the algorithm in Section 2. Apply the activation function: Simple examples The weights and biases in Examples 2.
A minor modification to Example 2. Weights that minimize the total squared error for the bipolar form of the OR function are. The delta rule for adjusting the Zth weight for each pattern is The nomenclature we use in the derivation is as follows: The gradient of E is the vector consisting of the partial derivatives of E with respect to each of the weights.
Fundamentals Of Neural Networks Pdf
E is a function of all of the weights. Weight cor- rections can also be accumulated over a number of training patterns so-called batch updating if desired. The gradient gives the direction of most rapid increase in E.
The aim is to minimize the error over all training patterns. We shall return to the more standard lowercase indices for weights whenever this distinction is not needed. In order to distinguish between the fixed but arbitrary index for the weight whose adjustment is being determined in the derivation that follows and the index of summation needed in the derivation. This vector gives the direction of most rapid increase in E.
The squared error for a particular training pattern is E is a function of all of the weights. The weights are changed to reduce the difference between the net input to the output unit. Weight corrections can also be accumulated over a number of training patterns so-called batch updating if desired. The delta rule for adjusting the weight from the input unit to the Jth output unit for each pattern is Derivation. The gradient of E is a vector consisting of the partial derivatives of E with respect to each of the weights.
The error can be reduced most rapidly by adjusting the weight in the direction of - We now find an explicit formula for for the arbitrary weight First. Delta rule for several output units The derivation given in this subsection allows for more than one output unit. The examples given for the perceptron and the deri- vation of the delta rule for several output units both indicate there is essentially no change in the process of training if several Adaline units are combined in a single-layer net.
Minimizing the error for the training set will also minimize the ex- pected value for the error of the underlying probability distribution. As mentioned earlier. The preceding two derivations of the delta rule can be generalized to the case where the training data are only samples from a larger data set. Gener- alizations to more hidden units. We consider first the MRI algorithm. The weights on the first hidden Adaline and and the weights on the second hidden Adaline W and are adjusted according to the algorithm.
Determine output of net: Step 7. For each bipolar training pair. Determine output of each hidden Adaline unit: The activation function for units and Y is 1 -1 0. Compute net input to each hidden Adaline unit: Initialize weights: Weights and and the bias are set as described.
Determine error and update weights: Set the learning rate a as in the Adaline training algorithm a small value. Step 7 is motivated by the desire to 1 update the weights only if an error occurred and 2 update the weights in such a way that it is more likely for the net to produce the desired response. Set the learning rate Step 1.
Compute output of net as in the MRI algorithm. Determine error and update weights if necessary: This is sometimes called the "don't rock the boat" principle. Training Algorithm for Madaline Step 0. If weight changes have stopped or reached an acceptable level. Madaline can also be formed with the weights on the output unit set to perform some other logic function such as AND or. Test stopping condition. Several output units may be used. Step The weights into Z1 and into Z2 are small random values.
Begin training. Only the computations for the first weight updates are shown. For the first training pair. Step 8. Start with the unit whose net input is closest to 0. Step 7a. The learning rate. Application Example 2. Similarly adaptation could then be attempted for triplets of units. The training patterns are: Step 7b.
If weight changes have stopped or reached an accept- able level. A further modification is the possibility of attempting to modify pairs of units at the first layer after all of the individual modifications have been attempted.
Recompute the response of the net. If the error is reduced: For hidden unit Z 1. Closed regions convex polygons can be bounded by taking the intersection of several half-planes bounded by the sepa- rating lines described earlier. A Model for Brain Functioning. It is possible to construct a net with 2p hidden units in a single layer that will learn p bipolar input training patterns each with an associated bipolar target value perfectly.
New York: Thus a net with one hidden layer with p units can learn a response region bounded by p straight lines. Bullinaria, 1. What is a Self Organizing Map? Download Fundamentals Of Neural Networks Laurene Fausett With our online resources, you can find fundamentals of neural networks laurene fausett solution easily without hassle, since there are more than , titles available. We have made it easy for you to find a PDF Ebooks without any digging.
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View larger cover. Our price: An exceptionally clear, thorough introduction to neural networks written at an elementary level. Written with the beginning student in mind, the text features systematic discussions of all major neural networks and fortifies the reader's understanding with many examples. Written with the beginning student in mind, the text features systematic discussions of all major neural networks and fortifies the reader's understudy with many examples.
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