Radial Basis Function Neural Networks in .NET framework

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52 Radial Basis Function Neural Networks
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where J is the number of centers (or hidden units), and dmax is the maximum Euclidean distance between centers Weight values of connections between the hidden and output layers are found by solving for wk in wk = ( T ) 1 T tk (512) where wk is the weight vector of output unit k, tk is the vector of target outputs, and RPT J is the matrix of RBF nonlinear mappings performed by the hidden layer Algorithm 51 Training an RBFNN with Fixed Centers Set J to indicate the number of centers; Choose the centers, j , j = 1, , J, as j = zp , p U (1, PT ) (513)
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Calculate the width, j , using equation (511); Initialize all wkj , k = 1, , K and j = 1, , J to small random values; Calculate the output for each output unit using equation (53) with Gaussian radial basis functions; Solve for the network weights using equation (512) for each k = 1, , K;
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Training an RBFNN using Gradient Descent Moody and Darken [605] and Poggio and Girosi [682] used gradient descent to adjust weights, centers, and widths The algorithm is summarized in Algorithm 52 In Algorithm 52, w , , and respectively indicate the learning rate for weights, centers, and widths In this algorithm, centers are initialized by sampling from the training set The next subsection shows that these centers can be obtained in an unsupervised training step, prior to training the weights between hidden units (radial basis) and output units
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Two-Phase RBFNN Training The training algorithms discussed thus far have shown slow convergence times [899] In order to increase training time, RBFNN training can be done in two phases [605, 881]: (1) unsupervised learning of the centers, j , and then, (2) supervised training of the wk weights between the hidden and output layers using gradient descent Algorithm 53 summarizes a training algorithm where the rst phase utilizes an LVQ-I to cluster input patterns [881]
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5 Radial Basis Function Networks
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Algorithm 52 Gradient Descent Training of RBFNN Select the number of centers, J; for j = 1, , J do p U (1, PT ); j (t) = zp ; max j (t) = d J ; end for k = 1, , K do for j = 1, , J do wkj U (wmin , wmax ); end end while stopping condition(s) not true do Select an input pattern, dp = (zp , tp ); for k = 1, , K do Compute ok,p using equation (53); for j = 1, , J do Compute weight adjustment step size, wkj (t) = w Adjust weights using wkj (t + 1) = wkj (t) + wkj (t) end end for j = 1, , J do for i = 1, , I do Compute center step size, ji (t) = Adjust centers using ji (t + 1) = ji (t) + ji (t) end Compute width step size, j (t) = Adjust widths using j (t + 1) = j (t) + j (t) end end (519) E (t) j (518) (517) E (t) ji (516) (515) E (t) wkj (514)
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52 Radial Basis Function Neural Networks Algorithm 53 Two-Phase RBFNN Training Initialize wkj , k = 1, , K and j = 1, , J; Initialize ji , j = 1, , J and i = 1, , I; Initialize j , j = 1, , J; while LVQ-I has not converged do Apply one epoch of LVQ-I to adjust j , j = 1, , J; Adjust j , j = 1, , J; end t = 0; while gradient descent has not converged do Select an input pattern, (zp , tp ); Compute the weight step sizes,
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wkj (t) = Adjust the weights,
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