Hidden nodes

Table 1 Blind test results for "Lower wing skin" using network with 2 hidden nodes and training for 2000 iterations...

The number of neurons to be used in the input/output layer are based on the number of input/output variables to be considered in the model. However, no algorithms are available for selecting a network structure or the number of hidden nodes. Zurada [16] has discussed several heuristic based techniques for this purpose. One hidden layer is more than sufficient for most problems. The number of neurons in the hidden layer neuron was selected by a trial-and-error procedure by monitoring the sum-of-squared error progression of the validation data set used during training. Details about this proce-... 

The output of these hidden nodes, o, is then forwarded to all output nodes through weighted connections. The output yj of these nodes consists of a linear combination of the kernel functions ... 

J. Zhang, J.-H. Jiang, P. Liu, Y.-Z. Liang and R.-Q. Yu, Multivariate nonlinear modelling of fluorescence data by neural network with hidden node pruning algorithm. Anal. Chim. Acta, 344(1997) 29 0. 

The PPR model usually has fewer hidden nodes than a BPN model and is easier and faster to train (Hwang et al, 1994). The PPR basis functions... 

One layer of input nodes and another of output nodes form the bookends to one or more layers of hidden nodes Signals flow from the input layer to the hidden nodes, where they are processed, and then on to the output nodes, which feed the response of the network out to the user. There are no recursive links in the network that could feed signals from a "later" node to an "earlier" one or return the output from a node to itself. Because the messages in this type of layered network move only in the forward direction when input data are processed, this is known as a feedforward network. 

With the introduction of a hidden layer, training becomes trickier because, although the target responses for the output nodes are still available from the database of sample patterns, there are no target values in the database for hidden nodes. Unless we know what output a hidden node should be generating, it is not possible to adjust the weights of the connections into it in order to reduce the difference between the required output and that which is actually delivered. 

The lack of a recipe for adjusting the weights of connections into hidden nodes brought research in neural networks to a virtual standstill until the publication by Rumelhart, Hinton, and Williams2 of a technique now known as backpropagation (BP). This offered a way out of the difficulty. 

The output of hidden nodes in the immediately preceeding layer that generated the input signals into the node. 

A hidden node that sends a large signal to an output node is more responsible for any error at that node than a hidden node that sends a small signal,... 

To determine the error at the hidden nodes, we backpropagate the error from the output nodes. We can expand this error in terms of the posterior nodes ... 

The degree to which it has learned general rules rather than simply learned to recognize specific sample patterns is then difficult to assess. In using a network on a new dataset, it is, therefore, important to try to estimate the complexity of the data (in essence, the number of rules that will be necessary to satisfactorily describe it) so that a network of suitable size can be used. If the network contains more hidden nodes than are needed to fit the rules that describe the data, some of the power of the network will be siphoned off into the learning of specific examples in the training set. 

The number of middle layers (hidden nodes) in a NN must be identified either through a particular choice or through an optimization procedure with careful monitoring of the predictive behavior of the derived model (see point 2). 

Unfortunately, the ANN method is probably the most susceptible to overfitting of the methods discussed thus far. For similar N and M, ANNs reqnire many more parameters to be estimated in order to define the model. In addition, cross validation can be very time-consuming, as models with varying complexity (nnmber of hidden nodes) mnst be trained individually before testing. Also, the execntion of an ANN model is considerably more elaborate than a simple dot product, as it is for MLR, CLS, PCR and PLS (Eqnations 12.34, 12.37, 12.43 and 12.46). Finally, there is very little, or no, interpretive value in the parameters of an ANN model, which eliminates one nseful means for improving the confidence of a predictive model. 

For the styrene-butadiene copolymer application, fit results of ANN models using one to six hidden nodes are shown in Table 12.7. Based on these results, it appears that only three, or perhaps four, hidden nodes are required in the model, and the addition of more hidden nodes does not greatly improve the fit of the... 

Suppose that we have hidden a number of the branch currents, and we now wish to display one of the hidden currents. Displaying hidden branch currents is a little different than displaying hidden node voltages. Node voltages are... 

Figure 8.17 shows a very specific case of a feed-forward network with four inputs, three hidden nodes, and one output. However, such networks can vary widely in their design. First of all, one can choose any number of inputs, hidden nodes, and number of outputs in the network. In addition, one can even choose to have more than one hidden layer in the network. Furthermore, it is common to perform scaling operations on both the inputs and the outputs, as this can enable more efficient training of the network. Finally, the transfer function used in the hidden layer (f) can vary widely as well. Many feed-forward networks use a non-linear function called the sigmoid function, defined as ... 

In the styrene—butadiene copolymer application, a series of quantitative ANN models for the as-butadiene content was developed. For each of these models, all of the 141 X-variables were used as inputs, and the sigmoid function (Equation 8.39) was used as the transfer function in the hidden layer. The X-data and Y-data were both mean-centered before being used to train the networks. A total of six different models were built, using one to six nodes in the hidden layer. The model fit results are shown in Table 8.7. Based on these results, it appears that only three, or perhaps four, hidden nodes are required in the model, and the addition of more hidden nodes does not greatly improve the fit of the model. Also, note that the model fit (RMSEE) is slightly less for the ANN model that uses three hidden nodes (1.13) than for the PLS model that uses four latent variables (1.25). 

Throughout the above discussion of quantitative modeling tools, a recurrent theme is the danger of overfitting a model, through the use of too many variables in MLR, too many estimated pure components in CLS, too many factors in PCR and PLS, or too many hidden nodes in ANN. This danger cannot be understated, not only because it is so... 

Figure6.25 Schematicdrawingofan artificial neural network with a multilayer perceptron topology, showing the pathways from the input Xj to the output y , and the visible and hidden node layers.

Fig. 2. Structure of an artificial neural network. The network consists of three layers the input layer, the hidden layer, and the output layer. The input nodes take the values of the normalized QSAR descriptors. Each node in the hidden layer takes the weighted sum of the input nodes (represented as lines) and transforms the sum into an output value. The output node takes the weighted sum of these hidden node values and transforms the sum into an output value between 0 and 1.

Fig. 5 a, b. Illustration of the computation principles for (a) artificial neural networks with input signals (array signals), hidden nodes chosen during the training of the net, and output signals (the parameters to be predicted) and (b) principal component analysis with two principal components (PC 1 and PC 2) based on three sensor signals (represented by the x, y and z axes). Normally reduces from approximately 100 signals down to two or three PCs... 

Radial basis function networks with more than one input unit have more parameters for each hidden node e.g.,. if there are two input units, then the basis function for each hidden unit j needs two location parameters, pij and p2j, for the center, and, optionally, two parameters, Oij and a2j, for variability. The dimension of the centers for each of the hidden units matches the dimension of the input vector. 

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