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Any study of the performance of NNs (or any other algorithm for that matter) and any conclusions based on just one simulation are incomplete and inconclusive. Conclusions on the performance of NNs must be based on the results obtained from several simulations. For each simulation the NN starts with new random initial weights and uses a different training, validation and generalization sets, independent of previous sets. Performance results are then expressed as averages over all the simulations, together with variances, or confidence intervals. Let Q denote the performance measure under consideration. Results are then reported as ~Q ae. The average ~Q is an indication of the average performance over all simulations, while ae gives an indication of the variance in performance. The ae parameter is very important in decision making. For example, if two algorithms A and B are compared where the MSE for A is 0.001 0.0001, and that of B is 0.0009 0.0006, then algorithm A will be preferred even though B has a smaller MSE. Algorithm A has a smaller variance, having MSE values in the range [0.0009,0.0011], while B has MSE values in a larger range of [0.0003,0.0015]. While the above approach to present results is sufficient, results are usually reported with associated confidence intervals. If a confidence level of a 0.01 is used, for example, then 99% of the observations will be within the calculated confidence interval. Before explaining how to compute the confidence intervals, it is important to

which means that the in-plane strains in adjacent layers I as: and the average in-plane stress can follow as:

d o r ^ Q - r : = r - Q { 0 < r < Q A (3d :: P = Qxd + r) } . Where specifications involve existential quantifications, it is a useful technique to introduce variables into the program in order to compute so-called witnesses to the existentially quantified dummies. These are called auxiliary variables. With care, the program can often be developed in such a way that the auxiliary variables play no role in the computation proper, and can be discarded once the development of the algorithm is complete. The technique is useful because calculating the implementation from the specification avoids direct manipulation of existential quantifications. We will use the technique of introducing, and then discarding, auxiliary variables several times throughout this chapter.

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The zero-stress (or strain-balance) condition implies that the in-plane stress is zero (a 0), and using = a\/a,k (k = 2,3,...) then equation (7.37) gives:

In Section 15.1, we showed that (15.1) defines at most one remainder r and quotient d for all integers P and all strictly positive natural numbers Q. The con-

Figure 1.2: Illustration of a biological neuron that the list below is by no means complete): single-layer NNs, such as the Hopfield network; multilayer feedforward NNs, including, for example, standard backpropagation, functional link and product unit networks; temporal NNs, such as the Elman and Jordan simple recurrent networks as well as time-delay neural networks; self-organizing NNs, such as the Kohonen self-organizing feature maps and the learning vector quantizer; combined feedforward and self-organizing NNs, such as the radial basis function networks. These NN types have been used for a wide range of applications, including diagnosis of diseases, speech recognition, data mining, composing music, image processing, forecasting, robot control, credit approval, classification, pattern recognition, planning game strategies, compression and many others.

which delivers the lattice constant of the substrate (or suitably grown buffer layer) necessary for strain-balancing the quantum well stack as:

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