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Input Figure 7.2: Effect of outliers Use a robust objective function that is not influenced by outliers. An example objective function is the Huber function as illustrated in Figure 7.4 [Huber 1981]. Patterns for which the error is larger than |a| have a constant value, and have a zero influence when weights are updated (the derivative of a constant is zero). Slade and Gedeon [Slade and Gedeon 1993] and Gedeon, Wong and Harris [Gedeon et al. 1995] proposed Bimodal Distribution Removal, where the aim is to remove outliers from training sets during training. Frequency distributions of pattern errors are analyzed during training to identify and remove outliers. If the original training set contains no outliers, the method simply reduces to standard learning.

which can be solved provided appropriate boundary conditions across the structure are defined. Though there are no constraints on the choice of the boundary condition, the natural choice are the 'hard wall' boundary conditions, which enforce the total

Note the use of range translation in the calculation. (See Exercise 11.5.7 for the statement of the rule.) Generally, when using (15.19) instead of (15.6), the use of range translation replaces explicit calculations of the relationship between quotients. This simplifies the calculations, particularly when range translation is used more than once, because it permits the omission of unwanted detail; we know that appropriate witnesses to the existentially quantified variables exist but their exact form is not required. Exercise 15.21. Express (3d :: P = Qxd + r) as an equation involving r and P, (That is, eliminate the existential quantification.) D

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te is used to represent the permittivity in this chapter to distinguish it from e which is the commonly accepted symbol for the components of strain.

A rule for reasoning about the -r function, without explicit mention of the mod function, is d = P Cl = Or :: 0 ^ r < Q A P = Qxd + r) . (15.22)

Scaling and Normalization Data needs to be scaled to the active range and domain of the activation functions used. While it is not necessary to scale input values, performance can be improved if inputs are scaled to the active domain of the activation functions. For example, consider the sigmoid activation function. Simple mathematical calculations show that the active domain of the sigmoid function is [ \/3, v/3], corresponding to the parts of the function for which changes in input values have relatively large changes in output. Values near the asymptotic ends of the sigmoid function have a very small influence on weight updates. Changes in these values result in very small changes in

potential drop across the structure to be zero. All other possibilities would imply that the quantum well structure acts as a voltage source in the closed circuit. For a periodic structure (e.g. a superlattice) the hard wall and the periodic boundary conditions are equivalent. Knowing the boundary condition the necessary nth equation of the system is defined and reads:

(This is the definition of integer division used in 6.)

where Li is the width of the i-th layer. The solution of the system gives a rather simple equation:

{ { (15.22) } arithmetic and trading (heading towards one-point rule) } (3r : r = P-Qxd : 0 ^ r < Q) = { one-point rule } 0 ^ P-Qxd < Q { arithmetic } P P Q-'^Q { for all real x and integer ra, [x\ = m == x -I < m ^ x Or :: 0 ^ r < Q A P = Qxd + r)

(t-o)

from which the intrinsic electric field can be easily calculated. It is interesting to see that in the quantum wells the electric field exists even in the layers with zero piezoelectric polarisation, this is a consequence of the charge induced at the interfaces with the adjacent piezoelectric layers.

A remainder after dividing by Q is called a remainder modulo Q. (Indeed, P mod Q is sometimes read as P modulo Q.) A modulo-Q number is any number in the range O..Q-1. In this section, we identify addition, multiplication and negation operations on modulo-Q numbers. We do so by seeking distributivity properties of the mod Q function over addition, multiplication and negation.

Figure 7.11 The induced piezoelectric field in a GaN well and AUxGa1-xN barrier layers grown on a GaN substrate as a function of the barrier layer composition x.

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