Convergence

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The convergence characteristics of a NN can be described by the ability of the network to converge to speci ed error levels (usually considering the generalization error) The ability of a network to converge to a speci c error is expressed as the number of times, out of a xed number of simulations, that the network succeeded in reaching that error While this is an empirical approach, rigorous theoretical analysis has been done for some network types

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Analysis of Performance

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Any study of the performance of NNs (or any other stochastic 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 di erent training, validation and generalization sets, independent of previous sets Performance results are then expressed as averages over all the simulations, together with variances, or con dence intervals Let denote the performance measure under consideration Results are then reported as The average is an indication of the average performance over all simulations,

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73 Performance Factors

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while gives an indication of the variance in performance The parameter is very important in decision making For example, if two algorithms A and B are compared where the MSE for A is 0001 00001, and that of B is 00009 00006, 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 [00009, 00011], while B has MSE values in a larger range of [00003, 00015] While the above approach to present results is su cient, results are usually reported with associated con dence intervals If a con dence level of = 001 is used, for example, then 99% of the observations will be within the calculated con dence interval Before explaining how to compute the con dence intervals, it is important to note that statistical literature suggests that at least 30 independent simulations are needed This allows the normality assumption as stated by the central limit theorem: the probability distribution governing the variable approaches a Normal distribution as the number of observations (simulations) tends to in nity Using this result, the con dence interval associated with con dence level can be estimated as t ,n 1 (78)

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where t ,n 1 is a constant obtained from the t-distribution with n 1 degrees of freedom (n is the number of simulations) and =

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n i=1 ( i

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)2 n(n 1)

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(79)

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It should be noted at this point that the t-test assumes that samples are normally distributed It is, however, not always the case that 30 samples will guarantee a normal distribution If not normally distributed, nonparametric tests need to be used

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Performance Factors

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This section discusses various aspects that have an in uence on the performance of supervised NNs These aspects include data manipulation, learning parameters, architecture selection, and optimization methods

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Data Preparation

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One of the most important steps in using a NN to solve real-world problems is to collect and transform data into a form acceptable to the NN The rst step is to decide on what the inputs and the outputs are Obviously irrelevant inputs should be excluded Section 735 discusses ways in which the NN can decide itself which inputs are irrelevant The second step is to process the data in order to remove outliers, handle missing data, transform non-numeric data to numeric data and to scale the data into the active range of the activation functions used Each of these aspects are discussed in the sections below

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100 Missing Values

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7 Performance Issues (Supervised Learning)

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It is common that real-world data sets have missing values for input parameters NNs need a value for each of the input parameters Therefore, something has to be done with missing values The following options exist: Remove the entire pattern if it has a missing value While pattern removal solves the missing value problem, other problems are introduced: (1) the available information for training is reduced which can be a problem if data is already limited, and (2) important information may be lost Replace each missing value with the average value for that input parameter in the case of continuous values, or with the most frequently occurring value in the case of nominal or discrete values This replacing of missing values introduces no bias For each input parameter that has a missing value, add an additional input unit to indicate patterns for which parameters are missing It can then be determined after training whether the missing values had a signi cant in uence on the performance of the network While missing values present a problem to supervised neural networks, SOMs do not su er under these problems Missing values do not need to be replaced The BMN for a pattern with missing values is, for example, calculated by ignoring the missing value and the corresponding weight value of the codebook vector in the calculation of the Euclidean distance between the pattern and codebook vector

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Coding of Input Values All input values to a NN must be numeric Nominal values therefore need to be transformed to numerical values A nominal input parameter that has n di erent values is coded as n di erent binary input parameters, where the input parameter that corresponds to a nominal value has the value 1, and the rest of these parameters have the value 0 An alternative is to use just one input parameter and to map each nominal value into an equivalent numerical value This is, however, not a good idea, since the NN will interpret the input parameter as having continuous values, thereby losing the discrete characteristic of the original data

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Outliers Outliers have severe e ects on accuracy, especially when gradient descent is used with the SSE as objective function An outlier is a data pattern that deviates substantially from the data distribution Because of the large deviation from the norm, outliers result in large errors, and consequently large weight updates Figure 73 shows that larger di erences between target and output values cause an exponential increase in the error if the SSE is used as objective function The tted function is then pulled toward the outliers in an attempt to reduce the training error As result, the generalization

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