avoiding overly complex models whereas P(W
r
) **is the** 
prior probability of W
r and *P(D)* **is the evidence term** 
that normalizes the posterior probability. The input is transformed from x to the middle layer, a**, using weights** 
wij and biases bi **as follows [8]:** 

$$a_{j}=\sum_{i=1}^{d}\bar{W}_{j i}x_{i}+b_{j}$$
 (2) 
where j = 1 and j = 2 **represent the first and second layer** respectively. The input is further transformed using the activation function such as the hyperbolic tangent (*tanh*) 
or the sigmoid in the hidden layer. More information on neural networks can be found in [10]. 

## 3.2 Genetic Algorithms

Genetic algorithms use the concept of survival of the fittest over consecutive generations to solve optimisation problems [11]. As in biological evolution, the fitness of each population member in a generation is evaluated to determine whether it will be used in the breeding of the next generation. In creating the next generation, the use of techniques (such as inheritance, mutation, natural selection, and recombination) common in the field of evolutionary biology are employed. The GA algorithm implemented in this paper uses a population of string chromosomes, which represent a point in the search space [11]. In this paper, all GA parameters were empirically determined. GA is implemented by following three main procedures which are selection, crossover and mutation. The algorithm listing in Figure 2 illustrates how GA operates. 

GA Algorithm 

1). Create an initial population P *, beginning at an initial* generation g = .0 2). for each population P, evaluate each population member (chromosome) using the defined fitness evaluation function possessing the knowledge of the competition environment. 

3). using genetic operators such as inheritance, mutation and crossover, alter P(g) *to* produce P(g + )1 **from the fit chromosomes in P** 
(g). 

4). repeat steps (2) and (3) for the number of generations G *required.* 

## 3.3 Fuzzy Artmap

$${\mathrm{(2)}}$$

Fuzzy ARTMAP is a neural network architecture developed by Carpernter *et al* **[12] and is based on** 
Adaptive Resonance Theory (ART). The Fuzzy ARTMAP has been used in condition monitoring by Javadpour and Knapp [13], but their application was not online. The Fuzzy ARTMAP architecture is capable of fast, online, supervised incremental learning, classification and prediction [12]. The fuzzy ARTMAP operates by dividing the input space into a number of hyperboxes, which are mapped to an output space. Instance based learning is used, where each individual input is mapped to a class label. Three parameters namely the vigilance ρ ∈[0, 1], the learning rate β ∈**[0, 1] and** 
the choice parameter α **, are used to control the learning** process. The choice parameter is generally made small and a value of 0.01 was used in this application. The parameter β **controls the adaptation speed, where 0** implies a slow speed and 1, the fastest. If β **= 1, the** 
hyperboxes get enlarged to include the point represented by the input vector. The vigilance represents the degree of belonging and it controls how large any hyperbox can become, resulting in new hyperboxes being formed. 

Larger values of ρ **lead to a case where smaller** 
hyperboxes are formed and this eventually lead to 'category proliferation', which can be viewed as overtraining. A complete description of the Fuzzy ARTMAP is provided in [12]. In this work, Fuzzy ARTMAP is preferred due to its incremental learning ability. As new data is sampled, there will be no need to retrain the network as would be the case with the MLP. 

## 4. **Neural Networks And Genetic** Algorithm For Missing Data

The method used here combines the use of autoassociative neural networks with genetic algorithms to approximate missing data. This method has been used by Abdella and Marwala [4] to approximate missing data in a database. A genetic algorithm is used in this work to estimate **the missing values by optimising an objective** 
function as presented shortly in this section. The complete vector combining the estimated and the observed values is input into the autoencoder as shown in Figure 3. Symbols Xk and Xu **represent the known** 
variables and the unknown (or missing) variables respectively. The combination of Xk and Xu **represent the** 
full input space. 