## 2 Data and Methods  
Two datasets were used. First, a simulated dataset was used to illustrate the issues and also to facilitate, if desired, replication. Second, a real dataset consisting of remotely-sensed data and associated ground reference data on land cover class labels was used. A very simple multi-class classification scenario was simulated. This comprised data on four classes acquired in two dimensions, or bands. The data for each class were formed using a random number generator using analyst provided values for the class mean and standard deviation on the assumption that the data for each class were normally distributed. For each class the standard deviation was set equal to 5 and the mean values used to generate the data in each band are shown in Table 1. In the scenario generated, class 1 was the most abundant class. Specifically, class 1 was five times more abundant than each of the other classes. Most attention was focused on class 1 and class 2, which exhibited a degree of overlap in their distributions with class 3 and, especially, class 4 was highly separable (Figure 1). For the analyses of the simulated dataset, training validation and testing sets were generated (Table 2). These were used in two series of analyses, one using a validation dataset formed via simple random sampling and the other formed via a stratified random sample design. In brief, a single training set was generated using a stratified random sample of 400 cases per-class. Similarly, a single testing set was used to evaluate the accuracy of the classifications from the neural networks selected as optimal in each series of analyses undertaken. This testing set was simulated to represent a sample acquired by simple random sampling and comprised 800 cases. Due to the way the scenario was designed, class 1 was five times more abundant in it than the other classes. The two validation datasets used each contained 800 cases, but one was formed via a simple random sample in which class abundance varied as in the testing set, while the other was formed with a stratified random sample in which each class was equally represented. Each series of neural network analyses used a software package that sought to generate an optimal network, with optimality defined as the maximization of the overall accuracy with which the validation set was classified. The remotely-sensed data were acquired by an airborne thematic mapper (ATM) sensor for a test site near the village of Feltwell, Norfolk, UK. The latter is located approximately 58 km to the northeast of the city of Cambridge. The land around the village was topographically flat and its land cover mosaic was characterized mainly by large agricultural fields. At the time of the ATM data acquisition these fields had also typically been planted with a single crop type (Figure 2). The ATM used was a basic multispectral scanning system that acquired data in 11 spectral wavebands from blue to thermal infrared wavelengths (Table 3). Given the relatively low altitude of airborne data acquisition (~2000 m), the spatial resolution of the imagery was very much smaller than the typical field size, approximately 5 m. As a result, image pixels tended to represent an area composed of a single class (i.e., pure pixels) and, hence, were appropriate for hard image classification analysis; boundary pixels were ignored. Attention focused here on the six crop classes that dominated the region at the time of the ATM data acquisition. These classes and their approximate coverage (%) of the study area at the time were: sugar beet (S, 30.3%), wheat (W, 30.0%), barley (B, 16.0%), carrot (C, 10.3%), potato (P, 7.8%), and grass (G, 5.4%). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Band** & **Class 1** & **Class 2** & **Class 3** & **Class 4** \\ \hline Band 1 & 50 & 35 & 83 & 20 \\ Band 2 & 50 & 60 & 50 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The classes in the simulated dataset; note: units are arbitrary. Figure 1: Simulated data (training, validation (random), and testing).