To recap, the model selection and forecasting system is designed around a three-phase approach. Phase 1 determines the time series characteristics for a given data set, which in turn provide the inputs for the neural network engine of Phase 2. Phase 3 is the selection of the forecast model, among the set of potential forecasting models under consideration, and subsequent forecasts.
The architecture of the neural network is shown in Figures 3 and 4. The values of the variables that represent the time series characteristics are the input for the stage I network (NN1), as shown in Figure 3. The stage I network has been trained and subsequently tested to select a forecasting group (with each group consisting of three speci®c forecasting methods) that would result in a forecast with minimum error.
Based on the forecasting group selected using the stage I network (NN1), the neural network to be used to determine a speci®c forecasting method is then selected from a pool of three networks, named NN2, NN3, and NN4 (see Figure 4). If the group selected using NN1 is GRP1, then for the stage II, NN2 is used. If the group selected using NN1 is GRP2, then NN3 is used for stage II and if the group selected using NN1 is GRP3, then NN4 is used for stage II. Each of these networks has the same set of time series characteristics as input but has three dierent speci®c forecasting methods as output. The values of the time series characteristics for the given data set are again input into the selected network, which gives one speci®c forecasting method as its output. The selection of a speci®c forecasting method, from the set of potential forecasting models under consideration, using the stage II network then leads to the application of the selected method to produce the forecast, using a statistical software package.
The selection of a forecasting method is accomplished in the above neural network in two stages, that is, stage I for the selection of a forecasting group and stage II for the selection of a speci®c forecasting method. It may be argued that the selection of a speci®c forecasting method can be achieved using just one neural network instead of two stages involving a total of four
Figure 3. Backpropagation neural network (NN1) for stage I. Note: For simplicity only the ®rst neuron in each layer is shown as connected to the next layer. In fact all the neurons in a layer are connected to all the neurons in the next layer (fully connected network).
networks, as done in this research. The ®rst attempt has been to develop a single neural network with the input neurons as the time series characteristics and the output neurons as the nine forecasting methods. The network training and subsequent testing results have been unsatisfactory. The highest percentage of good classi®cation achieved during training and testing the network were 41% and 18% respectively with a tolerance value of 0.5. The unsatisfactory results obtained in the development of a single neural network for the model selection problem thus led to the idea of developing a two-stage process, as described in this paper.
The stage I and stage II networks have been developed using the backpropagation neural network design. Each network is constructed with eleven input nodes and three output nodes. Backpropagation learning design was chosen since it has been used successfully by researchers for many prediction and classi®cation problems.
The input neurons represent the time series characteristics of the data set (see Table I). The output neurons for the NN1 (stage I) network represent the three forecasting groups, namely, GRP1, GRP2, and GRP3. The output neurons for each of the networks NN2, NN3, and NN4
(stage II) represent three speci®c forecasting methods which are grouped based on the similarities of the underlying methodologies (see Table II). In order to facilitate the estimation of the neural network and to achieve a reasonable discrimination among the input criteria the forecasting
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