Figure 4. Backpropagation neural networks (NN2, NN3 and NN4) for stage II. 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)
methods are grouped into three categories. In the absence of these categories it is questionable if a reasonable degree of reliability canbe attained in a model classi®cation scheme. Speci®cally, in the absence of the groups, one would need to classify forecasting methods according to several time series characteristics. The ability of a reasonable set of time series characteristics to achieve the necessary discrimination to result in a discrete and meaningful classi®cation of forecast models is questionable. It is precisely this inherent similarity among time series characteristics and the need to identify subtle dierences among the input/output combinations that provided the motivation for this research and the impetus for exploring the applicability of neural networks to aid in this classi®cation problem.
The forecasting methods and groups are brie¯y described below:
These are ¯exible models that are adaptive to a variety of trends.
(1) Holt's two-parameter linear exponential smoothing (HOL). Holt's linear model with the value of the two smoothing parameters chosen so as to minimize the mean square error.
(2) Brown's one-parameter quadratic exponential smoothing (BRT). Triple exponential smoothing with one smoothing parameter.
(3) Winter's three-parameter exponential smoothing (WIN). Winter's linear model with the three parameters chosen so as to minimize the mean square error.
Methods in this category are responsive to linear trends.
(1) Brown's one-parameter linear exponential smoothing (BRD). Double exponential smoothing with one smoothing parameter.
(2) Linear regression trend ®tting (LIN). Time series regression with t as the independent variable.
(3) Adaptive response rate exponential smoothing (AES). The same as single exponential smoothing, except that the smoothing constant is allowed to vary over time.
This category represents relatively simple models that are used in practice but are not necessarily optimal for business and economic data.
(1) Naive 1 (NAI). The forecast at time period t 1 is equal to the actual value at time period t.
(2) Simple moving average (SMA). The forecast is the average of the last N values of the time series, where N is chosen to minimize the sum of the squared error.
(3) Single exponential smoothing (SES). One parameter simple exponential smoothing.
NEURAL NETWORK DEVELOPMENTÐTRAINING AND TESTING
Figure 3 provides the details of the stage I (NN1) network. There are eleven input neurons, representing the time series characteristics. The number of hidden neurons is 23. The three output neurons represent the three forecasting groups. The network has been trained and tested with the number of hidden neurons ranging from 7 (average of sum of number of input and output neurons) to 23 (twice the number of input neurons plus one) with the maximum selection accuracy during testing occurring at 23.
In this research, the M-competition data (1982) has been used for the purposes of training and testing. A subset of the 1001 series consisting of 180 series is obtained by a strati®ed random sample to assure an adequate cross-section of the series and is used to train the neural net. The time series characteristics for each of the series in the training set are calculated using SAS and form the input vectors for the network. In this research Mean Absolute Percentage Error (MAPE) is used as the measure of forecast accuracy. A survey of the literature indicates the use of other measures of forecast accuracy. Examples include: Mean squared forecast error (see Swanson and White, 1995; Clements and Hendry, 1993; Diebold and Mariano, 1995; Mizrach, 1995), generalized forecast error second moment (Clements and Hendry, 1993), and absolute percentage error (Hill, O'Connor and Remus, 1996).
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