The Algorithm

The neural network sees the time series $X_1,\ldots,X_n$ in the form of many mappings of an input vector to an output value (see figure 6). This technique was presented by [CMMR92].

A number of adjoining data points of the time series (the input window $X_{t-s},X_{t-s+1}\ldots,X_t$) are mapped to the interval [0,1] and used as activation levels for the units of the input layer. The size s of the input window corresponds to the number of input units of the neural network. In a forward path, these activation levels are propagated over one hidden layer to one output unit. The error used for the backpropagation learning algorithm is now computed by comparing the value of the output unit with the transformed value of the time series at time t+1. This error is propagated back to the connections between output and hidden layer and to those between hidden and output layer. After all weights have been updated accordingly, one presentation has been completed. Training a neural network with the backpropagation algorithm usually requires that all representations of the input set (called one epoch) are presented many times. In our examples, we used 60 to 138 epochs.

For the learning of time series data, the representations were presented in a randomly manner: As reported by [CMMR92], choosing a random location for each representation's input window ensures better network performance and avoids local minima.

  

Figure 6: Learning a Time Series

The next section is concerned with the selection of the right parameters for the learning algorithm and the selection of a suitable topology for the forecasting network.