Comparison with ARIMA Modeling

 We compared our results with the results of the ARIMA procedure of the SAS software, an integrated system for data access, management, analysis and presentation. The implementation of the ARIMA procedure of SAS follows the programs described by Box and Jenkins in Part V of their classic [BJ76].

The ARIMA model is called an autoregressive integrated moving average process of order (p, d, q). It is described by the equation

$$a(z) \nabla^d X_t = b(z) U_t$$

where X_t stands for in time ordered values of a time series, $t=1, \ldots,n$ for n observations. U_t is a sequence of random values called ``white noise'' process. The backward difference operator $\nabla$ is defined as

$$\nabla X_t = X_t - X_{t-1} = (1 - z)X_t$$

The variable d states how often the difference should be calculated, z is the so called backward shift operator which is defined as z^m X_t = X_t-m. The autoregressive operator a(z) of order p is defined as

$$a(z) = 1 - a_1 z - a_2 z^2 - \ldots - a_p z^p$$

the moving average operator b(z) of order q is defined as

$$b(z) = 1 - b_1 z - b_2 z^2 - \ldots - b_q z^q$$

We fitted an ARIMA model for each time series using the SAS system and let it predict the next 20 observations of the time series. The last 20 observations were dropped from the time series and used to calculate the prediction error of the models.

The following ARIMA models were calculated for the airline passenger time series (after a logarithmic transformation):

(1-z)(1-z¹²)X_t = (1 - 0.24169z - 0.47962z¹²) U_t

and for the IBM time series:

(1-z) X_t = (1 - 0.10538z) U_t

As an opponent for the ARIMA modeling technique, we selected those networks that delivered the smallest forecast error s_f for the respective time series data:

 

$$\eta \alpha$$ series # input # hidden

units units

airline 0.1 0.9 70 45

IBM 0.1 0.9 80 30

In Table 2 the prediction errors for the artificial neural network (ANN), the artificial neural network using the logarithmic and $\nabla$ transformation (ANN log,$\nabla$) and the ARIMA model are compared: The artificial neural network using the logarithmic and $\nabla$ transformed time series outperformed the ARIMA models for both time series, whereas the ``simple'' artificial neural network predicted more accurately only for the IBM shares time series. This behavior can be explained as follows: the larger data range of the airline passenger time series leads to a loss of precision for the untransformed input set. Differencing and logarithmic transformations helped to eliminate the trend and mapped the time series data into a smaller range.

  

Figure: Forecasting errors for ANN and ARIMA model