In the evaluation of the precision of forecasting systems, a purely statistical measure is commonly used: the average deviation between the prediction and the actual observation. The most common measure for the determination of error in connection with neural networks is the so-called *mean squared error (MSE)*, which is calculated analogously to standard deviation.

For determining the *mean squared error* of a predicted time series ** y** compared to the actually observed time series

*with respectively*

**x****data points, the following formula can be used:**

*N*The result is consequently the average deviation between prediction and actual observation. With the use of such a measure of error, both models represented in Fig. 1 show exactly the same error. For at all of the three points, the deviation between the predictions of the models and the actual observations is of the same magnitude.

Fig. 1: The above illustration shows three data points of a time series (solid line) as well as the predictions made by two neural models (dashed and dotted line) at the respective points in time. Both models reveal precisely the same forecast error: at every data point, the deviation of both models from the actual observation is of the same magnitude. Nevertheless, Model 1 is obviously more exact, for it exactly traces the course of the curve and deviates from the observation only in terms of the level. The difference in quality is revealed by geometric error assessment.

Fig. 1 shows very clearly that, while such an error assessment is mathematically correct, it frequently fails to address the requirements of the user. Imagine that this series represents the price development of a stock, and that we want to act on the basis of the predictions. If we follow the predictions of Model 2, we incur a loss every day. Even though the MSE of Model 1 is of the same magnitude, with the forecasts of this model, by contrast, we would gain.

Especially in the strategic consideration of financial time series is it the case that the precision in the prediction of the course of the curve is more important than mathematical accuracy in the sense of the MSE. Hence, a geometrical consideration is more appropriate than the statistical assessment.

What is true of the establishment of the forecast error also applies to the modelling of the time series in preparation for the neural learning process. Common neural systems for financial forecasts use market indicators familiar from technical trading such as moving averages, rate of change (ROC), relative strength index (RSI) and others.

In addition, however, we have developed a method for modelling time series, which explicitly demonstrates the geometrical behavior of the time series. Normally, the interesting value is the percent deviation between two points in time:

This indicator provides the return from day 0 to day t as a factor, for example, 0.1 for a 10% gain or -0.08 for a loss of 8%. For the purpose of geometric modelling, alternatively, a pair of indicators can be calculated:

The first indicator provides the return smoothed out against the actual value through normalization, while the second indicator, proportional to the course of the curve and likewise normalized, delivers geometrical supplementary information. For the purpose of modelling, there are now various options. On the one hand, one must decide which series should be modelled, e.g. the series of the closing prices. Alternatively, one can also regard the sequences * open(n-1), close(n-1), open(n), close(n)* as a time series and model it accordingly. The span

*must also be chosen in a meaningful way.*

**n**The result is an indicator, which, on the one hand, is already normalized and which, on the other hand, extracts significantly more information from the time series in a way that is meaningful for an analysis system. A pure forecasting system could now be trained in such a way that it will, for example, predict a comparable indicator pair. The two independent predictions can then be added, for the sum of * I1* and

**yields the**

*I2**t-return*.

**I**A strategic neural system, of course, can likewise profit from these more precise predictions. From the * I2* indicator in particular, such a system will gain valuable information regarding the dynamic forces in the development of prices.