7.2.7 Anticipation performance

To test the anticipation performance, a separate set was used (section 7.2.2). The reason is that a theoretical prediction of the dependence of the anticipation error on the chain length relies on randomly independent errors for each prediction step (appendix C.3). Randomly independent errors are unlikely for series with a constant motor command. Therefore, the test patterns were collected during random walks.

However, the slight dependence of velocities, as mentioned in section 7.2.2, results in a deviation from a pure random walk. As the dashed curve in figure 7.8 indicates, the square distance (in the sensory representation) to the starting point of a sequence increases stronger than linear, which would be the expectation for a random walk (appendix C.3). In addition, the limited movement range within the circle of obstacles reduced the increase of the square distance for longer sequences. Still, for intermediate sequence lengths, the increase is roughly linear (figure 7.8, solid line). Therefore, for the comparison with the theory, and to compute the average linear increase of the square error, only the prediction intervals 2 to 6 were evaluated.

Figure 7.8: Increase of the mean squared distance per image sector (in pixels squared) from the sensory representation after interval l to the representation at the starting point of a sequence. The dashed curve is a quadratic function fitted to the intervals zero to five. The solid line shows the almost linear increase between interval one and six.

The square error E² of the anticipation was evaluated after each chain link l. The error E² is the squared difference between the output $\bf o$ of a chain with l links and the real sensory information $\bf r$ after l 2 sec intervals,

$$\sum_{{i=1}}^{{10}}$$ (7.1)