Abstract:Quantifying the behavior of robots which was generated autonomously by task-independent objective functions is an important prerequisite for objective comparisons of algorithms with each other or even with human or animal movements. The excess entropy (Shaw 1984), effective measure complexity (Grassberger 1986) or predictive information (Bialek et al.~2001) is usually considered as the most natural complexity measure for temporal sequences. It measures the amount of information that the past contains about the future (predictive information) which is equal to the non-extensive part of the entropy (excess entropy) of the sequence. However, using the excess entropy as a complexity measure for continuous valued time series one has to deal with the fact that its value will be different for different partitions and diverges with increasing resolution. We therefore propose a new decomposition of the excess entropy in an resolution dependent and a resolution independent part and discuss how they depend on the dimensionality of the dynamics, correlations and the noise level. For the practical estimation we propose to use estimates based on the correlation integral instead of the direct estimation of the mutual information using the algorithm by Kraskov et al.~(2004) which is based on next neighbor statistics because the latter allows less control of the scale dependencies. Using our algorithm we are able to show how autonomous learning generates behavior of increasing complexity with increasing learning duration.

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