Hilbert Entropy for the Simple and Precise Measurement of Complexity of Two or Higher Dimensional Arrays*

Measuring complexity of higher dimensional arrays has been an important way of quantifying information. For the measurement of complexity of 1D vectors, there are a variety of methods of calculating entropies such as sample, permutation, or Lempel-Ziv entropy. Unfortunately, for higher dimensional arrays, it is not possible to employ these entropies. This is due mainly to the information loss in the course of dimension reduction. To address this problem, we introduce space-filling curve (SFC)-based approach. Thanks to the fact that SFC allows information loss-free dimension reduction, we successfully measured complexity by calculating sample entropy of higher dimensoinal arrays. We found that developed algorithm precisely measured the complexity of higher dimensional arrays as well as detected the critical points in the cases of various phase transition experiments. We also observed that the developed algorithm (Hilbert entropy) can measure scale-invariance and fractal dimension of higher dimensional arrays. We proved that the Hilbert entropy for higher dimesional arrays exhibits power law dependence for arrays with self-similarity.