Computing multiscale entropy with orthogonal range search

Multi-scale entropy (MSE) is a measurement of a system's complexity. It has received a great deal of attention in recent years, and its effectiveness has been verified, and applied in a number of different fields. However, the algorithms proposed in past studies required O(N²), which represented a degree of execution time insufficient for on-line applications, or for applications with long-term correlations. In this study, we showed that the probability function in the entropy term could be transformed into an orthogonal range search in the field of computational geometry. We then developed an efficient new algorithm for computing multi-scale entropy. The execution time in the results of our experiments with electrocardiogram (ECG), electroencephalography (EEG), interbeat interval (RR), and mechanical and ecological signals showed a significant improvement from 10 to 70 times over that of conventional methods for N = 80,000. Because the execution time has been significant reduced, the new algorithm could be applied to online diagnosis, in the computation of MSE for long-term correlation of signal.