Bubble Entropy: An Entropy Almost Free of Parameters (2017)

<i>Objective</i>: A critical point in any definition of entropy is the selection of the parameters employed to obtain an estimate in practice. We propose a new definition of entropy aiming to reduce the significance of this selection. <i>Methods:</i> We call the new definition <i>Bubble Entropy</i>. Bubble Entropy is based on permutation entropy, where the vectors in the embedding space are ranked. We use the <i>bubble sort</i> algorithm for the ordering procedure and count instead the number of swaps performed for each vector. Doing so, we create a more coarse-grained distribution and then compute the entropy of this distribution. <i>Results:</i> Experimental results with both real and synthetic HRV signals showed that bubble entropy presents remarkable stability and exhibits increased descriptive and discriminating power compared to all other definitions, including the most popular ones. <i>Conclusion:</i> The definition proposed is almost free of parameters. The most common ones are the scale factor <i>r</i> and the embedding dimension <i>m</i> . In our definition, the scale factor is totally eliminated and the importance of <i>m</i> is significantly reduced. The proposed method presents increased stability and discriminating power. <i>Significance:</i> After the extensive use of some entropy measures in physiological signals, typical values for their parameters have been suggested, or at least, widely used. However, the parameters are still there, application and dataset dependent, influencing the computed value and affecting the descriptive power. Reducing their significance or eliminating them alleviates the problem, decoupling the method from the data and the application, and eliminating subjective factors.<i>Objective</i>: A critical point in any definition of entropy is the selection of the parameters employed to obtain an estimate in practice. We propose a new definition of entropy aiming to reduce the significance of this selection. <i>Methods:</i> We call the new definition <i>Bubble Entropy</i>. Bubble Entropy is based on permutation entropy, where the vectors in the embedding space are ranked. We use the <i>bubble sort</i> algorithm for the ordering procedure and count instead the number of swaps performed for each vector. Doing so, we create a more coarse-grained distribution and then compute the entropy of this distribution. <i>Results:</i> Experimental results with both real and synthetic HRV signals showed that bubble entropy presents remarkable stability and exhibits increased descriptive and discriminating power compared to all other definitions, including the most popular ones. <i>Conclusion:</i> The definition proposed is almost free of parameters. The most common ones are the scale factor <i>r</i> and the embedding dimension <i>m</i> . In our definition, the scale factor is totally eliminated and the importance of <i>m</i> is significantly reduced. The proposed method presents increased stability and discriminating power. <i>Significance:</i> After the extensive use of some entropy measures in physiological signals, typical values for their parameters have been suggested, or at least, widely used. However, the parameters are still there, application and dataset dependent, influencing the computed value and affecting the descriptive power. Reducing their significance or eliminating them alleviates the problem, decoupling the method from the data and the application, and eliminating subjective factors.