On the relation between correlation dimension, approximate entropy and sample entropy parameters, and a fast algorithm for their calculation

• Published in: Physica A Vol. 391; no. 24; pp. 6601 - 6610
• Main Authors: Zurek, Sebastian, Guzik, Przemyslaw, Pawlak, Sebastian, Kosmider, Marcin, Piskorski, Jaroslaw
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.12.2012
• Subjects: Algorithms; Approximate entropy; Approximation; Complexity; Correlation; Correlation dimension; Entropy; Fast numerical algorithms; Mathematical analysis; Medical time series; Sample entropy; Samples; Statistical analysis; Statistical methods; Time series analysis; Fast numerical algorithms; Time series analysis; Correlation dimension; Medical time series; Sample entropy; Approximate entropy; Complexity
• ISSN: 0378-4371; 1873-2119
• DOI: 10.1016/j.physa.2012.07.003

We explore the relation between correlation dimension, approximate entropy and sample entropy parameters, which are commonly used in nonlinear systems analysis. Using theoretical considerations we identify the points which are shared by all these complexity algorithms and show explicitly that the above parameters are intimately connected and mutually interdependent. A new geometrical interpretation of sample entropy and correlation dimension is provided and the consequences for the interpretation of sample entropy, its relative consistency and some of the algorithms for parameter selection for this quantity are discussed. To get an exact algorithmic relation between the three parameters we construct a very fast algorithm for simultaneous calculations of the above, which uses the full time series as the source of templates, rather than the usual 10%. This algorithm can be used in medical applications of complexity theory, as it can calculate all three parameters for a realistic recording of 104 points within minutes with the use of an average notebook computer. ► The theoretical relationship between CD, ApEn and SampEn techniques is explored. ► It is shown that all the above can be calculated with a single algorithm. ► The theoretical and practical consequences of the above are discussed. ► A fast algorithm for CD, ApEn and SampEn is introduced.