Coarse-graining and the Haar wavelet transform for multiscale analysis

Bioelectronic Medicine

Tobias Loddenkemper¹  Solveig Vieluf²  William J. Bosl³ 

[1]Department of Neurology, Division of Epilepsy and Clinical Neurophysiology, Boston Children’s Hospital, Harvard Medical School, Boston, MA, USA

[2]Department of Neurology, Division of Epilepsy and Clinical Neurophysiology, Boston Children’s Hospital, Harvard Medical School, Boston, MA, USA

[3]Institute of Sports Medicine, Paderborn University, Warburger Str. 100, 33098, Paderborn, Germany

[4]University of San Francisco, 2130 Fulton Street, 94117, San Francisco, CA, USA

[5]Department of Pediatrics, Harvard Medical School, Boston, USA

[6]Computational Health Informatics Program, Boston Children’s Hospital, Boston, USA

关键词: Haar;    Wavelet;    Multiscale;    Coarse-graining;    Multiscale entropy;    Fourier transform;   

DOI  :  10.1186/s42234-022-00085-z

来源: Springer

【 摘 要 】

BackgroundMultiscale entropy (MSE) has become increasingly common as a quantitative tool for analysis of physiological signals. The MSE computation involves first decomposing a signal into multiple sub-signal ‘scales’ using a coarse-graining algorithm.MethodsThe coarse-graining algorithm averages adjacent values in a time series to produce a coarser scale time series. The Haar wavelet transform convolutes a time series with a scaled square wave function to produce an approximation which is equivalent to averaging points.ResultsCoarse-graining is mathematically identical to the Haar wavelet transform approximations. Thus, multiscale entropy is entropy computed on sub-signals derived from approximations of the Haar wavelet transform. By describing coarse-graining algorithms properly as Haar wavelet transforms, the meaning of ‘scales’ as wavelet approximations becomes transparent. The computed value of entropy is different with different wavelet basis functions, suggesting further research is needed to determine optimal methods for computing multiscale entropy.ConclusionCoarse-graining is mathematically identical to Haar wavelet approximations at power-of-two scales. Referring to coarse-graining as a Haar wavelet transform motivates research into the optimal approach to signal decomposition for entropy analysis.

【 授权许可】

CC BY