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The sigmoidally transformed cosine curve: a mathematical model for circadian rhythms with symmetric non‐sinusoidal shapes
Author(s) -
Marler Matthew R.,
Gehrman Philip,
Martin Jennifer L.,
AncoliIsrael Sonia
Publication year - 2005
Publication title -
statistics in medicine
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.996
H-Index - 183
eISSN - 1097-0258
pISSN - 0277-6715
DOI - 10.1002/sim.2466
Subject(s) - non sinusoidal waveform , trigonometric functions , sine wave , mathematics , amplitude , harmonics , rhythm , inverse trigonometric functions , sigmoid function , fourier series , circadian rhythm , mathematical analysis , computer science , physics , acoustics , waveform , geometry , telecommunications , radar , quantum mechanics , voltage , machine learning , artificial neural network , neuroscience , biology
We introduce a family of non‐linear transformations of the traditional cosine curve used in the modelling of biological rhythms. The non‐linear transformation is the sigmoidal family, represented here by three family members: the Hill function, the anti‐logistic function, and the arctangent function. These transforms add two additional parameters that must be estimated, in addition to the acrophase, MESOR, and amplitude (and period in some applications), but the estimated curves have shapes requiring many more than two additional harmonics to achieve the same fit when modelled by harmonic regression. Particular values of the additional parameters can yield rectangular waves, narrow pulses, wide pulses, and for rectangular waves (representing alternating ‘on’ and ‘off’ states) the times of onset and offset (hence duration, as when modelling the duration of the large night‐time melatonin secretory epoch). We illustrate the sigmoidally transformed cosine curves, and compare them to harmonic regression modelling, in a sample of eight activity recordings made on patients in a nursing home. Copyright © 2006 John Wiley & Sons, Ltd.