The sigmoidally transformed cosine curve: A mathematical model for circadian rhythms with symmetric non-sinusoidal shapes
The sigmoidally transformed cosine curve: A mathematical model for circadian rhythms with symmetric non-sinusoidal shapes
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DOI：
10.1002/sim.2466
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发表时间：
2006-11-30
影响因子：
2
通讯作者：
Ancoli-Israel, Sonia
中科院分区：
文献类型：
--
作者：
Marler, Matthew R.;Gehrman, Philip;Ancoli-Israel, Sonia
我们介绍了生物节律建模中使用的传统余弦曲线的一系列非线性变换。非线性变换是 sigmoidal 系列，这里由三个系列成员代表：希尔函数、反对逻辑函数和反正切函数。除了顶相、MESOR 和振幅（以及某些应用中的周期）之外，这些变换还添加了两个必须估计的附加参数，但估计曲线的形状需要两个以上的附加谐波才能在通过谐波回归建模时实现相同的拟合。附加参数的特定值可以产生矩形波、窄脉冲、宽脉冲，并且对于矩形波（代表交替的“开”和“关”状态），产生开始和偏移的时间（因此持续时间，如在对夜间褪黑素大分泌时期的持续时间进行建模时）。我们展示了 S 形变换的余弦曲线，并将它们与调和回归模型进行比较，样本中对疗养院患者进行了八个活动记录。版权所有 (c) 2006 John Wiley & Sons, Ltd.
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 (c) 2006 John Wiley & Sons, Ltd.