Marler Matthew R, Gehrman Philip, Martin Jennifer L, Ancoli-Israel Sonia
Department of Psychiatry, University of California, San Diego, CA, USA.
Stat Med. 2006 Nov 30;25(22):3893-904. doi: 10.1002/sim.2466.
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.
我们引入了一族用于生物节律建模的传统余弦曲线的非线性变换。非线性变换为S形函数族，这里由三个族成员表示：希尔函数、反对数函数和反正切函数。这些变换除了要估计的峰相位、MESOR和振幅（在某些应用中还有周期）之外，还增加了两个必须估计的额外参数，但是当通过谐波回归建模时，估计曲线的形状需要不止两个额外的谐波才能达到相同的拟合效果。额外参数的特定值可以产生矩形波、窄脉冲、宽脉冲，对于矩形波（代表交替的“开”和“关”状态），可以产生起始和偏移时间（从而得到持续时间，如在模拟夜间大量褪黑素分泌时期的持续时间时）。我们在一家养老院对患者进行的八次活动记录样本中展示了S形变换的余弦曲线，并将它们与谐波回归建模进行了比较。
