The 24-hour signal curve data was smoothed using B-spline smoothing transform,
17,18 based on the least squares approach, to describe the overall shape characteristics of the 24-hour signal curve. The smoothed signal curve data were then analyzed by functional data analysis. Residual signal in terms of the difference between the observed signal data and the B-spline smoothed data was regarded as the fluctuation of signal against the overall shape. The frequency of fluctuation was quantified in terms of semivariogram (also known as semivariance)
19 and analyzed by functional data analysis. The semivariogram of a signal curve is defined mathematically as
Display Formula
, where
s(
t) represents the signal value at time
t, which describes the amplitude of fluctuation with frequency measured in terms of
Δt. The differences in signal patterns, gradients, and curvatures at any time point were compared between the progressive and stable groups by permutation tests on the functional
t-statistic for the smoothed signal data.
20 The functional
t-statistic is defined mathematically by
Display Formula
, where
Display Formula
and
Display Formula
represent the mean signal values of the progressive and stable groups at time
t,
Display Formula
and
Display Formula
represent the variance of signal value of the progressive and stable groups at time
t, and
n1 and
n2 represent the sample size of the progressive and stable groups, respectively. Overall differences in signal patterns, gradients, and curvatures of the signal data in the whole 24-hour period, as well as specific periods within the bedtime hours (1800–0100) and the wake-up hours (0300–1100), were compared by permutation tests on the supremum
t-statistic between the two groups, where the supremum
t-statistic is defined as the least upper bound of the functional
t-statistic across the specific time period.
20