Anti-Logistic (Extended) Cosinor Analysis (Marler et al. 2006)

Fits the sigmoidally transformed cosine ("anti-logistic" extended cosinor) model of Marler et al. (2006) to the averaged 24-hour activity profile. The extended model relaxes the symmetric shape of the ordinary cosinor by adding two shape parameters: alpha sets the relative width of the active versus rest phase, and beta sets the steepness of the rest-to-active transitions. It is the same model as ActCR::ActExtendCosinor and reproduces its parameter estimates numerically.

Usage

cosinor.antilogistic(counts, timestamps, period = 24)

Arguments

counts: Numeric vector of activity counts (one value per epoch).
timestamps: POSIXct vector of timestamps, the same length as counts.
period: Numeric period of the rhythm in hours (default 24).

Value

A list with class "actiRhythm_cosinor_ext" containing:

minimum: Lower asymptote of the fitted curve (rest-phase level).
amplitude: Vertical span of the sigmoidal transition (amp); the asymptotic ceiling is minimum + amplitude, with the fitted-curve maximum slightly below it at a finite steepness.
alpha: Width-asymmetry parameter in $[-1, 1]$.
beta: Steepness parameter ($\ge 0$).
acrophase: Time of peak activity in clock hours (acrotime).
acrotime: Alias of acrophase (ActCR naming).
peak: Fitted activity level at the acrophase (renormalized Marler peak).
UpMesor: Clock time of the rest-to-active (rising) transition, $-\arccos(\alpha) / (2\pi/T) + acrotime$.
DownMesor: Clock time of the active-to-rest (falling) transition, $\arccos(\alpha) / (2\pi/T) + acrotime$.
MESOR: Midline statistic minimum + amplitude / 2.
F_pseudo: Pseudo-F improvement of the extended fit over the ordinary cosinor: $((RSS_{cos} - RSS_{ext})/2)/(RSS_{ext}/(n-5))$.
rss_cosinor: Residual sum of squares of the ordinary cosinor.
rss_extended: Residual sum of squares of the extended cosinor.
converged: Logical; whether the nonlinear fit converged.
period: Period used (hours).
n_profile_hours: Number of profile hours used in the fit.

On non-convergence or insufficient data all numeric parameters are returned as NA with converged = FALSE.

Details

The model fitted to the averaged profile is $$f(t) = minimum + amplitude \cdot expit(\beta (cos(2\pi (t - acrotime)/T) - \alpha))$$ where $expit(z) = 1/(1+e^{-z})$. This is the ActCR/Marler parameterization, in which amplitude is the raw multiplier of the logistic transform (it is not renormalized by $expit(\beta(1-\alpha))$). The closely related "normalized" Marler form, $f(t) = min + amp \cdot expit(\beta(cos(\cdot) - \alpha)) / expit(\beta(1-\alpha))$, rescales amplitude so that the peak equals exactly minimum + amplitude; the two forms describe the same curve shape and the (alpha, beta, acrotime) parameters are identical. The normalized peak level is reported as peak.

Starting values are taken from an ordinary cosinor fit (minimum = max(mesor - amp, 0), amplitude = 2 * amp, alpha = 0, beta = 2, acrotime = ordinary acrophase). The nonlinear least-squares problem is solved with stats::optim() using the box-constrained L-BFGS-B method (bounds lower = c(0, 0, -1, 0, -3), upper = c(Inf, Inf, 1, Inf, 27), matching ActCR), so no extra package dependency is required.

References

Marler MR, Gehrman P, Martin JL, Ancoli-Israel S (2006). “The sigmoidally transformed cosine curve: a mathematical model for circadian rhythms with symmetric non-sinusoidal shapes.” Statistics in Medicine, 25(22), 3893–3904. doi:10.1002/sim.2466 .

Di J, Zipunnikov V (2021). ActCR: Extract Circadian Rhythms Metrics from Actigraphy Data. R package version 0.2.0, https://CRAN.R-project.org/package=ActCR.

Examples

# \donttest{
ts <- seq(as.POSIXct("2024-01-01 00:00:00"), by = 60, length.out = 1440 * 3)
hour <- as.numeric(format(ts, "%H")) + as.numeric(format(ts, "%M")) / 60
counts <- 50 + 300 * plogis(4 * (cos((hour - 14) * 2 * pi / 24) - 0.2))
fit <- cosinor.antilogistic(counts, ts)
print(fit)
#> $minimum
#> [1] 49.62602
#> 
#> $amplitude
#> [1] 301.3645
#> 
#> $alpha
#> [1] 0.2014903
#> 
#> $beta
#> [1] 3.903463
#> 
#> $acrophase
#> [1] 14.00833
#> 
#> $acrotime
#> [1] 14.00833
#> 
#> $peak
#> [1] 338.2087
#> 
#> $UpMesor
#> [1] 8.783275
#> 
#> $DownMesor
#> [1] 19.23339
#> 
#> $MESOR
#> [1] 200.3083
#> 
#> $F_pseudo
#> [1] 632333.8
#> 
#> $rss_cosinor
#> [1] 13776.46
#> 
#> $rss_extended
#> [1] 0.2069704
#> 
#> $converged
#> [1] TRUE
#> 
#> $period
#> [1] 24
#> 
#> $n_profile_hours
#> [1] 24
#> 
#> attr(,"class")
#> [1] "actiRhythm_cosinor_ext"
# }