Joint Amplitude-Acrophase Confidence Ellipse (Bingham et al. 1982) — cosinor.confidence.ellipse • actiRhythm

Computes the joint confidence region for the cosinor amplitude and acrophase, following Bingham et al. (1982). The region is an ellipse in the $(\beta_1, \beta_2)$ (cosine/sine coefficient) plane. When the ellipse excludes the origin, the population amplitude differs significantly from zero and a rhythm is detected.

Usage

cosinor.confidence.ellipse(cosinor_result, level = 0.95, n_points = 200)

Arguments

cosinor_result: A list returned by cosinor.analysis() (class "actiRhythm_cosinor"). It must contain amplitude, acrophase (clock hours), se_amplitude, period and n_profile_hours. If the object additionally carries an explicit vcov_beta (2x2 covariance of $(\beta_1, \beta_2)$) and/or beta1/beta2, those are used directly.
level: Confidence level for the region (default 0.95).
n_points: Number of points used to trace the ellipse boundary (default 200).

Value

A list with class "actiRhythm_cosinor_ellipse" containing:

ellipse: Data frame with columns x ($\beta_1$) and y ($\beta_2$) giving the ellipse boundary vertices.
center: Numeric vector c(beta1, beta2) of the point estimate.
excludes_origin: Logical; TRUE when the ellipse excludes (0, 0) (i.e. a rhythm is detected at the requested level).
rhythm_detected: Alias of excludes_origin.
distance_stat: Mahalanobis-type statistic of the origin relative to the fitted ellipse.
critical_value: Threshold $2 F_{2, df, level}$ the statistic is compared against.
level: Confidence level used.

When the input is missing required fields or is degenerate, the boundary is returned as NA and excludes_origin = NA.

Details

Under the cosinor model $Y(t) = M + \beta_1 cos(\omega t) + \beta_2 sin(\omega t)$, the $100(1-\alpha)\%$ joint confidence region for $(\beta_1, \beta_2)$ is the set of points $b$ satisfying $$(b - \hat b)^\top \Sigma^{-1} (b - \hat b) \le 2 F_{2, df, 1-\alpha}$$ where $\Sigma$ is the covariance of the estimated coefficients and $df$ the residual degrees of freedom. The averaged-profile cosinor design used by cosinor.analysis() has orthogonal cosine and sine columns, so $\Sigma$ is (very nearly) diagonal with equal variances $\sigma^2$; in that case se_amplitude equals the standard error of each coefficient $\sqrt{\mathrm{Var}(\beta)}$, which is how $\Sigma$ is reconstructed when it is not supplied explicitly. The point estimate is recovered from amplitude and acrophase as $\beta_1 = A cos(\phi)$, $\beta_2 = A sin(\phi)$ with $\phi = acrophase \cdot 2\pi / T$.

References

Bingham C, Arbogast B, Cornelissen Guillaume G, Lee JK, Halberg F (1982). “Inferential statistical methods for estimating and comparing cosinor parameters.” Chronobiologia, 9(4), 397–439.

Examples

# \donttest{
ts <- seq(as.POSIXct("2024-01-01", tz = "UTC"), by = 60, length.out = 1440 * 3)
counts <- 100 + 80 * cos(2 * pi * (as.numeric(format(ts, "%H")) - 14) / 24)
cos <- cosinor.analysis(counts, ts)
cosinor.confidence.ellipse(cos)$rhythm_detected
#> [1] NA
# }