Fig 1.
Circadian studies can optimize collection times to balance power and rest-window duration.
(A) Measurement times for optimal and naive designs plotted as phases of a 24hr cycle. Each panel corresponds to a different duration of the rest-window (shaded region), during which no samples can be collected. Optimal designs were generated using a brute-force search with Nt = 48 points in the temporal discretization (Sect 5.2.1). Naive designs were constructed by distributing all measurements outside the rest window at equal time-intervals. (B) Power (y-axis) as a function of acrophase (x-axis) for each duration of the rest-window. The color of each curve represents rest-window duration and the line-type represents if the design is optimal or naive. (C) The noncentrality parameter (y-axis) as a function of rest-window duration (x-axis) for naive and optimal designs. Noncentrality parameters are reported relative to the noncentrality parameter of an unconstrained equispaced design with the same sample size (N = 8 samples).
Fig 2.
Globally optimal designs for discrete period uncertainty.
(A) Measurement collection times of an equispaced design (top) and a bifrequency optimal design (bottom) for detecting 2hr and 24hr rhythms. (B) Power (y-axis) as a function of frequency (x-axis) for the equispaced and bifrequency optimal design. The two frequencies included in the optimization problem are indicated by the dashed vertical lines. (C) Bifrequency optimal design plotted as phases of a 24hr cycle. The values of the radial coordinate were chosen to emphasize that the design can be split into four equiphase designs, each containing three measurement times. (D) Bifrequency optimal design plotted as phases of the 2hr cycle. The radial position of each point represents its phase along the 24hr cycle. (E) Days marked with circles indicate that n = 4 equispaced circadian measurements are to be collected. Repeating this schedule four times over the course of a year, produces a trifrequency optimal design with measurements equispaced along circadian (24 hr), circalunar (28 day), and circannual ( day) cycles. (F) An alternative measurement schedule that distributes the measurements across all three months while maintaining the equiphase property.
Fig 3.
Irregular sampling improves power near the Nyquist rate in free-period rhythm detection.
(A) Permutation tests were performed with the amplitude test-statistic (Sect 5.1). Independent signals were generated across all included frequencies and acrophases for each frequency window and amplitude (panels). Curves report the minimum power over acrophase (y-axis) as a function of frequency (x-axis) for equispaced and irregular designs (color), each with N = 48 measurements. Irregular designs were generated either by maximizing the permutation bound or using the worst-case fixed-period power as a heuristic. (B) Signals were simulated with N = 12 equispaced measurements spanning the slowest cycle in the frequency window
. Each point represents an estimate of the fixed-period (x-axis) and free-period (y-axis) cosinor power for a signal of random amplitude, acrophases, and frequency (color). Parameters: (A) each frequency window was discretized using
and
and
independent samples of Gaussian white noise. Each of the
signals were then permuted
for the permutation test. The
test statistic was discretized using
frequencies. For the equispaced design,
becomes singular at the Nyquist rate so the frequency grid was restricted to only include frequencies
.The differential evolution design was generated using the same parameters as Fig 4. Parameters (B): n = 3000 signals were generated in each panel with amplitude
, acrophase
, frequency
. Power was estimated using
and
independent samples for each frequency. The
test statistic was discretized using
frequencies.
Fig 4.
Irregular sampling improves power in experiments with continuous period uncertainty.
Irregular designs were optimized by differential evolution to detect signals whose frequency is in a window . (A) Differences in power between equispaced designs and optimized irregular designs where the x and y axes represent the upper (
) and lower (
) limits of each frequency range of interest. The color of each circle represents the difference in worst-case power between an irregular and equispaced design with the same number of measurements (panels). Grey circles correspond to negative power differentials. (B) Heatmap of power as a function of frequency (x-axis) and acrophase (y-axis) where color represents the power. (A-B) Simulated cosinor parameters: amplitude A = 1 and noise strength
. Each differential evolution was run with 1hr of compute time with parameters
,
,
(see Sect 5.2.4). Details on power calculation at the Nyquist rate are given in S1 Text Sect S1-1.4.
Fig 5.
Irregular designs are robust to perturbations in measurement timing.
(A) Measurement schedules before (top) and after (bottom) jittering. Jittered designs were generated by randomly perturbing the measurement times of equispaced and irregular designs with Gaussian white noise. (B) Worst-case power of jittered designs (y-axis) as a function of noise intensity (x-axis) for various sample sizes (panels). The bars indicate the interquartile range for ensembles (n = 100) jittered designs. Irregular designs were optimized for the frequency window and
using differential evolution with same parameters as Fig 4. Power was calculated assuming an amplitude A = 1.
Fig 6.
Irregular designs improve simulated periodogram analysis at frequencies up to the Nyquist rate.
(A)-(B) Oscillations were detected using periodogram analysis with measurements (N = 40 samples) from either an equispaced or irregular design optimized for the frequencies . Each signal in the dataset was assigned an oscillatory (amplitude A = 2, noise strength
) or non-oscillatory (amplitude A = 0, noise strength
) state with equal probability. The acrophase of the oscillatory signals was assigned uniformly at random (
). (A) Performance of the irregular and equispaced designs at detecting oscillations across frequencies (x-axis) included in the optimization, summarized by the AUC score (y-axis) of a receiver operator characteristic curve with p-values generated from a Lomb-Scargle periodogram. The AUC score for each frequency was computed by testing for oscillations in a dataset of oscillatory and white-noise signals (n = 104 signals per dataset). The dashed line indicates the Nyquist rate of the equispaced design. (B) The same analysis as (A) but at frequencies above the Nyquist rate of the equispaced design. Periodogram analysis was performed using the lomb library [37] and AUC scores were computed using the pROC library [38]. Irregular designs were generated using the same differential evolution parameters as in Fig 4.