2.1 Harmonic regression

Harmonic regression is a popular statistical framework for studying systems with oscillatory features [21]. We perform harmonic regression using the fixed-period cosinor model. For a MESOR (Midline Estimating Statistic of Rhythm) Y₀, amplitude A, acrophase ϕ, and frequency f, the fixed-period cosinor model takes the form

(1)

in which is homoscedastic Gaussian white noise. Assuming the frequency f is fixed, Eq 1 can be rewritten as

(2)

so that all unknown parameters appear linearly. Given data measured at times , where N is the sample size of the experiment, the optimal least squares solution for estimating the coefficients appearing in Eq 2 is

(3)

We assume contains measurements at sufficiently many distinct phases for Eq 3 to be well-defined (S1 Text Lemma S1-1.1). For most of our analysis, we only consider signals with frequency where is the Nyquist rate of an equispaced design containing N measurements. Estimates of the signal’s amplitude and acrophase can be computed from to obtain

(4)

2.2 Power analysis

Using the fixed-period cosinor model, rhythm detection can be formulated as a hypothesis test with null hypothesis and alternative or . The statistical power quantifies how reliably oscillations are detected by the experimental design and hypothesis test.

Definition 2.1 (Statistical power of the fixed-period cosinor model). Given a parametric model with parameters , data measured at times , and a rejection region , the power of the hypothesis test is given by the probability of the data lying within the rejection region

(5)

For the fixed-period-cosinor-based hypothesis test, the rejection region is , in which is the F-statistic

(6)

, , , is the cosinor design matrix and is the least-squares estimate of the parameters from Eq 3.

Since the exact parameters of the signal are rarely well-known when the design is constructed, we seek designs that achieve high power across a range of parameter values. To this end, we quantify performance using the worst-case power of the design, meaning the lowest power across all signals of interest. Prioritization of worst-case rather than average power ensures that the power is above a known threshold for all relevant signals. A formal definition of worst-case power is given below. For the remainder of the paper we will simply use the terminology “optimal power designs” to refer to optimality with respect to a worst-case scenario.

Definition 2.2 (Optimal worst-case power). Given a domain in parameter space, a design matrix , and a power function , an experimental design achieves optimal worst-case power with respect to if it satisfies

(7)

Our results are organized based on the degree of period uncertainty at the time of study design. In the simplest setting, the period is treated as known and the design is optimized to detect signals of various acrophases and amplitudes for the predetermined period. The latter two contexts, defined below, assume a uniform prior distribution on the candidate periods, meaning that no period is prioritized higher than the others.

Definition 2.3 (Discrete period uncertainty). An experiment that investigates a finite list of periods is said to have discrete period uncertainty.

Definition 2.4 (Continuous period uncertainty). An experiment that investigates all periods in a given range is said to have continuous period uncertainty.

When considering continuous period uncertainty, we test for rhythmicity by applying permutation tests to the free-period cosinor model (Sect 5.1). The models of period uncertainty we consider are guided by the type of experiment we aim to optimize. We assume that the experiment is an exploratory investigation of a sparsely characterized system, with limited prior knowledge of the underlying parameters. This assumption is also relevant to our choice to prioritize the cosinor model. Several alternative approaches to the cosinor model have been introduced [13,14,22–25]. Alternative models can be particularly useful for systems that violate the assumptions of the cosinor model, for instance biological rhythms that display non-stationary peak-to-peak duration [26]. We focus on the cosinor model because it is simple, popular, and broadly applicable, making it well suited for initial studies of sparsely characterized systems. Data collected in an exploratory study could inform the optimization of a follow-up study focused on questions beyond rhythm detection.

3.1 Rhythms of known period

We construct optimal designs for known-period experiments by maximizing a closed-form expression for the statistical power. The power expression can be evaluated much faster than Monte Carlo power estimation and generalizes an earlier formula [27] which is only valid when measurements are equispaced (S2 Fig). A related generalization of [27] presented in [28] was obtained independently of our work. The reader is directed to S1 Text for a proof of Theorem 3.1 and the subsequent theorems.

Theorem 3.1 (Power of the one-frequency cosinor model). Consider the one-frequency cosinor model

(8)

applied to data collected at distinct times with N > 3. Suppose the following hypotheses are tested using an F-test

• null hypothesis ,
• alternative hypothesis or .

Given parameters , the power γ of this hypothesis test is given by

(9)

(10)

in which is the design matrix, is the type I error rate, is the noncentral F-distribution with degrees of freedom and noncentrality parameter λ, is the F-distribution, and

(11)

is the hypothesis matrix.

Analysis of Eqs 9 and 10 in S1 Text Sect S1-1.3 shows that worst-case power maximization in the sense of Definition 2.2 and Theorem 3.1 is equivalent to Elfving optimality [29]. As a consequence of this equivalence, we obtain a simple condition for equispaced designs to be optimal and provide the same power at all acrophases.

Theorem 3.2 (Optimality condition for equispaced designs). Suppose a study aims to detect signals of a specific frequency and unknown acrophase. Maximizing the worst-case power across all acrophases is equivalent to the following eigenvalue optimization problem

(12)

in which denotes the smallest eigenvalue and the matrix is given by

(13)

If it is possible to collect N > 3 equispaced measurements per cycle, then equispaced designs maximize Eq 12. Moreover, such designs maintain constant power as a function of acrophase (S3A Fig), at a constant value determined by the noncentrality parameter

(14)

It may seem strange that the formulation of power optimization in Theorem 3.2 does not require knowledge of the amplitude or noise strength. This is because the influence of measurement timing on power is essentially independent of amplitude and noise strength. To be precise, the ratio between amplitude and noise strength simply rescales the noncentrality parameter. Since the power function depends monotonically on this parameter, the rescaling therefore does not affect the location of its critical points with respect to measurement timing. While amplitude and noise strength are of course needed for determining the precise level of power, they are not needed for the identification of designs that maximize power.

While Theorems 3.1 and 3.2 are broadly applicable to cosinor rhythms, many biological rhythms do not strictly satisfy the assumptions of the cosinor model. To understand the extent to which equispaced optimality requires cosinor assumptions, we compared the performance of equispaced and random designs on simulated non-cosinor signals. We maintained the assumption that the study had no prior knowledge of the signal’s acrophase and found that equispaced designs remained optimal or at least comparable to random designs (S4 Fig). The connection between uniform acrophase uncertainty and equispaced optimality is discussed further in S1 Text Sect S1-1.6, where we demonstrate that acrophase invariance implies equispaced optimality for a more general class of optimality criteria.

In practice, studies may need to consider sub-optimal designs in order to balance power with other aspects of study design. For instance, a circadian study with human participants may aim to minimize the number of times a participant is awoken during the night for sample collection while ensuring that rhythms of all acrophases can still be reliably detected. Since strong acute disruptions of sleep have been shown to impact various biological measurements [30,31], accurate characterization of naturally occurring oscillations should avoid disruptions to a 6-12hr window of a subject’s regular sleep routine.

Incorporating a rest-window reduces the power relative to equispaced designs, so we aimed to determine how much power could be recovered by optimizing the measurement times under the timing constraint. To simplify the design space, we assumed that measurements could only be collected at times aligned with a half-hour grid. Consequently, it was possible to find optimal designs by running a brute-force search (Sect 5.2.1). To gauge the benefit of timing optimization, we compared the optimal designs to “naive” designs in which all measurements are equispaced outside the rest-window (Fig 1A). Relative to the naive designs, the power of the constrained-optimal designs was considerably less sensitive to the acrophase of the signal (Fig 1B). This difference in sensitivity can be observed in the peak-to-trough power variability of the two designs (, ; 8hr rest-window, amplitude A = 2.5). Comparing the noncentrality parameters of the constrained-optimal and naive designs to an equispaced design allowed us to estimate how much the constraints limit the power. Negligible power was lost in the constrained-optimal design when imposing the shortest window duration. As the window length increased, both the constrained-optimal and naive designs lost power (Fig 1C). Still, the constrained-optimal designs out-performed the naive designs across the windows under consideration, suggesting that timing optimization can help studies accommodate timing constraints without unnecessarily reducing their power.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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 N_t = 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).

https://doi.org/10.1371/journal.pcbi.1013662.g001

3.2 Discrete period uncertainty

Experiments may prioritize the detection of specific periods based on prior knowledge of the biological system. For instance, a study may investigate harmonics of a known rhythm [12] or assume that rhythms are entrained to environmental cues (i.e. circadian, circalunar, and circannual cycles [32]). To incorporate discrete period uncertainty in our design process, we score designs based on their lowest power across all acrophases and frequencies of interest, using an objective function of the form

(15)

in which is a collection of frequencies of interest, is the the matrix given in Eq 13, and maps a matrix to its smallest eigenvalue. Continuing to focus on worst-case optimality (Definition 2.2), we seek designs that achieve the greatest possible value of across all signals of interest. The connection between and power optimization is made explicit in Corollary 3.2.1.

Corollary 3.2.1. Maximizing the objective function in Eq 15 is equivalent to maximizing the worst-case power over frequencies and acrophase. The value achieved by this optimal solution is bounded by

(16)

The bound on in Eq 16 can be converted into a bound on the noncentrality parameter from Eq 10 so that the content of the corollary can be expressed in units of power (Sect 5.2). We also note that the worst-case power of the fixed-period model provides a lower bound on the power of free-period rhythm detection after Bonferroni correction, as clarified further in S1 Text Sect S1-1.7. As a result, designs that are constructed to improve the worst-case fixed-period power should be understood as indirectly improving the multiple-test corrected free-period power.

Since the study investigates multiple rhythms, we assume that it has a higher sample size than the studies considered in the previous section. The brute force searches in Sect 3.1 are not tractable at high sample size (S5 Fig), so we developed an alternative optimization approach, stated formally in Theorem 3.3 and derived in Sect 5.2.2.

Theorem 3.3 (Mixed-integer conic program for power optimization). Worst-case power maximization with discrete time

(17)

(18)

is equivalent to the following mixed-integer conic programming problem

(19)

(20)

(21)

in which is the design matrix of the one-frequency cosinor model evaluated at frequency f on the partition , and is given by

(22)

We illustrate the applicability of Theorem 3.3 to bifrequency design optimization, the simplest nontrivial example of discrete period uncertainty. In this setting, the study investigates a pair of frequencies and aims to maximize the worst-case power at both frequencies for a given sample size. The widest frequency separation corresponded to resolving a pair of periods and with a sample size of N = 12 measurements. Interestingly, the optimal solution contained repeating patterns in its measurement schedule (Figs 2A and S6A). While the equispaced design suffered from a low noncentrality parameter at the 2hr rhythm (Fig 2B), the optimal design reached its theoretical maximum value (Theorem 3.2) at both frequencies of interest. This ability to resolve both frequencies without sacrificing power at either frequency was not observed in 10,000 randomly generated designs (Fig S7A) or when additional frequencies were included in the design problem (Fig S7B).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1013662.g002

Closer examination of the bifrequency optimal designs revealed a simple explanation for their ideal power balance; they can be split into equispaced designs when visualized as phases of each period in the design problem (Fig 2C and 2D) and therefore inherit the optimality properties of equispaced designs (S1 Text Sect S1-1.5). Since optimal designs can be constructed by simply ensuring that they satisfy this equiphase property, we used this approach to construct optimal designs for detecting circadian, circalunar, and circannual rhythms in a 12 month experiment (Fig 2E and 2F; Fig S7C).

3.3 Continuous period uncertainty

3.3.1 Optimization of free-period power.

In experiments where the period of rhythmic activity must be estimated across a continuous range (i.e. longer than an hour and shorter than a day), the closed-form expression for statistical power from Theorem 3.1 is not directly available. We rely instead on a permutation test based on the amplitude estimate of the free-period cosinor model (Sect 5.1). Estimating the power of the permutation test is computationally expensive, so we instead derive a mean-squared bound that can be evaluated more efficiently (S1 Text Sect S1-2). We use this bound to construct candidate designs and then benchmark their performance using the power of the full permutation test.

Optimization of the bound requires specification of the frequency window and the sample size of the study. Improvements relative to equispaced designs were only observed when the frequency window included the Nyquist rate of the equispaced design () and when the signal amplitude was sufficiently high (Fig 3A). Although the permutation bound reduced the compute time compared with the full permutation test, it remained much slower than the objective functions considered in earlier sections (S8 Fig). This limitation motivated us to ask whether comparable designs could be obtained using a simpler heuristic approach. To this end, we evaluated the worst-case fixed-period power as a candidate heuristic. Fixed-period and free-period power were strongly correlated across a broad range of signals (Fig 3B). When designs were optimized to maximize fixed-period power across the frequency window, they maintained high free-period power at all frequencies including the Nyquist rate (Fig 3A). Our results suggest that our heuristic approach – generating designs using fixed-period power and benchmarking them with free-period power – is an practical and effective way to optimize power under continuous period uncertainty.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1013662.g003

3.3.2 Worst-case fixed-period power under continuous period uncertainty.

In this section we treat heuristic design optimization as a standalone problem and investigate the structure and robustness of candidate solutions. Designs were constructed to maximize the power heuristic on various frequency windows and sample sizes using PowerCHORD’s differential evolution method. The greatest improvements to worst case for each sample size N were observed when the Nyquist rate was included in the frequency window (Fig 4A). Optimization of other windows produced at best a slight change in power. In comparison to equispaced designs, the power of irregular designs had much weaker power fluctuations around the Nyquist rate (Figs 4B and S3B). Away from the Nyquist rate, the equispaced and irregular designs exhibited power fluctuations that diminish in intensity as the sample size increases. Hence, for a large enough sample size, irregular designs improve worst-case power while maintaining homogeneous power levels throughout the frequency-acrophase cylinder.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1013662.g004

Some of the measurements in irregular designs are spaced farther apart than in equispaced designs of the same sample size. These measurement gaps could make the design undesirably sensitive to the timing of specific measurements. To investigate this sensitivity, we generated “jittered” irregular and equispaced designs by adding Gaussian white noise to each measurement time (Fig 5A) and calculated the worst-case power of the increasingly perturbed designs. As the noise intensity increased, the power improvements of the irregular design declined as both designs converged in power as measurement times were effectively being sampled from the same distribution (Fig 5B). For a sample size of 24, the power difference between the irregular and equispaced designs reached half of its full difference when the measurement error was 9 minutes, while for a larger sample size of 48 this difference diminishes more quickly at a measurement error of 5 minutes. Irregular designs always outperformed the eventual random design ( for , respectively). Since timing error is likely to be on the order of minutes, our power improvements are robust to errors that occur on a realistic timescale.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1013662.g005

Our irregular designs were optimized for cosinor analysis, yet the data they produce may be used for other purposes such as spectral analysis. To determine the applicability of our irregular designs in such a context, we performed periodogram analysis on simulated datasets. We made use of the Lomb-Scargle periodogram, a common method developed for the study of irregularly sampled data [33,34], and applied a hypothesis test derived from this method [35]. We generated independent simulated datasets for a range of frequencies, each made up of white noise and oscillatory signals with uniformly random acrophases. We quantified performance using the area under an ROC (receiver operator characteristic) curve [36] because it can be calculated without knowing the false positive rate, and this rate could be affected by switching from cosinor to periodogram analysis. Relative to equispaced designs, irregular designs exhibited weaker fluctuations in their AUC (area under the curve) score at frequencies below the Nyquist rate (Fig 6A). At the Nyquist rate of the equispaced design (), the irregular designs dramatically outperformed the equispaced design. Irregular designs performed well only at the frequencies included in their optimization (Fig 6B), emphasizing the importance of choosing a realistic frequency range before optimizing an experimental design. Although they were constructed for cosinor-based analysis, PowerCHORD designs remain applicable in a broader statistical context.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1013662.g006

5.1 Permutation testing with the free-period cosinor model

The free-period cosinor model is given by

(23)

with parameters . Assuming that the frequency lies within a window , least squares regression with the free-period model reduces to linear regression, hence an optimal frequency f^* can be written as

(24)

in which and X_f is the cosinor design matrix evaluated at frequency f. Rhythm detection with the free-period model can be performed using a permutation test. In this framework, p-values are computed by determining how often the permuted data generates a value of a given test statistic that is more extreme than the observed value

(25)

in which denotes the indicator function

(26)

and is a given test statistic. We analyze two test statistics based on the amplitude estimator of the free-period model

(27)

(28)

in which and H is as given in Eq 11. Up to normalization, the T₂ test statistic corresponds to the mean squared amplitude across the frequencies of interest and corresponds to the largest amplitude. The latter quantity is likely to be more familiar to readers with experience in periodogram-based hypothesis testing. Both quantities are permutation invariant under the null distribution and are therefore amenable to permutation testing. While appears to be more sensitive to weak signals, T₂ is more analytically tractable. Consequently, we derive a lower bound based on the T₂ test statistic and use the test statistic to benchmark the performance of our designs.

5.2 PowerCHORD’s optimization methods

PowerCHORD optimizes power using using a brute-force search, mixed-integer conic programming, a genetic algorithm, or differential evolution. Brute-force searches are computationally tractable if the sample size is low and the measurements are confined to an underlying grid (e.g. only sampling on the hour). Mixed integer conic programming is intended for higher sample size experiments with discrete period uncertainty, and the latter two methods are intended for optimization under continuous period uncertainty. Our methods are designed for studies with limited prior knowledge of the rhythm’s parameters and therefore require minimal input data (sample size, frequencies under consideration, and timing constraints) for their optimization.

The optimization methods in PowerCHORD work directly with the eigenvalue problem from Eq 15 and their results can be converted back into power using the following identity

(29)

in which is the minimal eigenvalue from Eq 12 and is the lowest value of the noncentrality parameter across all acrophases for signals of amplitude A and frequency f. Justification for Eq 29 is given in the proof of Lemma S1-1.12. In our analysis, we set so that we may refer to the noncentrality parameter and minimum eigenvalue interchangeably. We also make use of the fact that the noncentral F distribution is a monotone function of its noncentrality parameter (see S1 Text Sect S1-1.2), which permits us to refer to the power of a design and its noncentrality parameter interchangeably.

5.2.1 Brute-force search.

Suppose measurements are confined to a single cycle and can only be collected at times belonging to a fixed partition

(30)

in which N_t is the coarseness of the partition. Assuming that at most one measurement can be collected at each time in the partition, such scenarios are naturally represented by binary vectors which satisfy . Provided that N and N_t are not too large, it is feasible to search the entire space of binary vectors.

The search over binary vectors can be accelerated by making use of the rotational symmetry inherent to rhythm detection. Since any design will achieve equal performance to all of its cyclic translations, we need only consider equivalence classes of such binary vectors up to rotational symmetry. Following the convention from combinatorics, these equivalence classes are referred to as fixed-density binary necklaces. For a partition of coarseness N_t and sample size N, the number of equivalence classes of such necklaces is given by a sum over the factors of the greatest common divisor of the sample size and the partition coarseness

(31)

in which is Euler’s totient function [51]. We recommend restricting and since this function grows rapidly with N and N_t (S5 Fig). An efficient algorithm for generating a representative from each equivalence class is given in [51]. We use their C implementation to generate the designs and search the design database to identify representatives of equivalence classes of optimal solutions.

There is an additional reflection symmetry present in the problem which could further improve the performance of the brute-force search. Notice that the time-reversal of any design will still have equivalent power, hence designs could be considered equivalent up to rotational and reflectional symmetry. In the terminology of combinatorics, these larger equivalence classes are known as “fixed-density binary bracelets”. Efficient algorithms for generating representatives of the bracelets have been proposed [40] and would improve the performance of PowerCHORD if they were included.

5.2.2 Mixed-integer conic programming.

Mixed-integer conic programming addresses discrete period uncertainty, meaning that multiple cycle lengths are considered and there is no longer a one-to-one mapping between linear time and phase (e.g. circular time vs linear time in [52]). While measurements are still encoded as binary vectors under the assumption of at most one measurement per linear time, multiple measurements may coincide when they are viewed as phases of a shorter cycle (e.g. timepoints and in a study that investigates 24hr and 6hr rhythms map to distinct phases of the 24hr cycle but coincide as phases of the 6hr cycle).

We provide a proof of Theorem 3.3 which justifies our use of mixed-integer conic programming. The main technical device in the proof is stated in the lemma below.

Lemma 5.1 (Schur positivity, [53, Sect A.5.5]). Let be a symmetric matrix partitioned as

(32)

with and let be the Schur complement of Y. Then we have if and only if and .

Proof of Theorem 3.3. Let and . We seek a binary vector that satisfies

(33)

in which can be expressed as

(34)

as justified by Lemma S1-1.13. Restricting frequency to a discrete set , we arrive at an optimization problem of the form

(35)

The quadratic dependence on in Eq 35 can be reduced to a linear dependence by applying Lemma 5.1 with the matrix Y given by

(36)

which gives

(37)

□

Optimal designs were generated by maximizing Eqs 19 and 21 using the CUTSDP method in YALMIP [54] together with Gurobi as a backend solver [55].

5.2.3 Permutation bound optimization using genetic algorithm.

Fig 3 compares an equispaced design to a design constructed using the permutation bound from S1 Text Sect S1-2 as an objective function. To optimize the bound numerically, MATLAB’s built-in genetic algorithm [56,57] was applied to the following constrained optimization problem

(38)(39)(40)

in which is the permutation bound from Eq S163. The parameter ensures samples are not placed too close together and the equality constraints in Eq 40 ensure that samples remain spread out across the study and reduce the degrees of freedom in the problem.

5.2.4 Differential evolution.

Differential evolution algorithms initialize a population of candidate solutions at random. Successive generations are constructed by taking component-wise linear combinations of the parents according to an algorithm-specific update rule. While many variations on this core idea have been studied [58], we chose to implement a relatively simple version of the algorithm.

Given an objective function , our implementation of differential evolution requires four hyper-parameters: the population size , the differential weight , the crossover probability , and the number of iterations . At initialization, the population is represented by a matrix , in which N is the sample size, whose entries are independently and identically distributed as . To produce the next generation, the state of the i-th member of the population is updated by generating a random vector , sampling three indices without replacement to obtain

(41)

If , then the new state is accepted. The system repeats this process until a total of generations have been produced. The highest scoring individual of the terminal population is then returned. A summary of the algorithm in pseudo-code is given below.

Algorithm 1 Differential evolution in PowerCHORD.

Require:      sample size

Require: , , ,       hyperparameters

Require:       objective function

 

  while do       loop over generations

   for to do

    for to N do       generate candidate y_i

    

    

     if u<CR then

            sample three distinct indices

     

     end if

    end for

    if then       determine if y_i should replace x_i

    

    end if

   end for

  

  end while