Abstract
Motivated by applications in physical and biological sciences, we developed a Frequency Modulated Möbius (FMM) model to describe rhythmic patterns in oscillatory systems. Unlike standard symmetric sinusoidal models, FMM is a flexible parametric model that allows deformations to sinusoidal shape to accommodate commonly seen asymmetries in applications. FMM model parameters are easy to estimate and the model is easy to interpret complex rhythmic data. We illustrate FMM model in three disparate applications, namely, circadian clock gene expression, corticoptropin levels in depressed patients and the temporal light intensity patterns of distant stars. In each case, FMM model is demonstrated to be flexible, scientifically plausible and easy to interpret. Analysis of synthetic data derived from patterns of real data, suggest that FMM model fits the data very well both visually as well as in terms of the goodness of fit measure total mean squared error. An R language based software for implementing FMM model is available.
Introduction
Periodic data arise in a variety of contexts, such as the circadian clock, cellcycle, hormone levels, astrophysics, although the scientific question of interest varies according to the application. In the case of gene expression studies involving cellcycle or circadian clock (chronobiology), researchers are typically interested in identifying genes with rhythmic patterns, as those shown in panels (a) and (b) of Fig. 1, and various statistical parameters associated with them, whereas astrophysicists are often interested in classification of stars using temporal patterns of light emitted from them (panels (c) and (d) in Fig. 1). Unlike the typical daily stock market or weather patterns data, these data are generally less dense. Secondly, the parameters and questions of interest in time series analysis are generally different from the parameters and questions asked in oscillatory data considered in this paper, such as cellcycle, circadian clock etc.
There are two classes of methods in the literature. One class of methods describes shapes using mathematical inequalities, called order restrictions^{1}. A strength of these methods is that they are very flexible because they do not rely on a mathematical function to describe shape^{1}. For example^{1} characterized the updownup (or downupdown) patterns using order restrictions around a unit circle called the circular signals. They demonstrated that these order restrictionsbased methods describe rhythmic patterns better than the existing methods. However, a weakness of these order restrictionsbased methods is that they are not designed to estimates important parameters of rhythmic patterns.
The second class of methods, the focus of this paper, are based on a mathematical functions such as variations of cosine function. A commonly used model is the Cosinor model (COS) ^{2}. Whenever they fit the data well, these methods are useful for describing various characteristics of rhythmic patterns.
The COS model is characterized by a phase and amplitude within a period and is member of a family of models called monocomponent models. Following^{3}, a rhythmic signal can be described by a sum of monocomponent models, each defined by a phase and an amplitude. A wide range of representations of rhythmic signals exist in the literature. Representations vary in the number of monocomponents and whether the amplitude and/or phase are considered to be fixed or variable. In particular, representations with constant amplitude and variable phase are known as Frequency Modulated (FM) representations. For a review on this subject one may refer to^{3,4,5,6}.
One of the popular and widely used representations is the Fourier Decomposition (FD) which is a multicomponent representation where each component has a fixed time amplitude. The COS model is a special form of FD with only one component and it is an appropriate model if the expected functional response is sinusoidal within a period. If a gene displays two peaks (or troughs) within a single period, such as a Quasicyclical pattern^{7}, then the twocomponent FD model, denoted by FD^{2}, is potentially a useful model. However, a problem with FD^{2} model is that it can potentially introduce two peaks (or troughs) even though scientifically only one peak within a period is justified.
Astrophysicists conduct temporal studies to investigate the properties of light patterns emitted by distant stars^{8} and classify them into groups. In some cases, temporal patterns of the observed light intensities from these stars display asymmetrical shapes (panels (c) and (d) in Fig. 1) that cannot be captured by COS or even two component models such as FD^{2}. Since the family of FD models is rigid, researchers use a large number of components, as many as 10 or even 15 in some cases, to capture the shape of light patterns. Despite using that many components, the asymmetries in the data cannot always be captured by FD models. Furthermore, with increase in the number of components one may lower the bias but increase variability, resulting in overfitting issues. On the other hand, fewer components may result in over smoothed curves with large bias but low variability. Observed light patterns from each star suggest one or at most two oscillations within a period. Thus, higher order FD models may not be ideal to describe these temporal data. Moreover, other widely used methods to analyse oscillatory signals such as JTK_Cycle^{9} and RAIN^{10} are nonparametric and do not help to describe the underlying physical phenomenon properly.
Motivated by these limitations of the existing methodologies and urgent need for flexible, scientifically interpretable, parametric models for rhythmic data, in this article we introduce a novel model called Frequency Modulated Möbius (FMM) model.
Suppose X(t_{i}), t_{1} < t_{2} < … < t_{n}, are real valued time course observations. We model these data using a Möbius phase, as follows.
Definition 1. FMM model.
 1.
\(M\in \Re ,\,A\in {\Re }^{+}\)
 2.
\(\varphi (t)=\beta +2\,\arctan (\omega \tan (\frac{t\alpha }{2}));\,\alpha ,\beta ,\in [0,2\pi ],\omega \in [0,1]\)
 3.
\((e({t}_{1}),\ldots ,e({t}_{n}))^{\prime} \sim {N}_{n}\mathrm{(0,}\,{\sigma }^{2}I\mathrm{)}.\)
Methodological details that justify the mathematical formulation of this model are included in Subsection 1.1 of the Supplementary Material. Note that rather than using the linear link function for the phase angle ϕ, as done in COS model, we use the Möbius link proposed in^{11,12} which allows for asymmetric shapes as seen in the examples provided in this paper. In particular, Proposition 1 in Subsection 3.1 demonstrates that, with the above choice of the link function, FMM is suitable for describing rhythmic updownup (or downupdown) patterns. The five parameters of the FMM model characterize various aspects of a rhythmic pattern. M and A are intercept and scale parameters measuring the baseline level and the amplitude of the signal, respectively. α is a phase translation parameter while β and ω are parameters describing the shape. Specifically, an extreme spiked signal corresponds to the case ω = 0 and a sinusoidal curve to ω = 1, thus to the COS model, and in that case φ = β − α is the well known acrophase. Subsection 1.2 in the Supplementary Material includes figures illustrating the deformations from sinusoidal to spiked shapes in terms of the parameters and a detailed discussion about the parameters, respectively.
Other important parameters that are of practical use are peak and trough times, denoted by t_{U} and t_{L}, respectively. They are derived from FMM as follows:
and the values of the signal at these points are derived as:
It is important to note that FMM is a nonlinear parametric regression model. Asymptotic properties of estimators of parameters of nonlinear models, such as asymptotic unbiasedness and consistency are wellknown in the literature^{13}. Thus, asymptotic likelihood ratio tests and confidence intervals (CI) for individual parameters can be derived using standard asymptotic statistical methods^{13}.
Results
We illustrate and discuss the performance of FMM model using real and synthetic temporal data. For real data, we used publicly available data on (i) circadian clock gene expression, (ii) corticoptropin hormonal measurements in clinically depressed patients, and (iii) light intensities from variable stars. Specifically, gene expression data were originally recorded along two periods and then they were averaged. Hormonal and star data were directly given along a unique period but corresponding to averaged values too. In each case, we compared FMM with FD based methodologies. In particular, we focus on COS and FD^{2}.
To further validate our findings, we generated synthetic data using parameters derived from the above real data. Due to space limitations, results of simulation study are relegated to Subsection 3.3 in the Supplementary Material. Results therein reinforce our findings of this section that FMM is indeed more flexible and a better fitting model than the existing models.
Circadian gene expression patterns
Several researchers have studied the twoperiod circadian clock gene expression data obtained from invivo experiments on mouse liver and pituitary gland, and invitro experiment data obtained from NIH3T3 (a mouse cellline) and U2OS (a human cellline). All four data are available from the NCBI GEO website (http://www.ncbi.nlm.nih.gov/geo/). These are very comprehensive data which are useful for evaluating the performance of a model fitting strategy.
We denote the average mean squared error (mse) of each data set by Mmse and its standard deviation by SDmse. Details regarding these performance measures are provided in Subsection 3.3. Since FMM is by design a single peak (trough) model with a more flexible shape than COS, we expect FMM to perform the best followed by COS model. As seen in Table 1, in all four data sets, FMM has the smallest Mmse compared to COS and FD^{2} models. In some cases the reduction in Mmse of FMM relative to COS was dramatic. We notice a 33% reduction in the case of U2OS cellline data and a 41% reduction in the case of mouse liver data.
The above dramatic performance of FMM relative to COS function is graphically illustrated in Fig. 2 for a sample of rhythmic circadian genes. In each case, not only FMM fits the data better than COS, but more importantly, the times to peak gene expression estimated by the two methods are dramatically different, the difference ranging from 4 to 7 hours approximately (see Fig. 2(a), (c)). In their seminal work^{14} noted that phases of circadian clock genes play a key role in drug delivery to patients, and that it is critical to estimate the phases of circadian clock genes as accurately as possible. In view of^{14}, an error in the range of 4 to 7 hours could potentially have important clinical and pharmacological effects. From the figures displayed, it is clear that FMM provides a better description of these genes. The performance of FMM was equally surprising in the case of quasicyclical shaped pattern (patterns with more than one local maximum or minimum within each period), by design, FD^{2} is expected to have the smallest Mmse. However, surprisingly, FMM was very competitive with FD^{2} in terms of Mmse. Apart from the mouse liver data, in all other cases FMM had smaller Mmse than FD^{2}. Again, we provided plots of a subset of genes in Fig. 3. As we see, FD^{2} imposes two peaks by virtue of its functional form when clearly the data does not display two peaks. Secondly, these peaks are not biologically interpretable. On the other hand FMM seems to fit the data better with a single peak.
Thus, these illustrations exemplify the performance of FMM to describe temporal patterns of circadian clock genes. Subsection 3.1 of the Supplementary Material includes more details about the distribution of the estimated values of α, β and ω.
Temporal patterns of corticoptropin levels in clinically depressed patients
In this section we illustrate the performance of FMM for modeling hourly corticoptropin levels during a day in patients suffering from major clinical depression. We used data from^{15} which consisted of 3 groups of subjects where 11 were patients with psychotic major depression (Pmd), 38 were patients with nonpsychotic major depression (Npmd), and 33 were healthy controls. From the fitted curves in Fig. 4 it is apparent that COS model does not fit the data as well as FMM. It is also apparent that FD^{2} performs nearly as well as FMM in the case of Pmd and control groups but does not fit as well as FMM in the case of Npmd group. Furthermore, among the three models, FMM is the best fitting model because it has the smallest mse in all three patient groups (Table 2). We also estimated two important parameters relevant for this hormonal study, namely, t_{U}: the peak time and Z_{U}: the mean hormone level corresponding to the peak time. These estimates are provided in Table 2 and confidence intervals for pairwise differences between groups are in Table 3.
Consistent with the plots in the Fig. 4, we notice that t_{U} values obtained from FMM are smaller than those of the other two methods.
Besides, 90% CI for pairwise differences in Table 3 derived using FMM show significantly different Z_{U} values between the three groups, which is not detected with the other approaches. This is a clinically relevant finding because it suggests that there are differences in the mean peak hormone levels among the three groups with control group having the smallest peak followed by nonpsychotic major depression group and psychotic major depression groups. The psychotic major depression group has the largest peak. Thus FMM model allows us to discover a trend in the peak levels of corticotropin with the disease severity.
Temporal patterns of light emitted by stars
Light intensities of stars from six different star groups ^{8}, namely, RR Lyraes (RRab and RRc), Cepheids [Fundamental, (FU) and Overtone (FO)], Mira, and Eclipsing Binary (EB), are investigated in this section. The data consisted of 17,606 variable stars with 100 time points on each. FMM is more flexible fitting a wide range of patterns seen in the six groups of stars. On the other hand FD based methods, such as COS and FD^{2}, fit well only when the data are approximately symmetric sinusoidal in shape.
Temporal plots of a sample of typical curves from each of these groups are provided in Fig. 5. A representative from the RRc group is not shown because RRc patterns are similar to those of FO group; two different representative patterns from EB are provided instead. We overlaid on each figure the fitted curves obtained from FMM FD^{2} and COS methodologies along with respective mse values. Except for one of the EB subclasses (panel (f)) where FD^{2} performs best, in all other cases, FMM displays great flexibility to fit the data. Moreover, as seen in panels (a), (b), (d) and (e) of Fig. 5, the COS and FD^{2} perform poorly to fit asymmetric patterns.
The estimated Mmse values for the three models are summarized in Table 4 for each star group. In almost all cases, FMM has the smallest estimated Mmse, suggesting that it fits the data best for almost all groups. The only slight exception is the star group EB, but even there, the Mmse for FMM is very slightly larger than that of FD^{2}, 0.021 versus 0.020, a difference of 0.001. In comparison to FD^{2} and COS, the performance of FMM is best in the cases of RRab and FU.
In addition to fitting models, researchers are typically interested in classifying stars into various groups. PCA (Principal Component Analysis) and FD have been the two most popular approaches until now^{8,16}.
We compared the performance of FMM, FD^{2} and PCA in classifying samples using standard canonical discriminant analysis with two variables from each model and classification errors estimated using leave one out crossvalidation. The variables used for discrimination were the first two principal components, PC1, PC2 from PCA; the two parameters with the highest discriminative power, those associated with the first component, denoted by A_{1} and B_{1}, from FD; and ω and A from FMM. The scatterplots for the three pairs of variables are shown in Fig. 6 where it is shown that A1, B1 and PC1, PC2 clearly separate EB from the rest, but they are not very successful in separating the remaining groups. On the other hand, from panel (b) in Fig. 6, it is very clear that the FMM model based parameters perform well in separating all groups of stars. In particular, the shape parameter ω plays a critical role in discriminating all groups of stars. In fact, smaller misclassification rates are obtained when FMM variables are used, as it is shown in Table S2 of the Supplementary Material.
Subsection 3.2 of the Supplementary Material provides graphical displays and comments on the distribution of the estimated values of α and β for the star set by groups.
Methods
We begin with some definitions and notations. In the following we assume t ∈ [0, 2π]; if observed times takes values on a real interval then t′ ∈ [t_{0}, T + t_{0}], \(t=\frac{(t^{\prime} {t}_{0})2\pi }{T}\), t ∈ [0, 2π].
Circular signal and FMM
It is generally accepted that for a given oscillatory phenomenon, there exists an underlying complex valued signal. Even more^{5}, among others, argues that a physical phenomenon is not entirely modelled unless the complex signal it is related to, has been defined. In this paper we deal with periodic signals, which are described as complex functions of time, which we denote as S(t), t ∈ [0, 2π].
Definition 2. A complexvalued signal S(t)
From the complex formulation, a model for a real signal is derived as:
The latter term in Eq. (1) is known as the quadrature form of the signal S(t), where ρ(t) and ϕ(t) are the signal’s amplitude and phase respectively. The derivative of ϕ(t) is known as Instantaneous Frequency (IF), which is expected to be nonnegative in most applications^{5}.
When ν(t) is unknown, there are infinite pairs ρ(t), ϕ(t) for which μ(t) may be equivalently described. An important subclass, is that of analytic signals. In particular, one of the elemental is the Fourier atom which is defined using the Möbius transform. Besides, analytic signals having a nonnegative IF and constant amplitude are often used by researchers in applications due to their interpretability and simplicity. Specifically, the real signal corresponding to these latter signals is a monocomponent. Definitions of these signals and additional theoretical details are given in Subsection 1 of the Supplementary Material.
We now introduce circular signals as follows:
Definition 3. Circular signal in the Euclidean space
μ(t) ∈ ℝ, t ∈ [0, 2π] is circular iff ∃ t_{U}, t_{L} such that
if t_{U} ≤ t_{L}: μ(t) ≥ μ(t′), t_{U} ≤ t ≤ t′ ≤ t_{L}, and μ(t) ≤ μ(t′), 0 ≤ t ≤ t′ ≤ t_{U}; t_{L} ≤ t ≤ t′ ≤ 2π.
or equivalently
if t_{U} ≥ t_{L}: : μ(t) ≤ μ(t′), t_{L} ≤ t ≤ t′ ≤ t_{U}, and μ(t) ≥ μ(t′), 0 ≤ t ≤ t′ ≤ t_{L}; t_{U} ≤ t ≤ t′ ≤ 2π.
Without loss of generality, we assume that t_{U} ≤ t_{L}. In the Euclidean space, such a signal is also called an updownup signal (resp. downupdown) ^{1}, as it monotonically increases (resp. decreases) to t_{U} (resp. t_{L}) and then decreases (resp. increases) to t_{L} (resp. t_{U}) before increasing (resp. decreasing) again. As illustrated in Subsection 2.2, t_{U} is an important parameter in applications because it is the time to first peak.
In addition, a circular signal on the unit circle is a signal that follows the circular order (see^{17} for a definition on circular order),
Definition 4. Circular signal on the unit circle
ϕ(t) ∈ [0, 2π], t ∈ [0, 2π] is circular iff ϕ(t) ≤ ϕ(t′), 0 ≤ t ≤ t′ ≤ 2π (resp. ϕ(t) ≥ ϕ(t′), 0 ≤ t ≤ t′ ≤ 2π)
The most popular circular signal, and also the simplest one, is the sinusoidal signal: μ(t) = cos(t + φ). Its corresponding circular signal is ϕ(t) = t + φ.
It is straight forward to derive that, if \({e}^{i{\varphi }_{a}(t)}\) is a Fourier atom, \(Re({e}^{i{\varphi }_{a}(t)})\) is a circular signal.
Next, we provide a useful characterization of FMM to demonstrate the relationship between FMM models and Fourier atoms. In particular, Proposition 1 demonstrates that FMM is restricted to circular signals and that the IF is nonnegative. Thus, the FMM model is appropriate for describing typical periodic updownup signals
The FMM phase can be equivalently derived from:
where, \(\phi \in {\rm{\Re }}\) and \(a=r{e}^{iv}\in {\mathbb{C}}\). Then, the relationship between this formulation and FMM model (see Definition 1) is given by:
The equivalence formulation above is also stated in the seminal papers of circular regression:^{11,12}.
Proposition 1. Let μ(t) = M + Acos(ϕ(t)), \(\varphi (t)=\beta +2\,\arctan (\omega \,\tan (\frac{t\alpha }{2}))\) and t ∈ [0, 2π], then:
 1.
μ(t) is a circular signal in the Euclidean space.
 2.
ϕ(t) is a circular signal in the unit circle.
 3.
\({\varphi }^{{\rm{^{\prime} }}}(t)=\frac{\omega }{2(1+{\omega }^{2}si{n}^{2}(\frac{t\alpha }{2}))}\)
The proofs follow immediately from the definitions.
Estimation algorithm
A twostep algorithm is developed to estimate FMM parameters. First, initial parameter estimation is given by solving a least square problem along the lines of^{2}. Second, we used NelderMead optimization method ^{18} to obtain the final FMM parameter estimates, see Subsection 2 in the Supplementary Material for details. The proposed methodology is not limited by the choice of the optimization method. Based on our experiences with complicated objective functions involving angular data^{19,20}, as well as the data analyzed in this paper, we find NelderMead to provide estimates that fit data well. It tends to successfully avoid local solutions. For example, see figures presented in this paper.
Model performance measures
To assess the performance of various models, we use the total mse over all observed times as a criterion. This is a common measure of goodness of fit used by statisticians when assessing the performance of an estimator or a model and routinely discussed and used in textbooks^{21}. Smaller values suggests better fit of the model to the data. More precisely, in simulations, mse is a measure of distance of the estimated signal from the true signal μ and is given by \(mse={\sum }_{i=1}^{n}\,{({\hat{\mu }}_{i}{\mu }_{i})}^{2}/n\). In practice, however, since the true signal is unknown, the empirical value of mse is given by: \(mse={\sum }_{i=1}^{n}\,{({\hat{\mu }}_{i}{X}_{i})}^{2}/n\).
In addition, mse estimates for a specific parameter θ are denoted as mse(θ) in simulations. Finally, when mse values are averaged across different scenarios or individuals, an M is added. Thus, Mmse is the average mse over all measurements available. The standard deviation of mse is denoted by SDmse.
Time to first peak and trough in COS and FD
COS model has well defined maximum and minimum t_{U} = −φ and t_{L} = π − φ respectively. However, the computation of extremes is not trivial for FD. In fact, FD^{N} model has multiplicity of N extrema (see^{22} for details). There are no close form expressions for t_{U} and t_{L} and they are numerically derived as the values where μ(t) reaches its maximum and minimum, see Section 2 for details.
Discussion
As seen in this paper, oscillatory systems arise naturally in a wide range of applications including biology, medicine, pharmacology, astronomy and so on. An oscillatory system consists of several components that display rhythmic temporal patterns. The temporal patterns and the associated parameters, such as the amplitude and phase, have critical scientific importance and implications. For example, as demonstrated in^{14} the efficacy of a drug in treating a patient may depend upon the time of the day the drug is delivered, and this determination is made based on the phases of some circadian clock genes. Thus, in all such applications it is not only important to determine all components (e.g. genes) that display a temporal rhythmic pattern, but it is critically important to derive an appropriate parametric model and estimate the associated parameters correctly. A poor choice of the model may result in wrong estimates of phase and amplitude that may have important downstream implications. For example, as we saw in the circadian clock genes example discussed in this paper, a poor model may result in a 4 to 7 hours difference in the phase estimate relative to what might be the true phase. In view of^{14} findings this might have major clinical and pharmacological impact on when patient receives a drug.
A common parametric model used in almost all applications to fit a temporal rhythmic data is the cosinor model (COS). While it is easy to fit and interpret, it is a very rigid model in the sense that the observed temporal signals are required to have a sinusoidal shape, which is intrinsically symmetric. As we saw in the examples presented in this paper, the temporal patterns of rhythmic signals do not always follow this rigid structure. In fact, as observed in^{11}, it is common to have a nonlinear relationship or link between an angular parameter and time. Although, Fourier decomposition (FD) was developed in the literature to provide some flexibility from COS, intrinsically it too has a symmetric shape. Secondly, because it is a linear combination of several sinusoidal functions, it may induce multiple peaks (or troughs) within a period. In many applications, especially in the circadian clock or cellcycle, those multiple peaks are hard to interpret.
The primary contribution of this paper is to derive a flexible parametric model that allows deformations to the sinusoidal shape and contains easy to interpret parameters. As demonstrated in this paper, the model performs extremely well in a very disparate types of applications. The model fits circadian clock data, hormonal data as well as light data from distant stars. The rhythmic patterns are very varied and yet in each case the model seems to outperform the existing models. Extensive simulations seem to confirm these findings. It is important to reiterate that we fill an important gap in the literature to derive a flexible parametric model for describing rhythmic patterns that are deformations to sinusoidal models.
Once an appropriate nonlinear model is derived, as noted in the paper, given decades of literature, statistical inference regarding the parameters of the nonlinear model is routine problem. In this paper we used bootstrap based methodology.
As frequently quoted by modelers, a quote attributed to George Box, “No model is perfect but some models are more useful”, the proposed basic FMM model has limitations. Firstly, we have not discussed here the problem of detecting if a component of an oscillatory system is rhythmic or not rhythmic and if it is rhythmic, then whether it is also a sinusoidal. However, as described in Subsection 1.3 of the Supplementary Material, parametric hypothesis testing problems to test the above hypotheses can be easily addressed using FMM.
Secondly, in many studies researchers are interested in fitting nonlinear models after adjusting for covariates. This is particularly true for modeling hormone data. For example, gender and age would be two important factors to consider when modeling hormonal data. The problem can be even more complex when potential interactions may be suspected. Specifically^{15}, used COS model but adjusted for important covariates, such as age and gender as additive effects in a linear model. The endocrine system for males and females is fundamentally different. This leads to differences in biological responses and hence it is reasonable to expect males and females to have curves with different shapes. A similar phenomena may occur with age. Compared to sinusoidal models such as COS, an advantage of using FMM in the above formulations is that it allows for deformations to sinusoidal shape. The current formulation of FMM requires further refinements and modifications to model interactions and covariates.
Finally, other important limitation of FMM is that it does not parametrize the period but takes it as a fixed known quantity. While in many examples the period of a cycle is determined by the experimental design, such as in a circadian clock or cellcycle experiment, there are also examples, such as the EB star data where the period may be poorly determined. However, the period can be formulated as an unknown parameter in the model, then the methodology can be suitably modified by designing a computational intensive algorithm that considers different period values and then chooses the period that results in a smaller total mse.
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Acknowledgements
The authors gratefully acknowledge the financial support received by the Spanish Ministerio de Ciencia e Innovación and European Regional Development Fund; Ministerio de Economía y Competitividad grant [MTM201571217R to CR] and Spanish Ministerio de Educación, Cultura y Deporte [FPU14/04534 to YL].
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C.R. Conceived aims, theoretical proporsal, conceptual design, data analysis, interpretation of the results, wrote and approved manuscript, proposed future applications; Y.L. Involved in the discussion about some aspect of the methodology, processed original data, generated simulations, analyzed data, interpreted the results, wrote and approved manuscript; and S.P. involved in the discussion about some aspect of the methodology, interpretated the results, wrote and approved manuscript, proposed future applications.
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Rueda, C., Larriba, Y. & Peddada, S.D. Frequency Modulated Möbius Model Accurately Predicts Rhythmic Signals in Biological and Physical Sciences. Sci Rep 9, 18701 (2019). https://doi.org/10.1038/s41598019545691
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