A Procedure of Multiple Period Searching in Unequally Spaced Time-Series with the Lomb-Scargle Method

Periodogram analysis of unequally spaced time-series, as part of many biological rhythm investigations, is complicated. The mathematical frameworkis scattered over the literature, and the interpretation of results is often debatable. In this paper, we show that the Lomb-Scargle method is the appropriate tool for periodogram analysis of unequally spaced data. A unique procedure of multiple period searching is derived, facilitating the assessment of the various rhythms that may be present in a time-series. All relevant mathematical and statistical aspects are considered in detail, and much attention is given to the correct interpretation of results. The use of the procedure is illustrated by examples, and problems that may be encountered are discussed. It is argued that, when following the procedure of multiple period searching, we can even benefit from the unequal spacing of a time-series in biological rhythm research.     

The Lomb-Scargle Periodogram in Biological Rhythm Research: Analysis of Incomplete and Unequally Spaced Time-Series

This paper investigates the utility of the Lomb-Scargle periodogram for the analysis of biological rhythms. This method is particularly suited to detect periodic components in unequally sampled time-series and data sets with missing values, but restricts all calculations to actually measured values. The Lomb-Scargle method was tested on both real and simulated time-series with even and uneven sampling, and compared to a standard method in biomedical rhythm research, the Chi-square periodogram. Results indicate that the Lomb-Scargle algorithm shows a clearly better detection efficiency and accuracy in the presence of noise, and avoids possible bias or erroneous results that may arise from replacement of missing data by interpolation techniques. Hence, the Lomb-Scargle periodogram may serve as a useful method for the study of biological rhythms, especially when applied to telemetrical or observational time-series obtained from free-living animals, i.e., data sets that notoriously lack points.     

Detecting periodic patterns in unevenly spaced gene expression time series using Lomb-Scargle periodograms

MOTIVATION: Periodic patterns in time series resulting from biological experiments are of great interest. The commonly used Fast Fourier Transform (FFT) algorithm is applicable only when data are evenly spaced and when no values are missing, which is not always the case in high-throughput measurements. The choice of statistic to evaluate the significance of the periodic patterns for unevenly spaced gene expression time series has not been well substantiated. METHODS: The Lomb-Scargle periodogram approach is used to search time series of gene expression to quantify the periodic behavior of every gene represented on the DNA array. The Lomb-Scargle periodogram analysis provides a direct method to treat missing values and unevenly spaced time points. We propose the combination of a Lomb-Scargle test statistic for periodicity and a multiple hypothesis testing procedure with controlled false discovery rate to detect significant periodic gene expression patterns. RESULTS: We analyzed the Plasmodium falciparum gene expression dataset. In the Quality Control Dataset of 5080 expression patterns, we found 4112 periodic probes. In addition, we identified 243 probes with periodic expression in the Complete Dataset, which could not be examined in the original study by the FFT analysis due to an excessive number of missing values. While most periodic genes had a period of 48 h, some had a period close to 24 h. Our approach should be applicable for detection and quantification of periodic patterns in any unevenly spaced gene expression time-series data.     

The Lomb-Scargle Periodogram in Biological Rhythm Research: Analysis of Incomplete and Unequally Spaced Time-Series

This paper investigates the utility of the Lomb–Scargle periodogram for the analysis of biological rhythms. This method is particularly suited to detect periodic components in unequally sampled time-series and data sets with missing values, but restricts all calculations to actually measured values. The Lomb-Scargle method was tested on both real and simulated time-series with even and uneven sampling, and compared to a standard method in biomedical rhythm research, the Chi-square periodogram. Results indicate that the Lomb–Scargle algorithm shows a clearly better detection efficiency and accuracy in the presence of noise, and avoids possible bias or erroneous results that may arise from replacement of missing data by interpolation techniques. Hence, the Lomb–Scargle periodogram may serve as a useful method for the study of biological rhythms, especially when applied to telemetrical or observational time-series obtained from free-living animals, i.e., data sets that notoriously lack points.     

Emphasizing difficulties in the detection of rhythms with Lomb-Scargle periodograms

The Lomb-Scargle periodogram was introduced in astrophysics to detect sinusoidal signals in noisy unevenly sampled time series. It proved to be a powerful tool in time series analysis and has recently been adapted in biomedical sciences. Its use is motivated by handling non-uniform data which is a common characteristic due to the restricted and irregular observations of, for instance, free-living animals. However, the observational data often contain fractions of non-Gaussian noise or may consist of periodic signals with non-sinusoidal shapes. These properties can make more difficult the interpretation of Lomb-Scargle periodograms and can lead to misleading estimates. In this letter we illustrate these difficulties for noise-free bimodal rhythms and sinusoidal signals with outliers. The examples are aimed to emphasize limitations and to complement the recent discussion on Lomb-Scargle periodograms.     

Emphasizing Difficulties in the Detection of Rhythms with Lomb-Scargle Periodograms

The Lomb-Scargle periodogram was introduced in astrophysics to detect sinusoidal signals in noisy unevenly sampled time series. It proved to be a powerful tool in time series analysis and has recently been adapted in biomedical sciences. Its use is motivated by handling non-uniform data which is a common characteristic due to the restricted and irregular observations of, for instance, free-living animals. However, the observational data often contain fractions of non-Gaussian noise or may consist of periodic signals with non-sinusoidal shapes. These properties can make more difficult the interpretation of Lomb-Scargle periodograms and can lead to misleading estimates. In this letter we illustrate these difficulties for noise-free bimodal rhythms and sinusoidal signals with outliers. The examples are aimed to emphasize limitations and to complement the recent discussion on Lomb-Scargle periodograms.     

A Procedure of Multiple Period Searching in Unequally Spaced Time-Series with the Lomb–Scargle Method

Periodogram analysis of unequally spaced time-series, as part of many biological rhythm investigations, is complicated. The mathematical frameworkis scattered over the literature, and the interpretation of results is often debatable. In this paper, we show that the Lomb–Scargle method is the appropriate tool for periodogram analysis of unequally spaced data. A unique procedure of multiple period searching is derived, facilitating the assessment of the various rhythms that may be present in a time-series. All relevant mathematical and statistical aspects are considered in detail, and much attention is given to the correct interpretation of results. The use of the procedure is illustrated by examples, and problems that may be encountered are discussed. It is argued that, when following the procedure of multiple period searching, we can even benefit from the unequal spacing of a time-series in biological rhythm research.     

Spectral resolution of cardio-circulatory variations in men measured by autorhythmometry over 2 years

An analogue of the periodogram method for unequally spaced data is presented with a view to resolving the frequency structure of the observations. The algorithm is explicitly based on the sequential least squares procedure. In particular, the key concept is that the with-in-plot spectral analysis can be augmented by the between-plot information to make inferences about common characteristics. It is also shown how the between-plot random variations can be incorporated into the multiple harmonic regression model. A detailed spectral analysis investigates the periodic fluctuations in four cardio-circulatory variables, measured by autorhythmometric observation by eight men at rest and extending over a time span of 2 years. The spectral curves show the existence of circadian and circaseptan rhythmicities. The amplitude modulation of the dian rhythm by circaseptan variation is assimilated with the rhythmicity of work during the week. The blood-pressure variables situate their maximum annual peak in the winter period. These quasi-periodic fluctuations appear to be related to the amount of physical activity performed in time by the subjects.     

Detection of Cyclic Trends in Incidence Data Using the Efficiency Score Method

In studies involving a cyclic regularity, researchers usually have a good working knowledge regarding the peak time in the cycle. Capitalizing on this information, we derive the asymptotically uniformly most powerful unbiased test for detecting a cyclic trend using the likelihood score, and present the asymptotic power function of the test and the approximate formula for sample size. Numerical studies demonstrate great advantages of the proposed test over the standard test in terms of power and sample size. Asymptotic power of the score test is satisfactorily close to actual power. We also generalize this method so that it is applicable for incidence data from unequally spaced intervals or risk populations of unequal size.     

Continuous time Markov models for binary longitudinal data

Longitudinal data usually consist of a number of short time series. A group of subjects or groups of subjects are followed over time and observations are often taken at unequally spaced time points, and may be at different times for different subjects. When the errors and random effects are Gaussian, the likelihood of these unbalanced linear mixed models can be directly calculated, and nonlinear optimization used to obtain maximum likelihood estimates of the fixed regression coefficients and parameters in the variance components. For binary longitudinal data, a two state, non-homogeneous continuous time Markov process approach is used to model serial correlation within subjects. Formulating the model as a continuous time Markov process allows the observations to be equally or unequally spaced. Fixed and time varying covariates can be included in the model, and the continuous time model allows the estimation of the odds ratio for an exposure variable based on the steady state distribution. Exact likelihoods can be calculated. The initial probability distribution on the first observation on each subject is estimated using logistic regression that can involve covariates, and this estimation is embedded in the overall estimation. These models are applied to an intervention study designed to reduce children's sun exposure.     

Analysis of longitudinal count data with serial correlation

We propose a state space model for analyzing equally or unequally spaced longitudinal count data with serial correlation. With a log link function, the mean of the Poisson response variable is a nonlinear function of the fixed and random effects. The random effects are assumed to be generated from a Gaussian first order autoregression (AR(1)). In this case, the mean of the observations has a log normal distribution. We use a combination of linear and nonlinear methods to take advantage of the Gaussian process embedded in a nonlinear function. The state space model uses a modified Kalman filter recursion to estimate the mean and variance of the AR(1) random error given the previous observations. The marginal likelihood is approximated by numerically integrating out the AR(1) random error. Simulation studies with different sets of parameters show that the state space model performs well. The model is applied to Epileptic Seizure data and Primary Care Visits Data. Missing and unequally spaced observations are handled naturally with this model.     

Tapered empirical likelihood for time series data in time and frequency domains

We investigate data tapering in two formulations of empiricallikelihood for time series. One empirical likelihood is formedfrom tapered data blocks in the time domain and a second isbased on the tapered periodogram in the frequency domain. Limitingdistributions are provided for both empirical likelihood versionsunder tapering. Theoretical and simulation evidence indicatesthat a data taper improves the coverage accuracy of empiricallikelihood confidence intervals for time series parameters,such as means and correlations.     

Predictive functional ANOVA models for longitudinal analysis of mandibular shape changes

In this paper, we introduce a Bayesian statistical model for the analysis of functional data observed at several time points. Examples of such data include the Michigan growth study where we wish to characterize the shape changes of human mandible profiles. The form of the mandible is often used by clinicians as an aid in predicting the mandibular growth. However, whereas many studies have demonstrated the changes in size that may occur during the period of pubertal growth spurt, shape changes have been less well investigated. Considering a group of subjects presenting normal occlusion, in this paper we thus describe a Bayesian functional ANOVA model that provides information about where and when the shape changes of the mandible occur during different stages of development. The model is developed by defining the notion of predictive process models for Gaussian process (GP) distributions used as priors over the random functional effects. We show that the predictive approach is computationally appealing and that it is useful to analyze multivariate functional data with unequally spaced observations that differ among subjects and times. Graphical posterior summaries show that our model is able to provide a biological interpretation of the morphometric findings and that they comprehensively describe the shape changes of the human mandible profiles. Compared with classical cephalometric analysis, this paper represents a significant methodological advance for the study of mandibular shape changes in two dimensions.     

The risks of using the chi-square periodogram to estimate the period of biological rhythms

The chi-square periodogram (CSP), developed over 40 years ago, continues to be one of the most popular methods to estimate the period of circadian (circa 24-h) rhythms. Previous work has indicated the CSP is sometimes less accurate than other methods, but understanding of why and under what conditions remains incomplete. Using simulated rhythmic time-courses, we found that the CSP is prone to underestimating the period in a manner that depends on the true period and the length of the time-course. This underestimation bias is most severe in short time-courses (e.g., 3 days), but is also visible in longer simulated time-courses (e.g., 12 days) and in experimental time-courses of mouse wheel-running and ex vivo bioluminescence. We traced the source of the bias to discontinuities in the periodogram that are related to the number of time-points the CSP uses to calculate the observed variance for a given test period. By revising the calculation to avoid discontinuities, we developed a new version, the greedy CSP, that shows reduced bias and improved accuracy. Nonetheless, even the greedy CSP tended to be less accurate on our simulated time-courses than an alternative method, namely the Lomb-Scargle periodogram. Thus, although our study describes a major improvement to a classic method, it also suggests that users should generally avoid the CSP when estimating the period of biological rhythms.     

The issue of significant features in random noise

With respect to the first example in Schimmel (2001), Van Dongen et al. (2001) conclude from their Lomb-Scargle analysis that the noise I used 'contains new periodicities that are added to the signal (these periodicities by themselves resemble a harmonic series of a 38-hour rhythm).' They infer that 'the variance of the added noise is about five times as large as the variance of the signal' causing the detection of the new significant periodicities in the noise prior to the 24-h bimodal rhythm. Moreover the 'example reflects a combination of an extremely non-sinusoidal signal with noise that is not independent, which results in a time series that is difficult to analyze with virtually any known method.' In the following, I briefly examine these concerns to avoid misunderstandings and to alert that with an inadequate use of the statistical significance test, misleading conclusions can be obtained. Although this paper further emphasizes difficulties in the detection with Lomb-Scargle periodograms, this should not be used as de-motivation. As stated in Schimmel (2001) Lomb-Scargle is a powerful technique but such as any other method one should be aware about its limitations, and use additional tools to constrain the true data characteristics.     

Unequally spaced longitudinal data with AR(1) serial correlation

This paper discusses longitudinal data analysis when each subject is observed at different unequally spaced time points. Observations within subjects are assumed to be either uncorrelated or to have a continuous-time first-order autoregressive structure, possibly with observation error. The random coefficients are assumed to have an arbitrary between-subject covariance matrix. Covariates can be included in the fixed effects part of the model. Exact maximum likelihood estimates of the unknown parameters are computed using the Kalman filter to evaluate the likelihood, which is then maximized with a nonlinear optimization program. An example is presented where a large number of subjects are each observed at a small number of observation times. Hypothesis tests for selecting the best model are carried out using Wald's test on contrasts or likelihood ratio tests based on fitting full and restricted models.     

Planktonic versus benthic foraminifera response to Milankovitch forcing (Late Jurassic,Betic Cordillera): testing methods for cyclostratigraphic analysis

Foraminiferal assemblages from the Bifurcatus Zone (Oxfordian, Upper Jurassic) are studied in the Navalperal section (Betic Cordillera, southern Spain). A total of 24 sampling stations and around 5,700 specimens of foraminifera were recognized on thin-section analysis and classified into two major categories: planktonic and benthic. The abundance of foraminifera (number of specimens/cm²) was calculated for both categories and the cyclic pattern was studied by spectral analysis, considering the autochthonous and para-autochthonous character of the studied assemblages. The Lomb-Scargle periodogram together with a permutation test were tested for performing the high-resolution spectral analysis, being particularly well suited for working with short time series and uneven sampling. Spectral analysis reveals the influence of orbital-scale Milankovitch cyclicity at the eccentricity, obliquity, and precession bands. Moreover, this incidence is significantly different depending on the analyzed group (benthic versus planktonic). Whereas the long-range eccentricity band is not distinguishable from a trend and the short-range eccentricity band is not statistically significant (at 90% confidence level), the obliquity band is better represented in the planktonic component and the precession band is better developed in the benthic group. Variations in temperature affecting upper waters, determined by obliquity-scale fluctuations, could be responsible for changes in the planktonic foraminiferal assemblage, while changes in nutrient availability and substrate oxygenation, as a consequence of input variations from source areas at the precession-scale cycles, could affect the benthic foraminiferal assemblage.     

A time-heterogeneous D-vine copula model for unbalanced and unequally spaced longitudinal data

In many longitudinal studies, the number and timing of measurements differ across study subjects. Statistical analysis of such data requires accounting for both the unbalanced study design and the unequal spacing of repeated measurements. This paper proposes a time-heterogeneous D-vine copula model that allows for time adjustment in the dependence structure of unequally spaced and potentially unbalanced longitudinal data. The proposed approach not only offers flexibility over its time-homogeneous counterparts but also allows for parsimonious model specifications at the tree or vine level for a given D-vine structure. It further provides a robust strategy to specify the joint distribution of non-Gaussian longitudinal data. The performance of the time-heterogeneous D-vine copula models are evaluated through simulation studies and by a real data application. Our findings suggest improved predictive performance of the proposed approach over the linear mixed-effects model and time-homogeneous D-vine copula model.