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.     

Searching for biological rhythms: peak detection in the periodogram of unequally spaced data

The classical power spectrum, computed in the frequency domain, outranks traditionally used periodograms derived in the time domain (such as the chi2 periodogram) regarding the search for biological rhythms. Unfortunately, classical power spectral analysis is not possible with unequally spaced data (e.g., time series with missing data). The Lomb-Scargle periodogram fixes this shortcoming. However, peak detection in the Lomb-Scargle periodogram of unequally spaced data requires some careful consideration. To guide researchers in the proper evaluation of detected peaks, therefore, a novel procedure and a computer program have recently become available. It is recommended that the Lomb-Scargle periodogram be the default method of periodogram analysis in future biomedical applications of rhythm investigation.     

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.     

Strengths and Limitations of Period Estimation Methods for Circadian Data

A key step in the analysis of circadian data is to make an accurate estimate of the underlying period. There are many different techniques and algorithms for determining period, all with different assumptions and with differing levels of complexity. Choosing which algorithm, which implementation and which measures of accuracy to use can offer many pitfalls, especially for the non-expert. We have developed the BioDare system, an online service allowing data-sharing (including public dissemination), data-processing and analysis. Circadian experiments are the main focus of BioDare hence performing period analysis is a major feature of the system. Six methods have been incorporated into BioDare: Enright and Lomb-Scargle periodograms, FFT-NLLS, mFourfit, MESA and Spectrum Resampling. Here we review those six techniques, explain the principles behind each algorithm and evaluate their performance. In order to quantify the methods'' accuracy, we examine the algorithms against artificial mathematical test signals and model-generated mRNA data. Our re-implementation of each method in Java allows meaningful comparisons of the computational complexity and computing time associated with each algorithm. Finally, we provide guidelines on which algorithms are most appropriate for which data types, and recommendations on experimental design to extract optimal data for analysis.     

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 chi square periodogram: its utility for analysis of circadian rhythms

It is proposed that chi-square statistic be employed in constructing periodograms for the analysis of hourly time series data obtained in studies of circadian rhythmicity. We show that even for relatively short (10 day) time series, the integral-valued chi-square periodogram can distinguish circadian-periodic from random series at a level of significance of about 0·01. In addition, we describe the effects of serial correlation and examine the resolving power of the method for two periodic components in the circadian range. We suggest how the method can be most profitably employed in the analysis of event-recorder data for detection of rhythmicity in the range 14 to 34 h., and for the estimation of period to ±0·2 h.     

A robust orthogonal algorithm for system identification and time-series analysis

We describe and illustrate methods for obtaining a parsimonious sinusoidal series representation or model of biological time-series data. The methods are also used to identify nonlinear systems with unknown structure. A key aspect is a rapid search for significant terms to include in the model for the system or the time-series. For example, the methods use fast and robust orthogonal searches for significant frequencies in the time-series, and differ from conventional Fourier series analysis in several important respects. In particular, the frequencies in our resulting sinusoidal series need not be commensurate, nor integral multiples of the fundamental frequency corresponding to the record length. Freed of these restrictions, the methods produce a more economical sinusoidal series representation (than a Fourier series), finding the most significant frequencies first, and automatically determine model order. The methods are also capable of higher resolution than a conventional Fourier series analysis. In addition, the methods can cope with unequally-spaced or missing data, and are applicable to time-series corrupted by noise. Fially, we compare one of our methods with a wellknown technique for resolving sinusoidal signals in noise using published data for the test time-series.     

Spectral Analysis of Binary Time Series: Square Waves vs. Sinusoidal Functions

The analysis of signals consisting of discrete and irregular data causes methodological problems for the Fourier spectral Analysis: Since it is based on sinusoidal functions, rectangular signals with unequal periodicities cannot easily be replicated. The Walsh spectral Analysis is based on the so called "Walsh functions", a complete set of orthonormal, rectangular waves and thus seems to be the method of choice for analysing signals consisting of binary or ordinal data. The paper compares the Walsh spectral analysis and the Fourier spectral analysis on the basis of simulated and real binary data sets of various length. Simulated data were derived from signals with defined cyclic patterns that were noised by randomly generated signals of the same length. The Walsh and Fourier spectra of each set were determined and up to 25% of the periodogram coefficients were utilized as input for an inverse transform. Mean square approximation error (MSE) was calculated for each of the series in order to compare the goodness of fit between the original and the reconstructed signal. The same procedure was performed with real data derived from a behavioral observation in pigs. The comparison of the two methods revealed that, in the analysis of discrete and binary time series, Walsh spectral analysis is the more appropriate method, if the time series is rather short. If the length of the signal increases, the difference between the two methods is less substantial.     

Spectral Analysis of Binary Time Series: Square Waves vs. Sinusoidal Functions

The analysis of signals consisting of discrete and irregular data causes methodological problems for the Fourier spectral Analysis: Since it is based on sinusoidal functions, rectangular signals with unequal periodicities cannot easily be replicated. The Walsh spectral Analysis is based on the so called "Walsh functions", a complete set of orthonormal, rectangular waves and thus seems to be the method of choice for analysing signals consisting of binary or ordinal data. The paper compares the Walsh spectral analysis and the Fourier spectral analysis on the basis of simulated and real binary data sets of various length. Simulated data were derived from signals with defined cyclic patterns that were noised by randomly generated signals of the same length. The Walsh and Fourier spectra of each set were determined and up to 25% of the periodogram coefficients were utilized as input for an inverse transform. Mean square approximation error (MSE) was calculated for each of the series in order to compare the goodness of fit between the original and the reconstructed signal. The same procedure was performed with real data derived from a behavioral observation in pigs. The comparison of the two methods revealed that, in the analysis of discrete and binary time series, Walsh spectral analysis is the more appropriate method, if the time series is rather short. If the length of the signal increases, the difference between the two methods is less substantial.     

Understanding cyclical thrombocytopenia: a mathematical modeling approach

Cyclical thrombocytopenia (CT) is a rare hematological disease characterized by periodic oscillations in the platelet count. Although first reported in 1936, the pathogenesis and an effective therapy remain to be identified. Since besides fluctuations in platelet levels the patients hematological profile have been consistently normal, a destabilization of a peripheral control mechanism might play an important role in the genesis of this disorder. In this paper, we investigate through computer simulations the mechanisms underlying the platelet oscillations observed in CT. First, we collected the data published in the last 40 years and quantified the significance of the platelet fluctuations using Lomb-Scargle periodograms. Our analysis reveals that the incidence of the statistically significant periodic data is equally distributed in men and women. The mathematical model proposed in this paper captures the essential features of hematopoiesis and successfully duplicates the characteristics of CT. With the same parameter changes, the model is able to fit the platelet counts and to qualitatively reproduce the TPO oscillations (when data is available). Our results indicate that a variation in the megakaryocyte maturity, a slower relative growth rate of megakaryocytes, as well as an increased random destruction of platelets are the critical elements generating the platelet oscillations in CT.     

Ciclograma: a tool for detection of rhythmicities in sleep/wake cycles

The Fourier spectral analysis of binary time series (or rectangular signals) causes methodological problems, due to the fact that it is based on sinusoidal functions. We propose a new tool for the detection of periodicities in binary time series, focusing on sleep/wake cycles. This methodology is based on a weighted histogram of cycle durations. In this paper, we compare our methodology with the Fourier spectral analysis on the basis of simulated and real binary data sets of various lengths. We also provide an approach to statistical validation of the periodicities determined with our methodology. Furthermore, we analyze the discriminating power of both methods in terms of standard deviation. Our results indicate that the Ciclograma is much more powerful than Fourier analysis when applied on this type of time series.     

CICLOGRAMA: A TOOL FOR DETECTION OF RHYTHMICITIES IN SLEEP/WAKE CYCLES

The Fourier spectral analysis of binary time series (or rectangular signals) causes methodological problems, due to the fact that it is based on sinusoidal functions. We propose a new tool for the detection of periodicities in binary time series, focusing on sleep/wake cycles. This methodology is based on a weighted histogram of cycle durations. In this paper, we compare our methodology with the Fourier spectral analysis on the basis of simulated and real binary data sets of various lengths. We also provide an approach to statistical validation of the periodicities determined with our methodology. Furthermore, we analyze the discriminating power of both methods in terms of standard deviation. Our results indicate that the Ciclograma is much more powerful than Fourier analysis when applied on this type of time series.     

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.     

山东省典型地表覆被NDVI时间序列谐波分析

NDVI时间序列的谐波分析可描述不同地表覆被的季节变化.本文运用谐波分析法对山东省一年的MDOIS NDVI时间序列数据进行分析处理,提取了该地区几种典型地表覆被的谐波特征.结果表明:利用谐波分析生成的前几个谐波分量就可重建原始时间序列,且重建的时间序列剖面为平滑曲线,因此谐波分析不仅可减少数据量,并可去除噪声的影响;...     

Detecting Periodic Genes from Irregularly Sampled Gene Expressions: A Comparison Study

Time series microarray measurements of gene expressions have been exploited to discover genes involved in cell cycles. Due to experimental constraints, most microarray observations are obtained through irregular sampling. In this paper three popular spectral analysis schemes, namely, Lomb-Scargle, Capon and missing-data amplitude and phase estimation (MAPES), are compared in terms of their ability and efficiency to recover periodically expressed genes. Based on in silico experiments for microarray measurements of Saccharomyces cerevisiae, Lomb-Scargle is found to be the most efficacious scheme. 149 genes are then identified to be periodically expressed in the Drosophila melanogaster data set.     

Semiparametric models and inference for biomedical time series with extra-variation

Biomedical trials often give rise to data having the form of time series of a common process on separate individuals. One model which has been proposed to explain variations in such series across individuals is a random effects model based on sample periodograms. The use of spectral coefficients enables models for individual series to be constructed on the basis of standard asymptotic theory, whilst variations between individuals are handled by permitting a random effect perturbation of model coefficients. This paper extends such methodology in two ways: first, by enabling a nonparametric specification of underlying spectral behaviour; second, by addressing some of the tricky computational issues which are encountered when working with this class of random effect models. This leads to a model in which a population spectrum is specified nonparametrically through a dynamic system, and the processes measured on individuals within the population are assumed to have a spectrum which has a random effect perturbation from the population norm. Simulation studies show that standard MCMC algorithms give effective inferences for this model, and applications to biomedical data suggest that the model itself is capable of revealing scientifically important structure in temporal characteristics both within and between individual processes.     

New method for simultaneous measurement of frequency spectra of complex dielectric constants of polyelectrolyte solutions

A high-speed and high-accuracy measurement of relaxational frequency spectra of complex dielectric constants of polyelectrolyte solutions with a high conductivity was realized by a new digital signal processing technique. In this method, a sum of sinusoidal waves of geometrical series of frequencies is utilized as a multifrequency excitation signal and demodulation of the resulting response is carried out simply by addition and subtraction of digital signals in a minicomputer. This new technique is superior to the conventional cross correlation method using the fast Fourier transform in that it greatly reduces the processing time and avoids effectively the influence of a quantization error. The result for a DNA solution obtained by this method is presented to demonstrate the utility of this method.     

A Hierarchical Bayesian Model to Quantify Uncertainty of Stream Water Temperature Forecasts

Providing generic and cost effective modelling approaches to reconstruct and forecast freshwater temperature using predictors as air temperature and water discharge is a prerequisite to understanding ecological processes underlying the impact of water temperature and of global warming on continental aquatic ecosystems. Using air temperature as a simple linear predictor of water temperature can lead to significant bias in forecasts as it does not disentangle seasonality and long term trends in the signal. Here, we develop an alternative approach based on hierarchical Bayesian statistical time series modelling of water temperature, air temperature and water discharge using seasonal sinusoidal periodic signals and time varying means and amplitudes. Fitting and forecasting performances of this approach are compared with that of simple linear regression between water and air temperatures using i) an emotive simulated example, ii) application to three French coastal streams with contrasting bio-geographical conditions and sizes. The time series modelling approach better fit data and does not exhibit forecasting bias in long term trends contrary to the linear regression. This new model also allows for more accurate forecasts of water temperature than linear regression together with a fair assessment of the uncertainty around forecasting. Warming of water temperature forecast by our hierarchical Bayesian model was slower and more uncertain than that expected with the classical regression approach. These new forecasts are in a form that is readily usable in further ecological analyses and will allow weighting of outcomes from different scenarios to manage climate change impacts on freshwater wildlife.     

An eleven-year cycle in human births

The prominent endogenous cycle of the sun, with a period of approximately 11 years, is correlated with human conceptions. Time series methods (periodograms and periodic regression analyses) established that an approximate 11-year period exists in the data on births in the 20th century (1909–1985) in the United States and in New Zealand. Statistical comparisons indicated a reliable and direct relationship of conceptions with the approximately 11-year sunspot cycle. These findings were discussed in terms of the possible mediating role of geomagnetic disturbances and other factors that have been suggested in the literature to mediate human conception.     

Long-Term Prediction of the Arctic Ionospheric TEC Based on Time-Varying Periodograms

Knowledge of the polar ionospheric total electron content (TEC) and its future variations is of scientific and engineering relevance. In this study, a new method is developed to predict Arctic mean TEC on the scale of a solar cycle using previous data covering 14 years. The Arctic TEC is derived from global positioning system measurements using the spherical cap harmonic analysis mapping method. The study indicates that the variability of the Arctic TEC results in highly time-varying periodograms, which are utilized for prediction in the proposed method. The TEC time series is divided into two components of periodic oscillations and the average TEC. The newly developed method of TEC prediction is based on an extrapolation method that requires no input of physical observations of the time interval of prediction, and it is performed in both temporally backward and forward directions by summing the extrapolation of the two components. The backward prediction indicates that the Arctic TEC variability includes a 9 years period for the study duration, in addition to the well-established periods. The long-term prediction has an uncertainty of 4.8–5.6 TECU for different period sets.