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.     

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.     

A Simple Approach for the Computation of Multiple Periodicities in Biological Time Series

We have described a simple approach for the analysis and isolation of multiple periodicities from a biological time series. For the estimation of the periodicities, we used simulated data and data from ongoing experiments in our laboratory. Two time series were simulated, one which consisted of only white noise and the other consisted white noise along with periodicities of 6, 11, 17 and 23 h, to demonstrate that our method can successfully isolate multiple patterns in a time series. Our method of analysis is objective, simple, flexible and adaptive since it distinctly delineates the individual contribution from an overlap of multiple periodicities. The key features of our method are: (i) identification of a reliable phase reference point, (ii) scanning the time series using a moving window in increments, (iii) use of Siegel's modification of Fisher's method to detect significant periodicit(y)ies in the time series. The use of window sizes of increasing length to examine the time series elegantly reduces noise while identifying periodicities that are otherwise not apparent. Finally, the periodogram can be smoothed in order to normalize the contribution by attendant frequency components within the waveform. A minimum critical value for relative contribution of various frequencies was calculated to delineate the periodicities that contributed significantly to the time series. We executed this method of time series analysis using MS Excel and C.     

A Simple Approach for the Computation of Multiple Periodicities in Biological Time Series

We have described a simple approach for the analysis and isolation of multiple periodicities from a biological time series. For the estimation of the periodicities, we used simulated data and data from ongoing experiments in our laboratory. Two time series were simulated, one which consisted of only white noise and the other consisted white noise along with periodicities of 6, 11, 17 and 23 h, to demonstrate that our method can successfully isolate multiple patterns in a time series. Our method of analysis is objective, simple, flexible and adaptive since it distinctly delineates the individual contribution from an overlap of multiple periodicities. The key features of our method are: (i) identification of a reliable phase reference point, (ii) scanning the time series using a moving window in increments, (iii) use of Siegel's modification of Fisher's method to detect significant periodicit(y)ies in the time series. The use of window sizes of increasing length to examine the time series elegantly reduces noise while identifying periodicities that are otherwise not apparent. Finally, the periodogram can be smoothed in order to normalize the contribution by attendant frequency components within the waveform. A minimum critical value for relative contribution of various frequencies was calculated to delineate the periodicities that contributed significantly to the time series. We executed this method of time series analysis using MS Excel and C.     

Infradian rhythmicity in sleep/wake ratio in developing infants

Although there are several reports on ultradian and circadian rhythms in newborns, we found only one report in which infradian periodicities are described for heart-rate measurements in the early stages of human development. Here, we report infradian rhythms in the monthly range in the sleep/wake cycle of four infants studied along 24 consecutive weeks. Our procedure was applied to sleep diary records from four healthy newborns. The data were arranged in binary time series representing sleep (-1) or wake (1) states. These time series were integrated in order to obtain the cumulative sleep/wake time. A measure of the sleep/wake ratio (SWR) was obtained by computing the average slope of the cumulative sleep/wake time. To extract periodicities we applied the Fourier periodogram to the temporal course of the SWR. We found a notorious difference in the SWR pattern among infants. In two infants the SWR showed a marked linear decay, spending more time asleep than awake, while in the two other infants oscillated near zero. We found robust oscillations in all children. In all cases the Fourier periodogram results present significant power in the infradian range. From these results, we suggest that sleep and wake durations are probably modulated by some internal stimuli.     

Infradian Rhythmicity in Sleep/Wake Ratio in Developing Infants

Although there are several reports on ultradian and circadian rhythms in newborns, we found only one report in which infradian periodicities are described for heart-rate measurements in the early stages of human development. Here, we report infradian rhythms in the monthly range in the sleep/wake cycle of four infants studied along 24 consecutive weeks. Our procedure was applied to sleep diary records from four healthy newborns. The data were arranged in binary time series representing sleep (?1) or wake (1) states. These time series were integrated in order to obtain the cumulative sleep/wake time. A measure of the sleep/wake ratio (SWR) was obtained by computing the average slope of the cumulative sleep/wake time. To extract periodicities we applied the Fourier periodogram to the temporal course of the SWR. We found a notorious difference in the SWR pattern among infants. In two infants the SWR showed a marked linear decay, spending more time asleep than awake, while in the two other infants oscillated near zero. We found robust oscillations in all children. In all cases the Fourier periodogram results present significant power in the infradian range. From these results, we suggest that sleep and wake durations are probably modulated by some internal stimuli.     

Spectral analysis of a Plio-Pleistocene multispecies time series using the Mantel periodogram

A simple method for the spectral analysis of multispecies microfossil data through time or stratigraphic level is presented. The method is based on the Mantel correlogram, allowing any ecological similarity measure to be used. The method can therefore be applied to binary (presence-absence) data as well as raw or normalized species counts. In contrast with spectral analysis of univariate ordination scores, this approach does not explicitly discard information. The method, referred to as the Mantel periodogram, is exemplified with a data set from the literature, demonstrating several astronomically forced periodicities in microfaunal data from the Plio-Pleistocene.     

Correlation between oscillations in ventilation and frequency content of the electroencephalogram.

Periodicities of ventilation are common in elderly subjects during stage 1/2 sleep. The mechanism producing these periodicities is unknown. We hypothesized that the oscillations in ventilation might be related to oscillations in sleep state. To address this hypothesis, we examined, using cross correlation, the relationship between the oscillations in ventilation and parameters (alpha power, mean frequency) derived from spectral analysis of the electroencephalogram. In wakefulness, although ventilation and mean frequency, and ventilation and alpha power, were related, there were no consistent patterns to these relationships. Both positive and negative correlations were found. Clearer relationships were found in stage 1/2 sleep. Correlation between mean frequency and ventilation was the most consistent. All correlations were positive; i.e., ventilation fell as mean frequency fell. The maximum correlation occurred at zero lag between the time series. Thus these oscillations are synchronous within the time resolution of our methodology. These data are compatible with the hypothesis that the initiation of apnea in stage 1/2 sleep is related to a reduction in the state-dependent input to the ventilatory control system.     

Sensitivity analysis for causal inference using inverse probability weighting

Evaluation of impact of potential uncontrolled confounding is an important component for causal inference based on observational studies. In this article, we introduce a general framework of sensitivity analysis that is based on inverse probability weighting. We propose a general methodology that allows both non‐parametric and parametric analyses, which are driven by two parameters that govern the magnitude of the variation of the multiplicative errors of the propensity score and their correlations with the potential outcomes. We also introduce a specific parametric model that offers a mechanistic view on how the uncontrolled confounding may bias the inference through these parameters. Our method can be readily applied to both binary and continuous outcomes and depends on the covariates only through the propensity score that can be estimated by any parametric or non‐parametric method. We illustrate our method with two medical data sets.     

A new algorithm for wavelet-based heart rate variability analysis

One of the most promising non-invasive markers of the activity of the autonomic nervous system is heart rate variability (HRV). HRV analysis toolkits often provide spectral analysis techniques using the Fourier transform, which assumes that the heart rate series is stationary. To overcome this issue, the Short Time Fourier Transform (STFT) is often used. However, the wavelet transform is thought to be a more suitable tool for analyzing non-stationary signals than the STFT. Given the lack of support for wavelet-based analysis in HRV toolkits, such analysis must be implemented by the researcher. This has made this technique underutilized.This paper presents a new algorithm to perform HRV power spectrum analysis based on the Maximal Overlap Discrete Wavelet Packet Transform (MODWPT). The algorithm calculates the power in any spectral band with a given tolerance for the band's boundaries. The MODWPT decomposition tree is pruned to avoid calculating unnecessary wavelet coefficients, thereby optimizing execution time. The center of energy shift correction is applied to achieve optimum alignment of the wavelet coefficients. This algorithm has been implemented in RHRV, an open-source package for HRV analysis. To the best of our knowledge, RHRV is the first HRV toolkit with support for wavelet-based spectral analysis.     

Re-cracking the nucleosome positioning code

Nucleosomes, the fundamental repeating subunits of all eukaryotic chromatin, are responsible for packaging DNA into chromosomes inside the cell nucleus and controlling gene expression. While it has been well established that nucleosomes exhibit higher affinity for select DNA sequences, until recently it was unclear whether such preferences exerted a significant, genome-wide effect on nucleosome positioning in vivo. This question was seemingly and recently resolved in the affirmative: a wide-ranging series of experimental and computational analyses provided extensive evidence that the instructions for wrapping DNA around nucleosomes are contained in the DNA itself. This subsequently labeled second genetic code was based on data-driven, structural, and biophysical considerations. It was subjected to an extensive suite of validation procedures, with one conclusion being that intrinsic, genome-encoded, nucleosome organization explains approximately 50% of in vivo nucleosome positioning. Here, we revisit both the nature of the underlying sequence preferences, and the performance of the proposed code. A series of new analyses, employing spectral envelope (Fourier transform) methods for assessing key sequence periodicities, classification techniques for evaluating predictive performance, and discriminatory motif finding methods for devising alternate models, are applied. The findings from the respective analyses indicate that signature dinucleotide periodicities are absent from the bulk of the high affinity nucleosome-bound sequences, and that the predictive performance of the code is modest. We conclude that further exploration of the role of sequence-based preferences in genome-wide nucleosome positioning is warranted. This work offers a methodologic counterpart to a recent, high resolution determination of nucleosome positioning that also questions the accuracy of the proposed code and, further, provides illustrations of techniques useful in assessing sequence periodicity and predictive performance.     

Hilbert-Huang transform for analysis of heart rate variability in cardiac health

This paper introduces a modified technique based on Hilbert-Huang transform (HHT) to improve the spectrum estimates of heart rate variability (HRV). In order to make the beat-to-beat (RR) interval be a function of time and produce an evenly sampled time series, we first adopt a preprocessing method to interpolate and resample the original RR interval. Then, the HHT, which is based on the empirical mode decomposition (EMD) approach to decompose the HRV signal into several monocomponent signals that become analytic signals by means of Hilbert transform, is proposed to extract the features of preprocessed time series and to characterize the dynamic behaviors of parasympathetic and sympathetic nervous system of heart. At last, the frequency behaviors of the Hilbert spectrum and Hilbert marginal spectrum (HMS) are studied to estimate the spectral traits of HRV signals. In this paper, two kinds of experiment data are used to compare our method with the conventional power spectral density (PSD) estimation. The analysis results of the simulated HRV series show that interpolation and resampling are basic requirements for HRV data processing, and HMS is superior to PSD estimation. On the other hand, in order to further prove the superiority of our approach, real HRV signals are collected from seven young health subjects under the condition that autonomic nervous system (ANS) is blocked by certain acute selective blocking drugs: atropine and metoprolol. The high-frequency power/total power ratio and low-frequency power/high-frequency power ratio indicate that compared with the Fourier spectrum based on principal dynamic mode, our method is more sensitive and effective to identify the low-frequency and high-frequency bands of HRV.     

Methods for comparison of parameters from longitudinal rhythmometric models with multiple components

Multiple components linear least-squares methods have been proposed for the detection of periodic components in nonsinusoidal longitudinal time series. However, a proper test for comparison of parameters obtained from this method for two or more time series is not yet available. Accordingly, we propose two methods, one parametric and one nonparametric, to compare parameters from rhythmometric models with multiple components. The parametric method is based on techniques commonly and generally employed in linear regression analysis. The comparison of parameters among two or more time series is accomplished by the use of so-called dummy variables. The nonparametric method is based on bootstrap techniques. This approach basically tests if the difference in any given parameter obtained by fitting a model with the same periods to two different longitudinal time series differs from zero. This method calculates a confidence interval for the difference in the tested parameter. If this interval does not contain zero, it can be concluded that the parameters obtained from the two time series are different with high probability. An estimation of the p-value for the corresponding test can also be calculated. By the use of similar bootstrap techniques, confidence intervals can also be obtained for any parameter derived from the multiple component fit of several periods to nonsinusoidal longitudinal time series, including the orthophase (peak time), bathyphase (trough time), and global amplitude (difference between the maximum and the minimum) of the fitted model waveform. These methods represent a valuable tool for the comparison of rhythm parameters obtained by multiple component analysis, and they render this approach as a generally applicable one for waveform representation and detection of periodicities in nonsinusoidal, sparse, and noisy longitudinal time series sampled with either equidistant or unequidistant observations.     

Anticodons, frameshifts, and hidden periodicities in tRNA sequences

Fourier analysis of the short-range periodicities for the complete set of sequences coding for tRNA genes in genome of Bacillus subtilis proves that periodicities with periods p = 2, 3, 4, and 6 sites are the inherent properties of tRNAs. The related periodicities should be understood in a broad statistical sense and their identifying needs the elaborate statistical methods. To improve the statistics, the analysis of significant periodicities was performed for the binary R-Y, S-W, and K-M sequences. Generally, such short-range periodicities are produced via biased positioning of particular nucleotides rather than via the tandem multiplication and subsequent modifications of repeats, though the latter mechanism may also be realized. Quasi-coherently piercing long segments of tRNA, the short-range periodicities create the effective long-range structural coupling between the acceptor stem and the anticodon loop and may participate in the mechanisms of molecular recognition. The periodicities with p = 2 and 4 provide the natural ground for the translation with spontaneous or programmed frameshifting and are present in tRNAs decoding the most frameshift-prone codons. The observation of short-range periodicities suggests that the mechanisms of amino-acylation of tRNAs and codon-anticodon pairing are not independent. Their study may also provide the important information related to the origin and evolution of the genetic code.     

An empirical link between the spectral colour of climate and the spectral colour of field populations in the context of climate change

1. The spectral colour of population dynamics and its causes have attracted much interest. The spectral colour of a time series can be determined from its power spectrum, which shows what proportion of the total variance in the time series occurs at each frequency. A time series with a red spectrum (a negative spectral exponent) is dominated by low-frequency oscillations, and a time series with a blue spectrum (a positive spectral exponent) is dominated by high-frequency oscillations. 2. Both climate variables and population time series are characterised by red spectra, suggesting that a population's environment might be partly responsible for its spectral colour. Laboratory experiments and models have been used to investigate this potential link. However, no study using field data has directly tested whether populations in redder environments are redder. 3. This study uses the Global Population Dynamics Database together with climate data to test for this effect. We found that the spectral exponent of mean summer temperatures correlates positively and significantly with population spectral exponent. 4. We also found that over the last century, temperature climate variables on most continents have become bluer. 5. Although population time series are not long or abundant enough to judge directly whether their spectral colours are changing, our two results taken together suggest that population spectral colour may be affected by the changing spectral colour of climate variables. Population spectral colour has been linked to extinction; we discuss the potential implications of our results for extinction probability.     

Periodicity in DNA coding sequences: implications in gene evolution

In this paper we have employed Fourier analysis of DNA coding and non-coding sequences in an attempt to identify possible patterns in gene sequences. It was found that while intronic sequences show a rather random pattern, coding sequences show periodicities and in particular a periodicity of 3. We were able to reconstruct such patterns by assuming a gene having one codon occurring in about 40% of the sequence. This could indicate that the predominant presence of codons all starting from the same base could confer the observed periodicities. Indeed, it was found that proteins do obey this rule. Implications of this finding in gene evolution are discussed.     

A technique for the measurement of synchronised feeding in miniature pigs: determination by means of Walsh - Fourier spectral analysis

In order to investigate feeding synchronization in miniature pigs, a computer-controlled laboratory setup has been developed, recording the feeding behavior of two pigs at a time for weeks. Since, this setup delivers time series with a binary data structure, and thus Fourier-Spectral analysis is difficult to perform, Walsh - Fourier Spectral analysis for ordinal or binary data was utilized. Synchronicity between pigs housed together was estimated by coherency values, determined for the highest sequences of the Walsh - Fourier power spectra. Feeding behavior was recorded in 12 pigs (26 - 50 kg) housed in pairs, in separate, but adjacent pens. Pigs were conditioned to operate feeders and feeding was recorded for two weeks. Pigs housed adjacent to one another showed an overlap in the dominant sequencies which was confirmed by high corresponding coherency values. Furthermore, pairs of pigs were matched according to age, weight and gender, and combined by chance. Compared to these pairings, coherencies were higher and more consistent in pigs housed in adjacent pens. If the data structure is binary, Walsh - Fourier Spectral analysis utilizing coherencies as a measure of correlation between the spectra has been shown to be a useful tool in the investigation of behavioral synchronization.     

Map-invariant spectral analysis for the identification of DNA periodicities

Many signal processing based methods for finding hidden periodicities in DNA sequences have primarily focused on assigning numerical values to the symbolic DNA sequence and then applying spectral analysis tools such as the short-time discrete Fourier transform (ST-DFT) to locate these repeats. The key results pertaining to this approach are however obtained using a very specific symbolic to numerical map, namely the so-called Voss representation. An important research problem is to therefore quantify the sensitivity of these results to the choice of the symbolic to numerical map. In this article, a novel algebraic approach to the periodicity detection problem is presented and provides a natural framework for studying the role of the symbolic to numerical map in finding these repeats. More specifically, we derive a new matrix-based expression of the DNA spectrum that comprises most of the widely used mappings in the literature as special cases, shows that the DNA spectrum is in fact invariable under all these mappings, and generates a necessary and sufficient condition for the invariance of the DNA spectrum to the symbolic to numerical map. Furthermore, the new algebraic framework decomposes the periodicity detection problem into several fundamental building blocks that are totally independent of each other. Sophisticated digital filters and/or alternate fast data transforms such as the discrete cosine and sine transforms can therefore be always incorporated in the periodicity detection scheme regardless of the choice of the symbolic to numerical map. Although the newly proposed framework is matrix based, identification of these periodicities can be achieved at a low computational cost.     

Mass extinctions over the last 500 myr: an astronomical cause?

A Fourier analysis of the magnitudes and timing of the Phanerozoic mass extinctions (MEs) demonstrates that many of the periodicities claimed in other analyses are not statistically significant. Moreover we show that the periodicities associated with oscillations of the Solar System about the galactic plane are too irregular to give narrow peaks in the Fourier periodograms. This leads us to conclude that, apart from possibly a small number of major events, astronomical causes for MEs can largely be ruled out.