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.     

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 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.     

Steps and Bumps: Precision Extraction of Discrete States of Molecular Machines

We report statistical time-series analysis tools providing improvements in the rapid, precision extraction of discrete state dynamics from time traces of experimental observations of molecular machines. By building physical knowledge and statistical innovations into analysis tools, we provide techniques for estimating discrete state transitions buried in highly correlated molecular noise. We demonstrate the effectiveness of our approach on simulated and real examples of steplike rotation of the bacterial flagellar motor and the F1-ATPase enzyme. We show that our method can clearly identify molecular steps, periodicities and cascaded processes that are too weak for existing algorithms to detect, and can do so much faster than existing algorithms. Our techniques represent a step in the direction toward automated analysis of high-sample-rate, molecular-machine dynamics. Modular, open-source software that implements these techniques is provided.     

CIRCADIAN PATTERN OF SIMULATED FLIGHT PERFORMANCE OF PILOTS IS DERIVED FROM ULTRADIAN COMPONENTS *

Studies suggest some physiologic, cognitive, and behavioral 24h rhythms are generated by cyclic components that are shorter in period than circadian. The aim of this study was [1] to examine the hypothesis that 24h human performance rhythms arise from the integration of high-frequency endogenous components and [2] to quantify the contribution of each higher frequency component to the phenotype of the rhythm. We monitored the performance of 9 experienced pilots by employing an array of cognitive-based tests conducted in a flight simulator so that, over the 6-day experiment, data were obtained for each 2h interval of the 24h. The activity-rest schedule of the subjects, no matter the exact clock time schedule of sleep and activity, always consisted of 14h activity (when they carried out regular professional duties) and 10h rest, with at least 8h of sleep. The simulated combat scenarios consisted of simple and complex tasks associated with target interception, aircraft maneuvering, and target shooting and downing. The results yielded two indices: the number of prominent periodicities in the time series and the relative magnitude of the amplitude of each relative to the construction of the composite 24h waveform. Three cyclic components (8h, 12h, and 24h) composed the observed 24h performance pattern. The dominant period and acrophase (peak time) of the compound output rhythm were determined by the interplay between the amplitudes of the various individual ultradian components. Task complexity (workload) increases the expression of the ultradian entities in the 24h pattern. We constructed a model composed of the multiple ultradian components; the composite output defined a “time span” (of 2h-4h duration) as opposed to an exact “time point” of high and low performance, endowing elevated functional capability. (Chronobiology International, 18(6), 987-1003, 2001)     

Does red noise increase or decrease extinction risk? Single extreme events versus series of unfavorable conditions

Recent theoretical studies have shown contrasting effects of temporal correlation of environmental fluctuations (red noise) on the risk of population extinction. It is still debated whether and under which conditions red noise increases or decreases extinction risk compared with uncorrelated (white) noise. Here, we explain the opposing effects by introducing two features of red noise time series. On the one hand, positive autocorrelation increases the probability of series of poor environmental conditions, implying increasing extinction risk. On the other hand, for a given time period, the probability of at least one extremely bad year ("catastrophe") is reduced compared with white noise, implying decreasing extinction risk. Which of these two features determines extinction risk depends on the strength of environmental fluctuations and the sensitivity of population dynamics to these fluctuations. If extreme (catastrophic) events can occur (strong noise) or sensitivity is high (overcompensatory density dependence), then temporal correlation decreases extinction risk; otherwise, it increases it. Thus, our results provide a simple explanation for the contrasting previous findings and are a crucial step toward a general understanding of the effect of noise color on extinction risk.     

Simple computer method for evaluation of lateral diffusion coefficients from fluorescence photobleaching recovery kinetics.

A method is presented for the analysis of fluorescence photobleaching recovery curves. Based on the simplified kinetic expression of Yguerabide, J., J.A. Schmidt, and E.E. Yguerabide (1982, Biophys. J., 40:69-75), a linearization procedure is described that permits unequivocal determination of all diffusion parameters. The presence of additional membrane flow or multiple diffusion coefficients can easily be detected by this method, and simple corrections for the presence of these alternative recovery processes can be made by the use of a regular mini-computer. The validity of the method is tested on simulated recovery curves, varying the contribution of flow, multiple diffusion coefficients, and statistical noise due to counting error.     

Stability of hemoglobin mass over 100 days in active men

The purpose of this study was to investigate the suggestion in a recent meta-analysis that variability in hemoglobin mass increases when time between measurements increases from days to months. Hemoglobin mass of six active men was measured with the carbon monoxide method every 1-6 days for 100-114 days (42 +/- 3 measurements, mean +/- SD). Measurement error for each individual's series was estimated from the standard deviation of consecutive pairwise changes and compared with his total error (standard deviation of all values). Linear trends and periodicities in each series were quantified by regression and spectral analysis. Series with known random error and periodicity were also simulated and analyzed. There were clear differences in the pairwise error of measurement between subjects (range 1.4-2.7%). For five men, there was little difference between the total and pairwise errors; their mean ratio (1.06, 90% confidence limits 0.96-1.17) was less than ratios for simulated sinusoidal series with random error of 2%, amplitude of 2%, and periods of 20-100 days (ratios 1.13-1.21). Spectral analysis clearly revealed such periodicities in the simulated series but not in the series of these subjects. The sixth man, who had donated blood 12 days before commencing measurements, showed errors, trend, and periodicity consistent with gradual restoration of hemoglobin mass. Measurement error of hemoglobin mass does not increase over 100 days. Consequently, hemoglobin mass may be suitable for long-term monitoring of small changes that might occur with training or erythropoietin abuse, taking into consideration the small differences between athletes in errors and trends.     

Modelling multiple time series via common factors

We propose a new method for estimating common factors of multipletime series. One distinctive feature of the new approach isthat it is applicable to some nonstationary time series. Theunobservable, nonstationary factors are identified by expandingthe white noise space step by step, thereby solving a high-dimensionaloptimization problem by several low-dimensional sub-problems.Asymptotic properties of the estimation are investigated. Theproposed methodology is illustrated with both simulated andreal datasets.     

Use of Phase for the Detection of Hidden Periodicites

Standard tests for the detection of hidden periodicities in time series are largely ignored by applied workers. Various simple but inappropriate methods are used instead. Therefore a method is suggested which is both simple and appropriate but which requires the prior knowledge of certain characteristics of the suspected periodicity. For illustration, this method is applied to a set of data from a chronobiological study.     

Black noise and population persistence

Biological populations are susceptible to random variation in environmental influences such as temperature and moisture. This variability (or noise) can determine population size and, ultimately, cause extinctions. Extinction risk depends on noise colour or the amount of short- and long-term variation. Most environmental noise is reddened: the variation is dominated by long-term fluctuations. Recent modelling has shown that moderately reddened noise affects populations differently from the white noise used in earlier studies. However, some geophysical phenomena, such as temperature and river height, can have deeply reddened ''brown'' or even ''black'' spectra. We find that, compared to environments characterized by red noise, very long population persistence times are more likely for black noise. Unlike previous work incorporating a simple autoregressive model of reddened noise, our model suggests that the large variation associated with persistence in a red-noise environment limits our ability to predict the fate of particular populations subject to this noise colour. Thus, we identify the colour of noise experienced by a population (red or black) as a crucial factor in any attempt to manage or conserve that population.     

Extinction risk, coloured noise and the scaling of variance

The impact of temporally correlated fluctuating environments (coloured noise) on the extinction risk of populations has become a main focus in theoretical population ecology. In this study we particularly focus on the extinction risk in strongly correlated environments. Here, we found that, in contrast to moderate auto-correlation, the extinction risk was highly dependent on the process of noise generation, in particular on the method of variance scaling. Such scaling is commonly applied to avoid variance-driven biases when comparing the extinction risk under white and coloured noise. We show that for strong auto-correlation often-used scaling techniques lead to a high variability in the variances of the resulting time series and thus to deviations in the subsequent extinction risk. Therefore, we present an alternative scaling method that always delivers the target variance, even in the case of strong auto-correlation. In contrast to earlier techniques, our very intuitive method is not bound to auto-regressive processes but can be applied to all types of coloured noises. We strongly recommend our method to generate time series when the target of interest is the effect of noise colour on extinction risk not obscured by any variance effects.     

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.     

Rescuing Rhythms from Noise: A New Method of Analysis

The analysis of a temporal series usually begins with a visual inspection of the raw data, from which a proper method for the detection of periodicities is chosen. Some of the methods currently used, as circular statistics, Cosinor, or spectral analyses, are useful when it comes to ascertain the existence of some periods, expected ‘a priori’ or to detect unknown frequencies. Even though some of the methods allow a wide scanning of possibilities, difficulties arise when signals are weak and concealed in larger amplitude noise. The register of the activity of a cave cricket, Strinatia brevipennis, under constant conditions, showed an intricate pattern of small peaks, interspersed with rare ones of much higher amplitudes. Attempts to analyse these data with the usual methods gave inconsistent results and sometimes did not detect rhythms. The results are mostly biased by the large amplitude components which hamper the detection of rhythms from weak signals. Schimmel and Paulssen (1997) proposed a noise-reduction method, which detects weak but coherent signals. This new tool was developed for the analysis of seismic data, being afterwards adapted to the analysis of temporal series of biological data. The method is called phase weighted stack (PWS) and performs a weighted summation of temporal series according to their coherence. The results are stacked time series which are cleaned up from incoherent noise, allowing the detection of weak signals that otherwise would be undistinguishable from noises. The method also enables the identification of the time (hour) of every periodic signal. The use of PWS in the analysis of cricketsÕ activity data cleared out frequencies, exposing a circadian component in all records.     

Extinction risk and the 1/f family of noise models

In order to predict extinction risk in the presence of reddened, or correlated, environmental variability, fluctuating parameters may be represented by the family of 1/f noises, a series of stochastic models with different levels of variation acting on different timescales. We compare the process of parameter estimation for three 1/f models (white, pink and brown noise) with each other, and with autoregressive noise models (which are not 1/f noises), using data from a model time-series (length, T) of population. We then calculate the expected increase in variance and the expected extinction risk for each model, and we use these to explore the implication of assuming an incorrect noise model. When parameterising these models, it is necessary to do so in terms of the measured ("sample") parameters rather than fundamental ("population") parameters. This is because these models are non-stationary: their parameters need not stabilize on measurement over long periods of time and are uniquely defined only over a specified "window" of timescales defined by a measurement process. We find that extinction forecasts can differ greatly between models, depending on the length, T, and the coefficient of variability, CV, of the time series used to parameterise the models, and on the length of time into the future which is to be projected. For the simplest possible models, ones with population itself the 1/f noise process, it is possible to predict the extinction risk based on CV of the observed time series. Our predictions, based on explicit formulae and on simulations, indicate that (a) for very short projection times relative to T, brown and pink noise models are usually optimistic relative to equivalent white noise model; (b) for projection timescales equal to and substantially greater than T, an equivalent brown or pink noise model usually predicts a greater extinction risk, unless CV is very large; and (c) except for very small values of CV, for timescales very much greater than T, the brown and pink models present a more optimistic picture than the white noise model. In most cases, a pink noise is intermediate between white and brown models. Thus, while reddening of environmental noise may increase the long-term extinction probability for stationary processes, this is not generally true for non-stationary processes, such as pink or brown noises.     

Parameters of exponentially damped sinusoids in noise and their application to the analysis of visual evoked potentials

The backward prediction and singular value (SV) truncation methods for estimating multiple exponentially damped real sinusoids in noise have been studied. The basic theory and algorithm are outlined, and the effect on the error of computational parameters, such as the sampling window, the sampling rate, the order of the prediction-error filter (PEF) and the truncated point of SV, were studied using simulated data. If the computational parameters are carefully chosen, the estimate of frequency and damping factor of a damped sinusoid is quite accurate (within a certain range of signal to noise (S/N) ratio). We use this method to estimate the parameters of visual evoked potentials (VEP) and then reconstructed them from estimated parameters according to the resonant model. The error between original VEP and the reconstructed waveform is within 5%. This method can be used in the analysis and recognition of various systems.     

Measuring kinetics of complex single ion channel data using mean-variance histograms.

The measurement of single ion channel kinetics is difficult when those channels exhibit subconductance events. When the kinetics are fast, and when the current magnitudes are small, as is the case for Na+, Ca2+, and some K+ channels, these difficulties can lead to serious errors in the estimation of channel kinetics. I present here a method, based on the construction and analysis of mean-variance histograms, that can overcome these problems. A mean-variance histogram is constructed by calculating the mean current and the current variance within a brief "window" (a set of N consecutive data samples) superimposed on the digitized raw channel data. Systematic movement of this window over the data produces large numbers of mean-variance pairs which can be assembled into a two-dimensional histogram. Defined current levels (open, closed, or sublevel) appear in such plots as low variance regions. The total number of events in such low variance regions is estimated by curve fitting and plotted as a function of window width. This function decreases with the same time constants as the original dwell time probability distribution for each of the regions. The method can therefore be used: 1) to present a qualitative summary of the single channel data from which the signal-to-noise ratio, open channel noise, steadiness of the baseline, and number of conductance levels can be quickly determined; 2) to quantify the dwell time distribution in each of the levels exhibited. In this paper I present the analysis of a Na+ channel recording that had a number of complexities. The signal-to-noise ratio was only about 8 for the main open state, open channel noise, and fast flickers to other states were present, as were a substantial number of subconductance states. "Standard" half-amplitude threshold analysis of these data produce open and closed time histograms that were well fitted by the sum of two exponentials, but with apparently erroneous time constants, whereas the mean-variance histogram technique provided a more credible analysis of the open, closed, and subconductance times for the patch. I also show that the method produces accurate results on simulated data in a wide variety of conditions, whereas the half-amplitude method, when applied to complex simulated data shows the same errors as were apparent in the real data. The utility and the limitations of this new method are discussed.     

Signal to noise analysis of multiple color fluorescence imaging microscopy

BACKGROUND: Various approaches that were recently developed demonstrate the ability to simultaneously detect all human (or other species) chromosomes by using combinatorial labeling and fluorescence in situ hybridization (FISH). With the growing interest in this field, it is important to develop tools for optimizing and estimating the accuracy of different experimental methods. METHODS: We have analyzed the principles of multiple color fluorescence imaging microscopy. First, formalism based on the physical principles of fluorescence microscopy and noise analysis is introduced. Next, a signal to noise (S/N) analysis is performed and summarized in a simple accuracy criterion. The analysis assumes shot noise to be the dominant source of noise. RESULTS: The accuracy criterion was used to calculate the S/N of multicolor FISH (M-FISH), spectral karyotyping, ratio imaging, and a method based on using a set of broad band filters. Spectral karyotyping is tested on various types of samples and shows accurate classifications. We have also tested classification accuracy as a function of total measurement time. CONCLUSIONS: The accuracy criterion that we have developed can be used for optimizing and analyzing different multiple color fluorescence microscopy methods. The assumption that shot noise is dominant in these measurements is supported by our measurements.     

Time-course analysis of cyanobacterium transcriptome: detecting oscillatory genes

The microarray technique allows the simultaneous measurements of the expression levels of thousands of mRNAs. By mining these data one can identify the dynamics of the gene expression time series. The detection of genes that are periodically expressed is an important step that allows us to study the regulatory mechanisms associated with the circadian cycle. The problem of finding periodicity in biological time series poses many challenges. Such challenge occurs due to the fact that the observed time series usually exhibit non-idealities, such as noise, short length, outliers and unevenly sampled time points. Consequently, the method for finding periodicity should preferably be robust against such anomalies in the data. In this paper, we propose a general and robust procedure for identifying genes with a periodic signature at a given significance level. This identification method is based on autoregressive models and the information theory. By using simulated data we show that the suggested method is capable of identifying rhythmic profiles even in the presence of noise and when the number of data points is small. By recourse of our analysis, we uncover the circadian rhythmic patterns underlying the gene expression profiles from Cyanobacterium Synechocystis.