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 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.     

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.     

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.     

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.     

Daily and estrous rhythmicity of body temperature in domestic cattle

Background

Rhythmicity in core body temperature has been extensively studied in humans and laboratory animals but much less in farm animals. Extending the study of rhythmicity of body temperature to farm animals is important not only from a comparative perspective but also from an economic perspective, as greater knowledge of this process can lead to improvements in livestock production practices. In this study in cattle, we investigated the maturation of the daily rhythm of body temperature in newborn calves, characterized the parameters of the daily rhythm in young cows, and studied the oscillation in body temperature associated with the estrous cycle in adult cows.

Results

We found that the daily rhythm of body temperature is absent at birth but matures fully during the first two months of life. The mature rhythm had a mean level of 38.3°C, a range of excursion of 1.4°C, and was more robust than that of any mammalian species previously studied (90% of maximal robustness). Sexually mature cows also exhibited a robust estrous rhythm of body temperature. An elevation of about 1.3°C was observed every 21 days on the day of estrus. Small seasonal variations in this pattern were observed.

Conclusion

In conclusion, calves exhibit a very robust daily rhythm of body temperature, although this rhythm is absent at birth and develops during the first two months of life. Adult cows exhibit also 21-day rhythmicity in body temperature reflecting the duration of the estrous cycle.

     

Non-stationary time series and the robustness of circadian rhythms

Circadian rhythms are regular oscillations in the value of behavioral and physiological variables of organisms that recur on a daily basis. The purpose of this study was to evaluate the extent of non-stationarity of circadian rhythms over several days, to determine how damaging is the violation of the assumption of stationarity in the analysis of circadian rhythms, and to formalize the concept of "rhythm robustness" as an index of oscillatory ("weak") stationarity. Simulated (computer-generated) and experimental data sets (rhythms of body temperature and running-wheel activity in several rodent species) were analysed. Tests of stationarity based on the variance of the daily means and the variance of the daily variances revealed that most experimental data sets are not stationary. Analysis of linear trends indicated that significant trends are rare in experimental data sets. Although the non-stationarity of the experimental data sets reduced the spectral energy of the Enright periodogram used to assess rhythmicity, detection of circadian rhythmicity was not prevented in any of the rhythmic data sets. The results of the various analyses allow the inference that, after high-frequency noise is filtered out, the value of the periodogram's Q(P) statistic reflects the extent of stationarity of the time series. Thus, the "robustness" of a circadian rhythm (i.e. the magnitude of the empirical Q(P) value as compared to the Q(P) value associated with a perfectly rhythmic time series) can serve as an index of the stationarity of the rhythm.     

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.     

Analogies between rhythms in plant material,in atmospheric pressure,and solar lunar periodicities

Low frequency fluctuations of the gaseous exchanges of seeds and of the length of the roots of seedlings, maintained in darkness and at constant temperature, were compared with the atmospheric pressure fluctuations and solar lunar rhythmicities. Several experiments were conducted in short and long series on seeds and seedlings of different species. Gaseous exchanges were measured with the Warburg apparatus. The experiments were conducted with or without renewing the series, on different conditions of temperature and humidity, with or without agitation. On the short series, the morning and afternoon series were compared. On the long series, a pooled variance spectrum or a periodogram were established. OnLycopersicum esculentum L. seeds, morning and afternoon gaseous consumptions oscillated with respect to one another with a greater morning consumption at intervals of 15 days in correlation with new and full moon. The same oscillation was detected on the inverse of the atmospheric pressure. OnHelianthus annuus L. seeds a rhythmicity of the gaseous exchanges of 15 days and the same rhythmicity for the inverse of the atmospherical pressure were observed. In a new experiment a periodicity of 15 days was detected with some other low frequency rhythms and a plurimonthly tendency almost correlated with the vernal equinox. OnLepidium sativum L. a monthly rhythm and a plurimonthly tendency of the growth of roots of 72-h seedlings were found. Analogies of rhythmicity between the biological material, the atmospherical pressure and the solar lunar rhythmicity are observed.     

Comparison of EEGs Before and After Stunning of Cattle Taking Account of Animal-to-Animal Variation

Data are available for the electroencephalograms (EEGs) of cattle before and after they received a mild electric shock. The purpose of the proposed statistical analysis is to extract from the data what differences there an in pre- and post-stun EEGs and possibly restrict attention to a small frequency band where there is a significant change. We study the log periodogram ratios for each animal and propose a stochastic model based method for smoothing these ratios. This method is novel in that it allows (i) for between animal variation, and (ii) the amount of local smoothing to adapt according to the data requirements. The smoothing method is implemented by utilizing a Kalman filter approach and the fixed interval smoothing algorithm, which allows us to obtain point-wise estimates and standard errors for the log periodogram ratio. Common animal effects which are intimated by animal-by-animal plots of log periodogram ratios against frequency are highlighted by this method.     

Feeble weekly rhythmicity in hematological, cardiovascular, and thermal parameters in the horse

The homeostatic control of physiological processes is affected by a variety of temporal programs, such as circadian rhythms, estrous cycles, and circannual rhythms. The existence of circaseptan rhythms (endogenous rhythms with the duration of a week) has been postulated but not properly verified. In this study, we compared plasma concentration of lactic acid, systolic and diastolic blood pressure, and rectal temperature in athletic horses (maintained under a weekly training schedule) with those in sedentary horses (maintained under a constant schedule throughout the week). Although exercise had robust acute effects on the measured parameters, measurements conducted early in the morning or in the evening showed significant weekly rhythmicity only in one of the parameters (plasma concentration of lactic acid) in a few animals. This rhythmicity was feeble, was present only in athletic horses, and vanished if rigorous statistical criteria were applied. In contrast, 24 h rhythmicity was significant in all parameters in all horses. We conclude that the overall 7-day pattern in physiological parameters of the horse is feeble and is caused by the weekly schedule of activity. The available evidence does not support the notion of endogenous, circaseptan rhythms.     

Feeble Weekly Rhythmicity in Hematological,Cardiovascular, and Thermal Parameters in the Horse

The homeostatic control of physiological processes is affected by a variety of temporal programs, such as circadian rhythms, estrous cycles, and circannual rhythms. The existence of circaseptan rhythms (endogenous rhythms with the duration of a week) has been postulated but not properly verified. In this study, we compared plasma concentration of lactic acid, systolic and diastolic blood pressure, and rectal temperature in athletic horses (maintained under a weekly training schedule) with those in sedentary horses (maintained under a constant schedule throughout the week). Although exercise had robust acute effects on the measured parameters, measurements conducted early in the morning or in the evening showed significant weekly rhythmicity only in one of the parameters (plasma concentration of lactic acid) in a few animals. This rhythmicity was feeble, was present only in athletic horses, and vanished if rigorous statistical criteria were applied. In contrast, 24 h rhythmicity was significant in all parameters in all horses. We conclude that the overall 7-day pattern in physiological parameters of the horse is feeble and is caused by the weekly schedule of activity. The available evidence does not support the notion of endogenous, circaseptan rhythms.     

Feeble Weekly Rhythmicity in Hematological, Cardiovascular, and Thermal Parameters in the Horse

The homeostatic control of physiological processes is affected by a variety of temporal programs, such as circadian rhythms, estrous cycles, and circannual rhythms. The existence of circaseptan rhythms (endogenous rhythms with the duration of a week) has been postulated but not properly verified. In this study, we compared plasma concentration of lactic acid, systolic and diastolic blood pressure, and rectal temperature in athletic horses (maintained under a weekly training schedule) with those in sedentary horses (maintained under a constant schedule throughout the week). Although exercise had robust acute effects on the measured parameters, measurements conducted early in the morning or in the evening showed significant weekly rhythmicity only in one of the parameters (plasma concentration of lactic acid) in a few animals. This rhythmicity was feeble, was present only in athletic horses, and vanished if rigorous statistical criteria were applied. In contrast, 24 h rhythmicity was significant in all parameters in all horses. We conclude that the overall 7-day pattern in physiological parameters of the horse is feeble and is caused by the weekly schedule of activity. The available evidence does not support the notion of endogenous, circaseptan rhythms.     

An Investigation on the Allelic Chi‐Square Test Used in Genetic Association Studies

Case‐control studies are primary study designs used in genetic association studies. Sasieni (Biometrics 1997, 53, 1253–1261) pointed out that the allelic chi‐square test used in genetic association studies is invalid when Hardy‐Weinberg equilibrium (HWE) is violated in a combined population. It is important to know how much type I error rate is deviated from the nominal level under violated HWE. We examine bounds of type I error rate of the allelic chi‐square test. We also investigate power of the goodness‐of‐fit test for HWE which can be used as a guideline for selecting an appropriate test between the allelic chi‐square test and the modified allelic chi‐square test, the latter of which was proposed for cases of violated HWE. In small samples, power is not large enough to detect the Wright's inbreeding model of small values of inbreeding coefficient. Therefore, when the null hypothesis of HWE is barely accepted, the modified test should be considered as an alternative method. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characteristics of Diarrheal Illnesses in Non-Breast Fed Infants Attending a Large Urban Diarrheal Disease Hospital in Bangladesh

Background

Lack of breast feeding is associated with higher morbidity and case-fatality from both bacterial and viral etiologic diarrheas. However, there is very limited data on the characteristics of non–breastfed infants attending hospital with diarrheal illnesses caused by common bacterial and viral pathogens. Our objective was to assess the impact of lack of breast feeding on diarrheal illnesses in infants living in urban Bangladesh.

Methods

We extracted data of infants (0–11 months) for analyses from the data archive of Diarrheal Disease Surveillance System (DDSS) of the Dhaka Hospital of icddr,b for the period 2008–2011.

Results

The prevalence of breastfeeding in infants attending the hospital with diarrhea reduced from 31% in 2008 to 17% in 2011, with corresponding increase in the prevalence of non-breastfed (chi square for trend <0.001). Among breastfed infants, the incidence of rotavirus infections was higher (43%) among the 0–5 months age group than infants aged 9–11 months (18%). On the other hand, among non-breastfed infants, the incidence of rotavirus infections was much higher (82%) among 9–11 months old infants compared to those in 0–5 months age group (57%) (chi square for trend <0.001). Very similar trends were also observed in the incidence of cholera and ETEC diarrheas among different age groups of breastfed and non-breastfed infants (chi square for trend 0.020 and 0.001 respectively). However, for shigellosis, the statistical difference remained unchanged among both the groups (chi square for trend 0.240).

Conclusion and Significance

We observed protective role of breastfeeding in infantile diarrhea caused by the major viral and common bacterial agents. These findings underscore the importance of promotion and expansion of breastfeeding campaigns in Bangladesh and elsewhere.     

The peripubertal golden hamster and the transition between daily and estrous cycle hormone rhythms

Transitional patterns of LH, FSH, and progesterone (P4) in the circulation were studied in peripubertal female golden hamsters. A daily rhythm, with afternoon surges of these hormones, is typical of the immature female, whereas 4-day rhythms characterize the estrous cycle of the adult. Blood samples were collected repeatedly from maturing individuals at either 1400 or 1700 h. Each animal was examined daily for the appearance of regular vaginal estrous cycles as indicated by a mucous exudate on the morning of ovulation. Between Days -10 and -5 relative to first vaginal estrus (FVE), afternoon surges of LH, FSH, and P4 were often observed. From Days -5 to -1 relative to FVE, afternoon surges of LH and FSH were less frequent, but P4 retained the daily rhythmicity until Day -2. A 4-day pattern of LH secretion, but not of FSH or P4, was established prior to FVE. To determine whether or not ovulations were occurring prior to the appearance of external vaginal estrous cycles, reproductive tracts were collected from 26-34 days of age and examined for evidence of ovulation. Of 124 females, concordance between the record of daily vaginal examinations and the examinations of the ovaries and oviducts was found in 103 cases (83%). The development of ovarian follicles was correlated with FVE in peripubertal hamsters by unilateral ovariectomy. Antral follicles were found only in the last 3 days prior to vaginal estrus.     

Repeatability of blood serum progesterone levels in dairy heifers on day 7 of the estrous cycle

Thirty normally cycling dairy heifers were used to determine the repeatability of blood serum progesterone levels on Day 7 ± 0.25 d of the estrous cycle. The experimental group consisted of 16 Holsteins and 14 dairy crossbreds ranging in age from 18 to 24 months. Day of the estrous cycle was determined from twice daily observations for standing heat (Day 0). Serum progesterone levels for Day 7 ± 0.25 d were determined by radioimmunoassay from blood samples collected by jugular venipuncture over three to four consecutive estrous cycles. Levels of blood serum progesterone for Day 7 ± 0.25 d ranged from 0.57 to 6.03 ng/ml. Least square means for the Holstein (2.74 ng/ml) and dairy crossbred (3.38 ng/ml) groups were different (P<0.006). The repeatability for levels of blood serum progesterone on Day 7 of the estrous cycle was low (0.0115).     

The description of spatial pattern using two-dimensional spectral analysis

Two-dimensional spectral analysis is a general interrogative technique for describing spatial patterns. Not only is it able to detect all possible scales of pattern which can be present in the data but it is also sensitive to directional components. Four functions are described: the autocorrelation function; the periodogram; and, the R- and Θ-spectra which respectively summarize the periodogram in terms of scale and directional components of pattern. The use of these functions is illustrated by their application to a simple wave pattern, a wave pattern with added noise, and patterns simulating competition and invasion processes.     

A Markov chain model for animal estrous cycling data

Estrous cycling data contain sequences of characters (e.g., DPEMD). Each sequence represents an animal's estrous cycle, with each character indicating the daily estrous cycle stage. Changes in the estrous cycle pattern, which is determined by estrous stage lengths, can provide information on adverse events. Stage lengths are not directly observable. However interval censored lengths for all but the first and the last stages in a sequence can be extracted from the data. We propose a Markov chain model to approximate the estrous cycling process. The transition probabilities from one stage to another can be derived by conditioning on stage lengths. Assuming Weibull distribution for stage lengths, with the second Weibull parameter depending upon treatment effects and animal-specific random effects, regression models on censored stage lengths are fitted. A Bayesian approach is used for inference on dose effects. The analysis is implemented with MCMC method in WinBUGS. An estrous cycling data set from a National Toxicology Program study is analyzed as an example.