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

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

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

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

Linear optimization of predictors for secondary structure. Application to transbilayer segments of membrane proteins

Sliding-window averaging of amino acid properties is a standard method for predicting protein secondary structure. For example, transmembrane segments are predicted to occur near the peaks in a hydropathy plot of a membrane protein. Such a scheme (linear convolutional recognizer, LCR) assigns a number (weight) to each type of monomer, and then convolutes some window function with the sequence of weights. The window has commonly been rectangular, and the weights derived from singlet amino acid frequencies in proteins of known secondary structure or from physical properties of amino acids. The accuracy of the windows and weights have remained unknown. We use linear optimization theory to develop a general method for approximating the optimal window and weights for a LCR. The method assumes that one knows the sequences of one or more chains and the locations of their "features", regions having the secondary structure of interest. We present formulae for quantifying the accuracy of predictors. We show why the optimal LCR is more accurate than methods based on the differences between singlet monomer frequencies inside and outside features. The advantage of an optimal LCR is that its weights inherently include correlations between nearby monomer positions. The optimal predictor is not perfect though. We argue that its inaccuracy is an intrinsic limitation of linear predictors based on monomer weights. As a practical example, we study predictors for transbilayer segments of membrane proteins. We estimate the optimal weights and windows for the two bacterial photosynthetic reaction centers whose three-dimensional structures are known. The resultant LCR, which is more accurate than previous ones, is still inexact. We apply it to bacteriorhodopsin and halorhodopsin. Several non-linear generalizations are examined as possible improvements to the LCR method: non-linear combinations of linear predictors and windowed Fourier transforms of the weight sequences. The former do not significantly increase the accuracy, while the latter reveal a weak negative correlation between the segments and periodic variations of the weights.     

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.     

Relative Efficiency of Estimators of the Mean of a Normal Distribution when Coefficient of Variation is Known

A biased but simple and consistent estimator of the parameter ? has been obtained for the normal distribution N(?, a?₂), ?>0 where a is a known constant. It is shown that the estimator is more efficient than the sample mean or any suitably chosen constant multiple of the sample standard deviation. It is also proved to be more efficient than the mimumum variance unbiased estimator among a typical class of unbiased estimators derived by RASUL KHAN (1968).     

Biased Brownian dynamics for rate constant calculation

An enhanced sampling method-biased Brownian dynamics-is developed for the calculation of diffusion-limited biomolecular association reaction rates with high energy or entropy barriers. Biased Brownian dynamics introduces a biasing force in addition to the electrostatic force between the reactants, and it associates a probability weight with each trajectory. A simulation loses weight when movement is along the biasing force and gains weight when movement is against the biasing force. The sampling of trajectories is then biased, but the sampling is unbiased when the trajectory outcomes are multiplied by their weights. With a suitable choice of the biasing force, more reacted trajectories are sampled. As a consequence, the variance of the estimate is reduced. In our test case, biased Brownian dynamics gives a sevenfold improvement in central processing unit (CPU) time with the choice of a simple centripetal biasing force.     

Extracting complexity waveforms from one-dimensional signals

Background  

Nonlinear methods provide a direct way of estimating complexity of one-dimensional sampled signals through calculation of Higuchi's fractal dimension (1<FD<2). In most cases the signal is treated as being characterized by one value of FD and consequently analyzed as one epoch or, if divided into more epochs, often only mean and standard deviation of epoch FD are calculated. If its complexity variation (or running fractal dimension), FD(t), is to be extracted, a moving window (epoch) approach is needed. However, due to low-pass filtering properties of moving windows, short epochs are preferred. Since Higuchi's method is based on consecutive reduction of signal sampling frequency, it is not suitable for estimating FD of very short epochs (N < 100 samples).     

Estimating Disease Prevalence Using Relatives of Case and Control Probands

Summary .  We introduce a method of estimating disease prevalence from case–control family study data. Case–control family studies are performed to investigate the familial aggregation of disease; families are sampled via either a case or a control proband, and the resulting data contain information on disease status and covariates for the probands and their relatives. Here, we introduce estimators for overall prevalence and for covariate-stratum-specific (e.g., sex-specific) prevalence. These estimators combine the proportion of affected relatives of control probands with the proportion of affected relatives of case probands and are designed to yield approximately unbiased estimates of their population counterparts under certain commonly made assumptions. We also introduce corresponding confidence intervals designed to have good coverage properties even for small prevalences. Next, we describe simulation experiments where our estimators and intervals were applied to case–control family data sampled from fictional populations with various levels of familial aggregation. At all aggregation levels, the resulting estimates varied closely and symmetrically around their population counterparts, and the resulting intervals had good coverage properties, even for small sample sizes. Finally, we discuss the assumptions required for our estimators to be approximately unbiased, highlighting situations where an alternative estimator based only on relatives of control probands may perform better.     

Oscillations in joint synchrony of reproductive hormones in healthy men

Negative-feedback (inhibitory) and positive-feedforward (stimulatory) processes regulate physiological systems. Whether such processes are themselves rhythmic is not known. Here, we apply cross-approximate entropy (cross-ApEn), a noninvasive measurement of joint (pairwise) signal synchrony, to inferentially assess hypothesized circadian and ultradian variations in feedback coupling. The data comprised simultaneous measurements of three pituitary and one peripheral hormone (LH, FSH, prolactin, and testosterone) in 12 healthy men each sampled every 10 min for 4 days (5,760 min). Ergodicity, due to the time series stationarity of the measurements over the 4 days, allows for effective estimation of parameters based upon the 12 subjects. Cross-ApEn changes were quantified via moving-window estimates applied to 4-day time series pairs. The resultant ordered windowed cross-ApEn series (in time) were subjected to power spectrum analysis. Rhythmicity was assessed against the null hypothesis of randomness using 1,000 simulated periodograms derived by shuffling the interpulse-interval hormone-concentration segments and redoing cross-ApEn windows and spectral analysis. By forward cross-ApEn analysis, paired LH-testosterone, LH-prolactin, and LH-FSH synchrony maintained dominant rhythms with periodicities of 18-22.5, 18, and 22.5 h, respectively (each P < 0.001). By reverse (feedback) cross-ApEn analysis, testosterone-LH, testosterone-prolactin, and testosterone-FSH synchrony cycles were 30, 18, and 30-45 h, respectively (each P ≤ 0.001). Significant 8- or 24-h rhythms were also detected in most linkages, and maximal bihormonal synchrony occurred consistently at ～0400-0500. Collectively, these analyses demonstrate significant ultradian (<24 h), circadian (～24 h), and infradian (>24 h) oscillations in pituitary-testis synchrony, wherein maximal biglandular coordination is strongly constrained to the early morning hours.     

Robustness of close‐kin mark–recapture estimators to dispersal limitation and spatially varying sampling probabilities

1. Close‐kin mark–recapture (CKMR) is a method for estimating abundance and vital rates from kinship relationships observed in genetic samples. CKMR inference only requires animals to be sampled once (e.g., lethally), potentially widening the scope of population‐level inference relative to traditional monitoring programs.
2. One assumption of CKMR is that, conditional on individual covariates like age, all animals have an equal probability of being sampled. However, if genetic data are collected opportunistically (e.g., via hunters or fishers), there is potential for spatial variation in sampling probability that can bias CKMR estimators, particularly when genetically related individuals stay in close proximity.
3. We used individual‐based simulation to investigate consequences of dispersal limitation and spatially biased sampling on performance of naive (nonspatial) CKMR estimators of abundance, fecundity, and adult survival. Population dynamics approximated that of a long‐lived mammal species subject to lethal sampling.
4. Naive CKMR abundance estimators were relatively unbiased when dispersal was unconstrained (i.e., complete mixing) or when sampling was random or subject to moderate levels of spatial variation. When dispersal was limited, extreme variation in spatial sampling probabilities negatively biased abundance estimates. Reproductive schedules and survival were well estimated, except for survival when adults could emigrate out of the sampled area. Incomplete mixing was readily detected using Kolmogorov–Smirnov tests.
5. Although CKMR appears promising for estimating abundance and vital rates with opportunistically collected genetic data, care is needed when dispersal limitation is coupled with spatially biased sampling. Fortunately, incomplete mixing is easily detected with adequate sample sizes. In principle, it is possible to devise and fit spatially explicit CKMR models to avoid bias under dispersal limitation, but development of such models necessitates additional complexity (and possibly additional data). We suggest using simulation studies to examine potential bias and precision of proposed modeling approaches prior to implementing a CKMR program.

     

163 Enhanced sampling of peptides and proteins with a new biasing replica exchange method

Classical MD simulations (cMD) are limited by the sampling of relevant states of the peptides. Replica exchange (REMD) methods aim to search the conformational space of proteins more efficiently (reviewed in Ostermeir & Zacharias, 2013). We have developed a Hamiltonian REMD method that takes advantage of an intrinsic property of proteins, the specific Φ ? dihedral angle combinations along the polymer backbone. By employing a coupled two-dimensional biasing potential the energy barriers along the polymer backbone are reduced more effectively than by a previous approach based on a one-D biasing potential (Kannan & Zacharias, 2007). Thus, adjacent amino acids along the polymers backbone can easily switch between favourable regions in the Ramachandran plot. Additionally, energy barriers of rotameric states of amino acid side chains of proteins are also biased in the replica runs. The method improves the sampling of conformational substates of proteins at a modest number of replicas (nine replicas in the standard set-up with one replica running without biasing potential) compared to much larger numbers necessary in the case of standard temperature (T)-REMD simulations. A further improvement is achieved by a dynamical adjustment of the penalty potential levels in the replicas such that high exchange rates and improved mixing of conformations between different replicas are guaranteed. The biasing potential (BP)-REMD method turns out to be suitable to speed up both the folding of spaghetti-like test peptides and the refinement of loop decoy structures. Starting from extended structures, an α-helical oligo-alanine and β-hairpin chignolin and the Trp-cage protein fold more rapidly in near-native structures than in cMD simulations. The BP-REMD simulations not only accelerate the folding process of test proteins but also enlarge the variety of sampled configurations in conformational space. Since flexible parts of the protein can be penalized selectively, this method provides a precise tool to investigate regions of interest of the protein.     

Unbiased estimator for genetic drift and effective population size

Amounts of genetic drift and the effective size of populations can be estimated from observed temporal shifts in sample allele frequencies. Bias in this so-called temporal method has been noted in cases of small sample sizes and when allele frequencies are highly skewed. We characterize bias in commonly applied estimators under different sampling plans and propose an alternative estimator for genetic drift and effective size that weights alleles differently. Numerical evaluations of exact probability distributions and computer simulations verify that this new estimator yields unbiased estimates also when based on a modest number of alleles and loci. At the cost of a larger standard deviation, it thus eliminates the bias associated with earlier estimators. The new estimator should be particularly useful for microsatellite loci and panels of SNPs, representing a large number of alleles, many of which will occur at low frequencies.     

Simultaneously model-unbiased, design-unbiased estimation

This paper proposes a class of inferential procedures (incorporating both design and estimation elements) that yield estimates of means that are simultaneously model unbiased and design unbiased. Classical regression procedures yield conditionally unbiased estimators for the mean (conditioning on the model and choice of observation points). In contrast, design-based methods yield estimators that are unconditionally unbiased on matter what the form of the underlying model. Variance properties of the proposed class are examined, and applications to bioavailability, water quality from mine run-off, and finite population regression estimation are considered. The proposed procedures perform well, especially in the typical case where a model is only approximately correct.     

种蛋开窗技术及其对鸡胚发育影响的研究

种蛋开窗对禽胚实施操作的技术在转基因禽类、利用多能细胞保存禽类种质资源以及禽胚实验模型等领域的研究具有重要价值。以家鸡新产蛋(含有禽胚胎干细胞)为材料,研究出了种蛋开窗切块自动复位技术,并在此基础上以接近禽胚实施操作为目的,进行了不同封口方法、不同开窗位置和添加液体排除因开窗形成的空气泡等对种蛋孵化率影响的实验。结果表明:种蛋赤道面开窗不同封口方式对孵化率影响的差异不显著,最高孵化率可达到65%;赤道面开窗,通过添加液体排除空气泡的影响不会提高孵化率,空气泡的大小与孵化率没有明显相关性;种蛋气室端开窗(针刺透内壳膜操作胚胎)方式,比赤道面开窗方式的孵化率明显提高,两种方法能见胚率均在90%以上;采用以上技术对种蛋开窗、找胚和封口,制作效率在每人每小时30个以上。本文系统地研究和报道了种蛋开窗操作禽胚简单而实用的技术,有助于解决目前种蛋开窗技术复杂且孵化率低的困境。     

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.     

Overestimates of maternity and population growth rates in multi-annual breeders

There has been limited attention to estimating maternity rate because it appears to be relatively simple. However, when used for multi-annual breeder species, such as the largest carnivores, the most common estimators introduce an upward bias by excluding unproductive females. Using a simulated dataset based on published data, we compare the accuracy of maternity estimates derived from standard methods against estimates derived from an alternative method. We show that standard methods overestimate maternity rates in the presence of unsuccessful pregnancies. Importantly, population growth rates derived from a matrix model parameterized with the biased estimates may indicate increasing populations although the populations are stable or even declining. We recommend the abandonment of the biased standard methods and to instead use the unbiased alternative method for population projections and assessments of population viability.     

A local evaluation of the individual state‐space to scale up Bayesian spatial capture–recapture

Spatial capture–recapture models (SCR) are used to estimate animal density and to investigate a range of problems in spatial ecology that cannot be addressed with traditional nonspatial methods. Bayesian approaches in particular offer tremendous flexibility for SCR modeling. Increasingly, SCR data are being collected over very large spatial extents making analysis computational intensive, sometimes prohibitively so. To mitigate the computational burden of large‐scale SCR models, we developed an improved formulation of the Bayesian SCR model that uses local evaluation of the individual state‐space (LESS). Based on prior knowledge about a species’ home range size, we created square evaluation windows that restrict the spatial domain in which an individual's detection probability (detector window) and activity center location (AC window) are estimated. We used simulations and empirical data analyses to assess the performance and bias of SCR with LESS. LESS produced unbiased estimates of SCR parameters when the AC window width was ≥5σ (σ: the scale parameter of the half‐normal detection function), and when the detector window extended beyond the edge of the AC window by 2σ. Importantly, LESS considerably decreased the computation time needed for fitting SCR models. In our simulations, LESS increased the computation speed of SCR models up to 57‐fold. We demonstrate the power of this new approach by mapping the density of an elusive large carnivore—the wolverine (Gulo gulo)—with an unprecedented resolution and across the species’ entire range in Norway (> 200,000 km²). Our approach helps overcome a major computational obstacle to population and landscape‐level SCR analyses. The LESS implementation in a Bayesian framework makes the customization and fitting of SCR accessible for practitioners working at scales that are relevant for conservation and management.     

The significance threshold for coherence when using the Welch's periodogram method: Effect of overlapping segments

A classical tool to identify coupling mechanisms between two biological signals is transfer function (TF) analysis, commonly estimated by the Welch's periodogram method. However, the reliability of TF estimation depends on the coherence function, an index of linear coupling in the frequency domain. Thus, a threshold value has to be determined before assuming a coupling between two signals at a given frequency.Koopmans (The spectral analysis of time series) proposed a theoretical expression in the case of non-overlapping segments. Overlapping segments, however, improves the estimation of TF.In the present study, a modification of the expression by Koopmans is proposed for taking into account the effect of the overlapping procedure, including the most common case of a 50% overlapping ratio. Simulation tests were done on non-coherent signals (using the Hanning window), and a Kolmogorov–Smirnov test was used to check for the adequacy with the proposed threshold formula. The results showed that for an overlapping ratio of 50%, no statistical difference could be found (P > 0.05), except when using less than seven segments.The method has been applied to the computation of the transfer function between systolic blood pressure and heart rate.     

Spectral Estimation by Variable Span Log Periodogram Smoothing: An Application to Annual Lynx Numbers

Spectral estimation is commonly based on smoothing the periodogram or its logarithm. In such a smoothing exercise, it is not entirely sensible to use the same bandwidth for all frequencies, since the smoothness of the underlying spectrum may vary. This note describes the use of a variable span smoother in estimating the log spectrum. An application to annual lynx data is presented.