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.     

A semi-parametric approach for comparing the estimated spectra of two stationary point processes

In this paper, we provide a semi-parametric test for the hypothesis that the spectra of two stationary point processes (SPPs) are the same. The estimates of the second-order spectral density functions of the SPPs are obtained by using two different approaches: (a) by smoothing the modified periodogram statistics using a moving average weighting scheme, (b) by employing the Welch's method on the modified periodogram statistics. The test is based on the likelihood ratio function under two alternative hypotheses. The first alternative hypothesis suggests that the ratio of the spectra is a constant (there is a shift to the power) while the second one adopts a quadratic model for the logarithmic ratio of the spectra. A comparison with a log-linear model indicates coincidence in the results. This is explained by using an illustrative example from the field of neurophysiology. It is shown that the information transferred to the spinal cord by the sensory axons, closely related with the complex physiological system called muscle spindle, under the influence of two different stimuli can be separated in two parts. The first part corresponds to the range of frequencies 0-19 Hz while the second to the range of frequencies 19-100 Hz. However, the dependence on the frequency in both parts is of a quadratic form.     

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.     

Oscillations in cell morphology and redox state

Fluctuations in the intensity of light scattered and absorbed by cells in suspension have been analysed by smoothing, periodogram and power spectrum methods to reveal oscillations attributed to changes in cell morphology and the redox state of NADH and FAD (periods 10 s to 30 min). The rhythms are themselves periodically modulated in amplitude at a similar frequency and exhibit burst characteristics. The low frequency scatter dynamics are provisionally attributed to oscillations in gross morphology and the high frequency variation to changes at the cell surface. Agents, such as insulin and transferrin, affect the dynamics. The scatter results suggest that rhythmic changes in cell morphology associated with locomotion are largely inherent in the cell and not due to periodic attachment and detachment from a surface.     

Evolutionary factor analysis of replicated time series

Summary In this article, we develop a novel method that explains the dynamic structure of multi-channel electroencephalograms (EEGs) recorded from several trials in a motor-visual task experiment. Preliminary analyses of our data suggest two statistical challenges. First, the variance at each channel and cross-covariance between each pair of channels evolve over time. Moreover, the cross-covariance profiles display a common structure across all pairs, and these features consistently appear across all trials. In the light of these features, we develop a novel evolutionary factor model (EFM) for multi-channel EEG data that systematically integrates information across replicated trials and allows for smoothly time-varying factor loadings. The individual EEGs series share common features across trials, thus, suggesting the need to pool information across trials, which motivates the use of the EFM for replicated time series. We explain the common co-movements of EEG signals through the existence of a small number of common factors. These latent factors are primarily responsible for processing the visual-motor task which, through the loadings, drive the behavior of the signals observed at different channels. The estimation of the time-varying loadings is based on the spectral decomposition of the estimated time-varying covariance matrix.     

Variance matters: The shape of a datum

In the quantitative analysis of behaviour, choice data are most often plotted and analyzed as logarithmic transforms of ratios of responses and of ratios of reinforcers according to the generalized-matching relation, or its derivatives such as conditional-discrimination models. The relation between log choice ratios and log reinforcer ratios has normally been found using ordinary linear regression, which minimizes the sums of the squares of the y deviations from the fitted line. However, linear regression of this type requires that the log choice data be normally distributed, of equal variance for each log reinforcer ratio, and that the x (log reinforcer ratio) measures be fixed with no variance. We argue that, while log transformed choice data may be normally distributed, log reinforcer ratios do have variance, and because these measures derive from a binomial process, log reinforcer ratio distributions will be non-normal and skewed to more extreme values. These effects result in ordinary linear regression systematically underestimating generalized-matching sensitivity values, and in faulty parameter estimates from non-linear regression to assume hyperbolic and exponential decay processes. They also lead to model comparisons, which assume equal normally distributed error around every data point, being incorrect. We describe an alternative approach that can be used if the variance in choice is measured.     

Calculating the distribution of ventilation-perfusion ratios from inert gas elimination data

It is well known that the major cause of hypoxemia in lung disease is ventilation-perfusion (VA/Q) inequality, but it has been extremely difficult to measure the distribution of ventilation-perfusion ratios except in terms of unrealistically simple (albeit useful) models. The multiple inert gas elimination technique provides considerable information concerning the shape, position, and dispersion of the VA/Q distribution, although it cannot precisely define all features of the distribution. Although there are many techniques for obtaining information about the distribution from inert gas elimination data, we have found the most flexible and useful approach to be a multicomponent analysis with enforced smoothing, sometimes known as ridge regression. This presentation describes in some detail the physiological and mathematical principles principles involved in the transformation of inert gas elimination data into a representative distribution of ventilation-perfusion ratios by enforced smoothing techniques. It is important to realize that with this approach and any other approach aimed at estimating the distribution of ventilation-perfusion ratios, the results must be properly interpreted.     

Confidence Regions for Spectral Peak Frequencies

A procedure is proposed to obtain confidence regions for spectral peak frequencies. The method is based on resampling the periodogram from the estimated spectrum in order to reestimate the spectrum and its peak frequency. We investigate the dependence of the results from the applied spectral estimator in three simulation studies and apply the method to tremor data.     

Animal movement rates as behavioural bouts

Johnson et al . ( Journal of Animal Ecology , 2002, 71 , 225–235) have proposed a new technique for identifying scales of movement in animals. Animals are located at certain time intervals, and movement rates between successive animal relocations are calculated. The null model of a nonscalar response predicts a decreasing linear relationship between log(frequency) vs. movement rate, while a scalar response predicts a monotonically decreasing curve with an inflection point at the separation between the processes. I tested this technique using three types of simulated movement paths: correlated random walks, directed walks, and movements in patchy habitat. None of the simulations showed the results expected by the technique. This occurs because the technique assumes that movement rates are exponentially distributed, which is highly unlikely. Thus before this technique can be applied to animal movement data we need to understand how spatial and temporal scale, as well as sampling interval, affect the frequency histogram of animal movement rates.     

Electroencephalography in the diagnosis of hydrocephalus in golden hamsters (Mesocricetus auratus)

The golden hamster (Mesocricetus auratus) is a popular laboratory animal and is used in a multitude of behavioural studies. However, it has been shown that it suffers from different forms of hereditary hydrocephalus, which may result in behavioural changes. This prospective study was designed to look into the usefulness of electroencephalography (EEG) measurements in the diagnosis of hydrocephalus in hamsters. The EEGs of the hydrocephalic hamsters were evaluated double-blind and showed a high-voltage slow wave activity, with a fast activity superimposed onto it. This pattern has already been well described in other hydrocephalic species and differed significantly from the EEGs that were obtained from the normal hamsters. It was concluded from our study that a background activity with an amplitude over 50 muV in combination with a frequency of < or =5 Hz was highly indicative of hydrocephalus in young hamsters. We believe that the EEG could be a very useful diagnostic tool in the screening for hydrocephalus in hamsters.     

Some Tools to Analyze Changes of Rhythms in Biological Time Series

The moving window principle applied to the khi-square periodogram allows, through local successive examinations, a comprehensive study of the biological time series. This method puts forward several cases of transition linked to environmental or physiological changes. Furthermore, we applied the Grassberger and Procaccia method (1983) for the analysis of more complex transition problems. The method helps to detect chaotic properties in behavioral activity rhythms.     

Some Tools to Analyze Changes of Rhythms in Biological Time Series

The moving window principle applied to the khi-square periodogram allows, through local successive examinations, a comprehensive study of the biological time series. This method puts forward several cases of transition linked to environmental or physiological changes. Furthermore, we applied the Grassberger and Procaccia method (1983) for the analysis of more complex transition problems. The method helps to detect chaotic properties in behavioral activity rhythms.     

Using semiparametric‐mixed model and functional linear model to detect vulnerable prenatal window to carcinogenic polycyclic aromatic hydrocarbons on fetal growth

Prenatal exposure to carcinogenic polycyclic aromatic hydrocarbons (c‐PAHs) through maternal inhalation induces higher risk for a wide range of fetotoxic effects. However, the most health‐relevant dose function from chronic gestational exposure remains unclear. Whether there is a gestational window during which the human embryo/fetus is particularly vulnerable to PAHs has not been examined thoroughly. We consider a longitudinal semiparametric‐mixed effect model to characterize the individual prenatal PAH exposure trajectory, where a nonparametric cyclic smooth function plus a linear function are used to model the time effect and random effects are used to account for the within‐subject correlation. We propose a penalized least squares approach to estimate the parametric regression coefficients and the nonparametric function of time. The smoothing parameter and variance components are selected using the generalized cross‐validation (GCV) criteria. The estimated subject‐specific trajectory of prenatal exposure is linked to the birth outcomes through a set of functional linear models, where the coefficient of log PAH exposure is a fully nonparametric function of gestational age. This allows the effect of PAH exposure on each birth outcome to vary at different gestational ages, and the window associated with significant adverse effect is identified as a vulnerable prenatal window to PAHs on fetal growth. We minimize the penalized sum of squared errors using a spline‐based expansion of the nonparametric coefficient function to draw statistical inferences, and the smoothing parameter is chosen through GCV.     

Dynamic logistic state space prediction model for clinical decision making

Prediction modeling for clinical decision making is of great importance and needed to be updated frequently with the changes of patient population and clinical practice. Existing methods are either done in an ad hoc fashion, such as model recalibration or focus on studying the relationship between predictors and outcome and less so for the purpose of prediction. In this article, we propose a dynamic logistic state space model to continuously update the parameters whenever new information becomes available. The proposed model allows for both time-varying and time-invariant coefficients. The varying coefficients are modeled using smoothing splines to account for their smooth trends over time. The smoothing parameters are objectively chosen by maximum likelihood. The model is updated using batch data accumulated at prespecified time intervals, which allows for better approximation of the underlying binomial density function. In the simulation, we show that the new model has significantly higher prediction accuracy compared to existing methods. We apply the method to predict 1 year survival after lung transplantation using the United Network for Organ Sharing data.     

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.     

Modeling longitudinal data with ordinal response by varying coefficients

This paper presents a smooth regression model for ordinal data with longitudinal dependence structure. A marginal model with cumulative logit link is applied to cope with the ordinal scale and the main and covariate effects in the model are allowed to vary with time. Local fitting is pursued and asymptotic properties of the estimates are discussed. In a second step, the longitudinal dependence of the observations is considered. Cumulative log odds ratios are fitted locally, which allows investigation of how the longitudinal dependence of the ordinal observations changes with time.     

Nutrient–chlorophyll relationships in tropical–subtropical lakes: do temperate models fit?

In tropical lakes relatively little is known about the general relationship between nutrient concentration and phytoplankton biomass. Using data from 192 lakes from tropical and subtropical regions we examine the relationship between total P (TP) and chlorophyll (Chl). The lakes are all located between 30° S to 31° N include systems in Asia, Africa, and North and South America but are dominated by Brazilian (n=79) and subtropical N. American (n=67) systems. The systems vary in morphometry (mean depth and lake area), trophic state as well total N (TN) to␣total P (TP) ratios and light extinction. Despite a nearly 500-fold range in TP concentrations (2–970 μg P l⁻¹), there was a poorer relationship between log TP and log Chl (r ²=0.42) than is generally observed for temperate systems from either narrow or broad geographic regions. N limitation is not a likely explanation for the relatively weak TP–Chl relationship in the tropical–subtropical systems. Systems had high average TN:TP ratios and neither a multiple regression with log TP and log TN nor separating systems with high TN:TP (>17 by weight) improved the predictive power of the log TP–log Chl relationship.     

High-frequency component of hippocampal theta activity

The signal corresponding to the second frequency group in spectra of the theta activity of a rat (a twofold theta-frequency harmonic with adjoning frequency components) and its regulation were studied. The level of the signal of the second frequency group was high in the theta activity diring natural orienting behavior and in desynchronized EEG during waking immobility. In some rats, the EEGs segments recorded during orienting behavior evoked by reticular stimulation contained an enhanced signal of the second frequency group (as compared to the signal of the first frequency group). The second frequency group could be also observed in the absence of the first. Estimation of signal dimension revealed two independent components in the theta activity with the high-level signal of the second frequency group. Time correlation between variations of signals of the first and second frequency groups in the theta activity accompanying natural orienting behavior was absent. These signals changed in different ways during transition from awake immobility of an animal to orienting behavior. It was concluded that in addition to the main component of the hippocampal theta activity, there exists the independently regulated twofold theta-freduency component.     

Bayesian semiparametric intensity estimation for inhomogeneous spatial point processes

In this work we propose a fully Bayesian semiparametric method to estimate the intensity of an inhomogeneous spatial point process. The basic idea is to first convert intensity estimation into a Poisson regression setting via binning data points on a regular grid, and then model the log intensity semiparametrically using an adaptive version of Gaussian Markov random fields to smooth the corresponding counts. The inference is carried by an efficient Markov chain Monte Carlo simulation algorithm. Compared to existing methods for intensity estimation, for example, parametric modeling and kernel smoothing, the proposed estimator not only provides inference regarding the dependence of the intensity function on possible covariates, but also uses information from the data to adaptively determine the amount of smoothing at the local level. The effectiveness of using our method is demonstrated through simulation studies and an application to a rainforest dataset.