Estimation of running frequency spectra using a Kalman filter algorithm

A method is suggested for the computation of running frequency spectra from non-stationary oscillations in a long time series. The method is based on an autoregressive model where the coefficients are assumed to vary slowly. The coefficients are updated using the Kalman filter technique. The method is shown to be superior to ordinary autoregressive spectral estimation based on stationary theory in recognizing rapid changes in the frequencies of oscillations.     

Reanalysis of oviposition and development rates in the Phytoseiidae using a phylogenetic autoregressive method

Life history patterns are usually identified by comparisons of extant species. Because of inferences regarding phylogenetic constraints, comparative data are often not statistically independent. In order to remove phylogenetic patterns embedded in life history data completely, we adopted a phylogenetic autoregressive method to reanalyse a data set of the ovipositional and developmental rates of 45 Phytoseiidae species. We first calculated the phylogenetic correlation in relation to different taxonomic levels using Moran's I statistics. Significant and positive phylogenetic correlations were found at the subgenus and subfamily levels. This indicates that some variation in both of these life-history traits could be accounted for by phylogeny. Phylogenetic associations, therefore, were removed by a phylogenetic autoregressive method. Using corrected data from this method, the specific components of the ovipositional rate are positively correlated with the specific components of th e developmental rate. The method that we have used obtains the same conclusion as others but differs from the phylogenetic effect in the way that it influences the relationship between comparative data. Because of no data reduction in the phylogenetic autoregressive method, the specific components are more useful than the mean values derived from the higher taxonomic nodes for testing ecological and evolutionary hypotheses about life history patterns. © Rapid Science Ltd. 1998     

AN EIGENVECTOR METHOD FOR ESTIMATING PHYLOGENETIC INERTIA

We propose a new method to estimate and correct for phylogenetic inertia in comparative data analysis. The method, called phylogenetic eigenvector regression (PVR) starts by performing a principal coordinate analysis on a pairwise phylogenetic distance matrix between species. Traits under analysis are regressed on eigenvectors retained by a broken-stick model in such a way that estimated values express phylogenetic trends in data and residuals express independent evolution of each species. This partitioning is similar to that realized by the spatial autoregressive method, but the method proposed here overcomes the problem of low statistical performance that occurs with autoregressive method when phylogenetic correlation is low or when sample size is too small to detect it. Also, PVR is easier to perform with large samples because it is based on well-known techniques of multivariate and regression analyses. We evaluated the performance of PVR and compared it with the autoregressive method using real datasets and simulations. A detailed worked example using body size evolution of Carnivora mammals indicated that phylogenetic inertia in this trait is elevated and similarly estimated by both methods. In this example, Type I error at α = 0.05 of PVR was equal to 0.048, but an increase in the number of eigenvectors used in the regression increases the error. Also, similarity between PVR and the autoregressive method, defined by correlation between their residuals, decreased by overestimating the number of eigenvalues necessary to express the phylogenetic distance matrix. To evaluate the influence of cladogram topology on the distribution of eigenvalues extracted from the double-centered phylogenetic distance matrix, we analyzed 100 randomly generated cladograms (up to 100 species). Multiple linear regression of log transformed variables indicated that the number of eigenvalues extracted by the broken-stick model can be fully explained by cladogram topology. Therefore, the broken-stick model is an adequate criterion for determining the correct number of eigenvectors to be used by PVR. We also simulated distinct levels of phylogenetic inertia by producing a trend across 10, 25, and 50 species arranged in “comblike” cladograms and then adding random vectors with increased residual variances around this trend. In doing so, we provide an evaluation of the performance of both methods with data generated under different evolutionary models than tested previously. The results showed that both PVR and autoregressive method are efficient in detecting inertia in data when sample size is relatively high (more than 25 species) and when phylogenetic inertia is high. However, PVR is more efficient at smaller sample sizes and when level of phylogenetic inertia is low. These conclusions were also supported by the analysis of 10 real datasets regarding body size evolution in different animal clades. We concluded that PVR can be a useful alternative to an autoregressive method in comparative data analysis.     

Analyzing multiple spike trains with nonparametric granger causality

Simultaneous recordings of spike trains from multiple single neurons are becoming commonplace. Understanding the interaction patterns among these spike trains remains a key research area. A question of interest is the evaluation of information flow between neurons through the analysis of whether one spike train exerts causal influence on another. For continuous-valued time series data, Granger causality has proven an effective method for this purpose. However, the basis for Granger causality estimation is autoregressive data modeling, which is not directly applicable to spike trains. Various filtering options distort the properties of spike trains as point processes. Here we propose a new nonparametric approach to estimate Granger causality directly from the Fourier transforms of spike train data. We validate the method on synthetic spike trains generated by model networks of neurons with known connectivity patterns and then apply it to neurons simultaneously recorded from the thalamus and the primary somatosensory cortex of a squirrel monkey undergoing tactile stimulation.     

Estimation of changes in instantaneous aortic blood flow by the analysis of arterial blood pressure

The purpose of this study was to introduce and validate a new algorithm to estimate instantaneous aortic blood flow (ABF) by mathematical analysis of arterial blood pressure (ABP) waveforms. The algorithm is based on an autoregressive with exogenous input (ARX) model. We applied this algorithm to diastolic ABP waveforms to estimate the autoregressive model coefficients by requiring the estimated diastolic flow to be zero. The algorithm incorporating the coefficients was then applied to the entire ABP signal to estimate ABF. The algorithm was applied to six Yorkshire swine data sets over a wide range of physiological conditions for validation. Quantitative measures of waveform shape (standard deviation, skewness, and kurtosis), as well as stroke volume and cardiac output from the estimated ABF, were computed. Values of these measures were compared with those obtained from ABF waveforms recorded using a Transonic aortic flow probe placed around the aortic root. The estimation errors were compared with those obtained using a windkessel model. The ARX model algorithm achieved significantly lower errors in the waveform measures, stroke volume, and cardiac output than those obtained using the windkessel model (P < 0.05).     

PHYLOGENIES,SPATIAL AUTOREGRESSION,AND THE COMPARATIVE METHOD: A COMPUTER SIMULATION TEST

Brownian motion computer simulation was used to test the statistical properties of a spatial autoregressive method in estimating evolutionary correlations between two traits using interspecific comparative data. When applied with a phylogeny of 42 species, the method exhibited reasonable Type I and II error rates. Estimation abilities were comparable to those of independent contrasts and minimum evolution (parsimony) methods, and generally superior to a traditional nonphylogenetic approach (not taking phylogenies into account at all). However, the autoregressive method performed extremely poorly with a smaller phylogeny (15 species) and with nearly independent (“star”) phylogenies. In both of these situations, any phylogenetic autocorrelation present in the data was not detected by the method. Results show how diagnostic techniques (e.g., Moran's I) can be useful in detecting and avoiding such situations, but that such techniques should not be used as definitive evidence that phylogenetic correlation is not present in a set of comparative data. The correction factor (α) proposed by Gittleman and Kot (1990) for use in weighting phylogenetic information had little effect in most analyses of 15 or 42 species with incorrect phylogenetic information, and may require much larger sample sizes before significant improvement is shown. With the sample sizes tested in this study, however, the autoregressive method implemented with this correction factor and correct phylogenetic information led to downwardly biased estimates of the absolute magnitude of the evolutionary correlation between two traits. Cautions and recommendations for implemention of the spatial autoregressive method are given; computer programs to conduct the analyses are available on request.     

Stochastic difference equation predictors of population fluctuations.

This paper studies the effectiveness of a class of linear statistical estimators called autoregressive and autoregressive-moving averages equations for mimicking and predicting the abundance fluctuations of three species of Drosophila censused at a pine plantation near Bogota, Colombia. A short introductory justification for the use of linear estimators is given followed by a brief discussion of the theoretical basis of statistical prediction. The assumptions of the method, fitting techniques, and use of the equations in forecasting are discussed. The mimicking ability of the equations is tested by comparing monte carlo simulations employing the fitted models to the observed fluctuations of the three species. Of the autoregressive equations fitted to each species two are judged successful and one less than successful. Autoregressive-moving averages models were found to be significantly worse predictors than the simpler autoregressive equations for these three species. The parameter estimates given by the preliminary estimation techniques are compared with the statistically efficient least squares estimates. The estimates compare well for most of the autoregressive models, but the parameter estimates for the autoregressive-moving averages models were misleading.     

A fast algorithm for the exact likelihood of stationary and partially nonstationary vector autoregressive-moving average processes

An expression for the likelihood function of a stationary vectorautoregressive-moving average process is developed. The expressionis very efficient numerically and applies to any stationarybut not necessarily invertible model. In particular, when themultivariate process is autoregressive, the exact likelihoodcan be evaluated with a small number of operations dependingon the order of the autoregressive operator and the processdimension, but not on the size of the observed series. The expressionalso provides an efficient method for the evaluation of theexact likelihood of a partially nonstationary vector autoregressive-movingaverage process, for which the determinant of the autoregressiveoperator has at least one unit root and the remaining rootsare outside the unit circle. This method does not require differencingthe series, so that complications caused by over-differencingthe series, such as noninvertibility and parameter identifiabilityproblems, are avoided. The results for autoregressive modelsare also applied to testing the stationarity and invertibilityof any autoregressive-moving average model with given parametervalues.     

Nonlinear methods in spectral analysis of NMR signals from living systems

Two autoregressive procedures, namely Burg's maximum entropy method and Prony's method, have been applied to the processing of 31P NMR signals (free induction decays) of human cells. A comparison of the main advantages obtained by using these methods in the field of the NMR spectral analysis of signals from living samples is here illustrated. Improvements in sensitivity and resolution are obtained with respect to Fourier transform method by using only very few data points.     

A vector autoregressive moving average time series approach for describing asymmetries of antennal control of two millipede species

This article presents a statistical method, vector autoregressive moving average time series analysis, which makes no initial assumptions about the controlling interactions between variables in the data beyond those of linear systems, and has been designed to be statistically valid without requiring several repetitions of data sets. It is therefore very useful for studying physiological and behavioral phenomena.This new methodology is applied to a kinematic analysis of antennal scanning movements in two species of millipede. The analysis demonstrates features of the generation of the antennal and head movements, the direction of information flow within the central nervous system and consequent asymmetric control relationships between bilaterally homologous body parts.     

The spectrum analysis of the electromyogram: the adaptation of autoregressive models to the interference electromyogram]

Power-density spectra of interference electromyograms of M. biceps brach. and triceps brach. from two subjects were estimated by the Tukey spectral window and by autoregressive series. By fitting autoregressive models parametric represeentations of spectra are obtained. The models for the electromyograms obtained from biceps (agonist) and triceps (antagonist) differed in the length of the autoregressive series. The intra-individual variation of the spectra from three time intervals of the same contraction was relatively small.     

Comparison of different methods of time shift measurement in EEG

Digital signal processing techniques are often used for measurement of small time shifts between EEG signals. In our work we tested properties of linear cross-correlation and phase/coherence method. The last mentioned method was used in two versions. The first version used fast Fourier transform (FFT) algorithm and the second was based on autoregressive modeling with fixed or adaptive model order. Methods were compared on several testing signals mimicking real EEG signals. The accuracy index for each method was computed. Results showed that for long signal segments all methods bring comparably good results. Accuracy of FFT phase/coherence method significantly decreased when very short segments were used and also decreased with an increasing level of the additive noise. The best results were obtained with autoregressive version of phase/coherence. This method is more reliable and may be used with high accuracy even in very short signals segments and it is also resistant to additive noise.     

On designing the capacity of a reservoir

Two methods of computing the capacity of a reservoir are discussed. In the first method suppose that inputs into the reservoir during successive periods of fixed lengths are independently and identically distributed and that the supplies or releases from it are at the unit rate. Assume that the input is a scarce commodity and that the input rate is less than the supply rate. We then use the mean emptiness period of the reservoir for computing its capacity. In particular the capacity has been determined for geometric, binomial and exponential inputs. The second method is that of simulation for gamma type Markov autoregressive inputs and content-dependent supplies.     

Partial directed coherence: a new concept in neural structure determination

 This paper introduces a new frequency-domain approach to describe the relationships (direction of information flow) between multivariate time series based on the decomposition of multivariate partial coherences computed from multivariate autoregressive models. We discuss its application and compare its performance to other approaches to the problem of determining neural structure relations from the simultaneous measurement of neural electrophysiological signals. The new concept is shown to reflect a frequency-domain representation of the concept of Granger causality. Received: 25 April 2000 / Accepted in revised form: 13 November 2000     

The use of a multi-channel kalman filter algorithm in structural analysis of the epileptic EEG

An extension of the Kalman filter algorithm to the multi-channel case is presented and its application as a segmenting procedure in the analysis of the epileptic EEG is discussed. An analytical example of structural analysis, using the segments extracted by the proposed filter, is presented for a particular set of 4-channel EEG recordings. This analysis is shown to be especially fruitful if the autoregressive coefficients - a by product of the filtering procedure - are used to estimate the information flow between the channels by the calculation of partial as well as directed coherences for the representative segments.     

Time domain analysis of diaphragmatic electromyogram during fatigue in men

Diaphragmatic fatigue has been correlated with a change in the electromyogram recorded from the diaphragm (EMGdi), which suggests that the electromyogram is a potential clinical tool to detect respiratory muscle fatigue. Changes in the EMGdi have previously been quantified by using the power spectral parameters high-low ratio or mean frequency. In this study, we developed an autoregressive model of the EMG in an attempt to improve the analysis of the EMGdi. This model was tested on recordings of the EMGdi that were obtained from an esophageal electrode in five normal subjects breathing to fatigue through an inspiratory resistor. The data obtained from the autoregressive model were directly compared with data from the high-low ratio and mean frequency techniques. The autoregressive model showed an excellent correlation with mean frequency. Both techniques were superior to the high-low ratio measurement. Because the autoregressive model requires much less computation than mean frequency and can be easily implemented in real time on a minicomputer, we propose this as a preferable approach.     

Conditionally autoregressive models improve occupancy analyses of autocorrelated data: An example with environmental DNA

Site occupancy‐detection models (SODMs) are statistical models widely used for biodiversity surveys where imperfect detection of species occurs. For instance, SODMs are increasingly used to analyse environmental DNA (eDNA) data, taking into account the occurrence of both false‐positive and false‐negative errors. However, species occurrence data are often characterized by spatial and temporal autocorrelation, which might challenge the use of standard SODMs. Here we reviewed the literature of eDNA biodiversity surveys and found that most of studies do not take into account spatial or temporal autocorrelation. We then demonstrated how the analysis of data with spatial or temporal autocorrelation can be improved by using a conditionally autoregressive SODM, and show its application to environmental DNA data. We tested the autoregressive model on both simulated and real data sets, including chronosequences with different degrees of autocorrelation, and a spatial data set on a virtual landscape. Analyses of simulated data showed that autoregressive SODMs perform better than traditional SODMs in the estimation of key parameters such as true‐/false‐positive rates and show a better discrimination capacity (e.g., higher true skill statistics). The usefulness of autoregressive SODMs was particularly high in data sets with strong autocorrelation. When applied to real eDNA data sets (eDNA from lake sediment cores and freshwater), autoregressive SODM provided more precise estimation of true‐/false‐positive rates, resulting in more reasonable inference of occupancy states. Our results suggest that analyses of occurrence data, such as many applications of eDNA, can be largely improved by applying conditionally autoregressive specifications to SODMs.     

Short-window spectral analysis using AMVAR and multitaper methods: a comparison

We compare two popular methods for estimating the power spectrum from short data windows, namely the adaptive multivariate autoregressive (AMVAR) method and the multitaper method. By analyzing a simulated signal (embedded in a background Ornstein–Uhlenbeck noise process) we demonstrate that the AMVAR method performs better at detecting short bursts of oscillations compared to the multitaper method. However, both methods are immune to jitter in the temporal location of the signal. We also show that coherence can still be detected in noisy bivariate time series data by the AMVAR method even if the individual power spectra fail to show any peaks. Finally, using data from two monkeys performing a visuomotor pattern discrimination task, we demonstrate that the AMVAR method is better able to determine the termination of the beta oscillations when compared to the multitaper method.