Autocorrelation Estimates Research Articles

On Outliers and Interventions in Count Time Series following GLMs

We discuss the analysis of count time series following generalised linear models in the presence of outliers and intervention effects. Different modifications of such models are formulated which allow to incorporate, detect and to a certain degree distinguish extraordinary events (interventions) of different types in count time series retrospectively. An outlook on extensions to the problem of robust parameter estimation, identification of the model orders by robust estimation of autocorrelations and partial autocorrelations, and online surveillance by sequential testing for outlyingness is provided.

Maximum likelihood Doppler frequency estimation under decorrelation noise for quantifying flow in optical coherence tomography.

Recent hardware advances in optical coherence tomography (OCT) have led to ever higher A-scan rates. However, the estimation of blood flow axial velocities is limited by the presence and type of noise. Higher acquisition rates alone do not necessarily enable precise quantification of Doppler velocities, particularly if the estimator is suboptimal. In previous work, we have shown that the Kasai autocorrelation estimator is statistically suboptimal under conditions of additive white Gaussian noise. In addition, for practical OCT measurements of flow, decorrelation noise affects Doppler frequency estimation by broadening the signal spectrum. Here, we derive a general maximum likelihood estimator (MLE) for Doppler frequency estimation that takes into account additive white noise as well as signal decorrelation. We compare the decorrelation MLE with existing techniques using simulated and flow phantom data and find that it has better performance, achieving the Cramer-Rao lower bound. By making an approximation, we also provide an interpretation of this method in the Fourier domain. We anticipate that this estimator will be particularly suited for estimating blood flow in in vivo scenarios.

The Autocorrelation Spectral Density for Doppler-Weather-Radar Signal Analysis

Time-domain autocovariance processing is widely accepted as a computationally efficient method to estimate the first three spectral moments of Doppler weather radar signals (i.e., mean signal power, mean Doppler velocity, and spectrum width). However, when signals with different frequency content (e.g., ground clutter) contaminate the weather signal, spectral processing using the periodogram estimator of the power spectral density (PSD) is the preferred tool of analysis. After spectral processing (i.e., filtering), a PSD-based autocorrelation estimator is typically employed to produce unbiased estimates of the weather-signal spectral moments. However, the PSD does not convey explicit phase information, which has the potential to aid in the spectral analysis of radar signals. In this paper, the autocorrelation spectral density (ASD) is introduced for spectral analysis of weather-radar signals as a generalization of the classical PSD, and an ASD-based autocorrelation estimator is proposed to produce unbiased estimates of the weather-signal spectral moments. A significant advantage of the ASD over the PSD is that it provides explicit phase information that can be exploited to identify and remove certain types of contaminant signals. Thus, the ASD provides an alternative means for spectral analysis, which can lead to improved quality of meteorological data from weather radars.

New Estimators and Guidelines for Better Use of Fetal Heart Rate Estimators with Doppler Ultrasound Devices

Characterizing fetal wellbeing with a Doppler ultrasound device requires computation of a score based on fetal parameters. In order to analyze the parameters derived from the fetal heart rate correctly, an accuracy of 0.25 beats per minute is needed. Simultaneously with the lowest false negative rate and the highest sensitivity, we investigated whether various Doppler techniques ensure this accuracy. We found that the accuracy was ensured if directional Doppler signals and autocorrelation estimation were used. Our best estimator provided sensitivity of 95.5%, corresponding to an improvement of 14% compared to the standard estimator.

A standardized approach to estimate life history tradeoffs in evolutionary ecology

As tradeoffs limit the maximum Darwinian fitness individuals can reach, measuring reliably the strength of tradeoffs using appropriate metrics is of prime importance to understand the evolution of traits under constraints. Tradeoffs involving phenotypic traits and fitness components, however, are difficult to quantify in free‐ranging populations because of confounding effects due to environmental variation and individual heterogeneity. Furthermore, although some methods have been used previously to quantify tradeoffs, these methods cannot be applied with respect to binary traits, which are common to describe life histories (e.g. probability of reproduction, nesting success, offspring survival). Here, we demonstrate how to measure reliably the strength of tradeoffs involving binary traits using (auto)correlation estimates obtained from generalized linear (mixed) models. We first propose a standardized approach that accounts for the variation in the nature of the tradeoffs being compared (e.g. continuous/binary traits, repeated/non‐repeated measures), and then apply this method to longitudinal data from two contrasting species of large herbivores. Empirical estimates of tradeoffs varied among traits, and between‐species comparisons suggested that reproductive tradeoffs between successive breeding attempts might only occur in capital breeders. The empirical results we obtained clearly demonstrate that the method we provide allows measuring reliably the strength of tradeoffs under most circumstances, including tradeoffs on binary traits. Our original approach therefore offers an important first step for comparing the strength and, hence, the relative importance of different tradeoffs, and opens the door to a better understanding of the evolution of life history traits in free‐ranging populations.

Asymptotically Unbiased Estimation of Autocovariances and Autocorrelations with Panel Data in the Presence of Individual and Time Effects

AbstractThis article proposes asymptotically unbiased estimators of autocovariances and autocorrelations for panel data with both individual and time effects. We show that the conventional autocovariance estimators suffer from the bias caused by the elimination of individual and time effects. The bias related to individual effects is proportional to the long-run variance, and it related to time effects is proportional to the value of the estimated autocovariance. For the conventional autocorrelation estimators, the elimination of time effects does not cause a bias while the elimination of individual effects does. We develop methods to estimate the long-run variance and propose bias-corrected estimators based on the proposed long-run variance estimator. We also consider the half-panel jackknife estimation for bias correction. The theoretical results are given by employing double asymptotics under which both the number of observations and the length of the time series tend to infinity. Monte Carlo simulations show that the asymptotic theory provides a good approximation to the actual bias and that the proposed bias-correction methods work well.

Asymptotically Unbiased Estimation of Autocovariances and Autocorrelations with Panel Data in the Presence of Individual and Time Effects

This paper proposes asymptotically unbiased estimators of autocovariances and autocorrelations for panel data with both individual and time effects. We show that the conventional autocovariance estimators suffers from the bias caused by the elimination of individual and time effects. The bias related to individual effects is proportional to the long-run variance, and that related to time effects is proportional to the value of the estimated autocovariance. For the conventional autocorrelation estimators, the elimination of time effects does not cause a bias while the elimination of individual effects does. We develop methods to estimate the long-run variance and propose bias-corrected estimators based on the proposed long-run variance estimator. The theoretical results are given by employing double asymptotics under which both the number of observations and the length of the time series tend to infinity. Monte Carlo simulations show that the asymptotic theory provides a good approximation to the actual bias and that the proposed bias correction works.

Nonparametric Multicycle Spectrum Sensing Method by Segmented Data Processing for Cognitive Radio

This paper proposes a novel multicycle spectrum sensing method. The method first divides the received data samples into several segments, and then gets cyclic autocorrelation estimates from these segments, constructs a test statistic from these estimates, and finally makes an F-test to determine whether there is a primary signal or not. The method can use multiple cyclic frequencies simultaneously; hence, it is a multicycle detector. The method is also nonparametric, and thus it is robust to noise uncertainty. Compared with existing cyclic spectrum sensing methods, the proposed method can reduce computational complexity significantly at the cost of a little performance loss. Simulation results are given to show the validity and the superiority of the proposed method.

Comparison of Kasai Autocorrelation and Maximum Likelihood Estimators for Doppler Optical Coherence Tomography

In optical coherence tomography (OCT) and ultrasound, unbiased Doppler frequency estimators with low variance are desirable for blood velocity estimation. Hardware improvements in OCT mean that ever higher acquisition rates are possible, which should also, in principle, improve estimation performance. Paradoxically, however, the widely used Kasai autocorrelation estimator's performance worsens with increasing acquisition rate. We propose that parametric estimators based on accurate models of noise statistics can offer better performance. We derive a maximum likelihood estimator (MLE) based on a simple additive white Gaussian noise model, and show that it can outperform the Kasai autocorrelation estimator. In addition, we also derive the Cramer Rao lower bound (CRLB), and show that the variance of the MLE approaches the CRLB for moderate data lengths and noise levels. We note that the MLE performance improves with longer acquisition time, and remains constant or improves with higher acquisition rates. These qualities may make it a preferred technique as OCT imaging speed continues to improve. Finally, our work motivates the development of more general parametric estimators based on statistical models of decorrelation noise.

Statistical Properties of Microstructure Noise

We study the estimation of moments and joint moments of microstructure noise based on high frequency data, under a nonparametric setting which allows the underlying price process to have jumps, the observation times to be irregularly spaced, and the noise to be dependent on the price process and to have diurnal features. Estimators of arbitrary orders of (joint) moments are provided, for which we establish consistency as well as central limit theorems. In particular, we provide estimators of autocovariances and autocorrelations of the noise. Simulation studies demonstrate excellent performance of our estimators in the presence of jumps, irregular observation times, and even rounding. Empirical studies reveal (moderate) positive autocorrelations of microstructure noise for the stocks tested.

Objective estimation versus subjective perceptions of earnings patterns and post‐earnings‐announcement drift

AbstractWe investigate how the market's subjective estimates of autocorrelation in quarterly earnings vary with objective time‐series estimates. Our results suggest that investors increasingly underestimate the correlation as the autocorrelation level increases, and as a result, the post‐earnings‐announcement drift (PEAD) increases with the level of autocorrelation. We further show that the ability of autocorrelation to explain variation in the PEAD is robust to alternative explanations based on risk and institutional factors. Additional analysis indicates that the market's inefficiency in assessing the existence and magnitude of autocorrelation (and the related impact on PEAD) is inversely related to the richness of the information environment.

Bayesian estimates of autocorrelations in single-case designs

Researchers in the single-case design tradition have debated the size and importance of the observed autocorrelations in those designs. All of the past estimates of the autocorrelation in that literature have taken the observed autocorrelation estimates as the data to be used in the debate. However, estimates of the autocorrelation are subject to great sampling error when the design has a small number of time points, as is typically the situation in single-case designs. Thus, a given observed autocorrelation may greatly over- or underestimate the corresponding population parameter. This article presents Bayesian estimates of the autocorrelation that greatly reduce the role of sampling error, as compared to past estimators. Simpler empirical Bayes estimates are presented first, in order to illustrate the fundamental notions of autocorrelation sampling error and shrinkage, followed by fully Bayesian estimates, and the difference between the two is explained. Scripts to do the analyses are available as supplemental materials. The analyses are illustrated using two examples from the single-case design literature. Bayesian estimation warrants wider use, not only in debates about the size of autocorrelations, but also in statistical methods that require an independent estimate of the autocorrelation to analyze the data.

Multilevel models for multiple-baseline data: modeling across-participant variation in autocorrelation and residual variance

Multilevel models (MLM) have been used as a method for analyzing multiple-baseline single-case data. However, some concerns can be raised because the models that have been used assume that the Level-1 error covariance matrix is the same for all participants. The purpose of this study was to extend the application of MLM of single-case data in order to accommodate across-participant variation in the Level-1 residual variance and autocorrelation. This more general model was then used in the analysis of single-case data sets to illustrate the method, to estimate the degree to which the autocorrelation and residual variances differed across participants, and to examine whether inferences about treatment effects were sensitive to whether or not the Level-1 error covariance matrix was allowed to vary across participants. The results from the analyses of five published studies showed that when the Level-1 error covariance matrix was allowed to vary across participants, some relatively large differences in autocorrelation estimates and error variance estimates emerged. The changes in modeling the variance structure did not change the conclusions about which fixed effects were statistically significant in most of the studies, but there was one exception. The fit indices did not consistently support selecting either the more complex covariance structure, which allowed the covariance parameters to vary across participants, or the simpler covariance structure. Given the uncertainty in model specification that may arise when modeling single-case data, researchers should consider conducting sensitivity analyses to examine the degree to which their conclusions are sensitive to modeling choices.

Statistical evaluation of parameters estimating autocorrelation and individual heterogeneity in longitudinal studies

Summary1. Autocorrelation and individual heterogeneity are now considered to reflect biological processes rather than simply being a nuisance requiring to be accounted for. Before using parameter estimates that represent autocorrelation and individual heterogeneity to infer biological processes, a statistical evaluation of their precision and accuracy is required to validate their use.2. Using simulated data, we evaluated accuracy and precision of temporal autocorrelation and individual heterogeneity estimates provided by different statistical models. We compared estimates across different intensity of individual variation and life histories, and sampling effort. We focused on recurrent binary variables because statistical evaluations of models describing binary processes have often been overlooked although several evolutionary and ecological processes are expressed as binary variables (e.g. probability of annual reproduction, plant annual flowering and detection, seasonal migration decision).3. Our results showed that autocorrelation and individual heterogeneity were generally better estimated using a ‘time series’ modelling approach, but that a ‘state dependence’ modelling approach also provided fair estimates in most cases. The latter method was even more robust when data sets included missing values. Data sets including missing values or consisting of very short times series resulted in important bias in some instances.4. Models ignoring either individual heterogeneity or autocorrelation performed poorly, illustrating the fundamental association between these two processes, and demonstrating that the complex structure of autocorrelation and individual heterogeneity patterns is difficult to tackle using simple models.5. Our work’s major finding is the demonstration that autocorrelation and individual heterogeneity need to be both accounted for to provide reliable estimates even in studies focusing on only one of these processes. Our study also offers a set of practical recommendations for helping researchers modelling these two processes depending on their scientific aims and the structure of their data. Finally, our results illustrate that more research is required for estimating individual heterogeneity when positive temporal autocorrelation is expected because none of the models evaluated provided suitable estimates.

U.S. stock market crash risk, 1926–2010

This paper examines how well alternate time-changed Lévy processes capture stochastic volatility and the substantial outliers observed in U.S. stock market returns over the past 85 years. The autocorrelation of daily stock market returns varies substantially over time, necessitating an additional state variable when analyzing historical data. I estimate various one- and two-factor stochastic volatility/Lévy models with time-varying autocorrelation via extensions of the Bates (2006) methodology that provide filtered daily estimates of volatility and autocorrelation. The paper explores option pricing implications, including for the Volatility Index (VIX) during the recent financial crisis.

An Online Adaptive Cooperation Scheme for Spectrum Sensing Based on a Second-Order Statistical Method

Spectrum sensing is one of the most important features of cognitive radio (CR) systems. Although spectrum sensing can be performed by a single CR, it is shown in the literature that cooperative techniques, including multiple CRs/sensors, improve the performance and reliability of spectrum sensing. Existing cooperation techniques usually assume a static communication scenario between the unknown source and sensors along with a fixed propagation environment class. In this paper, an online adaptive cooperation scheme is proposed for spectrum sensing to maintain the level of sensing reliability and performance under changing channel and environmental conditions. Each cooperating sensor analyzes second-order statistics of the received signal, which undergoes both correlated fast and slow fading. Autocorrelation estimation data from sensors are fused together by an adaptive weighted linear combination at the fusion center. Weight update operation is performed online through the use of orthogonal projection onto convex sets. Numerical results show that the performance of the proposed scheme is maintained for dynamically changing characteristics of the channel between an unknown source and sensors, even under different physical propagation environments. In addition, it is shown that the proposed cooperative scheme, which is based on second-order detectors, yields better results compared with the same fusion mechanism that is based on conventional energy detectors.

Cyclostationarity-based soft cooperative spectrum sensing for cognitive radio networks

Reliable detection of primary users (PUs) in the presence of interference and noise is a crucial problem in cognitive radio networks. To address the above issue, cooperative spectrum sensing based on cyclostationary feature detection that can robustly detect weak primary signals has been proposed in the literature. Among different cooperative techniques, in this study the authors focus on combination of soft decisions derived based on asymptotic properties of cyclic autocorrelation estimates. The objective is to maximise deflection coefficient performance metric at the fusion centre, where linear combination of cyclostationary soft decisions is performed. Since the proposed method allows for distributed cyclostationarity spectrum sensing, it is more reliable and faster than non-cooperative traditional multi-cycle cyclostationarity detection schemes. To reduce the computational complexity of the exact distribution of proposed test statistic at fusion centre, the authors derive efficient approximations for the distribution under null and alternative hypotheses. Extensive simulation results in different scenarios demonstrate the advantage of the proposed method and confirm the analytic performance characterisations. In addition, the authors study the impact of mobility of cognitive devices on the cyclostationarity of received signals and verify our analysis via simulation.

Recurrent statistics of nonstationary time series

Basic statistics of a nonstationary time series are estimated from its single realization. The estimates are represented in the form of a recurrent procedure forming residual time series and smoothing them with the help of effective models of digital data filtering or a locally weighted polynomial regression. The concept of local time weighting and robust weighting of residual series is generalized on the basis of a rational combination of models of distance-weighted least squares and an exponentially weighted regression. The estimation of trends, volatility, and autocorrelation for time series of companies’ sales volumes and the price dynamics of stock assets is simulated, and the results of simulation are presented.

Abstract: A Meta-Analysis of the Autocorrelation in Single Case Designs

Single case design (SCD) experiments in the behavioral sciences utilize just one participant from whom data is collected over time. This design permits causal inferences to be made regarding various intervention effects, often in clinical or educational settings, and is especially valuable when between-participant designs are not feasible or when interest lies in the effects of an individualized treatment. Regression techniques are the most common quantitative practice for analyzing time series data and provide parameter estimates for both treatment and trend effects. However, the presence of serially correlated residuals, known as autocorrelation, can severely bias inferences made regarding these parameter estimates. Despite the severity of the issue, few researchers test or correct for the autocorrelation in their analyses. Shadish and Sullivan (in press) recently conducted a meta-analysis of over 100 studies in order to assess the prevalence of the autocorrelation in the SCD literature. Although they found that the meta-analytic weighted average of the autocorrelation was close to zero, the distribution of autocorrelations was found to be highly heterogeneous. Using the same set of SCDs, the current study investigates various factors that may be related to the variation in autocorrelation estimates (e.g., study and outcome characteristics). Multiple moderator variables were coded for each study and then used in a metaregression in order to estimate the impact these predictor variables have on the autocorrelation. This current study investigates the autocorrelation using a multilevel meta-analytic framework. Although meta-analyses involve nested data structures (e.g., effect sizes nested within studies nested within journals), there are few instances of meta-analysts utilizing multilevel frameworks with more than two levels. This is likely attributable to the fact that very few software packages allow for meta-analyses to be conducted with more than two levels and those that do allow this provide sparse documentation on how to implement these models. The proposed presentation discusses methods for carrying out a multilevel meta-analysis. The presentation also discusses the findings from the metaregression on the autocorrelation and the implications these findings have on SCDs.

Inferential Precision in Single-Case Time-Series Data Streams: How Well Does the EM Procedure Perform When Missing Observations Occur in Autocorrelated Data?

The case-based time-series design is a viable methodology for treatment outcome research. However, the literature has not fully addressed the problem of missing observations with such autocorrelated data streams. Mainly, to what extent do missing observations compromise inference when observations are not independent? Do the available missing data replacement procedures preserve inferential integrity? Does the extent of autocorrelation matter? We use Monte Carlo simulation modeling of a single-subject intervention study to address these questions. We find power sensitivity to be within acceptable limits across four proportions of missing observations (10%, 20%, 30%, and 40%) when missing data are replaced using the Expectation-Maximization Algorithm, more commonly known as the EM Procedure (Dempster, Laird, & Rubin, 1977). This applies to data streams with lag-1 autocorrelation estimates under 0.80. As autocorrelation estimates approach 0.80, the replacement procedure yields an unacceptable power profile. The implications of these findings and directions for future research are discussed.