Short-range Dependent Processes Research Articles

Huber-Dutter estimation of linear models with dependent errors

In this article, we investigate a linear model that incorporates stationary causal processes, with a focus on utilizing Huber-Dutter methods to investigate the estimators of unknown parameters and a scale for the errors. Our results indicate that the Huber-Dutter methods can effectively be utilized for linear models that feature errors, which are short-range dependent linear processes, heavy-tailed linear processes, as well as some commonly used non linear time series.

Significance testing of rank cross-correlations between autocorrelated time series with short-range dependence

Statistical dependency measures such as Kendall’s Tau or Spearman’s Rho are frequently used to analyse the coherence between time series in environmental data analyses. Autocorrelation of the data can, however, result in spurious cross correlations if not accounted for. Here, we present the asymptotic distribution of the estimators of Spearman’s Rho and Kendall’s Tau, which can be used for statistical hypothesis testing of cross-correlations between autocorrelated observations. The results are derived using U-statistics under the assumption of absolutely regular (or β-mixing) processes. These comprise many short-range dependent processes, such as ARMA-, GARCH- and some copula-based models relevant in the environmental sciences. We show that while the assumption of absolute regularity is required, the specific type of model does not have to be specified for the hypothesis test. Simulations show the improved performance of the modified hypothesis test for some common stochastic models and small to moderate sample sizes under autocorrelation. The methodology is applied to observed climatological time series of flood discharges and temperatures in Europe. While the standard test results in spurious correlations between floods and temperatures, this is not the case for the proposed test, which is more consistent with the literature on flood regime changes in Europe.

A spectral approach to estimate the autocovariance function

This paper proposes two novel alternative estimators for the autocovariance function of a short-range dependent stationary process based on a spectral approach. One is based on the classical periodogram, and the other one uses the M-periodogram. The theoretical properties of both estimators are investigated for linear processes. Monte-Carlo experiments are conducted to compare these estimators to the standard sample autocovariance function in autoregressive moving average (ARMA) models with and without additive outliers. In the non-contaminated scenario, the empirical investigation shows that the three estimators display similar performance. However, in the contaminated case, the estimator based on the M-periodogram remains unaffected in the presence of additive outliers, while the two other estimators are corrupted.

Frequency domain bootstrap for ratio statistics under long-range dependence

A frequency domain bootstrap (FDB) is a common technique to apply Efron’s independent and identically distributed resampling technique (Efron, 1979) to periodogram ordinates – especially normalized periodogram ordinates – by using spectral density estimates. The FDB method is applicable to several classes of statistics, such as estimators of the normalized spectral mean, the autocorrelation (but not autocovariance), the normalized spectral density function, and Whittle parameters. While this FDB method has been extensively studied with respect to short-range dependent time processes, there is a dearth of research on its use with long-range dependent time processes. Therefore, we propose an FDB methodology for ratio statistics under long-range dependence, using semi- and nonparametric spectral density estimates as a normalizing factor. It is shown that the FDB approximation allows for valid distribution estimation for a broad class of stationary, long-range (or short-range) dependent linear processes, without any stringent assumptions on the distribution of the underlying process. The results of a large simulation study show that the FDB approximation using a semi- or nonparametric spectral density estimator is often robust for various values of a long-memory parameter reflecting magnitude of dependence. We apply the proposed procedure to two data examples.

Local polynomial estimation of regression operators from functional data with correlated errors

This article considers the problem of nonparametric estimation of the regression operator r in a functional regression model Y=r(x)+ε with a scalar response Y, a functional explanatory variable x, and a second order stationary error process ε. We construct a local polynomial estimator of r together with its Fréchet derivatives from functional data with correlated errors. The convergence in mean squared error of the constructed estimator is studied for both short and long range dependent error processes. Simulation studies on the performance of the proposed estimator are conducted, and applications to independent data and El Niño time series data are given.

Convergence rates in the law of large numbers for long-range dependent linear processes

Baum and Katz (Trans. Am. Math. Soc. 120:108-123, 1965) obtained convergence rates in the Marcinkiewicz-Zygmund law of large numbers. Their result has already been extended to the short-range dependent linear processes by many authors. In this paper, we extend the result of Baum and Katz to the long-range dependent linear processes. As a corollary, we obtain convergence rates in the Marcinkiewicz-Zygmund law of large numbers for short-range dependent linear processes.

An Empirical Study on Evolution of the Connectivity for VANETs Based on Taxi GPS Traces

Network connectivity is a fundamental requirement for intervehicle communication and services in VANETs. The fast movement of the vehicles results in rapid changes in network topology generating dynamic variation in network connectivity, which causes dynamic variation in network connectivity. In this paper, we analyze the evolution of the VANETs connectivity based on the realistic data collected by the GPS devices installed on taxis in Shanghai. The study of the evolution characteristics shows that the number of connected components, the size of the giant component, and the connectivity length time series possess strong temporal correlation and long-range dependence. As a result, the short-range dependent processes such as the exponential, normal, or Poisson distributions cannot capture the statistical distribution characteristics. Moreover, all connectivity parameters cannot be described by unimodal distributions. We also conclude that although the whole topology of the VANETs is broken into a large number of small-sized connected components under small transmission radius, the size of the giant component keeps stable when transmission radius reaches the threshold. These results will be helpful to optimize the RSUs deployment and transmission radius to keep the giant component stable and design more efficient strategies to forward the information in VANETs.

Short-range dependent processes subordinated to the Gaussian may not be strong mixing

There are all kinds of weak dependence. For example, strong mixing. Short-range dependence (SRD) is also a form of weak dependence. It occurs in the context of processes that are subordinated to the Gaussian. Is a SRD process strong mixing if the underlying Gaussian process is long-range dependent? We show that this is not necessarily the case.

Hierarchical Multilevel on/off Source Traffic Modeling for a Warship Combat System

For successful migration to network-centric warfare (NCW), it is necessary to control and manage sensor, weapons, and communication systems. Each of these systems must be incorporated in a combat management (CM) system through connection via the combat system databus (CSDB). This synergy between critical features is termed a warship combat system (WCS). For optimization of WCS, performance evaluation before real war should be conducted to measure the reliability of the system. This is accomplished through traffic modeling according to specific scenarios of warships. In practice, the traffic of each system within the WCS exhibits self-similar behavior. This is different from the behavior of short-range dependence (SRD) processes. In this paper, we propose a traffic model called the hierarchical multilevel on/off source (HMLOS) model, which aggregates a number of on/off sources in accordance with the hierarchical structure of the WCS after taking into account the individual characteristics of traffic in each system. The duration of time for the on state of each module is described using the Pareto distribution with a different Pareto index. For two different scenarios, including general navigation and training (GNT) and combat and emergency (CE), the proposed model demonstrates characteristics of self-similarity that are closer to the original characteristics of traffic than conventional traffic modeling.

Effects of Different Practice Task Constraints on Fluctuations of Player Heart Rate in Small-Sided Football Games

This paper analyzes effects of different practice task constraints on heart rate (HR) variability during 4v4 smallsided football games. Participants were sixteen football players divided into two age groups (U13, Mean age: 12.4±0.5 yrs; U15: 14.6±0.5). The task consisted of a 4v4 sub-phase without goalkeepers, on a 25x15 m field, of 15 minutes duration with an active recovery period of 6 minutes between each condition. We recorded players’ heart rates using heart rate monitors (Polar Team System, Polar Electro, Kempele, Finland) as scoring mode was manipulated (line goal: scoring by dribbling past an extended line; double goal: scoring in either of two lateral goals; and central goal: scoring only in one goal). Subsequently, %HR reserve was calculated with the Karvonen formula. We performed a time-series analysis of HR for each individual in each condition. Mean data for intra-participant variability showed that autocorrelation function was associated with more short-range dependence processes in the “line goal” condition, compared to other conditions, demonstrating that the “line goal” constraint induced more randomness in HR response. Relative to inter-individual variability, line goal constraints demonstrated lower %CV and %RMSD (U13: 9% and 19%; U15: 10% and 19%) compared with double goal (U13: 12% and 21%; U15: 12% and 21%) and central goal (U13: 14% and 24%; U15: 13% and 24%) task constraints, respectively. Results suggested that line goal constraints imposed more randomness on cardiovascular stimulation of each individual and lower inter-individual variability than double goal and central goal constraints.

Fractional Black-Scholes Model and Technical Analysis of Stock Price

In the stock market, some popular technical analysis indicators (e.g., Bollinger bands, RSI, ROC, etc.) are widely used to forecast the direction of prices. The validity is shown by observed relative frequency of certain statistics, using the daily (hourly, weekly, etc.) stock prices as samples. However, those samples are not independent. In earlier research, the stationary property and the law of large numbers related to those observations under Black-Scholes stock price model and stochastic volatility model have been discussed. Since the fitness of both Black-Scholes model and short-range dependent process has been questioned, we extend the above results to fractional Black-Scholes model with Hurst parameterH>1/2, under which the stock returns follow a kind of long-range dependent process. We also obtain the rate of convergence.

How are rescaled range analyses affected by different memory and distributional properties? A Monte Carlo study

In this paper, we present the results of Monte Carlo simulations for two popular techniques of long-range correlation detection — classical and modified rescaled range analyses. A focus is put on an effect of different distributional properties on an ability of the methods to efficiently distinguish between short-term memory and long-term memory. To do so, we analyze the behavior of the estimators for independent, short-range dependent, and long-range dependent processes with innovations from eight different distributions. We find that apart from a combination of very high levels of kurtosis and skewness, both estimators are quite robust to distributional properties. Importantly, we show that R/S is biased upwards (yet not strongly) for short-range dependent processes, while M-R/S is strongly biased downwards for long-range dependent processes regardless of the distribution of innovations.

Long-Range Dependence and Climate Noise Characteristics of Antarctic Temperature Data

Abstract This study examines the long-range dependency, climate noise characteristics, and nonlinear temperature trends of eight Antarctic stations from the Reference Antarctic Data for Environmental Research (READER) dataset. Evidence is shown that Antarctic temperatures are long-range dependent. To identify possible nonlinear trends, the ensemble empirical mode decomposition (EEMD) method is used, and then the question of whether the observed trends can arise from internal atmospheric fluctuations is examined. To answer this question, surrogate data are generated from two paradigmatic null models: a standard first-order autoregressive process representing a short-range dependent process and a fractional integrated process representing a long-range dependent process. It is found that three of the eight stations show statistically significant trends when tested against the short-range dependent process while only the Faraday–Vernadsky station temperature time series shows a significant trend when tested against the long-range dependent null model. All other considered stations show no trends that are statistically significant against the two null models, and thus they can be explained by internal atmospheric variability. These results imply that more attention should be given to assessing the correlation structure of climate time series.

Nonparametric regression for dependent data in the errors-in-variables problem

We consider the nonparametric estimation of the regression functions for dependent data. Suppose that the covariates are observed with additive errors in the data and we employ nonparametric deconvolution kernel techniques to estimate the regression functions in this paper. We investigate how the strength of time dependence affects the asymptotic properties of the local constant and linear estimators. We treat both short-range dependent and long-range dependent linear processes in a unified way and demonstrate that the long-range dependence (LRD) of the covariates affects the asymptotic properties of the nonparametric estimators as well as the LRD of regression errors does.

Effects of Different Practice Task Constraints on Fluctuations of Player Heart Rate in Small-Sided Football Games~!2009-07-05~!2009-12-05~!2010-04-09~!

This paper analyzes effects of different practice task constraints on heart rate (HR) variability during 4v4 smallsided football games. Participants were sixteen football players divided into two age groups (U13, Mean age: 12.4±0.5 yrs; U15: 14.6±0.5). The task consisted of a 4v4 sub-phase without goalkeepers, on a 25x15 m field, of 15 minutes duration with an active recovery period of 6 minutes between each condition. We recorded players’ heart rates using heart rate monitors (Polar Team System, Polar Electro, Kempele, Finland) as scoring mode was manipulated (line goal: scoring by dribbling past an extended line; double goal: scoring in either of two lateral goals; and central goal: scoring only in one goal). Subsequently, %HR reserve was calculated with the Karvonen formula. We performed a time-series analysis of HR for each individual in each condition. Mean data for intra-participant variability showed that autocorrelation function was associated with more short-range dependence processes in the “line goal” condition, compared to other conditions, demonstrating that the “line goal” constraint induced more randomness in HR response. Relative to inter-individual variability, line goal constraints demonstrated lower %CV and %RMSD (U13: 9% and 19%; U15: 10% and 19%)compared with double goal (U13: 12% and 21%; U15: 12% and 21%) and central goal (U13: 14% and 24%; U15: 13% and 24%) task constraints, respectively. Results suggested that line goal constraints imposed more randomness on cardiovascular stimulation of each individual and lower inter-individual variability than double goal and central goal constraints.

Steady-State Simulation of Long-Range Dependent Queuing Processes

Long-range dependence was first noted in hydrology by H. E. Hurst from a study of the water levels of the Nile river which showed a tendency for a flood year to be followed by another flood year. Long-range dependent processes are relevant not only in telecommunication networks but also in such areas of scientific activity as econophysics, climatology, economics, environmental sciences, geology, geophysics, hydrology, computer science, and computer engineering. They provide good models of packet trac in telecommunication networks, for example, in local area networks and video trac, and good models of persistence analysis in econophysics, especially for short-term interest rates. In this paper, the eects of long-range dependence on the behaviors of queuing systems have been investigated. We have also investigated how the long-range dependence in arrival processes aects the length of sequential steady-state simulations being executed to obtain simulation results with the required level of statistical error. Our results show that the finite buer overflow probability of a queuing system with a long-range dependent input is much greater than the equivalent queuing system with a Poisson or a short-range dependent input process and that the overflow probability increases as the Hurst parameter approaches one.

The effect of long-range dependence on modelling extremes with the generalised extreme value distribution

Two effects arise for the modelling of block maxima from dependent time series: a reduced rate of convergence for the block maxima probability distribution towards the generalised extreme value distribution, and an increase in uncertainty of the parameter estimates compared to independent or short range dependent records. These effects are exemplified with a simulation study using a white noise, a short-range and a long-range dependent process. The two issues raised turned out to be relatively unproblematic for short-range dependent processes. For long-range dependent processes, especially the increased parameter uncertainty poses a problem. Incautious use of standard procedures would lead to a severe underestimation of the parameter uncertainty which implies a misconception of accuracy for derived quantities, such as return levels which are frequently used for risk assessment and dimensioning of hydraulic structures.

Logarithmic asymptotics for a single-server processing distinguishable sources

We consider a single-server first-in-first-out queue fed by a finite number of distinct sources of jobs. For a large class of short-range dependent and light-tailed distributed job processes, using functional large deviation techniques we prove a large deviation principle and logarithmic asymptotics for the joint waiting time and queue lengths distribution. We identify the paths that are most likely to lead to the rare events of large waiting times and long queue lengths. A number of examples are presented to illustrate salient features of the results.

M-estimation of linear models with dependent errors

We study asymptotic properties of M-estimates of regression parameters in linear models in which errors are dependent. Weak and strong Bahadur representations of the M-estimates are derived and a central limit theorem is established. The results are applied to linear models with errors being short-range dependent linear processes, heavy-tailed linear processes and some widely used nonlinear time series.

Asymptotic theory for curve-crossing analysis

We consider asymptotic properties of curve-crossing counts of linear processes and nonlinear time series by curves. Central limit theorems are obtained for curve-crossing counts of short-range dependent processes. For the long-range dependence case, the asymptotic distributions are shown to be either multiple Wiener–Itô integrals or integrals with respect to stable Lévy processes, depending on the heaviness of tails of the underlying processes.