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The specification of a covariance function is of paramount importance when employing Gaussian process models, but the requirement of positive definiteness severely limits those used in practice. Designing flexible stationary covariance functions is, however, straightforward in the spectral domain, where one needs only to supply a positive and symmetric spectral density. In this work, we introduce an adaptive integration framework for efficiently and accurately evaluating covariance functions and their derivatives at irregular locations directly from any continuous, integrable spectral density. In order to make this approach computationally tractable, we employ high-order panel quadrature, the nonuniform fast Fourier transform, and a Nyquist-informed panel selection heuristic, and derive novel algebraic truncation error bounds which are used to monitor convergence. As a result, we demonstrate several orders of magnitude speedup compared to naive uniform quadrature approaches, allowing us to evaluate covariance functions from slowly decaying, singular spectral densities at millions of locations to a user-specified tolerance in seconds on a laptop. We then apply our methodology to perform gradient-based maximum likelihood estimation using a previously numerically infeasible long-memory spectral model for wind velocities below the atmospheric boundary layer. [ABSTRACT FROM AUTHOR]

River water quality often follows a long-memory stochastic process with power-type autocorrelation decay, which can only be reproduced using appropriate mathematical models. The selection of a stochastic process model, particularly its memory structure, is often subject to misspecifications owing to low data quality and quantity. Therefore, environmental risk assessment should account for model misspecification through mathematically rigorous and efficiently implementable approaches; however, such approaches have been still rare. We address this issue by first modeling water quality dynamics through the superposition of an affine diffusion process that is stationary and has a long memory. Second, the worst-case upper deviation of the water quality value from a prescribed threshold value under model misspecifications is evaluated using either the divergence risk or Wasserstein risk measure. The divergence risk measure can consistently deal with the misspecification of the memory structure to the worst-case upper deviation. The Wasserstein risk measure is more flexible but fails in this regard, as it does not directly consider the memory structure information. We theoretically compare both approaches to demonstrate that their assumed uncertainties differed substantially. From the application to the 30-year water quality data of a river in Japan, we categorized the water quality indices to be those with truly long memory (Total nitrogen, NO3-N, NH4-N, and SO 4 2 - ), those with moderate power-type memory (NO2-N, PO4-P, and Total Organic Carbon), and those with almost exponential memory (Total phosphorus and Chemical Oxygen demand). The risk measures are successfully computed numerically considering the seasonal variations of the water quality indices. [ABSTRACT FROM AUTHOR]

The modeling and identification of time series data with a long memory are important in various fields. The streamflow discharge is one such example that can be reasonably described as an aggregated stochastic process of randomized affine processes where the probability measure, we call it reversion measure, for the randomization is not directly observable. Accurate identification of the reversion measure is critical because of its omnipresence in the aggregated stochastic process. However, the modeling accuracy is commonly limited by the available real‐world data. We resolve this issue by proposing the novel upper and lower bounds of a statistic of interest subject to ambiguity of the reversion measure. Here, we use the Tsallis value‐at‐risk (TsVaR) as a convex risk functional to generalize the widely used entropic value‐at‐risk (EVaR) as a sharp statistical indicator. We demonstrate that the EVaR cannot be used for evaluating key statistics, such as mean and variance, of the streamflow discharge due to the blowup of some exponential integrand. We theoretically show that the TsVaR can avoid this issue because it requires only the existence of some polynomial moment, not exponential moment. As a demonstration, we apply the semi‐implicit gradient descent method to calculate the TsVaR and corresponding Radon–Nikodym derivative for time series data of actual streamflow discharges in mountainous river environments. [ABSTRACT FROM AUTHOR]

Long-memory process has been widely studied in classical financial time series analysis, which has merely been reported in the field of interval-valued financial time series. The aim of this paper is to explore long-memory process in the prediction of interval-valued time series (IvTS). To model the long-memory process, two novel interval-valued time series prediction models named as interval-valued vector autoregressive fractionally integrated moving average (IV-VARFIMA) and ARFIMAX-FIGARCH were established. In the developed long-memory pattern, both of the short term and long-term influences contained in IvTS can be included. As an application of the proposed models, interval-valued form of WTI crude oil futures price series is predicted. Compared to current IvTS prediction models, IV-VARFIMA and ARFIMAX-FIGARCH can provide better in-sample and out-of-sample forecasts. [ABSTRACT FROM AUTHOR]

Long-memory models are frequently used in finance and other fields to capture long-range dependence in time series data. However, correctly identifying whether a process has long memory is crucial. This paper highlights a significant limitation in using the sample autocorrelation function (ACF) to identify long-memory processes. While the ACF establishes the theoretical definition of a long-memory process, it is not possible to determine long memory by summing the sample ACFs. Hassani's − 1 2 theorem demonstrates that the sum of the sample ACF is always − 1 2 for any stationary time series with any length, rendering any diagnostic or analysis procedures that include this sum open to criticism. The paper presents several cases where discrepancies between the empirical and theoretical use of a long-memory process are evident, based on real and simulated time series. It is critical to be aware of this limitation when developing models and forecasting. Accurately identifying long-memory processes is essential in producing reliable predictions and avoiding incorrect model specification. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Dickey-Fuller difference statistic to sequentially detect the change-point that shift from an unit root process to a long-memory process. The limiting distribution of monitoring statistic under the unit root process null hypothesis as well as its consistency under the alternative hypothesis are proved. Simulations indicate that the new method can control the empirical size well even for the heavy-tailed unit root process when using the sieve bootstrap method computing its critical values. In particular, it performs significantly better than the available method in the literature under the alternative hypothesis. Finally, we illustrate the new monitoring procedure by a set of foreign exchange rate data.

Long-memory models are frequently used in finance and other fields to capture long-range dependence in time series data. However, correctly identifying whether a process has long memory is crucial. This paper highlights a significant limitation in using the sample autocorrelation function (ACF) to identify long-memory processes. While the ACF establishes the theoretical definition of a long-memory process, it is not possible to determine long memory by summing the sample ACFs. Hassani’s −12 theorem demonstrates that the sum of the sample ACF is always −12 for any stationary time series with any length, rendering any diagnostic or analysis procedures that include this sum open to criticism. The paper presents several cases where discrepancies between the empirical and theoretical use of a long-memory process are evident, based on real and simulated time series. It is critical to be aware of this limitation when developing models and forecasting. Accurately identifying long-memory processes is essential in producing reliable predictions and avoiding incorrect model specification.

This study examines the problem of robust spectral analysis of long-memory processes. We investigate the possibility of using Laplace and quantile periodograms for a non-Gaussian distribution structure. The Laplace periodogram, derived by the least absolute deviations in the harmonic regression procedure, demonstrates its superiority in handling heavy-tailed noise and nonlinear distortion. In this study, we discuss an asymptotic distribution of the Laplace periodogram for long-memory processes. We also derive an asymptotic distribution of the quantile periodogram. Through numerical experiments, we demonstrate the robustness of the Laplace periodogram and the usefulness of the quantile periodogram in detecting the hidden frequency for the spectral analysis of the long-memory process under non-Gaussian distribution. Moreover, as an application of robust periodograms under the long-memory process, we discuss the long-memory parameter estimation based on a log periodogram regression approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose two ratio-type statistics to sequentially detect the memory parameter change-points in the long-memory time series. The limiting distributions of monitoring statistics under the no-change-point null hypothesis as well as their consistency under the alternative hypothesis are proved. In particular, a sieve bootstrap approximation method is proposed to determine the critical values. Extensive simulations indicate that the new monitoring procedures perform well in finite samples. Finally, we illustrate our monitoring procedures by two sets of real data. [ABSTRACT FROM AUTHOR]

This paper investigates the linear relationships between two time-series in the frequency domain, termed coherence analysis. It is widely used in various fields, including signal processing, engineering, and meteorology. However, conventional coherence analysis tends to be sensitive to outliers. Laplace cross-periodogram and a corresponding robust coherence analysis based on the least-absolute deviation (LAD) regression have recently been developed to improve this shortcoming. In this paper, to extend the scope of Laplace cross-periodogram, we study a robust cross periodogram for long-memory processes and derive its asymptotic distribution. Through numerical studies, we demonstrate the usefulness of the proposed robust coherence analysis for long-memory processes. [ABSTRACT FROM AUTHOR]

We consider fractional stochastic volatility models that extend the classic Black–Scholes model for asset prices. The models are general and motivated by recent empirical results regarding the behavior of realized volatility. While such models retain the semimartingale property for the asset price the associated European option pricing problem becomes complex, with no explicit solution. In a number of canonical scaling regimes it is possible, however, to derive asymptotic and sparse representations for the option price and the associated implied volatility, that are parameterized by a few effective parameters and that involve power law dependencies on time to maturity. These effective parameters may depend in a complicated way on the volatility model, but they can be easily estimated from the observation of a few option prices. The effective parameters associated with a particular underlying asset can be calibrated with respect to liquid contracts written on this asset and then used for pricing less liquid contracts written on the same underlying asset. Therefore, the effective parameters provide a robust link between financial products written on a particular underlying asset.

In this paper the properties of the modified EWMA control chart for detecting changes in the mean of an ARFIMA process are discussed. The central question is related to the false alarm probability and its behavior for different autocorrelation structures and parameters of the underlying process. It is shown under which conditions the false alarm probability of an ARFIMA(p,d,q) process is larger than that of the pure ARFIMA(0,d,0) process. Furthermore, it is shown that the false alarm probability for ARFIMA(0,d,0) and ARFIMA(1,d,1) is monotonic in d for common parameter values of the processes.

Bitcoin price prediction is a substantial challenge for cryptocurrency investors. This study offers an innovative scheme to predict Bitcoin returns and volatilities using a hybrid model that incorporates the autoregressive fractionally integrated moving average (ARFIMA), empirical wavelet (EW) transform, and local linear wavelet neural network (LLWNN) approaches to produce an ARFIMA-EWLLWNN model. Our methodologies integrate the advantages of the long-memory model, EW decomposition technique, artificial neural network structure, and backpropagation and particle swarm optimization learning algorithms. The experimental results of the optimized hybrid approach outperform some classic models by providing more accurate out-of-sample forecasts over longer horizons. The model proves to be the most appropriate Bitcoin forecasting technique. Moreover, the implemented method produces smaller prediction errors than other computing techniques. • This paper provides a hybrid method to forecast the Bitcoin price. • The method combines dual long-memory estimation process, wavelet decomposition, and learning algorithms. • Empirical wavelet time-frequency analysis is used to improve the forecasting accuracy of the artificial neural-network structure. • Techno-economic feasibility analysis for CIP degradation using RHC. • We argue that this approach is a promising tool of modelling and forecasting Bitcoin prices. [ABSTRACT FROM AUTHOR]

In this paper EWMA control charts are introduced for detecting changes in the mean of a long-memory process. Besides the modified EWMA scheme the EWMA residual chart is also considered. The control design of the charts is calculated for an ARFIMA(p,d,q) process. In order to assess the introduced charts, the average run length is used as a performance criterion. Using an extensive simulation study, the control charts are compared with each other. The target process is assumed to be an ARFIMA(1,d,1) process. [ABSTRACT FROM AUTHOR]

The purpose of this paper is the estimation of the self-similarity index of the Rosenblatt process by using the Whittle estimator. Via chaos expansion into multiple stochastic integrals, we establish a non-central limit theorem satisfied by this estimator. We illustrate our results by numerical simulations. [ABSTRACT FROM AUTHOR]

In our research we examine the behaviour of both Thailand’s and India’s international tourism market by using long-memory analysis. The international tourism market of Thailand combined with seven groups such as East Asia, Europe, The Americas, South Asia, Oceania, Middle East and Africa. Similarly, the international tourism market of India combined with nine countries: USA, UK, Canada, Germany, France, Japan, Malaysia, Australia and Sri Lanka. Moreover, three statistical tests for long-memory process such as R/S test, Modified R/S test and GPH-test are employed to study these markets. The empirical findings in general provide more support for long memory process in international tourism market of Thailand and evidence for short-term dependence in international tourism market of India. Therefore, the policy makers of each country should understand the behaviour of long memory process in international tourism market before launching any stimulating campaign to this industry.

There have been growing interest in studying behavior of long memory process in tourism market. In this research examine the behavior of India’s international tourism market based on long-memory analysis. The international tourism market of India combined with nine countries: USA, UK, Canada, Germany, France, Japan, Malaysia, Australia and Sri Lanka. Moreover, three statistical tests for longmemory process such as R/S test, Modified R/S test and GPH-test are employed to test in these market. The empirical findings in general provide more support for no long memory process or no long-term dependence in international tourism market of India.

There have been growing interest in studying behavior of long memory process in tourism market. In this research examine the behavior of India’s international tourism market based on long-memory analysis. The international tourism market of India combined with nine countries: USA, UK, Canada, Germany, France, Japan, Malaysia, Australia and Sri Lanka. Moreover, three statistical tests for longmemory process such as R/S test, Modified R/S test and GPH-test are employed to test in these market. The empirical findings in general provide more support for no long memory process or no long-term dependence in international tourism market of India.

The asymptotic decorrelation of the discrete wavelet transform for long-memory processes (such as the fractionally differenced (FD) process, the autoregressive fractionally integrated moving average (ARFIMA) process, and the Gegenbauer autoregressive moving average (GARMA) process) as well as the statistical inference techniques based on this property, have received much attention nowadays. In this paper, we investigate the asymptotic decorrelation property of the discrete wavelet packet transform (DWPT) for two classes of discrete-time long-memory processes containing the ARFIMA and the GARMA processes. Especially, we prove theoretically that the covariance across between-packet DWPT coefficients decays hyperbolically or exponentially fast as the width of the underlying Daubechies scaling and wavelet filters to generate the DWPT gets large. Meanwhile, we show that the covariance between within-packet DWPT coefficients converges hyperbolically fast to its corresponding counterpart when the underlying scaling and wavelet filters to generate the DWPT are the Shannon's ideal low- and high-pass filters. [ABSTRACT FROM AUTHOR]

Abstract: The detection of long-range dependence in time series analysis is an important task to which this paper contributes by showing that whilst the theoretical definition of a long-memory (or long-range dependent) process is based on the autocorrelation function, it is not possible for long memory to be identified using the sum of the sample autocorrelations, as usually defined. The reason for this is that the sample sum is a predetermined constant for any stationary time series; a result that is independent of the sample size. Diagnostic or estimation procedures, such as those in the frequency domain, that embed this sum are equally open to this criticism. We develop this result in the context of long memory, extending it to the implications for the spectral density function and the variance of partial sums of a stationary stochastic process. The results are further extended to higher order sample autocorrelations and the bispectral density. The corresponding result is that the sum of the third order sample (auto) bicorrelations at lags , is also a predetermined constant, different from that in the second order case, for any stationary time series of arbitrary length. [Copyright &y& Elsevier]

Abstract: Series of motor outputs generated by cyclic movements are typically complex, suggesting that the correlation function of the time series spans over a large number of consecutive samples. Famous examples include inter-stride intervals, heartbeat variability, spontaneous neural firing patterns or motor synchronization with external pacing. Long-range correlations are potentially important for fundamental research, as the neural and biomechanical mechanisms generating these correlations remain unknown, and for clinical applications, given that the loss of long-range correlation may be a marker of disease. However, no systematic approach or robust analysis methods have yet been used to support the study of correlation functions in physiological series. This study investigates four selected methods (the Hurst exponent, the power spectral density analysis, the rate of moment convergence and the multiscale entropy methods). We present the result of each analysis performed on artificial computer-generated series in which the auto-correlation function is known, and then on time series extracted from gait and upper limb rhythmic movements. Our results suggest that combined analysis using the Hurst exponent and the power spectral density is suitable for rather short series (512 points). The rate of moment convergence directly supports the power spectral density analysis, and the multiscale entropy further confirms the presence of long-range correlation, although this method seems more appropriate for longer series. The proposed methodology increases the level of confidence in the hypothesis that physiological series are long-memory processes, which is of prime importance for future fundamental and clinical research. [Copyright &y& Elsevier]

For a class of time series regression models with long-memory disturbance, we are interested in estimation of a subset of the regression coefficient vector and spectral parameter of the residual process when the complementary subset is suspected to be close to 0. In this situation, we evaluate the mean square errors of the restricted and unrestricted MLE and a preliminary test estimator when the complementary parameters are contiguous to zero vector. The results are expressed in terms of the regression spectra and the residual spectra. Since we assume long-memory dependence for the disturbance, the asymptotics are much different from the case of i.i.d. disturbance. Numerical studies elucidate some interesting features of regression and long-memory structures. [ABSTRACT FROM AUTHOR]

Abstract: This paper analyzes the approximation of a general long-memory ARFIMA process by a short-memory ARMA(1, 1) process. To validate this approximation, a mean square error forecast criterion is considered, and the calculation of the mean square error between the observation of an ARFIMA process and the -step-ahead forecast of the ARMA(1, 1) process is presented. The performance of the ARMA(1, 1) approximation to an ARFIMA model is illustrated by using an application to a DNA sequence of orf virus. This paper gives some theoretical justification on the mean square error forecast criterion based on a selected ARMA(1, 1) model when compared to the general ARFIMA() process. The paper also provides a DNA sequence of orf virus analysis when the time series is originated from the tangent values of the Chaos Game Representation coordinates for each nucleotide. The conclusions of this paper work well because the estimator value of is small (). The paper also gives the other parameter estimate of the fitted ARFIMA (0, , 1) model and one-step predictions using ARMA(1, 1) model. [Copyright &y& Elsevier]

Wilcoxon-type rank statistics are considered for testing a long-memory time-series model with a common distribution against the alternatives involving a change in the distribution at an unknown time point. The asymptotic properties of the test statistics and the change-point estimators are studied. Finite-sample behaviours are investigated in a small Monte Carlo simulation study. Data examples from hydrology and telecommunications illustrate the method. [ABSTRACT FROM AUTHOR]

Novel periodic extensions of dynamic long-memory regression models with autoregressive conditional heteroscedastic errors are considered for the analysis of daily electricity spot prices. The parameters of the model with mean and variance specifications are estimated simultaneously by the method of approximate maximum likelihood. The methods are implemented for time series of 1,200-4,400 daily price observations in tour European power markets. Apart from persistence, heteroscedasticity, and extreme observations in prices, a novel empirical finding is the importance of day-of-the-week periodicity in the autocovariance function of electricity spot prices. In particular, the very persistent daily log prices from the Nord Pool power exchange of Norway are effectively modeled by our framework, which is also extended with explanatory variables to capture supply-and-demand effects. The daily log prices of the other three electricity markets--EEX in Germany, Powernext in France, and APX in The Netherlands are less persistent, but periodicity is also highly significant. The dynamic behavior differs from market to market and depends primarily on the method of power generation: hydro power, power generated from fossil fuels, or nuclear power. The article improves on existing models in capturing the memory characteristics, which are important in derivative pricing and real option analysis. [ABSTRACT FROM AUTHOR]

We investigate the mean squared error of the Stein–James estimator for the mean when the observations are generated from a Gaussian vector stationary process with dimension greater than two. First, assuming that the process is short-memory, we evaluate the mean squared error, and compare it with that for the sample mean. Then a sufficient condition for the Stein–James estimator to improve upon the sample mean is given in terms of the spectral density matrix around the origin. We repeat the analysis for Gaussian vector long-memory processes. Numerical examples clearly illuminate the Stein–James phenomenon for dependent samples. The results have the potential to improve the usual trend estimator in time series regression models. [ABSTRACT FROM AUTHOR]

We give the limiting distribution of the least-squares estimator in the general autoregressive model driven by a long-memory process. We prove that with an appropriate normalization the estimation error converges, in distribution, to a random vector which contains: (1) a stochastic component, due to the presence of the unstable roots, which are multiple Wiener–Itô integrals and a non-linear functionals of stochastic integrals with respect to a Brownian motion; (2) a constant component due to the stable roots; (3) a stochastic component, due to the presence of the explosive roots, which is a mixture of normal distributions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the estimation of the coefficient of a stochastic regression model whose explanatory variables and disturbances are permitted to exhibit short-memory or long-memory dependence. Three estimators of the coefficient are proposed. A variety of their asymptotics are illuminated under various assumptions on the explanatory variables and the disturbances. Numerical studies of the theoretical results are given. They show some unexpected aspects of the asymptotics of the three estimators. [ABSTRACT FROM AUTHOR]