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Spectral density of NARCCAP data climate data (circles), fitted ARTFIMA spectrum (solid line) and untempered ARFIMA spectrum (dotted line).
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The ARTFIMA model applies a tempered fractional difference to the standard ARMA time series. This paper develops parameter estimation methods for the ARTFIMA model, and demonstrates a new R package. Several examples illustrate the utility of the method.
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Context 1
... fit an ARTFIMA(0, d, λ, 0) model to the standardized time series with d = 0.933 and λ = 0.300 using the artfima package. The fitted spectral density and the pe- riodogram are shown in Figure 6. The spectral density levels off at low frequencies, consistent with the periodogram. ...
Context 2
... spectral density levels off at low frequencies, consistent with the periodogram. The spectrum of the untempered ARFIMA model in Figure 6 shows a lack of fit at low frequencies. As there is no evidence of non- stationarity in the standardized time series, the stationary ARTFIMA model seems highly preferable to the stationary increment ARFIMA model, which is mis-specified due to its lack of stationarity. ...
Citations
... For seasonal long memory process X t , the autocorrelation function for lag h denoted by ρ(h) behaves asymptotically as ρ(h) cos(hω 0 )h −α as h → ∞ for some positive α ∈ (0, 1) and ω 0 ∈ (0, π) (see [10]). In literature, many tempered distributions and processes are studied using the exponential tempering in the original distribution or process see e.g. and references therein [24,28,29,32,35,16,5]. The fractionally integrated process with seasonal components are studied and maximum likelihood estimation is done by Reisen et al. [27]. ...
... The parametric spectral density with power-law behaviour about a fractional pole at the unknown frequency ω is analysed and Gaussian estimates and limiting distributional behavior of estimate is studied by Giraitis and Hidalgo [14]. The autoregressive tempered fractionally integrated moving average (ARTFIMA) process is obtained by using exponential tempering in the original ARFIMA process [29]. The ARTFIMA process is semi LRD and has a summable autocovariance function. ...
... Apart from this, Pincherle, Horadam and Horadam-Pethe random fields will be interest of study on the line of Gegenbauer random fields [12]. Moreover, one can study the tempered versions of Humbert, Pincherle, Horadam and Horadam-Pethe ARMA processes similar to Sabzikar et al. [29]. ...
In this article, we use the generating functions of the Humbert polynomials to define two types of Humbert generalized fractional differenced ARMA processes. We present stationarity and invertibility conditions for the introduced models. The singularities for the spectral densities of the introduced models are obtained. In particular, Pincherle ARMA, Horadam ARMA and Horadam-Pethe ARMA processes are studied.
... These processes are obtained by exponential tempering in the original process. The ARTFIMA models were introduced as a generalization of ARFIMA model in [21]. According to [21], it is more convenient to study ARTFIMA time series as the covariance function is absolutely summable in finite variance cases and the spectral density converges to zero [9]. ...
... The ARTFIMA models were introduced as a generalization of ARFIMA model in [21]. According to [21], it is more convenient to study ARTFIMA time series as the covariance function is absolutely summable in finite variance cases and the spectral density converges to zero [9]. Also in the spectral density of ARFIMA process is unbounded as the frequency approaches to 0. In many scenarios, there exist many datasets for which the power spectrum is bounded as frequency approaches to 0 and these types of datasets cannot be models using ARFIMA process. ...
... The efficacy of the estimation procedure is checked on the simulated data. The ARTFIMA(p, d, λ, q) process defined by [21] generalizes the ARFIMA process defined in (2.4). The model is defined by introducing a tempering parameter λ in ARFIMA model, that is instead of taking the fractional shift operator (1 − B) d the authors have considered a tempered fractional shift operator (1 − e −λ B) d , where λ > 0. The series exhibits semi-long range dependence structure that is for λ close to 0 the autocovariance function of the process behaves similar to a long memory process and for large λ the autocovariance function decays exponentially [21]. ...
In this article, we introduce a Gegenbauer autoregressive tempered fractionally integrated moving average (GARTFIMA) process. We work on the spectral density and autocovariance function for the introduced process. The parameter estimation is done using the empirical spectral density with the help of the nonlinear least square technique and the Whittle likelihood estimation technique. The performance of the proposed estimation techniques is assessed on simulated data. Further, the introduced process is shown to better model the real-world data in comparison to other time series models.
... Many R Packages developed in the literatures are synonyms to the targeted models and they could be classified as mini software (see R Package arfima in Veenstra (2012), artfima in McLeod et al., (2016) and arfurima package of Jibrin and Rahman (2019)). There are other few packages that implement the mean, volatility and hybrid models. ...
... For example, the package fracdiff of Fraley et al. (2012) and the package arfima developed by Veenstra (2012). In addition, the artfima by McLeod et al., (2016) exaggerates the fractional difference value of a times series greater than unity, > 2. To the best of our knowledge and in the long memory time series literature, similar size of fractional differencing value were never discovered nor estimated using the available long memory estimation methods. The rugarch by Ghalanos (2022) for hybrid models estimation also restricted fractional difference value in the interval of 0 < < 1. ...
This paper introduces the R package arfurimaaparch version 0.1.0 for time series computations, big data analytics and estimation of Autoregressive Fractional Unit Root Integral Moving Average-Asymmetric Power Autoregressive Conditional Heteroscedasticity (ARFURIMA-APARCH) model. The fdr, arfurimaaparch, arfurimaaparchforecast, arfurimaaparchdiagnostic and arfurimaaparch.sim are the main functions of the package. An improved version of the arfurima package version 1.1.0 of Jibrin and Rahman (2019) for implementing Monte Carlo simulation is also presented. Daily Nigeria all share index and West Texas Intermediate (WTI) crude oil prices for the period 26th January 2004 to 31st December 2018 were used to explained the usage of the packages. When the arfurimaaparch package is compared with other long memory packages, It would produce better stationary process after transformation, appropriate fractional differencing values in the interval of , minimum Akaike Information Criteria values, larger log-likelihood values, minimum p-values of the ARFURIMA-APARCH parameters estimates and large p-values of the Ljung-Box, ARCH-LM and Jarque-Bera test. Findings show that both R packages and their functions are robust, simple and user-friendly. As conclusion, the R packages are suitable, good and reliable for time series analysis computations, statistical analysis and big data analytics.
... where d , ~ 0, and is decomposed as ∝ , where ∝ 1,2,3, … … … and | | . In this case, can be in the range of 1 ∞ while most time series usually have a fractional difference value, d, in the interval of 1 2 (see Gil-Alana et al., (2018) and Sabzikar et al. (2019)). However, Hurvich and Chen (2000) and Erfani and Samimi (2009) have highlighted the repercussion of overdifferencing including loss of information, negative values of differenced series, 0.5 and estimation of complex models. ...
This article defines the Autoregressive Fractional Unit Root Integrated Moving Average (ARFURIMA) model for modelling ILM time series with fractional difference value in the interval of 1 < d < 2. The performance of the ARFURIMA model is examined through a Monte Carlo simulation. Also, some applications were presented using the energy series, bitcoin exchange rates and some financial data to compare the performance of the ARFURIMA and the Semiparametric Fractional Autoregressive Moving Average (SEMIFARMA) models. Findings showed that the ARFURIMA outperformed the SEMIFARMA model. The study’s conclusion provides another perspective in analysing large time series data for modelling and forecasting, and the findings suggest that the ARFURIMA model should be applied if the studied data show a type of ILM process with a degree of fractional difference in the interval of 1 < d < 2.
... The discrete-time process corresponding in the limit sense to continuous-time tempered fractional processes is the autoregressive tempered fractionally integrated moving average (ARTFIMA) [31][32][33]. It is an important model to consider when describing data that were traditionally modeled by the ARFIMA. ...
... Moreover, its covariance function is absolutely summable. The ARTFIMA model has been already applied to various datasets to model climate, stock returns, water turbulence and hydrology [33,34]. In [35] two tempered linear and non-linear time series models were introduced, namely ARTFIMA with α-stable noise and ARTFIMA with generalized autoregressive conditional heteroskedasticity (GARCH) noise (ARTFIMA-GARCH). ...
... In [41] the authors investigated the parameter estimation for stable ARFIMA(p, d, q) when d ∈ (−1/2, 0) and α ∈ (2/3, 2]. The authors in [33] considered statistical inference, including the parameter estimation and asymptotic normality for ARTFIMA time series but with finite second moments innovations. Here, we deal with the ARTFIMA model with non-Gaussian α-stable innovations, which means the ARTFIMA model does not have a finite second moment. ...
We present here the autoregressive tempered fractionally integrated moving average (ARTFIMA) process obtained by taking the tempered fractional difference operator of the non-Gaussian stable noise. The tempering parameter makes the ARTFIMA process stationary for a wider range of the memory parameter values than for the classical autoregressive fractionally integrated moving average, and leads to semi-long range dependence and transient anomalous behavior. We investigate ARTFIMA dependence structure with stable noise and construct Whittle estimators. We also introduce the stable Yaglom noise as a continuous version of the ARTFIMA model with stable noise. Finally, we illustrate the usefulness of the ARTFIMA process on a trajectory from the Golding and Cox experiment.
... These processes have a semi-long memory property in the sense that their autocovariance functions initially resemble that of a long memory process but eventually decay fast at an exponential rate. One special case of such processes is the autoregressive tempered fractionally integrated moving average ARTFIMA (p, d, λ, q) process, which has been studied by Meerschaert et al. (2014) and Sabzikar et al. (2019). ...
... The range of our data do not exactly match with those of Chan and Wang (2015) as they modified the data set by transformations. We fit an ARTFIMA model with p = q = 0 to the data set using the artfima package from the statistical software R developed by Sabzikar et al. (2019). The Whittle estimators of fitted parameters d and λ for log(x k ) and log(y k ) of 3 countries are given in Table 4. Also, we fit an ARFIMA model with p = q = 0 to the data. ...
This article focuses on cointegrating regression models in which covariate processes exhibit long range or semi-long range memory behaviors, and may involve endogeneity in which covariate and response error terms are not independent. We assume semi-long range memory is produced in the covariate process by tempering of random shock coefficients. The fundamental properties of long memory processes are thus retained in the covariate process. We modify a test statistic proposed for the long memory case by Wang and Phillips (2016) to be suitable in the semi-long range memory setting. The limiting distribution is derived for this modified statistic and shown to depend only on the local memory process of standard Brownian motion. Because, unlike the original statistic of Wang and Phillips (2016), the limit distribution is independent of the differencing parameter of fractional Brownian motion, it is pivotal. Through simulation we investigate properties of nonparametric function estimation for semi-long range memory cointegrating models, and consider behavior of both the modified test statistic under semi-long range memory and the original statistic under long range memory. We also provide a brief empirical example.
... The effects of long-term memory are found in many sequence datasets across a wide range of applications, such as natural language and music [5], financial market [1] data. Recently, there have been a number of studies [1,5,19] based on statistical methods to analyze and model long-term memory. Since it is difficult for these statistical methods to flexibly transfer the effort of long-term memory to the neural network model, the novel definition of long-term memory suitable for the neural network model is provided in the study [23]. ...
... obey a non-independent identical distribution, then the expectation of long-term memory decay rate of Eqn. (19) is 1 +1 when the total time steps increases from to + 1. ...
... When the decay rate of Eqn. (19) is its expectation. Then LSTM network process with attention mechanism decays linearly and the network has long memory. ...
... Long memory phenomenon & statistical models Long memory effect can be observed in many datasets, including language and music (Greaves-Tunnell & Harchaoui, 2019), financial data (Ding et al., 1993;Bouri et al., 2019), dendrochronology and hydrology (Beran et al., 2016). There are many statistical results and applications on modeling datasets with long memory (Grange & Joyeux, 1980;Hosking, 1981;Ding et al., 1993;Torre et al., 2007;Bouri et al., 2019;Musunuru et al., 2019;Sabzikar et al., 2019). Admittedly, these models enjoy rigorous statistical properties. ...
The LSTM network was proposed to overcome the difficulty in learning long-term dependence, and has made significant advancements in applications. With its success and drawbacks in mind, this paper raises the question - do RNN and LSTM have long memory? We answer it partially by proving that RNN and LSTM do not have long memory from a statistical perspective. A new definition for long memory networks is further introduced, and it requires the gradient to decay hyperbolically. To verify our theory, we convert RNN and LSTM into long memory networks by making a minimal modification, and their superiority is illustrated in modeling long-term dependence of various datasets.
... Such processes have empirical relevance for modelling time series that are known to display various degrees of long memory with autocovariances that decay slowly at first but ultimately decay much faster, such as the magnitude or certain powers of financial returns (see, for example, [20]). For an empirical example, we refer to [54], where the ARTFIMA(0, 0.3, 0.025, 0) is used to model the log returns for AMZN stock price from 1/3/2000 to 12/19/2017. The advantage of using ARTFIMA is the fact that we can capture aspects of the low frequency activity better than the ARFIMA time series in part of the long-range dependence scenario. ...
... In fact, those estimators are strongly consistent under quite general conditions. For estimation of the parameter λ along these lines, see [54]. But when the parameter λ is sample size dependent and tends to zero as N → ∞, the problem is much more complex, just as it is in local to unity (LTU) and local to zero cases in simpler autoregressive models. ...
In an early article on near-unit root autoregression, Ahtola and Tiao (1984) studied the behavior of the score function in a stationary first order autoregression driven by independent Gaussian innovations as the autoregressive coefficient approached unity from below. The present paper develops asymptotic theory for near-integrated random processes and associated regressions including the score function in more general settings where the errors are tempered linear processes. Tempered processes are stationary time series that have a semi-long memory property in the sense that the autocovariogram of the process resembles that of a long memory model for moderate lags but eventually diminishes exponentially fast according to the presence of a decay factor governed by a tempering parameter. When the tempering parameter is sample size dependent, the resulting class of processes admits a wide range of behavior that includes both long memory, semi-long memory, and short memory processes. The paper develops asymptotic theory for such processes and associated regression statistics thereby extending earlier findings that fall within certain subclasses of processes involving near-integrated time series. The limit results relate to tempered fractional processes that include tempered fractional Brownian motion and tempered fractional diffusions of the second kind. The theory is extended to provide the limiting distribution for autoregressions with such tempered near-integrated time series, thereby enabling analysis of the limit properties of statistics of particular interest in econometrics, such as unit root tests, under more general conditions than existing theory. Some extensions of the theory to the multivariate case are reported.
... On the other hand, Sabzikar et al., [6] introduced the artfima package for fitting tempered fractionally integrated time series that possesses or displays LM. They introduced parameter estimation procedures for the Auto-Regressive Tempered Fractional Integral Moving Average (ARTFIMA) model. ...
This paper introduces the arfurima package for Interminable Long Memory (ILM) time series that exhibits strong hyperbolic decay Auto Correlations Function (ACF) and large spectrum at zero frequency. The package has facilities for Auto Regressive Fractional Integral Moving Average (ARFIMA) and Auto Regressive Fractional Unit Root Integral Moving Average (ARFURIMA) models of a mean of a long memory process. The mean of this LM process can be described in each ARFIMA and ARFURIMA models with fractional differencing values of 0 < d < 1 and 1 < d < 2 respectively. The arfurima.sim, furd, arfurima and arfurima.forecast are the main functions of the package. The first function simulates the fractional unit root integral series for d such that 1 < d < 2 so the simulated series has the form of ARFURIMA (p,d,q). The second function fractionally differences the Fractional Unit Root Integral (FURI) time series and returns stationary series. The third function estimates ARFIMA or ARFURIMA model while the fourth function undertakes the forecasts. Finally, daily Malaysia ringgit to United State dollar exchange rate and Malaysia average temperature data is used to illustrate the usage of the package.
