J. R. M. Hosking, "Fractional differencing, " Biometrika, vol. 68, no. 1, pp. 163-176, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....spectrum; even in these cases, though, the FSD model did a good job of summarizing dependence for the purpose of studying long range dependence. An FSD model consists of the sum of two Gaussian time series, independent of one another a dependent time series that is either a fractional ARIMA [31] or short range dependent, and a white noise time series. Suppose z u is an FSD time series. The dependent component, s u ,is #I B# s u # u # u## where u is Gaussian white noise with mean zero and variance # d## d# #### #d# (which makes the variance of s u equal to 1) B ....

J. R. M. Hosking, "Fractional Differencing," Biometrika, vol. 68, pp. 165--176, 1981.

....the joint distributions of all finite subsets of the time series are multivariate normal. Let # s u # # n u ; where s u and n u are independent of one another and each has mean 0 and variance 1. n u is white noise, that is, an uncorrelated time series. s u is a fractional ARIMA (FARIMA) model [31] s u # u # u## where Bs u # s u## , # d #:#,and u is white noise with mean 0 and variance # d## d# #### #d# to make the variance of s u equal to 1. z u is an FSD model. We coined this term because the model for z u can be written as a combination of fractional and ....

J. R. M. Hosking, "Fractional Differencing," Biometrika, vol. 68, pp. 165--176, 1981.

....ARMA model that can capture the longrange dependence of self similar signals. In some sense, these signals are borderline stationary. The AR, MA, ARMA, and ARIMA models are classical time series models described by Box, et al. [3] ARFIMA models are well covered in more recent literature [7, 6, 2]. The RPS technical report [5] also provides an explanation of these models as well as a detailed description of the implementations we use here. The same implementations are used for offline and online analysis in RPS. It is important to point out that we do not follow the Box Jenkins ....

HOSKING, J. R. M. Fractional differencing. Biometrika 68, 1 (1981), 165--176.

....for the predictions to the variance of the second half of the signal. This is basically the noise to signal ratio of the predictor. The smaller the ratio, the better the predictability. We use a wide range of models, including the classical AR, MA, ARMA, and ARIMA models [7] fractional ARIMAs [20], 18] and simple models such as LAST and a windowed average. Our prediction tools are currently available as part of our RPS Toolbox [15] Our Tsunami wavelet toolbox, which we describe further in Sections IV and V, will also soon be released. The earliest work in predicting network ....

....The ARFIMA(4, 1,4) model is a fractionally integrated ARMA model that can capture the long range dependence of self similar signals. AR, MA, ARMA, and ARIMA models are classical time series models well covered by Box, et al. [7] ARFIMA models are well covered in more recent literature [20], 18] 5] The RPS technical report [15] also provides an explanation of these models as well as a detailed description of the implementations we use here. The same implementations are used for offline and online analysis in RPS. For each of the 34 AUCKLAND traces, we performed the analysis ....

HOSKING, J. R. M. Fractional differencing. Biometrika 68,1 (1981), 165--176.

....[6] AR, MA, ARMA, and ARIMA classes might be appropriate for predicting load. On the other hand, the existence of self similarity induced long range dependence suggested that such models might require an impracti cal number of parameters or that the much more com plex ARFIMA model class [20,16,4], which explicitly captures long range dependence, might be more appro priate. Since it is not obvious which model is best, we empirically evaluated the predictive power of the AR, MA, ARMA, ARIMA, and ARFIMA model classes, as well as that of a simple ad hoc windowed mean predic tor called BM ....

....parameters in the range of 0.5 to 1.0 indicate self similarity with positive near neighbor correlations. This result tells us that load varies in complex ways on all time scales and has long range dependence. Long range dependence suggests that using the fractional ARIMA (ARFIMA) modeling approach [20,16,4] may be appropriate. 6) The traces display what we term epochal behavior. The local frequency content (measured by using a spectrogram) of the load signal remains quite stable Unpredictable Random Signal a, WhtteNotse (0. cr ) Fixed Linear Filter Partially Predictable Signal a ....

[Article contains additional citation context not shown here]

HOSKING, J. R. M. Fractional differencing. Biometrika 68, i (1981), 165-176.

....spectrum; even in these cases, though, the FSD model did a good job of summarizing dependence for the purpose of studying long range dependence. An FSD model consists of the sum of two Gaussian time series, independent of one another a dependent time series that is either a fractional ARIMA [31] or short range dependent, and a white noise time series. Suppose z u is an FSD time series. The dependent component, s u ,is #I # B# d s u # u # u## where u is Gaussian white noise with mean zero and variance # # ## # d## # ## # d# #### # #d# (which makes the variance ....

J. R. M. Hosking, "Fractional Differencing," Biometrika, vol. 68, pp. 165--176, 1981.

....loess to choose a parametric family that approximates the true spectrum locally was well as possible. FARIMA models have an additive term in the log spectrum consisting of a constant times A which introduces substantial curvature in the log spectrum near the origin, but is smooth elsewhere [15]. Since our packet spectra exhibit this behavior, the P F are smoothed by loess as a function of P F , using a quadratic as the local regression function. The fit, though, is then plotted against P F . This results in a much better fit than smoothing directly as a quadratic function ....

J. R. M. Hosking. Fractional Differencing. Biometrika, 68:165--176, 1981.

....are zero if k d and d is an integer. The representation in Eq. 25 for d is equivalent to an infinite order autoregressive process (all pole filter with an infinite order) F ARIMA(0, d, 0) process with 0 d 1 2, is stationary and long range dependent, with an auto correlation function [29] (26) Observe that for 0 d 1 2, the hyperbolic decay will produce persistence. By comparing Eq. 26 to Eq. 21, d = 1 a) 2 = H 0.5. F ARIMA processes can model short range and long range dependence. If Gaussian white noise is used, then the F ARIMA has a Gaussian distribution. This ....

J. Hosking, "Fractional Differencing," Biometrica, 1981, pp. 165--76.

....a parametric family that approximates the true spectrum locally was well as possible. FARIMA models have an additive term in the log spectrum consisting of a constant times g(u) log(4 sin 2 ( u) which introduces substantial curvature in the spectrum near the origin, but is smooth elsewhere [16]. Since our packet spectra exhibit this behavior, the L( fk ) are smoothed by loess as a function of g( fk ) using a quadratic as the local regression function. The fit, p(f) is then plotted against f . This results in a much better fit than smoothing directly as a quadratic function ....

J. R. M. Hosking. Fractional Differencing. Biometrika, 68:165--176, 1981.

....GPH test is sometimes designated as a semiparametric test. The dominant parametric discrete time model that exhibits hyperbolic decay of its autocorrelation function is the fractional integrated autoregressive moving average model (ARFIMA) introduced independently by Granger and Joyeux (1980) and Hosking (1981) see Viano, Deniau, and Oppenheim (1994) for a continuous time version. For Gamma0:5 d :5, X t is said to follow an ARFIMA(p,d,q) model if it is the unique stationary solution to the equation (1 Gamma B) d OE(B)X t = B)j t ; j t iidN(0; oe 2 j ) where B is the backshift operator ....

Hosking, J. (1981),"Fractional differencing," Biometrika 68, 165-176.

....originally introduced in [83] for social measurement purposes. However, initial sizeable empirical success of the long range dependence (LRD) concept in economics is certainly related to the autoregressive fractionally integrated moving average model (hereafter ARFIMA) proposed simultaneously by [64]and [51] and the increased flexibility it brings to the Box Jenkins linear time series methodology, in particular for modeling macroeconomic time series since [30] The relative success of the LRD concept in economics may also be attributed to the development of a rationale for its presence in ....

Hosking, J. (1981): "Fractional differencing," Biometrika, 68, 165--176.

.... goes to infinity.To obtain the results of this paper we formally define a stationary process X t as having long memory if the process s theoretical autocorrelation function ae X ( j j 2d;1 L( # where d 2 ( 1# 1=2) and L( isaslowvarying function at infinity [Granger and Joyeux (1980) Hosking (1981), Brockwell and Davis (1993) and Baillie (1996) 4 In contrast to a short memory ARMA process (d = 0) whose autocorrelation function decays geometrically fast to zero, the notable feature of long memory is a autocorrelation function that decays hyperbolically to zero when d 6=0. When d 0, ....

Hosking, J.R. (1981) "Fractional Differencing," Biometrika, 68, 165-176.

....as is typical for long range processes. The best known models for long range correlated processes are the increments of selfsimilar processes as mentioned above which are called fractional noise, and fractional ARIMA(p,d,q) processes, where d is between 0 and 1 2 (Granger and Joyeux 1980, Hosking 1981). For near Gaussian fractional noise, statistical methods have been worked out and programmed, including confidence and prediction intervals for the mean ( onesample t test for long range correlated data , see Beran 1986, 1989) For a more detailed survey of these statistical methods see Beran ....

....variation and bounded away from zero. 3:7) Condition (3.7) implies (3.6) see Zygmund (1959, Chap. V.2) In (3.6) stationarity is not required. The two most common models with long range correlations, fractional noise (Mandelbrot and van Ness, 1968) and fractional ARIMA (p,d,q) processes (Hosking, 1981), satisfy (3.7) see Sinai (1976) in the former and Hosking (1981) in the latter case. If the treatments are allocated in space as for instance in agriculture, the index t becomes multidimensional, i.e. the t 0 s are then a random field. We expect our results to generalize to this case. 3.4 ....

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Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68, 165-176.

....family that approximates the true spectrum locally was well as possible. FARIMA models have an additive term in the log spectrum consisting of a constant times g(u) log 10 (4 sin 2 ( u) which introduces substantial curvature in the log spectrum near the origin, but is smooth elsewhere [15]. Since our packet spectra exhibit this behavior, the L( fk ) are smoothed by loess as a function of g( fk ) using a quadratic as the local regression function. The fit, though, is then plotted against fk . This results in a much better fit than smoothing directly as a quadratic ....

J. R. M. Hosking. Fractional Differencing. Biometrika, 68:165--176, 1981.

....and Ebens (2000) 2.2. Fractionally integrated white noise 6 An important example of a long memory process is a stochastic process t y that requires fractional differencing to obtain a set of independent and identically distributed residuals t e . Following Granger and Joyeux (1980) and Hosking (1981), such a process is defined using the filter ( 3 ) 2 ) 1 ( 2 ) 1 ( 1 1 3 2 = L d d d L d d dL L d (6) where L is the usual lag operator, so that 1 = t t y Ly . Then a fractionally integrated white noise (FIWN) process t y is defined by ( t t d y L e = 1 ....

....e = 1 (7) with the t e assumed to have zero mean and variance 2 e s . Throughout this paper it is assumed that the differencing parameter d is constrained by 1 0 d . The mathematical properties of FIWN are summarised in Baillie (1996) and were first derived by Granger and Joyeux (1980) and Hosking (1981). The process is covariance stationary if 2 1 d and then the following results apply. First, the autocorrelations are given by ( 3 2 1 2 1 , 2 1 1 , 1 3 2 1 d d d d d d d d d d d d = r r r (8) or, in terms of the Gamma function, ....

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Hosking, J., 1981, Fractional differencing, Biometrika 68, 165-176.

.... = 1; 2; denote a covariance stationary, purely nondeterministic time series with mean zero and with autocovariance function, fl Z (k) cov(Z t ; Z t Gammak ) As is discussed by Beran (1994) many long memory processes such as the FGN (Mandelbrot, 1983) and FARMA (Granger and Joyeux, 1980; Hosking, 1981) may be characterized by the property that k ff fl Z (k) c ff;fl as k 1, for some ff 2 (0; 1) and c ff;fl 0. Equivalently, fl Z (k) c ff;fl k Gammaff : 1) As noted in Box and Jenkins (1976) the usual stationary ARMA models on the other hand are exponentially damped since fl Z (k) ....

....asymptotically smaller than fl k . This establishes the asymptotic equivalence of the lefthand side and the right hand side of eq. 3) and the theorem since fl Z (k) uniquely determines the coefficients in the inverted model. Pi The FARMA model of order (p; q) Granger and Joyeux, 1980; Hosking, 1981) may be defined by the equation, OE(B) 1 Gamma B) d Z t = B)A t ; 4) where jdj 0:5, A t is white noise with variance oe 2 A , OE(B) 1 Gamma OE 1 B Gamma : Gamma OE p B p , and (B) 1 Gamma 1 B Gamma : Gamma q B q . For stationarity and invertibility it is ....

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Hosking, J.R.M. (1981). Fractional differencing. Biometrika 68, 165--176.

....a process y t is said to be fractionally integrated when is stationary, with d being between 0 and 1. An ARFIMA (1 L) d y t process would admit autoregressive and moving average components after having been applied the fractional difference (1 L) d , with a white noise innovation [see Hosking (1981) and Granger and Joyeux (1980) Mills(1990) Geweke and Porter Hudak (1983) The process is stationary and ergodic for 0.5 d 0.5, with a bounded and positively valued spectrum at all frequencies. When 0 d 0.5, the sum of the absolute values of the autocorrelation coefficients goes to infinity ....

Hosking, J.R.M., 1981, Fractional differencing, Biometrika, 68, 165-176.

....of such a long memory time series is not summable and several authors, for example Hipel and McLeod (1978) have used this as the definition of long memory. In fact the spectral viewpoint provides a much better definition and understanding of long memory in time series. As first suggested by Hosking (1981), we will say that a covariance stationary time series exhibits long memory if and only if lim f 0 p(f) 1: 9) From eq. 8) it follows that the autocovariance function is not summable. However for the same reason it also follows that any time series such that lim f f0 p(f) 1 also has a ....

.... stationary model suggested was Mandelbrot s fractional Gaussian noise (FGN) model described by Mandelbrot and Van Ness (1968) and advocated for hydrological time series by Mandelbrot and Wallis (1969) A more flexible approach to long memory models was initiated by Granger and Joyeux (1980) and Hosking (1981) who suggested what is now referred to as the fractional ARMA model. This provides a comprehensive family of stationary and ergodic models which generalize the usual ARMA model. Beran (1992) gives a recent review of long memory time series models and several other researchers have enthusiastically ....

Hosking, J.R.M. (1981), Fractional differencing, Biometrika, 68(1), 165--176.

....Box Jenkins [6] AR, MA, ARMA, and ARIMA classes might be appropriate for predicting load. On the other hand, the existence of selfsimilarity induced long range dependence suggested that such models might require an impractical number of parameters or that the much more expensive ARFIMA model class [18, 14, 4], which explicitly captures long range dependence, might be more appropriate. Since it is not obvious which model is best, we empirically evaluated the predictive power of the AR, MA, ARMA, ARIMA, and ARFIMA model classes, as well as a simple ad hoc windowed mean predictor called BM and a ....

....parameters in the range of 0.5 to 1.0 indicate self similarity with positive near neighbor correlations. This result tells us that load varies in complex ways on all time scales and has long range dependence. Long range dependence suggests that using the fractional ARIMA (ARFIMA) modeling approach [18, 14, 4] may be appropriate. 6) The traces display what we term epochal behavior. The local frequency content (measured by using a spectrogram) of the load signal remains quite stable for long periods of time (150 450 seconds mean, with high standard deviations) but changes abruptly at the ....

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HOSKING, J. R. M. Fractional differencing. Biometrika 68, 1 (1981), 165--176.

....all the autocorrelation from the series so that the residuals from the ARMA model will still be autocorrelated, specifically long memoried. That is, if we use an ARMA(p,q) to filter a stationary ARFIMA(p,d,q) process the errors will be an I(d) process with the ACF: 4) Granger and Joyeux 1980; Hosking 1981). Thus, using ARMA filters to prewhiten ARFIMA processes in the Haugh Pierce test can produce nonzero covariance of the sample crosscorrelation estimates and lead to erroneous statistical inference. Note that since no finite ARMA model can fully capture the long memory implied by the fractional ....

Hosking, J. R. M. 1981. "Fractional Differencing." Biometrika 68: 165-76.

....b is the modal instantaneous volatility and without any loss of generality is assumed to equal 2 one. L is the lag operator, x t Gammas = L s x t , and oe 2 j is the variance of the log volatility. The long memory process was originally introduced by both Granger and Joyeux (1980) and Hosking (1981) (for a excellent review of the work performed in economics with longmemory processes see Baillie, 1996) Since its inception is has been well known that the long memory process h t is associated with highly persistent behavior as quantified by the slow hyperbolic decay of its autocovariance ....

Hosking, J.R.M. (1981) "Fractional differencing," Biometrika 68, 165-176.

....operator, OE(x) 1 Gamma P p j=1 OEx j is a polynomial with roots outside the unit circle and ffl i (i = Gamma1; 0; 1; 2; are iid zero mean normal with var (ffl i ) oe 2 ffl . Here, the fractional difference (1 Gamma B) ffi introduced by Granger and Joyeux (1980) and Hosking (1981) is defined by (1 Gamma B) ffi = 1 X k=0 b k (ffi)B k (2) with b k (ffi) Gamma1) k Gamma(ffi 1) Gamma(k 1) Gamma(ffi Gamma k 1) 3) The main motivation for introducing fractional autoregressive models (Hosking 1981, Granger and Joyeux 1980) was to model stationary time ....

.... B) ffi introduced by Granger and Joyeux (1980) and Hosking (1981) is defined by (1 Gamma B) ffi = 1 X k=0 b k (ffi)B k (2) with b k (ffi) Gamma1) k Gamma(ffi 1) Gamma(k 1) Gamma(ffi Gamma k 1) 3) The main motivation for introducing fractional autoregressive models (Hosking 1981, Granger and Joyeux 1980) was to model stationary time series with long range dependence (or long memory) and to avoid the problem of overdifferencing. Here, long range dependence is defined as follows (see, e.g. Mandelbrot 1983, Cox 1984, Hampel 1987, K unsch 1986, and Beran 1994 and references ....

Hosking, J.R.M. (1981) Fractional differencing. Biometrika 68 165-176.

.... Gamma(1 Gamma 2d) Gamma 2 (1 Gamma d) 1) for 0, we have s Y; s Y; Gamma ) When 0 d 1 2 , the SDF has a pole at zero, in which case the process exhibits slowly decaying autocovariances and constitutes a simple example of a long memory process; see Granger and Joyeux (1980) Hosking (1981), and Beran (1994, Sec. 2.5) In this case, d is called the long memory parameter. An important assumption behind any stationary process is that its variance is a constant independent of the time index t. In the context of short memory models, such as stationary autoregressive and moving average ....

Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68 (1), 165--176.

....increased from 0.01 to 0.05, while for larger outliers (i = 5; 7) the empirical size decreases as becomes larger. 3. 5 Long memory We now consider the case where y t is generated according to the ARFIMA(0,d,0) or fractionally differenced noise process introduced by Granger and Joyeux (1980) and Hosking (1981), 1 Gamma L) d y t = t ; 14) where d is a real number. For non integer values of d, the fractional differencing filter (1 Gamma L) d can be expanded as (1 Gamma L) d = 1 Gamma dL Gamma d(1 Gamma d)L 2 2 Gamma d(1 Gamma d) 2 Gamma d)L 3 3 Gamma : 15) For d 2 ....

Hosking, J.R.M., 1981, Fractional differencing, Biometrika 68, 165--176.

.... to model 1 time series include work by Diebold and Rudebusch (1989) Geweke and Porter Hudak (1983) Hosking (1984) Lo (1989) and Sowell (1992) A time series fz t g is generated by an autoregressive fractionally integrated moving average (ARFIMA) process (Granger and Joyeaux, 1980 and Hosking, 1981) if OE(B) 1 0B) d z t = B) t ; 1) where OE(B) 1 0 OE 1 B 0 1 1 1 0 OE p B p and (B) 1 0 1 B 0 1 1 1 q B q are polynomials in B of degrees p and q respectively, p and q are integers, B is the backshift operator, i.e. Bz t = z t01 ; d is a real number denoting the fractional ....

....d; q) process (1. 1) the exact likelihood function has a multivariate normal form: L(z n ; fi ) 2) 0n=2 j 6 j 0 1 2 expf0 1 2 z 0 n 6 01 z n g (3) where 6 = Cov(z n ) and is a function of fi with elements given by fl z k ; the autocovariances of fz t g of lag k: Hosking (1981) explicitly derived fl z k directly, while Sowell (1992) computed the elements fl z k as the inverse Fourier 2 transforms of the spectral density function of fz t g which has the form: s fi (w) oe 2 2 (e iw ) e 0iw ) OE(e iw )OE(e 0iw ) f(1 0 e iw ) 1 0 e 0iw )g 0d ....

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Hosking. J.R.M. (1981) Fractional differencing. Biometrika, 68, 165-176.

....P k= Gamma1 jae(k)j = 1 and a short memory if 1 P k= Gamma1 jae(k)j 1. See Granger (1980) Lawrance and Kottegoda (1977) and Greene and Fietlitz (1977, 1979) for applications of long memory models. Fractionally differenced autoregressive models provide a rich class of long memory models. Hosking (1981), Yajima (1985, 1991) and Fox and Taqqu (1986) have studied various aspects of the asymptotic estimation problem for fractionally differenced models. In this paper, we introduce a multivariate process based on fractionally differenced autoregressive process. In Section 2, we introduce the model ....

....direct calculation of j Sigma X j and Sigma Gamma1 X can be avoided by expressing Sigma X in terms of the one step predictors and their mean squared errors which are easily calculated recursively using the algorithms given in Brockwell and Davis (1991) To this end, we proceed as follows. Hosking (1981) showed that the best linear predictor of X js in terms of (X 1s ; X j Gamma1;s ) is X js = E(X js jX 1s ; X j Gamma1;s ) 8 : 0 for j = 1 and s = 1; n P j Gamma1 i=1 OE (d) j Gamma1;i X j Gammai;s for j 2 and s = 1; n 9 = 2.5) where OE (d) k;i = ....

Hosking, J. R. M. (1981). Fractional differencing. Biometrika, 68 , 165-176.

....is also presented. Keywords. Discrete wavelet packet transform, Gegenbauer process, long memory, multitaper spectral estimation, periodogram, wavelet variance. 1 Introduction A simple generalization of the fractional difference process (or fractional ARIMA model) was mentioned, in passing, by Hosking (1981) and allows the singularity in the spectral density function (SDF) of the process to be located at any frequency 0 f 1=2. Such a process has been referred to as a Gegenbauer process (Gray et al. 1989) and also a seasonal persistent process (SPP) Andel 1986) We prefer the latter term because ....

....for jOEj = 1 and Gamma1=4 ffi 1=4 or jOEj 1 and Gamma1=2 ffi 1=2. Clearly, the definition of an SPP also includes a fractional difference process; a simple example of a long memory process. When OE = 1 we have that fY t g is a fractional difference process with parameter d = 2ffi (Hosking 1981). If fffl t g is a Gaussian white noise process, then fY t g is also called a Gegenbauer process since Equation (1) may be written as an infinite moving average process via Y t = 1 X k=0 C (ffi) k;OE ffl t Gammak ; where C (ffi) k;OE is a Gegenbauer polynomial (Rainville 1960, Ch. 17) ....

Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68 (1), 165--176.

....traffic measurements have revealed that traffic in high speed networks exhibits selfsimilarity, i.e. long range dependence [2] that can t be captured by previous models. Hence, self similar models, such as FGN (fractional Gaussian noise) model [3] and FDN (fractional differencing noise) model [4] have been developed. Unfortunately, these models can t be used to describe the short range dependence. Recent real traffic measurements found the co existence of both long range and short range dependence in traffic traces [5] Therefore, models are required to describe both longrange and ....

....traffic trace. Section 5 studies the feasibility of FARIMA(p,d,q) models. Section 6 is the concluding remarks. 2. THE MODEL FARIMA processes are the natural generalizations of standard ARIMA (p,d,q) processes defined in [1] when the degree of differencing d is allowed to take nonintegeral values [4]. A FARIMA(p,d,q) process X t : t= 1, 0, 1, is defined to be , t d a B X B Q = D F (2 1) where a t is a white noise and d ( 0.5, 0.5) 1 ) 2 2 1 p p B B B B f f f = F # , 1 ) 2 2 1 q q B B B B q q q = Q # Q(B) F(B) have no common zeroes, ....

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. J.R.M. Hosking, "Fractional differencing," Biometrika, vol. 68, pp. 165-176, 1981.

....mean. This assumption will later be repeated at the point where it is needed. A. Discrete Time Power Laws Let the two sided spectral density of the sampled sequence be given by (11) Then as . These so called fractional difference processes were described by Granger and Joyeux [8] and by Hosking [9]. Because the first difference of the process defined by (4) is just , we know that is also a fractional difference process, with spectral density (12) This frequency domain description of has an equivalent time domain description, the generalized autocovariance (GACV) sequence , where runs ....

....values of needed in this study. Bear in mind that the noise type label applies to , a power law process with exponent , while applies to , a power law process with exponent . The formula for nonintegral in Table I is the same as the one derived for fractional difference processes by others [8] [9], 11] It has been verified that this formula actually does extend to the nonstationary situation. Because passage to the limit of the GACV as approaches an integer is unfortunately not straightforward in general, the formulas for integral were derived from known ACV s of stationary ....

J. R. M. Hosking, "Fractional differencing," Biometrika, vol 68, pp. 165--176, 1981.

.... methods have become a widely used tool to study the components (Riedi and Vehel 1997; Feldman, Gilbert, and Willinger 1998; Ribeiro, Riedi, Crouse, and Baraniuk 1999) Another approach is to build time domain models and include long range persistent components as is done in other disciplines (Hosking 1981; Haslett and Raftery 1989) This latter approach was used in the analysis of the Bell Labs HTTP start times. Nonstationarity Nonstationarity is as pervasive in Internet traffic data as long range persistence. Cleveland, Lin, and Sun 2000a; Cao, Cleveland, and Sun 2000) However, ....

Hosking, J. R. M. (1981). Fractional Differencing. Biometrika 68, 165--176.

....have the asymptotic proportionality strengthened to an exact equality for all k and all m. An example of an exactly self similar process with self similarity parameter H is fractional Gaussian noise with parameter 1=2 H 1 (see (see [31] for more details) fractional ARIMA(p; d; q) models (see [15, 19]) are examples of asymptotically self similar processes with selfsimilarity parameter H = d 1=2; 0 d 1=2. For other approaches to characterizing the fractal properties of measured packet traffic that make explicit use of certain fractal dimension descriptors, see [9, 7, 10] 3 Experimenting ....

J.R.M. Hosking, "Fractional Differencing", Biometrika 68, pp. 165-176, 1981.

.... standard normal random variables, and the coefficients f j g are given by j = 8 : d(d 1) Delta Delta Delta (d j Gamma 1) j ; j 1; 1; j = 0: Some authors distinguish the behavior of fX t g for different values of d 2 ( Gamma0:5; 0:5) It is well known that, see for example, Hosking (1981), Samorodnitsky and Taqqu (1994) or Brockwell and Davis (1991) for d 2 ( Gamma0:5; 0) the process is negative dependent and it is often referred as an intermediate memory process since its autocorrelation function ae(k) is always negative, of order k 2d Gamma1 as k 1, and P k jae(k)j 1. ....

Hosking, J.R.M. (1981). Fractional Differencing. Biometrika 68 165-176.

....model for long memory processes is given by the fractional difference process fX t g such that (1 Gamma B) d X t = ffl t ; 1) where Gamma1=2 d 1=2, B is the backward shift operator and fffl t g is Gaussian white noise with variance oe 2 ffl . This process is stationary and invertible (Hosking 1981). The spectral density function (SDF) of fX t g has a singularity at zero and is approximately linear on the log scale. The long memory aspect is seen through the fact that the autocovariance sequence (ACVS) decays at a hyperbolic rate. A simple generalization of this model was given, in passing, ....

....spectral density function (SDF) of fX t g has a singularity at zero and is approximately linear on the log scale. The long memory aspect is seen through the fact that the autocovariance sequence (ACVS) decays at a hyperbolic rate. A simple generalization of this model was given, in passing, by Hosking (1981) via Gamma 1 Gamma 2OEB B 2 Delta ffi Y t = ffl t ; where OE = cos(2f G ) and fffl t g is defined as before. The SDF of fY t g is given by S Y (f) oe 2 ffl f4[cos(2f) Gamma OE] 2 g Gammaffi ; for Gamma 1 2 f 1 2 ; 2) and hence, exhibits a singularity at jf G j 1=2 ....

Hosking, J. R. M. (1981). Fractional differencing. Biometrika, 68, 165--176.

....(SDF) of fX t g is given by SX (f) j oe 2 ffl j2 sin(f)j 2d for Gamma 1 2 f 1 2 ; so that SX (f) 1 as f 0 and, thus, the SDF diverges at zero frequency. Further introductions to fractional difference and related processes can be found in, e.g. Granger and Joyeux (1980) and Hosking (1981). Both time and frequency domain techniques have been established for the simulation of such long memory processes (see Hosking (1984) Percival (1992) and references therein) the partitioning of the time frequency plane by the discrete wavelet transform (DWT) makes it a natural alternative to ....

....time is illustrated. The wavelet based procedure is also extended to the case of two spectral asymptotes. Conclusions are presented and generalizations given at the end. 2 Seasonal Persistent Processes A simple generalization of the model given in Equation 1. 1 was mentioned, in passing, by Hosking (1981) and allows the singularity in the spectrum to be located at any frequency 0 f 1 2 . Such 3 a process has been referred to as a Gegenbauer process (Gray et al. 1989) and also a seasonal persistent process (SPP) Andel 1986) We prefer the latter term because it more accurately and concisely ....

Hosking, J. R. M. (1981), "Fractional differencing," Biometrika, 68, 165--176.

.... traffic models based on fractional autoregressive integrated moving average (FARIMA) processes are generalizations of the well known ARIMA(p; d; q) models of Box Jenkins [4] Fractional ARIMA processes allow generalizations of the parameter d to take on non integer values and were proposed in [8]. These processes are asymptotically second order self similar with self similarity parameter d 1=2 with 0 d 1=2. We now describe fractional ARIMA processes in greater detail and show their convergence to fractional Brownian motion. A Preliminaries The FARIMA(0; d; 0) can be heuristically ....

....d x t = a t (16) or equivalently, x t = 1 X j=0 b j ( Gammad)a t Gammaj ; t = Delta Delta Delta ; Gamma1; 0; 1; Delta Delta Delta (17) where the operator r d is defined in Equation (14) and the noise process a t consists of i.i.d. random variables with zero mean and finite variance [8]. In a more general case, though, the innovations a t may be from stable distributions with infinite variance. The generalized FARIMA(p; d; q) process where p and q are integers and d real is then defined as the stochastic process fy t g with OE(B)r d y t = B)a t (18) where r d is defined in ....

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J. R. M. Hosking, "Fractional differencing, " Biometrika, vol. 68, no. 1, pp. 163-176, 1981.

....# 0, v t is independent of # t , # t is a sequence of independent and identically distributed (i.i.d. random variables with mean zero and variance one, and v t is a long memory process having parameter d, with 0 d 0.5. The fractionally integrated 9 noise model of Hosking (1981) and Granger and Joyeux (1980) i.e. 1 B) d v t = # t , where B denotes the backward shift operator (B j x t = x t j ) and # t is a short memory process, can be used to parameterize the long memory process. Typically, v t is assumed to be Gaussian. The conditional variance # 2 t ....

Hosking, J. R. M. (1981): "Fractional differencing", Biometrika 68, 1, 165-176.

.... periods (depending on the short memory parameters in the autoregressive and moving average parts) and where the effect of a shock lasts forever in a unit root [I(1) process, the fractionally integrated model [FI(d) with d 2 (0; 1) takes up an intermediate position, see Granger Joyeux (1980) Hosking (1981), and more recently, Baillie (1996) and Beran (1994) The ARFIMA(p; d; q) model is written as Phi(L) 1 Gamma L) d (z t Gamma z ) Theta(L)ffl t t = 1; T (1) where z t is the time series at time t, z its mean, and Phi(L) 1 Gamma OE 1 L Gamma : Gamma OE p L p is the stable ....

Hosking, J. R. M. (1981), `Fractional differencing', Biometrika 68, 165--176.

....of fundamental importance to study the effects of SRD on the wavelet based estimator. 4 Experiments with FARIMA processes In order to investigate the impact of SRD on the performance of this estimator, we use the fractionally differenced autoregressive integrated moving average (FARIMA) process [5, 6]. This process possesses the LRD and SRD properties simultaneously. We apply wavelet based estimator [1] on large sets of data generated by FARIMA(1; d; 0) model in order to evaluate its performance. 4.1 FARIMA process FARIMA processes are the generalizations of standard ARIMA processes when the ....

....for the presence of LRD. The autoregressive coefficient OE 1 can be used as an asymptotic measure of the SRD structure. FARIMA(0; d; 0) process can be regarded as a special form of FARIMA(1; d; 0) with OE 1 = 0. OE 1 = 0:3 OE 1 = 0:5 [j 1 , j 2 ] H [j 1 , j 2 ] H [2, 7] 0.786 [3, 10] 0. 764 [3, 5] 0.736 [4, 8] 0.698 [3, 6] 0.694 [4, 9] 0.695 [3, 7] 0.707 [4, 10] 0.693 Table 2: Estimated Hurst parameters for FARIMA(1; d; 0) with d = 0:15 (H = 0:65) OE 1 = 0:3 and 0.5. We generated FARIMA(1; d; 0) traces with four values of d (0.15, 0.25, 0.35, 0.45) and four values of OE 1 (0, 0.1, ....

[Article contains additional citation context not shown here]

J. Hosking, "Fractional differencing," Biometrika, vol. 68, pp. 165--176, 1981.

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J. R. M. Hosking, "Fractional differencing, " Biometrika, vol. 68, no. 1, pp. 163-176, 1981.

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Hosking, J.R.M. (1981). Fractional differencing, Biometrika, 68, 165-176.

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J. R. M. Hosking (1981). Fractional Differencing. Biometrika, 68:165--176.

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HOSKING, J. R. M. Fractional differencing. Biometrika 68,1 (1981), 165--176.

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J. R. M. Hosking. Fractional differencing. Biometrika, 68(1):165-176, 1981.

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HOSKING, J. R. M. Fractional differencing. Biometrika 68, 1 (1981), 165--176.

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J. R. M. Hosking. Fractional Differencing. Biometrika,

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Hosking, J. R. (1981) Fractional differencing . Biometrika 68, 165--176.

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J.R.M. Hosking (1981). "Fractional differencing," Biometrika 68, 165-176.

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Hosking, J. (1981): "Fractional Differencing", Biometrika, 68, 165-176.

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J.R.M. Hosking (1981). "Fractional differencing," Biometrika 68, 165-176.

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Hosking, J.R.M. (1981). Fractional Differencing. Biometrika, 68:165--176.

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