On fast generation of fractional Gaussian noise

Purczyński, Jan; Włodarski, P.

doi:10.1016/j.csda.2005.04.021

Cited by 9 publications

(5 citation statements)

References 19 publications

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“…This natural extension of ARIMA(0, 1, 0) can be seen as an advantage of ARFIMA(0, d, 0) models compared with fGn (Graves et al 2017). On the other hand, many asymptotic relations of fGn processes also hold for finite samples (Taqqu et al 1995) and fGn-based models are very popular due to their analytic tractability (Purczyński and Wlodarski 2006).…”

Section: Introductionmentioning

confidence: 99%

“…We note that by using algorithms based on the fast Fourier transform, ARFIMAmodels can be fitted with O(n log(n)) computational cost (Jensen and Nielsen 2014) when the process is observed directly. The fast Fourier transform can also be used to give a computationally efficient infinite sum approximation of fGn, reducing the computational cost of the Whittle estimator (Purczyński and Wlodarski 2006). Chan and Palma (2006) gives a review of different likelihood-based methods to fit ARFIMA-models.…”

Section: Introductionmentioning

confidence: 99%

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An approximate fractional Gaussian noise model with $$\mathcal {O}(n)$$ O ( n ) computational cost

2018

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Fractional Gaussian noise (fGn) is a stationary time series model with long memory properties applied in various fields like econometrics, hydrology and climatology. The computational cost in fitting an fGn model of length n using a likelihood-based approach is O(n 2), exploiting the Toeplitz structure of the covariance matrix. In most realistic cases, we do not observe the fGn process directly but only through indirect Gaussian observations, so the Toeplitz structure is easily lost and the computational cost increases to O(n 3). This paper presents an approximate fGn model of O(n) computational cost, both with direct or indirect Gaussian observations, with or without conditioning. This is achieved Sigrunn H. Sørbye

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An approximate fractional Gaussian noise model with $$\mathcal {O}(n)$$ O ( n ) computational cost

2018

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“…Furthermore, the use of the fast Fourier transform to calculate the periodogram results in relatively short computation times. However, a difficulty with implementation of the method is that the theoretical spectral density of FGN has the form of an infinite series so that some sort of approximation must be used [11].…”

Section: Introductionmentioning

confidence: 99%

On maximum likelihood estimation of the long-memory parameter in fractional Gaussian noise

Robbertse

Lombard

2012

Journal of Statistical Computation and Simulation

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Approximate normality and unbiasedness of the maximum likelihood estimate (MLE) of the long-memory parameter H of a fractional Brownian motion hold reasonably well for sample sizes as small as 20 if the mean and scale parameter are known. We show in a Monte Carlo study that if the latter two parameters are unknown the bias and variance of the MLE of H both increase substantially. We also show that the bias can be reduced by using a parametric bootstrap procedure. In very large samples, maximum likelihood estimation becomes problematic because of the large dimension of the covariance matrix that must be inverted. To overcome this difficulty, we propose a maximum likelihood method based upon first differences of the data. These first differences form a short-memory process. We split the data into a number of contiguous blocks consisting of a relatively small number of observations. Computation of the likelihood function in a block then presents no computational problem. We form a pseudo-likelihood function consisting of the product of the likelihood functions in each of the blocks and provide a formula for the standard error of the resulting estimator of H. This formula is shown in a Monte Carlo study to provide a good approximation to the true standard error. The computation time required to obtain the estimate and its standard error from large data sets is an order of magnitude less than that required to obtain the widely used Whittle estimator. Application of the methodology is illustrated on two data sets.

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“…(2) while keeping high accuracy. Recently, another method, based on the Euler-Maclaurin summation, was proposed by Purczynsky and Wlodarski (2006). All these approximation methods have considerably reduced the direct calculation of the infinite summation of Eq.…”

Section: Summary and Future Workmentioning

confidence: 99%

“…An excellent method for approximating (2) was presented by Purczynsky and Wlodarski (2006). This novel method is based on the Euler-Maclaurin summation formula as clearly shown in (12).…”

Section: Euler Maclaurin-based Approximationmentioning

confidence: 99%

Two approximation methods to synthesize the power spectrum of fractional Gaussian noise

Ledesma

Liu

Hernández

2007

Computational Statistics & Data Analysis

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The simplest models with long-range dependence (LRD) are self-similar processes. Self-similar processes have been formally considered for modeling packet traffic in communication networks. The fractional Gaussian noise (FGN) is a proper example of exactly self-similar processes. Several numeric approximation methods are considered and reviewed, two methods are found that are able to provide a better accuracy and less running time than previous approximation methods for synthesizing the power spectrum of FGN. The first method is based on a second-order approximation. It is demonstrated that a parabolic curve can be indirectly used to approximate the power spectrum of FGN. The second method is based on cubic splines. Despite the fact that splines cannot be used directly to approximate the power spectrum of FGN, they can, however, considerably simplify the calculations while maintaining high accuracy. Both of the methods proposed can be used to estimate the Hurst parameter using Whittle's estimator. Additionally, they can be used on synthesis of LRD sequences.

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On fast generation of fractional Gaussian noise

Cited by 9 publications

References 19 publications

An approximate fractional Gaussian noise model with $$\mathcal {O}(n)$$ O ( n ) computational cost

An approximate fractional Gaussian noise model with $$\mathcal {O}(n)$$ O ( n ) computational cost

On maximum likelihood estimation of the long-memory parameter in fractional Gaussian noise

Two approximation methods to synthesize the power spectrum of fractional Gaussian noise

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