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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/09/14 23:24:27」(JST)

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In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gaussian process B_H(t) on [0, T], which starts at zero, has expectation zero for all t in [0, T], and has the following covariance function:

where H is a real number in (0, 1), called the Hurst index or Hurst parameter associated with the fractional Brownian motion. The Hurst exponent describes the raggedness of the resultant motion, with a higher value leading to a smoother motion. It was introduced by Mandelbrot & van Ness (1968).

The value of H determines what kind of process the fBm is:

if H = 1/2 then the process is in fact a Brownian motion or Wiener process;
if H > 1/2 then the increments of the process are positively correlated;
if H < 1/2 then the increments of the process are negatively correlated.

The increment process, X(t) = B_H(t+1) − B_H(t), is known as fractional Gaussian noise.

There is also a generalization of fractional Brownian motion: n-th order fractional Brownian motion, abbreviated as n-fBm.^[1] n-fBm is a Gaussian, self-similar, non-stationary process whose increments of order n are stationary. For n = 1, n-fBm is classical fBm.

Like the Brownian motion that it generalizes, fractional Brownian motion is named after 19th century biologist Robert Brown; fractional Gaussian noise is named after mathematician Carl Friedrich Gauss.

Background and definition

Prior to the introduction of the fractional Brownian motion, Lévy (1953) used the Riemann–Liouville fractional integral to define the process

where integration is with respect to the white noise measure dB(s). This integral turns out to be ill-suited to applications of fractional Brownian motion because of its over-emphasis of the origin (Mandelbrot & van Ness 1968, p. 424).

The idea instead is to use a different fractional integral of white noise to define the process: the Weyl integral

for t > 0 (and similarly for t < 0).

The main difference between fractional Brownian motion and regular Brownian motion is that while the increments in Brownian Motion are independent, the opposite is true for fractional Brownian motion. This dependence means that if there is an increasing pattern in the previous steps, then it is likely that the current step will be increasing as well. (If H > 1/2.)

Properties

Self-similarity

The process is self-similar, since in terms of probability distributions:

This property is due to the fact that the covariance function is homogeneous of order 2H and can be considered as a fractal property. Fractional Brownian motion is the only self-similar Gaussian process.

Stationary increments

It has stationary increments:

Long-range dependence

For H > ½ the process exhibits long-range dependence,

Regularity

Sample-paths are almost nowhere differentiable. However, almost-all trajectories are Hölder continuous of any order strictly less than H: for each such trajectory, there exists a constant c such that

for every ε > 0.

Dimension

With probability 1, the graph of B_H(t) has both Hausdorff dimension and box dimension of 2−H.^{[citation needed]}

Integration

As for regular Brownian motion, one can define stochastic integrals with respect to fractional Brownian motion, usually called "fractional stochastic integrals". In general though, unlike integrals with respect to regular Brownian motion, fractional stochastic integrals are not semimartingales.

Sample paths

Practical computer realisations of an fBm can be generated,^[2] although they are only a finite approximation. The sample paths chosen can be thought of as showing discrete sampled points on an fBm process. Three realisations are shown below, each with 1000 points of an fBm with Hurst parameter 0.75.

"H" = 0.75 realisation 1

"H" = 0.75 realisation 2

"H" = 0.75 realisation 3

Two realisations are shown below, each showing 1000 points of an fBm, the first with Hurst parameter 0.95 and the second with Hurst parameter 0.55.

"H" = 0.95

"H" = 0.55

Method 1 of simulation

One can simulate sample-paths of an fBm using methods for generating stationary Gaussian processes with known covariance function. The simplest method relies on the Cholesky decomposition method of the covariance matrix (explained below), which on a grid of size has complexity of order . A more complex, but computationally faster method is the circulant embedding method of Dietrich & Newsam (1997).

Suppose we want to simulate the values of the fBM at times using the Cholesky decomposition method.

Form the matrix where .

Compute the square root matrix of , i.e. . Loosely speaking, is the "standard deviation" matrix associated to the variance-covariance matrix .

Construct a vector of n numbers drawn according a standard Gaussian distribution,

If we define then yields a sample path of an fBm.

In order to compute , we can use for instance the Cholesky decomposition method. An alternative method uses the eigenvalues of :

Since is symmetric, positive-definite matrix, it follows that all eigenvalues of satisfy , ().

Let be the diagonal matrix of the eigenvalues, i.e. where is the Kronecker delta. We define as the diagonal matrix with entries , i.e. .

Note that the result is real-valued because .

Let an eigenvector associated to the eigenvalue . Define as the matrix whose -th column is the eigenvector .

Note that since the eigenvectors are linearly independent, the matrix is inversible.

It follows then that because .

Method 2 of simulation

It is also known that^{[citation needed]}

where B is a standard Brownian motion and

Where is the Euler hypergeometric integral.

Say we want simulate an fBm at points .

Construct a vector of n numbers drawn according a standard Gaussian distribution.

Multiply it component-wise by √(T/n) to obtain the increments of a Brownian motion on [0, T]. Denote this vector by .

For each , compute

The integral may be efficiently computed by Gaussian quadrature. Hypergeometric functions are part of the GNU scientific library.

Notes

^ Perrin et al., 2001.
^ Kroese, D.P.; Botev, Z.I. (2014). "Spatial Process Generation". Lectures on Stochastic Geometry, Spatial Statistics and Random Fields, Volume II: Analysis, Modeling and Simulation of Complex Structures, Springer-Verlag, Berlin. arXiv:1308.0399.

References

Beran, J. (1994), Statistics for Long-Memory Processes, Chapman & Hall, ISBN 0-412-04901-5 .
Craigmile P.F. (2003), "Simulating a class of stationary Gaussian processes using the Davies–Harte Algorithm, with application to long memory processes", Journal of Times Series Analysis, 24: 505–511.
Dieker, T. (2004). Simulation of fractional Brownian motion (PDF) (M.Sc. thesis). Retrieved 29 December 2012.
Dietrich, C. R.; Newsam, G. N. (1997), "Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix.", SIAM Journal on Scientific Computing 18 (4): 1088–1107, doi:10.1137/s1064827592240555 .
Lévy, P. (1953), Random functions: General theory with special references to Laplacian random functions, University of California Publications in Statistics 1, pp. 331–390 .
Mandelbrot, B.; van Ness, J.W. (1968), "Fractional Brownian motions, fractional noises and applications", SIAM Review 10 (4): 422–437, doi:10.1137/1010093, JSTOR 2027184 .
Perrin E. et al. (2001), "nth-order fractional Brownian motion and fractional Gaussian noises", IEEE Transactions on Signal Processing, 49: 1049-1059. doi:10.1109/78.917808
Samorodnitsky G., Taqqu M.S. (1994), Stable Non-Gaussian Random Processes, Chapter 7: "Self-similar processes" (Chapman & Hall).

English Journal

Partial deletion of the αC-domain in the Fibrinogen Perth variant is associated with thrombosis, increased clot strength and delayed fibrinolysis.

Westbury SK, Duval C, Philippou H, Brown R, Lee KR, Murden SL, Phillips E, Reilly-Stitt C, Whalley D, Ariëns RA, Mumford AD.Author information Dr. Sarah Westbury, University of Bristol, Level 7 Bristol Royal Infirmary, Bristol, BS2 8HW, United Kingdom, Tel +44 117 3423152; Fax +44 117 3424036, E-mail sarah.westbury@bristol.ac.uk.AbstractGenetic fibrinogen (FGN) variants that are associated with bleeding or thrombosis may be informative about fibrin polymerisation, structure and fibrinolysis. We report a four generation family with thrombosis and heritable dysfibrinogenaemia segregating with a c.[1541delC];[=] variation in FGA (FGN-Perth). This deletion predicts a truncated FGN αC-domain with an unpaired terminal Cys at residue 517 of FGN-Aα. In keeping with this, SDS-PAGE of purified FGN-Perth identified a truncated FGN-Aα chain with increased co-purification of albumin, consistent with disulphide bonding to the terminal Cys of the variant FGN-Aα. Clot visco-elastic strength in whole blood containing FGN-Perth was greater than controls and tPA-mediated fibrinolysis was delayed. In FGN-Perth plasma and in purified FGN-Perth, there was markedly reduced final turbidity after thrombin-mediated clot generation. Consistent with this, FGN-Perth formed tighter, thinner fibrin fibres than controls indicating defective lateral aggregation of protofibrils. Clots generated with thrombin in FGN-Perth plasma were resistant to tPA-mediated fibrinolysis. FGN-Perth clot also displayed impaired tPA-mediated plasmin generation but incorporated α2-antiplasmin at a similar rate to control. Impaired fibrinolysis because of defective plasmin generation potentially explains the FGN-Perth clinical phenotype. These findings highlight the importance of the FGN αC-domain in the regulation of clot formation and fibrinolysis.
Thrombosis and haemostasis.Thromb Haemost.2013 Nov 27;110(6):1135-44. doi: 10.1160/TH13-05-0408. Epub 2013 Sep 19.
Genetic fibrinogen (FGN) variants that are associated with bleeding or thrombosis may be informative about fibrin polymerisation, structure and fibrinolysis. We report a four generation family with thrombosis and heritable dysfibrinogenaemia segregating with a c.[1541delC];[=] variation in FGA (FGN-
PMID 24048413

Recurrence of fibrillary glomerulonephritis in a renal transplant recipient.

Mitwalli A, Shah I, Hammad D, Kafoury H.Author information King Khalid University Hospital, King Saud University, Riyadh, Saudi Arabia, amituualli@ksu.edu.sa.AbstractFibrillary glomerulonephritis (FGN) is a rare glomerular deposition disease and a rare cause of nephrotic syndrome. The patients usually present with renal insufficiency, nephrotic range proteinuria and microscopic hematuria. The electron microscopy study is the only means of diagnosis. The clinical course of the disease is generally unpredictive and the patients inevitably progress to ESRD. Here, we describe a case of FGN, which presented with nephrotic syndrome and impaired renal function. Renal biopsy showed that 26 out of 30 glomeruli were completely sclerosed. Remaining showed mesangial expansion and double contour consistent with a membranoproliferative pattern, with 70 % interstitial fibrosis and tubular atrophy. Immunofluorescence revealed C3 (2+) diffuse mesangial deposits. Electron microscopic showed subendothelial dense deposits with organized tubular structures. During follow-up, the patient underwent renal transplantation from a living unrelated kidney donor. Later on, as the renal allograft function showed deterioration, renal biopsy was performed and showed recurrence of FGN in the renal allograft.
International urology and nephrology.Int Urol Nephrol.2013 Oct;45(5):1527-32. doi: 10.1007/s11255-012-0238-5. Epub 2012 Jul 24.
Fibrillary glomerulonephritis (FGN) is a rare glomerular deposition disease and a rare cause of nephrotic syndrome. The patients usually present with renal insufficiency, nephrotic range proteinuria and microscopic hematuria. The electron microscopy study is the only means of diagnosis. The clinical
PMID 22826142

Japanese Journal

Coexistence of Fibrillary Glomerulonephritis in a Patient with Multiple Myeloma

Park Jin Hee,Kim Bo Ra,Chun Bong Geon,Kim Mi Hyang,Shin Ho Sik,Jung Yeon Soon,Rim Hark
Internal Medicine 51(11), 1379-1381, 2012
… The result of the patient's renal biopsy showed fibrillary glomerulonephritis (FGN). … FGN is a rare form of glomerulopathy that is diagnosed using electron microscopy when deposits of amyloid-like fibrils are observed on the mesangium and microvascular wall. …
NAID 130002062114

<論文>短期・長期記憶課程に基づく株式収益率モデルのナイーブなパラメータ推定法

宮崎浩一
電気通信大学紀要 16(2), 111-116, 2004-01-31
NAID 110000491114

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