View source for Fractional Brownian motion

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 stochastic process|continuous-time]] [[Gaussian process]] ''B<sub>H</sub>''(''t'') on [0,&nbsp;''T''], that starts at zero, has [[expectation (mathematics)|expectation]] zero for all ''t'' in [0,&nbsp;''T''], and has the following [[covariance function]]:

:<math>E[B_H(t) B_H (s)]=\tfrac{1}{2} (|t|^{2H}+|s|^{2H}-|t-s|^{2H}),</math>

where ''H'' is a real number in (0,&nbsp;1), called the [[Hurst exponent|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 {{harvtxt|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'' &gt; 1/2 then the increments of the process are positively [[correlation|correlated]];
* if ''H'' &lt; 1/2 then the increments of the process are negatively correlated.

The increment process, ''X''(''t'') = ''B<sub>H</sub>''(''t''+1) &minus; ''B<sub>H</sub>''(''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.<ref>Perrin et al., 2001.</ref> n-fBm is a [[Gaussian]], self-similar, non-stationary process whose increments of order ''n'' are stationary. For ''n''&nbsp;=&nbsp;1, n-fBm is classical fBm.

Like the Brownian motion that it generalizes, fractional Brownian motion is named after 19th century biologist [[Robert Brown (Scottish botanist from Montrose)|Robert Brown]]; fractional Gaussian noise is named after mathematician [[Carl Friedrich Gauss]].

==Background and definition==
Prior to the introduction of the fractional Brownian motion, {{harvtxt|Lévy|1953}} used the [[Riemann&ndash;Liouville integral|Riemann&ndash;Liouville fractional integral]] to define the process
:<math>B_H(t) = \frac{1}{\Gamma(H+1/2)}\int_0^t (t-s)^{H-1/2} \, dB(s)</math>
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 {{harv|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]]
:<math>B_H (t) = B_H (0) + \frac{1}{\Gamma(H+1/2)}\left\{\int_{-\infty}^0\left[(t-s)^{H-1/2}-(-s)^{H-1/2}\right]\,dB(s) + \int_0^t (t-s)^{H-1/2}\,dB(s)\right\}</math>
for ''t''&nbsp;> 0 (and similarly for ''t''&nbsp;< 0).

The main difference between fractional Brownian motion and regular Brownian motion is that while the increments in Brownian Motion are independent, increments for fractional Brownian motion are not. If H > 1/2, then there is positive autocorrelation: 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, the autocorrelation is negative.

==Properties==

===Self-similarity===

The process is [[self-similarity|self-similar]], since in terms of [[probability distribution]]s:

: <math>B_H (at) \sim |a|^{H}B_H (t).</math>

This property is due to the fact that the covariance function is homogeneous of order 2H and can be considered as a [[fractal]] property. FBm can also be defined as the unique mean-zero [[Gaussian process]], null
at the origin, with stationary and self-similar increments.

=== Stationary increments ===

It has stationary increments:

: <math>B_H (t) - B_H (s)\;   \sim \;   B_H (t-s). </math>

===Long-range dependence===

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

: <math>\sum_{n=1}^\infty E[B_H (1)(B_H (n+1)-B_H (n))] = \infty.</math>

===Regularity===

Sample-paths are [[almost nowhere|almost]] [[nowhere differentiable]]. However, [[Almost surely|almost-all]] trajectories are locally [[Hölder condition|Hölder continuous]] of any order strictly less than ''H'': for each such trajectory, for every ''T''&nbsp;>&nbsp;0 and for every&nbsp;''ε''&nbsp;>&nbsp;0 there exists a (random) constant ''c'' such that

::<math> |B_H (t)-B_H (s)| \le c |t-s|^{H-\varepsilon}</math>
for  0&nbsp;<&nbsp;''s'',''t''&nbsp;<&nbsp;''T''.

===Dimension===
With probability 1, the graph of ''B<sub>H</sub>''(''t'') has both [[Hausdorff dimension]]<ref>Orey, 1970.</ref> and [[box dimension]]{{Citation needed|reason=Orey paper discusses just Hausdorff dimension|date=July 2018}} of 2&minus;''H''.

===Integration===

As for regular Brownian motion, one can define [[ito calculus|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 [[semimartingale]]s.

===Frequency-domain interpretation===

Just as Brownian motion can be viewed as white noise filtered by <math>s^{-1}</math> (i.e. integrated), fractional Brownian motion is white noise filtered by <math>s^{-H-1/2}</math> (corresponding to [[fractional integration]]).

==Sample paths==

Practical computer realisations of an [http://www.mathworks.com/matlabcentral/fileexchange/38935 ''fBm'' can be generated],<ref>{{cite journal |author1=Kroese, D.P. |authorlink1=Dirk Kroese|author2=Botev, Z.I.|year=2014 |title=Spatial Process Generation |journal=Lectures on Stochastic Geometry, Spatial Statistics and Random Fields, Volume II: Analysis, Modeling and Simulation of Complex Structures, Springer-Verlag, Berlin  |arxiv=1308.0399|bibcode=2013arXiv1308.0399K}}</ref> 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 realizations are shown below, each with 1000 points of an ''fBm'' with Hurst parameter&nbsp;0.75.

{|
| [[Image:BrownFractionalH75Seed0.svg|thumb|200px|"H" = 0.75 realisation 1]] 
| [[Image:BrownFractionalH75Seed1.svg|thumb|200px|"H" = 0.75 realisation 2]]
| [[Image:BrownFractionalH75Seed2.svg|thumb|200px|"H" = 0.75 realisation 3]]
|}

Realizations of three different types of ''fBm'' are shown below, each showing 1000 points, the first with Hurst parameter 0.15, the second with Hurst parameter 0.55, and the third with Hurst parameter 0.95. The higher the Hurst parameter is, the smoother the curve will be.

{|
| [[Image:BrownFractionalH15.svg|thumb|200px|"H" = 0.15]]
| [[Image:BrownFractionalH55.svg|thumb|200px|"H" = 0.55]] 
| [[Image:BrownFractionalH95.svg|thumb|200px|"H" = 0.95]]
|}

=== 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|Cholesky decomposition method]] of the covariance matrix (explained below), which on a  grid of size <math> n</math>
has complexity of order <math>O(n^3) </math>. A more complex, but computationally faster method is the [[circulant embedding]] method of {{harvtxt|Dietrich|Newsam |1997}}.

Suppose we want to simulate the values of the ''fBM'' at times <math>t_1, \ldots, t_n</math> using the  [[Cholesky decomposition|Cholesky decomposition method]].

* Form the matrix <math>\Gamma=\bigl(R(t_i,\, t_j), i,j=1,\ldots,\, n\bigr)</math> where <math>\,R(t,s)=(s^{2H}+t^{2H}-|t-s|^{2H})/2</math>.
* Compute <math>\,\Sigma</math> the square root matrix of <math>\,\Gamma</math>, i.e. <math>\,\Sigma^2 = \Gamma</math>. Loosely speaking, <math>\,\Sigma</math> is the "standard deviation" matrix associated to the variance-covariance matrix <math>\,\Gamma</math>.
* Construct a vector <math>\,v</math> of ''n'' numbers drawn independently according to a standard Gaussian distribution,
* If we define <math>\,u=\Sigma v</math> then <math>\,u</math> yields a sample path of an ''fBm''.

In order to compute <math>\,\Sigma</math>, we can use for instance the [[Cholesky decomposition|Cholesky decomposition method]]. An alternative method uses the [[eigenvalues]] of <math>\,\Gamma</math>:

* Since <math>\,\Gamma</math> is [[symmetric matrix|symmetric]], [[positive-definite matrix|positive-definite]] matrix, it follows that all [[eigenvalues]] <math>\,\lambda_i</math> of <math>\,\Gamma</math> satisfy <math>\,\lambda_i\ge0</math>, (<math>i=1,\dots,n</math>).
* Let <math>\,\Lambda</math> be the diagonal matrix of the eigenvalues, i.e. <math>\Lambda_{ij} = \lambda_i\,\delta_{ij}</math> where <math>\delta_{ij}</math> is the [[Kronecker delta]]. We define <math>\Lambda^{1/2}</math> as the diagonal matrix with entries <math>\lambda_i^ {1/2}</math>, i.e. <math>\Lambda_{ij}^{1/2} = \lambda_i^{1/2}\,\delta_{ij}</math>.
Note that the result is real-valued because <math>\lambda_i\ge0</math>.

* Let <math>\,v_i</math> an eigenvector associated to the eigenvalue <math>\,\lambda_i</math>. Define <math>\,P</math> as the matrix whose <math>i</math>-th column is the eigenvector <math>\,v_i</math>.
Note that since the eigenvectors are linearly independent, the matrix <math>\,P</math> is invertible.

* It follows then that <math>\Sigma = P\,\Lambda^{1/2}\,P^{-1}</math> because <math>\Gamma= P\,\Lambda\,P^{-1}</math>.

=== Method 2 of simulation ===
It is also known that <ref>Stochastic Analysis of the Fractional Brownian
Motion, [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.201.5698&rep=rep1&type=pdf]</ref>

: <math>B_H (t)=\int_0^t K_H(t,s) \, dB(s)</math>

where ''B'' is a standard Brownian motion and

: <math>K_H(t,s)=\frac{(t-s)^{H-\frac{1}{2}}}{\Gamma(H+\frac{1}{2})}\;_2F_1\left (H-\frac{1}{2};\, \frac{1}{2}-H;\; H+\frac{1}{2};\, 1-\frac{t}{s} \right).</math>

Where <math> _2F_1</math> is the [[Euler hypergeometric integral]].

Say we want to simulate an ''fBm'' at points <math>0=t_0< t_1< \cdots < t_n=T</math>.

* Construct a vector of ''n'' numbers drawn according to a standard Gaussian distribution.
* Multiply it component-wise by {{radic|''T''/''n''}} to obtain the increments of a Brownian motion on [0,&nbsp;''T'']. Denote this vector by <math> (\delta B_1, \ldots, \delta B_n)</math>.
* For each <math> t_j</math>, compute
::<math> B_H (t_j)=\frac{n}{T}\sum_{i=0}^{j-1}  \int_{t_i}^{t_{i+1}} K_H(t_j,\, s)\, ds \ \delta B_i.</math>

The integral may be efficiently computed by [[Gaussian quadrature]]. Hypergeometric functions are part of the [[GNU Scientific Library|GNU scientific library]].

== See also ==
* [[Brownian surface]]
* [[Autoregressive fractionally integrated moving average]]
* [[Multifractal]]: The generalized framework of fractional Brownian motions.
* [[Pink noise]]
* [[Tweedie distributions]]

==Notes==
{{reflist}}

==References==
* {{citation|last=Beran|first=J.|year=1994|title=Statistics for Long-Memory Processes|publisher=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&ndash;511.
* {{cite thesis|last=Dieker|first=T.|year=2004|title=Simulation of fractional Brownian motion|url=http://www2.isye.gatech.edu/~adieker3/fbm/thesis.pdf|degree=M.Sc.|accessdate=29 December 2012}}
*{{citation|first1=C. R.|last1=Dietrich|first2=G. N.|last2=Newsam|title= Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix.|journal=SIAM Journal on Scientific Computing|year=1997|pages=1088&ndash;1107|volume=18|issue=4|doi=10.1137/s1064827592240555}}.
* {{citation|first=P.|last=Lévy|title=Random functions: General theory with special references to Laplacian random functions|series=University of California Publications in Statistics|volume=1|year=1953|pages=331&ndash;390}}.
*{{citation|first1=B.|last1=Mandelbrot|authorlink1=Benoit Mandelbrot|first2=J.W.|last2=van Ness|title=Fractional Brownian motions, fractional noises and applications|journal=SIAM Review|year=1968|pages=422&ndash;437|volume=10|issue=4|jstor=2027184|doi=10.1137/1010093}}.
*{{citation|first1=Steven|last1=Orey|year=1970|title=Gaussian sample functions and the Hausdorff dimension of level crossings|journal=Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete|volume=15|issue=3|pages=249–256|doi=10.1007/BF00534922}}.
*Perrin E. et al. (2001), "[http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=917808 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&nbsp;7: "Self-similar processes" (Chapman & Hall).

==Further reading==
*{{citation| last= Sainty | first= P. | title= Construction of a complex‐valued fractional Brownian motion of order ''N'' | journal= [[Journal of Mathematical Physics]] | volume= 33 | issue= 9 | page= 3128 | year= 1992 | doi= 10.1063/1.529976| bibcode= 1992JMP....33.3128S }}.

{{Stochastic processes}}

[[Category:Autocorrelation]]