In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a
continuous-time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. e ...
''B
H''(''t'') on
, ''T'' that starts at zero, has
expectation zero for all ''t'' in
, ''T'' and has the following
covariance function In probability theory and statistics, the covariance function describes how much two random variables change together (their ''covariance'') with varying spatial or temporal separation. For a random field or stochastic process ''Z''(''x'') on a doma ...
:
:
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 .
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
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
or
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
;
* if ''H'' > 1/2 then the increments of the process are positively
correlated
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
;
* 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. n-fBm is a
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
, 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
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
.
Background and definition
Prior to the introduction of the fractional Brownian motion, 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 .
The idea instead is to use a different fractional integral of white noise to define the process: the
Weyl integral
In mathematics, the Weyl integral (named after Hermann Weyl) is an operator defined, as an example of fractional calculus, on functions ''f'' on the unit circle having integral 0 and a Fourier series. In other words there is a Fourier series for ...
:
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, 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-similar
__NOTOC__
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
, since in terms of
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s:
:
This property is due to the fact that the covariance function is homogeneous of order 2H and can be considered as a
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
property. FBm can also be defined as the unique mean-zero
Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. e ...
, null
at the origin, with stationary and self-similar increments.
Stationary increments
It has
stationary increments:
:
Long-range dependence
For ''H'' > ½ the process exhibits
long-range dependence,
:
Regularity
Sample-paths are
almost
In set theory, when dealing with sets of infinite size, the term almost or nearly is used to refer to all but a negligible amount of elements in the set. The notion of "negligible" depends on the context, and may mean "of measure zero" (in a me ...
nowhere differentiable
In mathematics, a differentiable function of one Real number, real variable is a Function (mathematics), function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differ ...
. However,
almost-all trajectories are locally
Hölder continuous Hölder:
* ''Hölder, Hoelder'' as surname
* Hölder condition
* Hölder's inequality
* Hölder mean
* Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modul ...
of any order strictly less than ''H'': for each such trajectory, for every ''T'' > 0 and for every ''ε'' > 0 there exists a (random) constant ''c'' such that
::
for 0 < ''s'',''t'' < ''T''.
Dimension
With probability 1, the graph of ''B
H''(''t'') has both
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
and
box dimension
A box (plural: boxes) is a container used for the storage or transportation of its contents. Most boxes have flat, parallel, rectangular sides. Boxes can be very small (like a matchbox) or very large (like a shipping box for furniture), and can ...
of 2−''H''.
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
semimartingale
In probability theory, a real valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the l ...
s.
Frequency-domain interpretation
Just as Brownian motion can be viewed as white noise filtered by
(i.e. integrated), fractional Brownian motion is white noise filtered by
(corresponding to
fractional integration
In fractional calculus, an area of mathematical analysis, the differintegral (sometime also called the derivigral) is a combined differentiation/ integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted ...
).
Sample paths
Practical computer realisations of a
''fBm'' can be generated 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 0.75.
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.
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
In linear algebra, a circulant matrix is a square matrix in which all row vectors are composed of the same elements and each row vector is rotated one element to the right relative to the preceding row vector. It is a particular kind of Toeplitz ...
method of .
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 independently according to 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
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of
:
* Since
is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
,
positive-definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite fu ...
matrix, it follows that all
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of
satisfy
, (
).
* Let
be the diagonal matrix of the eigenvalues, i.e.
where
is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
. 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 invertible.
* It follows then that
because
.
Method 2 of simulation
It is also known that
[Stochastic Analysis of the Fractional Brownian
Motion]
/ref>
:
where ''B'' is a standard Brownian motion and
:
Where is the Euler hypergeometric integral
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
.
Say we want to simulate an ''fBm'' at points .
* Construct a vector of ''n'' numbers drawn according to a standard Gaussian distribution.
* Multiply it component-wise by to obtain the increments of a Brownian motion on , ''T'' Denote this vector by .
* For each , compute
::
The integral may be efficiently computed by Gaussian quadrature
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more ...
.
See also
* Brownian surface
A Brownian surface is a fractal surface generated via a fractal elevation function.
As with Brownian motion, Brownian surfaces are named after 19th-century biologist Robert Brown.
Example
For instance, in the three-dimensional case, where two v ...
* Autoregressive fractionally integrated moving average
In statistics, autoregressive fractionally integrated moving average models are time series models that generalize ARIMA (''autoregressive integrated moving average'') models by allowing non-integer values of the differencing parameter. These model ...
* Multifractal
A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed. ...
: The generalized framework of fractional Brownian motions.
* Pink noise
Pink noise or noise is a signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequency of the signal. In pink noise, each octave interval (halving ...
* Tweedie distributions
In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the cla ...
Notes
References
* .
* 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.
*
*.
* .
*.
*.
*Perrin E. et al. (2001),
nth-order fractional Brownian motion and fractional Gaussian noises
, ''IEEE Transactions on Signal Processing
The ''IEEE Transactions on Signal Processing'' is a biweekly peer-reviewed scientific journal published by the Institute of Electrical and Electronics Engineers covering research on signal processing. It was established in 1953 as the ''IRE Transac ...
'', 49: 1049-1059.
*Samorodnitsky G., Taqqu M.S. (1994), ''Stable Non-Gaussian Random Processes'', Chapter 7: "Self-similar processes" (Chapman & Hall).
Further reading
*.
{{Stochastic processes
Autocorrelation