Multifractal system
Encyclopedia
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.
Multifractal systems are common in nature, especially geophysics
. They include fully developed turbulence
, stock market
time series, real world scenes, the Sun’s magnetic field time series, heartbeat
dynamics, human gait, and natural luminosity time series. Models have been proposed in various contexts ranging from turbulence in fluid dynamics
to internet traffic, finance, image modeling, texture synthesis, meteorology, geophysics and more.
In a multifractal system
, the behavior around any point is described by a local power law
:
The exponent
is called the singularity exponent, as it describes the local degree of singularity
or regularity around the point
.
The ensemble formed by all the points that share the same singularity exponent is called the singularity manifold of exponent h, and is a fractal set of fractal dimension
D(h). The curve D(h) versus h is called the singularity spectrum and fully describes the (statistical) distribution of the variable
.
In practice, the multifractal behaviour of a physical system
is not directly characterized by its singularity specrum D(h). Data analysis rather gives access to the multiscaling exponents 
. Indeed, multifractal signals generally obey a scale invariance property which yields power law behaviours for multiresolution quantities depending on their scale 
. Depending on the object under study, these multiresolution quantities, denoted by 
in the following, can be local averages in boxes of size 
, gradients over distance 
, wavelet coefficients at scale 
... For multifractal objects, one usually observes a global power law scaling of the form:
at least in some range of scales and for some range of orders
. When such a behaviour is observed, one talks of scale invariance, self-similarity or multiscaling.
and the multiscaling exponents 
through a Legendre transform. While the determination of 
calls for some exhaustive local analysis of the data, which would result difficult and numerically unstable, the estimation of the 
relies on the use of statistical averages and linear regressions in log-log diagrams. Once the 
are known, one can deduce an estimate of 
thanks to a simple Legendre transform.
Multifractal systems are often modeled by stochastic processes such as multiplicative cascade
s. Interestingly, the
receives some statistical interpretation as they characterize the evolution of the distributions of the 
as 
goes from larger to smaller scales. This evolution is often called statistical intermittency and betrays a departure from Gaussian
models.
Modelling as a multiplicative cascade
also leads to estimation of multifractal properties for relatively small datasets . A maximum likelihood fit of a multiplicative cascade to the dataset not only estimates the complete spectrum, but also gives reasonable estimates of the errors (see the web service http://www.maths.adelaide.edu.au/anthony.roberts/multifractal.php).
Fractal dimension
In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...
) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed.
Multifractal systems are common in nature, especially geophysics
Geophysics
Geophysics is the physics of the Earth and its environment in space; also the study of the Earth using quantitative physical methods. The term geophysics sometimes refers only to the geological applications: Earth's shape; its gravitational and magnetic fields; its internal structure and...
. They include fully developed turbulence
Turbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
, stock market
Stock market
A stock market or equity market is a public entity for the trading of company stock and derivatives at an agreed price; these are securities listed on a stock exchange as well as those only traded privately.The size of the world stock market was estimated at about $36.6 trillion...
time series, real world scenes, the Sun’s magnetic field time series, heartbeat
Cardiac cycle
The cardiac cycle is a term referring to all or any of the events related to the flow or blood pressure that occurs from the beginning of one heartbeat to the beginning of the next. The frequency of the cardiac cycle is described by the heart rate. Each beat of the heart involves five major stages...
dynamics, human gait, and natural luminosity time series. Models have been proposed in various contexts ranging from turbulence in fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
to internet traffic, finance, image modeling, texture synthesis, meteorology, geophysics and more.
In a multifractal system
, the behavior around any point is described by a local power lawPower law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...
:
The exponent
is called the singularity exponent, as it describes the local degree of singularityMathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
or regularity around the point
.The ensemble formed by all the points that share the same singularity exponent is called the singularity manifold of exponent h, and is a fractal set of fractal dimension
Fractal dimension
In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...
D(h). The curve D(h) versus h is called the singularity spectrum and fully describes the (statistical) distribution of the variable
.In practice, the multifractal behaviour of a physical system
is not directly characterized by its singularity specrum D(h). Data analysis rather gives access to the multiscaling exponents . Indeed, multifractal signals generally obey a scale invariance property which yields power law behaviours for multiresolution quantities depending on their scale . Depending on the object under study, these multiresolution quantities, denoted by in the following, can be local averages in boxes of size , gradients over distance , wavelet coefficients at scale ... For multifractal objects, one usually observes a global power law scaling of the form:at least in some range of scales and for some range of orders
. When such a behaviour is observed, one talks of scale invariance, self-similarity or multiscaling.Estimation
Thanks to the so-called multifractal formalism, it can be shown that, under some well-suited assumptions, there exists a correspondence between the singularity spectrum and the multiscaling exponents through a Legendre transform. While the determination of calls for some exhaustive local analysis of the data, which would result difficult and numerically unstable, the estimation of the relies on the use of statistical averages and linear regressions in log-log diagrams. Once the are known, one can deduce an estimate of thanks to a simple Legendre transform.Multifractal systems are often modeled by stochastic processes such as multiplicative cascade
Multiplicative Cascade
In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process.Model I :Model II :Model III :...
s. Interestingly, the
receives some statistical interpretation as they characterize the evolution of the distributions of the as goes from larger to smaller scales. This evolution is often called statistical intermittency and betrays a departure from GaussianGAUSSIAN
Gaussian is a computational chemistry software program initially released in 1970 by John Pople and his research group at Carnegie-Mellon University as Gaussian 70. It has been continuously updated since then...
models.
Modelling as a multiplicative cascade
Multiplicative Cascade
In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process.Model I :Model II :Model III :...
also leads to estimation of multifractal properties for relatively small datasets . A maximum likelihood fit of a multiplicative cascade to the dataset not only estimates the complete spectrum, but also gives reasonable estimates of the errors (see the web service http://www.maths.adelaide.edu.au/anthony.roberts/multifractal.php).
See also
- Multifractal Model of Asset Returns (MMAR)
- Multifractal Random Walk model (MRW)
- Fractional Brownian motion
- Mandelbrot cascade, continuous cascade and lognormal cascade