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Probability distribution mapping function

MAPPING A DISTRIBUTION

Informally, the probability distribution mapping function (DMF) is a mapping of the probability distribution of points in n-dimensional space to the distribution of points in one-dimensional space of the distances . The distribution density mapping function (DDMF) is a one-dimensional analogy to the probability density function. The power approximation of the probability distribution mapping function has the form of $r^{-q}$, where r is a distance and exponent q is the distribution mapping exponent (DME), see reference cited. Function $z=r^{-q}$ transforms the true distribution of points so that the distribution density mapping function as a function of variable z is constant, at least in the neighborhood of the fixed point. For exact definitions see ^{[ 1 ]}. These notions are local, i.e. are related to a particular fixed point. The distribution mapping exponent q is something like a local value of the correlation dimension according to Grassberger and Procaccia . It can be also viewed as the local dimension of the attractor or singularity exponent eventually scaling exponent in the Multifractal system.

DECOMPOSITION OF THE CORRELATION INTEGRAL TO LOCAL FUNCTIONS

Correlation integral $C_{I}(r)$ was defined by Grassberger and Procaccia^{[ 2 ]} and can be written in the form

$C_{I}(r)=\lim _{N\rightarrow \infty }{\frac {2}{N(N-1)}}\sum _{\stackrel {i,j=1}{j>i}}^{N}h(r-||{x}(i)-{x}(j)||),\quad {x}(i)\in \mathbb {R} ^{m},$

where h(.) is the Heaviside step function, and considering all pairs of points of set of N points. It holds^{[ 1 ]}

$C_{I}(r)=\lim _{N\rightarrow \infty }{\frac {1}{N}}\sum _{i=1}^{N}D(x_{i},r),$

where $D(x_{i},r)$ is the distribution mapping function related to point $x_{i}$.

THE DISTRIBUTION MAPPING EXPONENT IN CLASSIFICATION METHODS

The DME can be used for constructing a classifier. Methods are described in ^{[ 1 ]}, and each individual method in more detail in and in freely available reports .