Measure of chaotic signal complexity

estimates the correlation dimension of the uniformly sampled time-domain signal
`corDim`

= correlationDimension(`X`

)`X`

. Correlation dimension is the measure of
dimensionality of the space occupied by a set of random points.
`corDim`

is estimated as the slope of the correlation
integral versus the range of radius of similarity. Use
`correlationDimension`

as a characteristic measure to
distinguish between deterministic chaos and random noise, to detect potential
faults.[1]

`[`

additionally estimates the range of radius of similarity and correlation
integral of the uniformly sampled time-domain signal `corDim`

,`rRange`

,`corInt`

] = correlationDimension(___)`X`

.
Correlation integral is the mean probability that the states of a system are
close at two different time intervals, which reflects self-similarity.

`___ = correlationDimension(___,`

estimates the correlation dimension with additional options specified by one or
more `Name,Value`

)`Name,Value`

pair arguments.

`correlationDimension(___)`

with no output
arguments creates a correlation integral versus neighborhood radius plot.

Correlation dimension is computed in the following way,

The

`correlationDimension`

function first generates a delayed reconstruction*Y*with embedding dimension_{1:N}*m*, and lag*τ*.The software then calculates the number of with-in range points, at point

*i*, given by,$${N}_{i}\left(R\right)={\displaystyle \sum _{i=1,i\ne k}^{N}1\left(\Vert {Y}_{i}-{Y}_{k}\Vert <R\right)}$$

where

**1**is the indicator function, and*R*is the radius of similarity, given by,*R*= exp(linspace(log(*r*), log(_{min}*r*),_{max}*N*)). Here,*r*is_{min}`MinRadius`

,*r*is_{max}`MaxRadius`

, and*N*is`NumPoints`

.The correlation dimension

`corDim`

is the slope of*C(R)*vs.*R*where, the correlation integral*C(R)*is defined as,$$C\left(R\right)=\frac{2}{N\left(N-1\right)}{\displaystyle \sum _{i=1}^{N}{N}_{i}\left(R\right)}$$

[1] Caesarendra, Wahyu &
Kosasih, P & Tieu, Kiet & Moodie, Craig. "An application of nonlinear feature
extraction-A case study for low speed slewing bearing condition monitoring and
prognosis." *IEEE/ASME International Conference on Advanced Intelligent
Mechatronics: Mechatronics for Human Wellbeing, AIM 2013*.1713-1718.
10.1109/AIM.2013.6584344.

[2] Theiler, James. "Efficient
algorithm for estimating the correlation dimension from a set of discrete points".
American Physical Society. *Physical Review A* 1987/11/1. Volume 36.
Issue 9. Pages 44-56.