# Correlation dimension

### Description

Estimates the correlation dimension by forming a delay embedding
of a time series, calculating correlation summation curves (one per embedding
dimension), and subsequently fitting the slopes of these curves on a log-log scale using a robust
linear regression model. If the slopes converge at a given embedding dimension *E*, then
*E* is the correct embedding dimension and the (convergent) slope value is an estimate
of the correlation dimension for the data.

### Usage

1 2 |

### Arguments

`x` |
a vector containing a uniformly-sampled real-valued time series or
a matrix containing an embedding with each column representing a different coordinate.
If the latter, the |

`dimension` |
the maximal embedding dimension. Default: |

`olag` |
the number of points along the trajectory of the
current point that must be exceeded in order for
another point in the phase space to be considered
a neighbor candidate. This argument is used
to help attenuate temporal correlation in the
the embedding which can lead to spuriously low
correlation dimension estimates. The orbital lag
must be positive or zero. Default: |

`resolution` |
an integer representing the spatial resolution factor.
A value of P increases the number of effective
scales by a factor of P at a cost of raising
the |

`tlag` |
the time delay between coordinates.
Default: |

### Details

To estimate the correlation dimension, correlation summation
curves must be generated and subsequently fit with a
robust linear regression model to obtain the slopes of these
curves on a log-log plot. The dimension at which these
slope estimates (appear to) converge reveals the proper
embedding dimension for the data and the slope at this
(and higher) embedding dimensions is an estimate of the
correlation dimension. The function used to fit the
correlation summation curves is `lmsreg`

which fits a robust
linear model to the data using the method of least median of squares
regression. See the on-line help documentation for help on the `lmsreg`

function: in R, `lmsreg`

is found in the `MASS`

package while in S-PLUS it is indigenous
and appears in the `splus`

database.

The correlation summation at scale *eps*
for a given embedding dimension is defined as

*C2(eps)=2 / (N - gamma)/(N - gamma - 1) * sum{i=1:N}sum{j=i+1+gamma:N} H(eps - || Xi - Xj ||),*

where *H* is the Heavyside function

*H(x)=0 if x <= 0 and H(x)=1 for x > 1,*

and *Xi* is the *i*th point of a
collection of `N`

points in the phase space. The parameter
*gamma* is the orbital lag.

The algorithm used to calculate the correlation summation is made computationally efficient by using:

- 1
The

*L-infinity*norm to calculate the distance between neighbors in the phase space as opposed to (say) the*L2*norm which involves taking computationally intense square root and power of two operations. The*L-infinity*norm of the distance between two points in the phase space is the absolute value of the maximal difference between any of the points' respective coordinates, i.e. if*X ={z1, z2, z3}*then*||X|| sub infinity=max{i}(zi)*.- 2
Bitwise masking and shift operations to reveal the radix-2 exponent of the

*L-infinity*norm. This direct means of obtaining the exponent immediately yields the associated scale of the distance between neighbors in the phase space while avoiding costly log operations. The bitwise mask and shift factors are based on the IEEE standard 754 for binary floating-point arithmetic. Initial tests are performed in the code to verify that the current machine follows this standard.- 3
a computationally efficient routine to calculate the resulting value of a float raised to a positive integer power. Specifically, the

*L-infinity*norm is raised to an integer power (`p`

) to effectively increase the spatial resolution by a factor of`p`

.

The correlation summation curves *C2(E,eps)*
where `E`

is the embedding dimension and
*eps* is the scale, the correlation dimension curves
*D2(E,eps)* can be calculated by

*D2(E,eps)=log2 (C2(E,2*eps) / C2(E,eps / 2)) / 2*

This formulation is used to help suppress numerical instabilities that are present in other numerical derivative schemes such as a first order difference.

As a caveat to the user, the slope estimates of the correlation summation curves will typically display a fair amount of variability and the range of scales over which the slopes are approximately linear may be small. Inasmuch, the correlation dimension estimate should always be interpretted as a subjective summary statistic, even when the original times series is representative of a truly noise-free chaotic response.

### Value

an object of class `chaoticInvariant`

.

### S3 METHODS

- eda.plot
plots an extended data analysis plot, which graphically summarizes the process of obtaining a correlation dimension estimate. A time history, phase plane embeddding, correlation summation curves, and the slopes of correlation summation curves as a function of scale are plotted.

- plot
plots the correlation summation curves on a log-log scale. The following options may be used to adjust the plot components:

- type
Character string denoting the type of data to be plotted. The

`"stat"`

option plots the correlation summation curves while the`"dstat"`

option plots a 3-point estimate of the derivatives of the correlation summation curves. The`"slope"`

option plots the estimated slope of the correlation summation curves as a function of embedding dimension. Default:`"stat"`

.- fit
Logical flag. If

`TRUE`

, a regression line is overlaid for each curve. Default:`TRUE`

.- grid
Logical flag. If

`TRUE`

, a grid is overlaid on the plot. Default:`TRUE`

.- legend
Logical flag. If

`TRUE`

, a legend of the estimated slopes as a function of embedding dimension is displayed. Default:`TRUE`

.- ...
Additional plot arguments (set internally by the

`par`

function).

prints a qualitiative summary of the results.

### References

Peter Grassberger and Itamar Procaccia (1983),
Measuring the strangeness of strange attractors,
*Physica D*, **9**, 189–208.

Holger Kantz and Thomas Schreiber (1997),
*Nonlinear Time Series Analysis*,
Cambridge University Press.

Peter Grassberger and Itamar Procaccia (1983),
Characterization of strange attractors,
*Physical Review Letters*, **50**(5), 346–349.

Rousseeuw, P. J. (1984). Least median of squares regression. *Journal
of the American Statistical Association*, **79**, 871–88.

### See Also

`infoDim`

, `embedSeries`

, `timeLag`

, `chaoticInvariant`

, `lyapunov`

, `poincareMap`

, `spaceTime`

, `findNeighbors`

, `determinism`

.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
## calculate the correlation dimension estimates
## for chaotic beam data using a delay
## embedding for dimensions 1 through 10, a
## orbital lag of 10, and a spatial resolution
## of 4.
beam.d2 <- corrDim(beamchaos, olag=10, dim=10, res=4)
## print a summary of the results
print(beam.d2)
## plot the correlation summation curves
plot(beam.d2, fit=FALSE, legend=FALSE)
## plot an extended data analysis plot
eda.plot(beam.d2)
``` |