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Function File: output = boxcount (S)
Function File: output = boxcount (S, paramName, paramValue, …)

Estimates the Renyi entropy of Qth order using a partition of the phase space instead of using the Grassberger-Procaccia scheme.

The program also can handle multivariate data, so that the phase space is build of the components of the time series plus a temporal embedding, if desired. Also, note that the memory requirement does not increase exponentially like 1/epsilon^M but only like M*(length of series). So it can also be used for small epsilon and large M. No finite sample corrections are implemented so far.

Input

S

This function always assumes that each time series is along the longer dimension of matrix S. It also assumes that every dimension (counting along the shorter dimension) of S is considered a component of the time series.

Parameters

m

The maximum embedding dimension [default = 10].

d

The delay used [default = 1].

q

Order of the entropy [default = 2.0].

rlow

Minimum length scale [default = 1e-3].

rhigh

Maximum length scale [default = 1].

eps_no

Number of length scale values [default = 20].

Output

The output is alligned with the input. If the input components where column vectors then the output is a maximum-embedding-dimension x number-of-components struct array with the following fields:

dim

Holds the embedding dimension of the struct.

entropy

The entropy output. Contains three columns which hold:

  1. epsilon
  2. Qth order entropy (Hq (dimension,epsilon))
  3. Qth order differential entropy (Hq (dimension,epsilon) - Hq (dimension-1,epsilon))

See also: demo boxcount, d2, c1.

Algorithms

The algorithms for this functions have been taken from the TISEAN package.

Demonstration 1

The following code

 res = boxcount (henon (1000),'m',5);

 do_plot_entrop = @(x) semilogx (x{1}(:,1),x{1}(:,3),'g');
 hold on
 # Show only for first component
 arrayfun (do_plot_entrop, {res(:,1).entropy});
 hold off
 axis tight
 xlabel ("Epsilon")
 ylabel ("Differential Entropies");
 title ("Entropies")

Produces the following figure

Figure 1

Package: tisean