Generalised dimensions and f-alpha spectrum
Brief Description:
There are two codes in this package, "dim_cube_dq.f" and "fit_DQA.f". The
two codes together compute the generalised dimensions D_q and the f-alpha
spectrum of a chaotic time series, by implementing the GP algorithm in a
non-subjective manner. Once the time series and the embedding dimension is
given, the codes compute D_q and f-alpha for the embedding dimension
without requiring a subjective analysis of the scaling region. More details
can be obtained from the following research paper, published based on
these codes:
"Computing the multifractal spectrum from time series: An algorithmic
approach", Chaos, 2009, Vol.19, 043129
Authors: K. P. Harikrishnan, R. Misra, G. Ambika and R. E. Amritkar
Important Note: These codes cannot be used for computing the D_q and
f-alpha of one dimensional multifractal sets such as, Cantor sets, Logistic
Attractor, etc. For that, the codes are to be modified and a different
algorithmic approach is required.
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Usage:
First compile and execute the code "dim_cube_dq.f" using
> g77 dim_cube_dq.f
> ./a.out
The file name and the embedding dimension (M) should be typed in.
For example, for time series from Henon, M=2 and that from Lorenz, M=3, etc.
For a practical time series, first one has to compute D_2, and the next
higher embedding dimension should be used as M.
The code will compute the spectrum of dimensions D_q for the given
embedding dimension and gives the output file "Out_<filename>".
The first column of this file contains the embedding dimension, the second
column contains the q values from -20 to +20, the third column the
corresponding D_q value, etc. It should be noted that in most cases,
the negative q values may not extend up to -20 and may stop well below that
depending on the nature and number of time series used.
This file should be used as the input file to compile the second code
"fit_DQA.f". This is done through an input file named "Inp_fit_DQ" which
should be present in the same folder, which contains the following
informations:
Input File Name
Embedding Dimension
D(20) D(-20) D(1)
The values D(20) and D(-20) are chosen from the extreme bottom and top values
respectively from the "Out_" file and D(1) as the value corresponding to q=1.
With these informations, compile the code using:
> g77 fit_DQA.f
> ./a.out
The code will now use a chi square fitting of the D_q spectrum using the
input parameters and the best fit D_q curve is chosen based on the
chi square minimum. The f-alpha curve is then computed from the best fit
D_q curve. For details, see the above reference.
The following are the output files:
1. DQ_of_..., which gives the spectrum of D_q
2. Fit_DQ_of_..., which is the best fit curve of D_q spectrum
3. Fal_DQ_of_..., which is the main output file that gives the f-alpha spectrum
4. Par_DQ_of_..., which gives the parameters used for the best fit.
As an example, one data file "Duffing", which is the time series from the
standard Duffing Oscillator is given, along with the corresponding
output files obtained by applying these codes, for verification.