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.