Fortran 90 Routine: LAMBDA_LOG_DISP

By: Jianbo Gao and Johnny Lin

If you send me [Johnny] <air_jlin@yahoo.com> your email address,
I'll let you know of any major bugs that are found in this library.

Note last updated: 26 Jun 2002

In this note we describe criteria for choosing a number of the parameters for the routine:

- Number of points to use in calculations:
The choice of the number of points (local variable nn) to use in the calculations is one informed by experience, a balance of accuracy and computational expense. If you are studying chaotic behavior, if the dimension is not too high, then a few thousand points of nn gives similar results as that with much larger nn. For diffusional scaling laws, depending on how long the scaling is, nn is preferred to be larger, if possible. But it is limited by nxp (nxp is the length of the input timeseries):

- Removing tangential motion:
As Gao and Zheng (1994) points out, it is best to remove tangential motion effects before calculating lambda. In that study, |i-j| < w (where w = (m-1)L, the embedding window) were removed. However, one can use |i-j| < 10 as a removal criteria, which is the default setting for the routine. If one can choose w, it's better for it to be very large so that the effect of any kind of correlation (more precisely, orbital or tangential motions) can be fully eliminated.

- mstep (step of evolution time k) and
mmap (number of evolution time k values):
For chaos-like signals, set mstep to be 1,
mmap to be 250;
for checking diffusional scaling laws from complicated
oscillatory motions, set mstep to be 10, mmap 250.
Often a smaller value
of mmap will do. But it may be more economic to set it to be the
maximal value of 250.
- md (embedding dimension) and mt (delay time):
This choice is VERY IMPORTANT!
See Gao and Zheng (1994) for details. A rule of thumb: make the
lambda(k) curves as smooth as possible. Often
md=3 or 4 will give instructive results.
- ie_value (ruler initialization):
Typically this can be chosen to be 5 or 7.
- For EEG analysis, "mt" may be chosen to be 3 or 4 if sampling
frequency is 128 Hz, and "in" chosen to be a few hundreds.

References:

`
Gao, J. and Z. Zheng (1994): Direct dynamical test for deterministic
chaos and optimal embedding of a chaotic time series, Phys. Rev. E,
Vol. 49, No. 5, pp. 3807-3814.
`