RECURRENCE PLOTS

Hi Norbert!

I try to understand why the m-dim. embedding uses m observations instead
of numerical approximations for the first m derivatives.
After all, the "natural" choice for the original phase space is (y(t),
dy(t)/dt, d²y(t)/dt², ...), right?

E.g., a much more natural 2-dim. embedding than (y_i, y_{i-1}) seems to
be (y_i, y_i - y_{i-1}) or (y_i, (y_{i+1} - y_{i-1})/2) or some other
numerical approximation of the derivative.

Of course, the information is basically the same, but when you apply a
Euclidean distance measure after the embedding to determine the
nearness, the results can be different since the distance between (y_i,
y_{i-1}) and (y_j, y_{j-1}) is not the same as that between (y_i, y_i -
y_{i-1}) and (y_j, y_j - y_{j-1}).

When we would use the derivatives, we can also apply some more robust
approximation methods like, e.g., spline interpolation:
For the k-th entry in an m-dim. embedding, with k=1...m, interpolate the
observations with a spline of degree k and use its (k-1)-th derivative
at i as the k-th entry at time i.
This would probably simplify the embedding in particular when the times
are irregularly spaced, and may be fruitful when both time and y have error.

Has this been tried before?

Jobst

Hi Jobst,

heitzig-j wrote:I try to understand why the m-dim. embedding uses m observations instead
of numerical approximations for the first m derivatives.
After all, the "natural" choice for the original phase space is (y(t),
dy(t)/dt, d²y(t)/dt², ...), right?

E.g., a much more natural 2-dim. embedding than (y_i, y_{i-1}) seems to
be (y_i, y_i - y_{i-1}) or (y_i, (y_{i+1} - y_{i-1})/2) or some other
numerical approximation of the derivative.

this derivative based embedding was already proposed, and, as far as I remember it is also included in several textbooks.
E.g. Mindlin & Gilmore, Physica D 58, 1992

When we would use the derivatives, we can also apply some more robust
approximation methods like, e.g., spline interpolation:
For the k-th entry in an m-dim. embedding, with k=1...m, interpolate the
observations with a spline of degree k and use its (k-1)-th derivative
at i as the k-th entry at time i.
This would probably simplify the embedding in particular when the times
are irregularly spaced, and may be fruitful when both time and y have error.

Has this been tried before?

This is a nice idea and should by tried, of course. Nevertheless, I'm not so happy with it, because it adds further parameters to be chosen (order and type of the spline), thus, adding potential pitfalls.

Best regards
Norbert

Hi Both,

This is quite a traditional problem for attractor reconstruction from one scalar time series. I'd like to draw your attention on the paper published by
Louis M. Pecora, Linda Moniz, Jonathan Nichols, and Thomas L. Carroll
"A unified approach to attractor reconstruction"
Chaos 17, 013110 (2007); doi:10.1063/1.2430294

This method determines the time delay and dimension in a unified manner, based on the functional independent relationship. This paper could be interesting for us.

What do you think?

P.S. I send you a copy via emails.
Best,
Yong