1. Dimension Review
Many of the geometric structures generated by chaotic map or differential dynamic systems are extremely complex.
Fractal :
hard to define mathmatically
images that can be divided into parts, each of which is similar to the original object.
have self-similarity
example)
Cantor set
Dimension :
There are many ways to define the dimension
Capacity dimension,
Correlation dimension ,
Information dimension,

2. Dimension Review, Capacity dimension
Consider a two-dimensional square of side L can be covered by boxes of size on a side as shown in the right.
If L is the length of a side, then
Taking logarithms one obtains
The capacity dimension is defined as L 1 L 4
L/2 16
L/4

3. Dimension Review, Correlation dimension
more efficient to compute than capacity dimension
Computed from correlation function
where and are points on the attractor, is the Heavyside function, ( 1 if and ), and N is the number of points randomly chosen from entire data set.
The correlation dimension is defined by the variation of C(R) with R: as
The correlation dimension is the slope of a graph of log C(R) versus log R
When R approaches the size of the phase, C(R) saturates at unity
When R is smaller than the spacing between the data points, C(R) levels off at

4. Dimension Review, Information dimension
related to the entropy
depends on the distribution of points on the attractor
The attractor is covered by a set of n boxes of size , and let the probability that a point is in the i-th box be
The metric entropy or missing information
The information dimension is defined by the equation:

5. 7. Testing for nonlinearity with surrogate data
Is the apparent structure in the data most likely due to nonlinearity or rather due to linear correlations?
Is the irregularity of the data most likely due to nonlinear determinism or rather due to random inputs to the system or fluctuations in the parameters?
Only nonlinear deterministic signals are characterized by
finite dimensional attractor
finite, positive Lyapunov exponents
good linear-short term predictability
The main idea for developing statistical significance test
Compute some nonlinear observable from the data
may be a dimension, a prediction error or whatever.
Does the value found for suggest that the data are indeed nonlinear?
What distribution of values for could we get from a comparable linear model?
Is the value we found perhaps consistent with a linear description?
If not, the data might be nonlinear