Representing linear or sequential data in a non-sequential format can lead to fascinating results. My first exposure to this form of analysis was from Gleick's most excellent book Chaos, where he described using a three dimentional plot of single dimentional data to extract a strange atractor from a dripping faucet. This method is not used overly much in undergraduate physics, so I did not use it again.
Much to my happy amazement, Michal Zalewski published a paper at Bindview which explores Strange Attractors and TCP/IP Sequence Number Analysis using this same method. Quoting from the page:
We wanted to generate clean, three-dimensional representations of one-dimensional input data. The method used is known as "delayed coordinates", and is well-known and widely used in the analysis of dynamic systems, especially nonlinear systems and deterministic chaos [2]. This method assumes that we can reconstruct missing dimensions using previous, delayed function values as additional coordinates. Instead of using function values, we decided to calculate the first-order difference for the input data to generate more suggestive and useful results to show the function dynamics. So if s stands for the input set, and x, y and z are the point coordinates we are looking for, the equations are:
The following is an example of input data from a sequence of ISNs. Looking at the example doesn't allow us to determine what kind of underlying function was used to generate the data. It appears that these numbers are random with no dependencies between subsequent results:
Here is what we would see when using the attractor reconstruction technique:
I am now wondering if we can use this powerful technique to explore the data generated from entropy or statistical processes. Pushing raw data through preprocessing and a statistical engine will most likely squeeze out any interesting stuff, but the entropy shifts may show somthing.
![[ curly function ]](funct.jpg)
