Recurrence quantification analysis

RQA is a system for anylizing nonlinear systems that works by looking at the pattern and number of recurrances present in a data signal. RQA can be thought of as a formal system for understanding Recurrence plots.

Method of RQA

RQA begins by representing the data in an N dimensional space. In order to do this RQA selects samples of the signal at a given time delay away from each other. Each one of these numbers a coordinate for a vector in the D space. In order to do an RQA on a signal, several parameters must be set including

DELAY- the number of units skipped (how many number to skip before putting one in the vector)
EMBED- the number of dimensions (how many numbers the vector will hold)
FIRST- what number vector to begin to include in the analysis
LAST- what number vector to end including them in the analysis

LAST - FIRST + 1 should equal the total number of vectors used in the anylsis

Example

Given a signal [13, 25, 67, 34, 56, 76, 34, 23, 65, 26, 36, 58, 21, 14, 47, 35, 14, 36, 25, 25, 58, 25, 36, 47, 47, 25]

and given the parameters DELAY=5, EMBED = 4, FIRST=1. LAST=6. With these parameters we won't even use the entire length of the signal the vectors are:

V1 = [13, 76, 36, 35]
V2 = [25, 34, 58, 14]
V3 = [67, 23, 21, 47]
V4 = [34, 65, 14, 35]
V5 = [56, 26, 47, 25]
V6 = [76, 36, 35, 25]

If the FIRST parameter were set to 2 we would ignore the the first vector. Now that we have put the data into 4 dimensional space we can calculate the distances between these vectors either by using Pythagorean theoremor approximations. These values are often divided by the maximum distance to yeild values of the percent maximum distance. These distances are plotted in a matrix.

Failed to parse (Conversion error. Server ("/media/api/rest_") reported: "Cannot get mml. upstream connect error or disconnect/reset before headers. reset reason: connection termination"): {\displaystyle {\begin{bmatrix}[1,6]&[2,6]&[3,6]&[4,6]&[5,6]&[6,6]\\[1,5]&[2,5]&[3,5]&[4,5]&[5,5]&-\\[1,4]&[2,4]&[3,4]&[4,4]&-&-\\[1,3]&[2,3]&[3,3]&-&-&-\\[1,2]&[2,2]&-&-&-&-\\[1,1]&-&-&-&-&-\\\end{bmatrix}}}