# Recurrence Quantification Analysis as a Form of Postural Control Assessment: A Systematic Review
## 1 Introduction
### Embedding Parameters for RQA Analysis
#### 1.2.2 Embedding Dimension Determination Methods  
This section introduces the false nearest neighbor algorithm according to the idea proposed by [PERSON] et al. [32]. Let us assume that two data points in a one-dimensional time series are close to each other--then they are neighbors. Their difference in magnitude gives us the distance between these neighbors. If we embed this time series (first in two dimensions) with a certain time delay, tau, then we can use the coordinates of these data points to examine whether the distance between them has changed significantly. If this embedding has changed the distance between the neighbors significantly, they are false neighbors, indicating the need for further data embedding in subsequent dimensions. In case the distance between them does not change significantly, they are called true neighbors. At that point, the embedding leaves the attractor shape unchanged, which means that the current embedding dimension is sufficient. The procedure repeats successively by increasing the embedding dimension D. The value of D is chosen at the point where the number of FNN falls to 0 (Figure 2).  
Figure 2: Example of a plot of percentage of false nearest neighbors as a function of the embedding dimension for the 30 s of CoP data. In this Figure, the embedding dimension is equal to 3 (as shown by the arrow).  
[PERSON] and [PERSON] [30; 39] in 1983 introduced a fractal dimension measurement called the correlation dimension (\(D_{2}\)). \(D_{2}\) is created by covering the set with boxes of a given size (\(r\)) and then calculating the probability \(p_{i}(r)\) of having a point of the set in the \(i\)-th box. The correlation dimension is defined as:  
\[D_{2}=\lim_{r\to 0}\sum_{i=1}^{N}\frac{log\left(\sum_{i}p_{i}(r)^{2}\right)}{ logr}\]  
The \(\sum_{i}p_{i}(r)^{2}\) is the probability of finding a pair of points in a box of size \(r\). The authors established that for small values of the separation distance (\(r\)) and sufficiently large \(N\), the correlation sum grows like a power:  
\[C(r)\,\sim\,r^{D_{2}}\Rightarrow D_{2}\,\sim\frac{log(C(r))}{log(r)}\]  
Here, time series are embedded at increasing embedding dimensions (\(D\)) until the measured \(D_{2}\) reaches a plateau (plot of \(D_{2}\) versus \(D\)). One of the most significant limitations of these applications is that the time series must be large enough. [PERSON] and [PERSON] [40] suggested that the upper limit of a reliable \(D_{2}\) measurement is \(D_{2}=2 log_{10}N\). Thus, any plateau in the \(D_{2}\) vs. \(D\) plot at \(D_{2}\) values higher than this limit should be discounted as artifacts of finite data size.  
After mentioning the general principles required for the RQA method and describing the individual parameters and features assessed by them, it is time to specify the aim of this paper. The purpose of this review was to show how the values of the above-discussed RQA measures look in various groups of subjects during the assessment of postural stability in typical free-standing tasks and how the authors interpret them.