# Multi-Scale Recurrence Quantification Measurements for Voice Disorder Detection
## 2 Extraction of Multi-Scale Recurrence Quantification Measurements (MRQMs)
### Recurrence Quantification Analysis (RQA)  
Nonlinear dynamics theory has become an effective tool for studying the characteristics of speech production mechanisms. Recurrence quantification analysis is a nonlinear method used to quantitatively study the recurrence characteristics of the state in the phase space of a dynamic system [35]. The recurrence quantification analysis method has an excellent ability to deal with non-stationary, nonlinear and relatively short data series. Compared with traditional signal processing methods, it can describe the sensitivity of some changing dynamic characteristics in detail. Generally speaking, pathological voice detection uses vowels to record voice signals. RQA can analyze the complex system behind the chaotic sound based on the vibration of the vocal folds caused by the continuous vowel. The first step in analyzing a signal using nonlinear dynamics theory is to reconstruct the multi-dimensional phase space trajectory of the signal. Assuming that a time series \(\{x(1),\ x(2),\ldots,\ x(N)\}\) of length \(N\) is obtained from the microphone, the phase space can be reconstructed by [PERSON] embedding theorem [36].  
\[X=\begin{bmatrix}X_{1}\\ X_{2}\\ X_{2}\\ \vdots\\ X_{n}\end{bmatrix}=\begin{bmatrix}x_{1}&x_{1+\tau}&\cdots&x_{1+(m-1)\tau}\\ x_{2}&x_{2+\tau}&\cdots&x_{2+(m-1)\tau}\\ \vdots&\vdots&\ddots&\vdots\\ x_{n}&x_{n+\tau}&\cdots&x_{n+(m-1)\tau}\end{bmatrix} \tag{4}\]  
where \(\tau\) is the time delay and \(m\) is the best embedding dimension. The total number of points, represented by vectors \(\{X_{1},X_{2},X_{3}\ldots,\ X_{n}\}\), in the reconstructed phase space is \(n=N-(m-1)\ \tau\).  
As long as m and \(\tau\) are selected appropriately, the dynamic characteristics of the original system can be restored in the sense of the topological equivalence. Determining the time delay, \(\tau\), and the best embedding dimension, m, better represents the speech signal's time sequence in the reconstructed phase space. In this article, the traditional mutual information method is used to obtain the time delay, \(\tau\), of the reconstruction of the chaotic time series [37], and the false nearest neighbors (FNN) algorithm is used to find the best embedding dimension [38]. The non-stationary nature of the high-dimensional phase space studied by the RPs is used to describe the fundamental dynamics of the system.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Channel** & \(f_{i}\) (**Hz**) & **\(\mathbf{Bw_{i}}\) (Hz)** & **Channel** & \(f_{i}\) (Hz) & **\(\mathbf{Bw_{i}}\) (Hz)** \\ \hline
1 & 0 & 25 & 13 & 1634 & 205 \\
2 & 44 & 30 & 14 & 1990 & 244 \\
3 & 96 & 36 & 15 & 2413 & 291 \\
4 & 158 & 43 & 16 & 2917 & 346 \\
5 & 232 & 51 & 17 & 3518 & 412 \\
6 & 319 & 60 & 18 & 4233 & 491 \\
7 & 424 & 72 & 19 & 5085 & 584 \\
8 & 549 & 86 & 20 & 6096 & 696 \\
9 & 697 & 102 & 21 & 7308 & 829 \\
10 & 874 & 121 & 22 & 8746 & 987 \\
11 & 1085 & 144 & 23 & 10,460 & 1176 \\
12 & 1335 & 172 & 24 & 12,500 & 1400 \\ \hline \end{tabular}
\end{table}
Table 1: The center frequency \(f_{i}\) and bandwidth \(\mathbf{Bw_{i}}\) of the gammatone filter bank.