# Multi-Scale Recurrence Quantification Measurements for Voice Disorder Detection
## 2 Extraction of Multi-Scale Recurrence Quantification Measurements (MRQMs)
### Recurrence Quantification Analysis (RQA)
#### 2.2.2 Recurrence Quantification Measurements  
The recurrence properties of time series depend on the geometric properties of the RP. RQA is a method of quantifying the dynamics of the system based on the RP [42]. A series of statistical parameters can be obtained based on the analysis of the density of recurrence points, diagonal lines, vertical lines or horizontal lines. This work uses 13 measurement RQMs, such as mean diagonal line length, maximal diagonal line length, clustering coefficient and transitivity. \begin{table}
\begin{tabular}{c c} \hline \hline
**RP Structure** & **Interpretation** \\ \hline Diagonal Parallel Lines & Deterministic Process, Periodic aspect \\ Dispersion Isolated Points & Anti-correlation process, weighty fluctuations \\ Long diagonal lines & Periodic/quasi-periodic patterns \\ White bands & Non-stationary, brusque transitions \\ Vertical and horizontal line & Laminar states, unchangeable states or slowly changes \\ Long bowed line & Dynamic of the system could change \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Interpretation of recurrent plot structures. Figure 3: Phase spaces and RPs of normal glottal signal and pathological glottal signal. _Recurrence rate (RR)_ indicates the ratio of the number of recurrence points that appear in the RP, compared to the total number of \(n\times n\) matrix types. \[RR=\frac{1}{n^{2}}\sum\limits_{i,j=0}^{n}R_{ij} \tag{6}\]  
_Determinism (DET)_ indicates the ratio of the recurrence points forming the diagonal line segment appearing in the RP to the total recurrence points. It owns the function of distinguishing individual recurrence points that diverge in disorder from other recurrence points that regularly form specific patterns. \[DET=\frac{\sum\limits_{i=l_{\min}}^{n}IP^{e}(l)}{\sum\limits_{i,j}^{n}R_{ij}} \tag{7}\]  
where \(l\) is the length of the diagonal line segment, and \(l_{\min}\) is its minimum value. The frequency distribution of the lengths, \(l\), of the diagonal lines is represented by \(P^{e}(l)\). \(P^{e}(l)=\{l_{l};i=1\ldots n_{l}\}\) and \(n_{l}\) are the absolute number of diagonal lines. _Maximal diagonal line length (\(L_{\max}\))_ refers to the length of the longest diagonal line in the RP structure. \[L_{\max}=\max(\{l_{i};i=1\ldots n_{l}\}) \tag{8}\]  
_Entropy of the diagonal line lengths (ENTR)_ refers to the Shannon entropy of the diagonal structure length distribution on the RP, which measures the amount of information contained in the RP structure. \[ENTR=-\sum\limits_{l=l_{\min}}^{n}P(l)\ln P(l) \tag{9}\]  
and \(P(l)\) can be defined as \(P(l)=\frac{P^{e}(l)}{\sum\limits_{l=l_{\min}}^{n}P^{e}(l)}\). _Mean diagonal line length (\(<\)L\(>\))_ is highly correlated with the mean prediction time of the dynamic system and the divergence of the system. \[<L>\ =\frac{\sum\limits_{i=l_{\min}}^{n}IP(l)}{\sum\limits_{i,j}^{n}Pl} \tag{10}\]  
_Laminarity (LAM)_ refers to the ratio of recurrence points forming a vertical structure to all recurrence points in the RP, which can reflect the complexity of the dynamic system. \[LAM=\frac{\sum\limits_{v=v_{\min}}^{n}vP^{e}(v)}{\sum\limits_{v=l}^{n}vP^{e}(v)} \tag{11}\]  
where \(v\) is the length of the vertical line segment and \(P^{e}(v)=\{l_{j};i=1\ldots nv\}\). _Trapping time (TT)_ indicates the mean length of vertical lines in the RP structure. It measures the average time that the system is in a very slowly changing state. \[TT=\frac{\sum\limits_{v=v_{\min}}^{n}v^{R}(v)}{\sum\limits_{v=l}^{n}\overline{P^{ e}(v)}} \tag{12}\]  
_Maximal vertical line length (\(V_{\max}\))_ indicates the maximal length of vertical lines in the RP structure. \[V_{\max}=\max(\{v_{l};i=1\dots n_{v}\}) \tag{13}\]  
Recurrence time of 1 st type (\(T_{1}\)) and Recurrence time of 2 nd type (\(T_{2}\)):  
\[ \begin{array}{l}T_{1}(i)=t_{i+1}-t_{i},t=1,2,K\\ T_{2}(i)=t_{i+1}-t_{i},t=1,2,K\end{array}  \tag{14}\]  
_Recurrence time entropy (RPDE)_ has been successfully applied in biomedical testing.