# Multi-Scale Recurrence Quantification Measurements for Voice Disorder Detection
## 2 Extraction of Multi-Scale Recurrence Quantification Measurements (MRQMs)
### Recurrence Quantification Analysis (RQA)
#### 2.2.1 Recurrence Plots (RPs)  
Recurrence plot is a tool for analyzing signal recurrence phenomenon in two-dimensional space graphs. The recurrence characteristics of the time series can be represented by black and blue dots and two-time axes. RP is defined in classical mathematics as:  
\[ \begin{split}\text{R}_{ij}&=\theta\big{(}\varepsilon- \big{|}|X_{i}-X_{j}|\big{|}\big{|}\big{)}\\ &\quad i,j=1,2\cdots n\end{split}  \tag{5}\]  
where \(\theta(x)\) is a Heaviside function. \(\varepsilon\) is a threshold parameter, an empirical value. \(||\cdot||\) is an Euclidean norm. Recurrence plots determine a field with \(X_{i}\) as the center and \(\varepsilon\) as the radius. Through formula a, an n*n distance matrix is converted into a 0-1 matrix. If \(X_{j}\) is in this field, under the action of the Heaviside function, \(\text{R}_{ij}=1\). At this time, the one-dimensional time series is considered to have recurrence properties. If \(X_{j}\) is not in this field, \(\text{R}_{ij}=0\). Blue dots represent the recurrence characteristics of the dynamic system, and a blue dot is scored when the dynamic system is non-recurrent. The recurrence threshold is an essential parameter of RPs. Different systems will correspond to different types of signals. With the transformation of signal properties, the choice of recurrence threshold should also be different due to non-stationarity, vibration characteristics or noise influence. At present, the essential methods for the selection of recurrence threshold are as follows:  
1. Phase space diameter--The threshold is set as a percentage of the phase space diameter. Usually, the value is 10% phase space diameter [39]. 2. The standard deviation of noise--Take the standard deviation of the observation noise as the benchmark, choose a multiple of the standard deviation, and generally take 5\(\sigma\) as the threshold [40]. 3. Recurrence rate (rr)--Measure rr value based on the percentage of the recurrence rate, 1%rr is usually used as the recurrence threshold [41]. The above commonly used threshold selection methods require experiments on the signal. For pathological voice signals, it is not appropriate to use the 5\(\sigma\) of the observed signal noise as the threshold because the speech signals were recorded in the acoustic laboratory, which is a quiet place. We have done related research, and the results of this threshold are not ideal. The abnormal waveform in the time series contains different characteristics of the pathological voice. The recurrence threshold is an empirical parameter. Too large or too small a recurrence threshold will destroy the structure of the recurrence plot. Too large a selection of recurrence threshold results in too many adjacency error points, which makes the black points in the recurrence plot too dense. When the recurrence threshold is too small, many normal adjacency points are not described, so the blue area of the recurrence plot becomes larger. As shown in [27], [PERSON] considers the most appropriate recurrence threshold is 1%rr. Building on his work, we continue to explore the best suited recurrence threshold for pathological voices. Figure 3 shows the phase spaces and RPs of normal and pathological glottal signals. The m and \(\tau\) are obtained using the FNN and mutual information method. It can be seen that Figure 3, which uses a typical 1%rr as the recurrence threshold, shows a diagonal recurrence structure. The phase space of the normal glottal signal is regular, but the phase space of the pathological glottal signal is chaotic. The regularity of healthy vowels is approximately a quasi-periodic process. A regular long diagonal structure is formed in RP, with parallel diagonal lines, no vertical and horizontal lines and no scattered, isolated points. Non-periodic vibrations of the vocal folds, incomplete glottis closure and interstitial vibrations can all lead to pathological voices.