# Recurrence Quantification Analysis (RQA) of Toroidal End Tool Milling Process
## 2 Methodology of the Experiment
### Method of Analysis
In this paper, the following RQA indicators were used:  
* the percentage of recurrence points in a recurrence plot (RR): \[\text{RR}=\frac{1}{N^{2}}\sum_{i,j=1}^{N}R_{i,j},\] (2)
* the determinism (DET): \[\text{DET}=\frac{\sum_{l=lmin}^{N}IP(l)}{\sum_{l=1}^{N}IP(l)},\] (3)
* the average diagonal length (L): \[\text{L}=\frac{\sum_{l=l_{\text{min}}}^{N}IP(l)}{\sum_{l=l_{\text{min}}}^{N}P( l)},\] (4)
* the longest diagonal line length (LMAX): \[\text{LMAX}=max(\{l_{j};i=1,\;\ldots,N_{l}\}),\] (5)
* the Entropy (ENT) is Shannon's entropy of the diagonal line segment distribution: \[\text{ENT}=-\sum_{l=lmin}^{N}P(l)\;lnP(l),\] (6)
* the trapping time (TT) is the average length of the vertical lines: \[\text{TT}=\frac{\sum_{l=l_{\text{min}}}^{N}vP(v)}{\sum_{l=l_{\text{min}}}^{N} P(v)},\] (7)
* the length of the longest vertical line (VMAX): \[\text{VMAX}=max(\{v_{i};i=1,\;\ldots,N_{v}\}),\] (8)  
where \(P(l)\) is the histogram of the lengths \(l\) of the diagonal lines, \(P(v)\) is the histogram of the vertical lines' lengths \(v\), and \(N\) denotes the quantity of points on the phase space trajectory. To calculate RP and RQA indicators in this paper, the application to make recurrence plots and conduct recurrence quantification analysis was created by the authors. Response Surface Methodology (RSM) was used to determine the relationship between the technological parameters and each of the RQA indicators studied. RSM represents a sophisticated statistical approach employed to systematically design experiments aimed at optimizing various parameters [43; 44; 45]. The data obtained through RSM facilitates the prediction of relationships among variables, derived from the statistical models constructed during the analysis. Furthermore, RSM allows for the simultaneous examination of multiple variables in relation to a desired outcome, thus significantly reducing the number of experiments required for a complete analysis. A predictive model is formulated, and the statistical significance of each variable is rigorously assessed using Analysis of Variance(ANOVA). In addition, three-dimensional response surface plots are generated utilizing specialized software packages, such as DesignExpert (v.3.3), Minitab (v.22.1.0), and Matlab (2024a). These visual representations enable the elucidation of the synergistic effects exhibited by pairs of process variables on the outcome of interest [46]. The most commonly used mathematical form in RMS analysis is the second-order polynomial [44], which can be expressed as follows:  
\[Y=\beta_{0}+\sum\
olimits_{i=1}^{k}\beta_{i}X_{i}+\sum\
olimits_{i=1}^{k} \beta_{ii}X_{i}^{2}+\sum\
olimits_{i=1}^{k}\sum\
olimits_{j>i}^{k}\beta_{ij}X _{i}X_{j}+\epsilon, \tag{9}\]  
The independent variables, denoted as \(X_{i}\), and the output response, denoted as \(Y\), are used in the model, with \(\beta\) representing the coefficients of the model and \(k\) indicating the number of independent variables. In Equation (9), the terms correspond to the following: the constant term (\(\beta_{0}\)), the individual effects of each variable (\(X_{i}\)), the interaction effects between variables (\(X_{i}X_{j}\)), and the quadratic effects of each variable (\(X_{i}^{2}\)), in that order. The second-order polynomial is relatively simple and effectively models the behavior of the experimental data, with a limited number of measurements, as is the case with toroidal milling tests. This model can provide accurate results when uncontrolled measurement errors and complex interdependencies between independent variables and the output response are eliminated. In order to analyze and assess the quality and fit of the model, statistical parameters such as the Contribution, F-value, and \(p\)-value are most commonly used. Contribution means the percentage share of the factor in the model.