# Evaluation of Geometric Attractor Structure and Recurrence Analysis in Professional Dancers

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Participants and Data Collection

#### 2.2. Phases of Pirouettes and Parameters

#### 2.3. Phase Space Reconstruction

#### 2.3.1. New Time Series Reconstruction

- (1)
- The ${P}_{i}=\left[\begin{array}{ccc}{x}_{{i}_{1}}& {y}_{{i}_{1}}& {z}_{{i}_{1}}\\ \vdots & \vdots & \vdots \\ {x}_{{i}_{n}}& {y}_{{i}_{n}}& {z}_{{i}_{n}}\end{array}\right],$ where ${P}_{i}$—is the matrix of LKNE or CoM coordinates for the analyzed pirouettes of the i-th person (i = 1, …, 15); n—is the length of the recorded time series. The n values were in the range of 101 to 250 frames. The x-coordinate describes motion in the anterior-posterior direction, y—along a vertical axis (inferior-superior), and z—in the mediolateral direction.
- (2)
- The signals from each matrix P
_{i}were resampled to obtain 300 samples. Its new length was longer than the maximum length of the recorded time series (250). The new signals were normalized by their maximal value.${\tilde{P}}_{i}=\left[\begin{array}{c}\begin{array}{ccc}{\tilde{x}}_{{i}_{1}}& {\tilde{y}}_{{i}_{1}}& {\tilde{z}}_{{i}_{1}}\end{array}\\ \begin{array}{ccc}\vdots & \vdots & \vdots \\ {\tilde{x}}_{{i}_{300}}& {\tilde{y}}_{{i}_{300}}& {\tilde{z}}_{{i}_{300}}\end{array}\end{array}\right],$ where ${\tilde{P}}_{i}$—matrix of LKNE or CoM after transformation, for the analyzed pirouettes of the i-th person (i =1, …, 15). - (3)
- Next, the new time series for LKNE and CoM for the x, y, and z coordinates were created. Individuals in the new time series have been shuffled to avoid bias based on their order.

#### 2.3.2. Hurst Exponent

#### 2.3.3. Test of Non-Stationarity

#### 2.3.4. Nonlinearity of Time Series

#### 2.3.5. Detection of Chaos Based on Largest Lyapunov Exponent

#### 2.3.6. Embedding Dimension, Time Delay, and Phase Space Reconstruction

#### 2.4. Recurrence Quantification Analysis (RQA)

## 3. Results

#### 3.1. New Time Series Reconstruction

#### 3.2. Hurst Exponent Analysis and the Largest Lyapunov Exponent

#### 3.3. Determination of the Embedding Dimension and Time Delay

#### 3.4. Phases Space Reconstruction and Convex Hull Calculation

#### 3.5. Recurrence Quantification Analysis

## 4. Discussion

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Participant model during a pirouette. (

**A**) Motion sequences; (

**B**) the location of the center of mass (CoM) and position of the marker on the knee’s lateral epicondyle of the supporting leg (LKNE) for the classic and jazz pirouette. The coordinate system, where: x—horizontal axis (anterior-posterior), y—vertical axis (inferior-superior), and z—horizontal axis (mediolateral).

**Figure 2.**Time series for LKNE marker and CoM for x, y, and z coordinates for jazz and classic pirouettes (

**A**,

**C**) for a single individual after the resampling procedure; (

**B**,

**D**) reconstruction of the time series before applying the quantiles function.

**Figure 3.**The global (all D) and directional (xD, yD, zD) embedding dimensions as an output of the mdFnn function for the (

**A**) LKNE marker for jazz pirouettes; (

**B**) CoM for jazz pirouettes; (

**C**) LKNE marker for classic pirouettes, and (

**D**) CoM for classic pirouettes.

**Figure 4.**The global (all tau) and directional (xtau, ytau, ztau) time delays as the output of the mdDelay function for (

**A**) LKNE marker for jazz pirouette; (

**B**) CoM for jazz pirouette; (

**C**) LKNE marker for classic pirouette; (

**D**) CoM for classic pirouette. The threshold was set at 1/e.

**Figure 5.**The reconstructed phase space (odd lines) with a global time lag (τ = all tau), convex hulls, and their volumes (even lines) for the (

**A**) LKNE marker and (

**B**) CoM x, y, and z coordinates for jazz and classic pirouettes (x-anterior-posterior, y-inferior-superior, z-mediolateral). The tables show convex hull volumes for global and directional parameters (D and tau).

**Figure 6.**The illustration of the multiscale recurrence analysis for the (

**A**) LKNE and (

**B**) CoM time series separately for the x-anterior-posterior, y-inferior-superior, z-mediolateral directions for jazz and classic pirouette. Graphs obtained for the global time delay and embedding dimensions.

**Table 1.**Methods for calculating the time delay τ (tau) and the embedding dimension (D) required for phase space reconstruction.

Phase Space Reconstruction State Vector: y(t) = (x(t), x(t + τ), x(t + 2τ), …, x(t + (D − 1)τ)) from Original Time Series Data x(t) by Using Time Delay (τ) and Dimension of the Attractor D | |
---|---|

Possible ways of τ selection [24]: (1) Autocorrelation function (2) Mutual information | Possible ways of D selection: (1) Principal Component Analysis (PCA) [25] (2) Correlation dimension [26] (3) Box-counting [27] (4) False Nearest Neighbor (FNN) [28] |

Group | Age [Years] | Body Mass [kg] | Body Height [m] | Training Period [Years] |
---|---|---|---|---|

N = 15 | 22.13 ± 2.73 | 57.56 ± 6.76 | 1.68 ± 0.62 | 12.19 ± 3.04 |

**Table 3.**The Hurst exponent values calculated for the x, y, and z coordinates of the reconstructed LKNE and CoM time series for classic and jazz pirouettes.

LKNEx | LKNEy | LKNEz | CoMx | CoMy | CoMz | |
---|---|---|---|---|---|---|

Classic | 0.75 | 0.71 | 0.71 | 0.75 | 0.72 | 0.70 |

Jazz | 0.76 | 0.66 | 0.70 | 0.73 | 0.69 | 0.75 |

LKNEx | LKNEy | LKNEz | CoMx | CoMy | CoMz | |
---|---|---|---|---|---|---|

Classic | 1.53 | 1.43 | 2.05 | 2.11 | 1.44 | 1.3 |

Jazz | 1.43 | 1.13 | 2.57 | 2.04 | 1.79 | 1.69 |

**Table 5.**Recurrence variables calculated for global and directional time lag and the embedding dimension for the LKNE marker and CoM motion in x-, y-, and z-directions for J—jazz and C—classic pirouettes. REC—% of recurrence, DET—%determinism, LMAX—the length of the longest diagonal line segment in the plot, ENT—Shannon information entropy, LAM—%laminarity, TT—trapping time. The prefix ‘all’ means global, and d denotes directional.

allREC | dREC | allDET | dDET | allLMAX | dLMAX | allENT | dENT | allLAM | dLAM | allTT | dTT | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

LKNEx_J | 8.61 | 8.08 | 99.59 | 99.60 | 1018 | 1019 | 5.75 | 5.83 | 99.74 | 99.77 | 16.54 | 19.95 |

LKNEy_J | 43.34 | 41.2 | 99.8 | 99.85 | 2212 | 2243 | 6.9 | 7.1 | 99.9 | 99.93 | 54.08 | 65.31 |

LKNEz_J | 90.41 | 82.46 | 99.93 | 99.96 | 3935 | 3972 | 7.54 | 8.17 | 99.97 | 99.98 | 163.39 | 223.12 |

LKNEx_C | 1.6 | 1.62 | 99.6 | 99.61 | 1246 | 1305 | 5.69 | 5.75 | 98.33 | 99.65 | 9.7 | 15.36 |

LKNEy_C | 16.3 | 20.1 | 99.72 | 99.86 | 2109 | 2210 | 6.46 | 7.05 | 99.87 | 99.94 | 43.45 | 77.38 |

LKNEz_C | 10.5 | 18.51 | 99.81 | 99.92 | 2425 | 2514 | 6.44 | 7.23 | 99.9 | 99.97 | 36.36 | 67.22 |

CoMx_J | 2 | 2.87 | 99.73 | 99.76 | 915 | 945 | 6.29 | 6.30 | 99.74 | 99.85 | 16.2 | 20.82 |

CoMy_J | 11.97 | 18.16 | 99.91 | 99.93 | 3939 | 3967 | 7.23 | 7.37 | 99.95 | 99.97 | 37.3 | 46.92 |

CoMz_J | 86.35 | 86.52 | 99.98 | 99.98 | 3939 | 3963 | 8.23 | 8.26 | 99.99 | 99.99 | 261.59 | 262 |

CoMx_C | 1.12 | 1.63 | 99.65 | 99.74 | 1536 | 1600 | 6.25 | 6.32 | 99.64 | 99.85 | 12.36 | 20.96 |

CoMy_C | 9.84 | 12.74 | 99.87 | 99.93 | 3927 | 3987 | 6.83 | 7.14 | 99.95 | 99.98 | 38.04 | 55.04 |

CoMz_C | 32.59 | 42.18 | 99.93 | 99.94 | 3927 | 3954 | 7.15 | 7.32 | 99.97 | 99.98 | 76.2 | 89.77 |

**Table 6.**The mean percentage difference between recurrence measures calculated for the global and directional settings.

REC | DET | LMAX | ENT | LAM | TT | |
---|---|---|---|---|---|---|

Jazz | 19.47 | 0.02 | 1.17 | 2.52 | 0.03 | 22.07 |

Classic | 34.22 | 0.07 | 3.27 | 5.08 | 0.29 | 58.9 |

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**MDPI and ACS Style**

Błażkiewicz, M.
Evaluation of Geometric Attractor Structure and Recurrence Analysis in Professional Dancers. *Entropy* **2022**, *24*, 1310.
https://doi.org/10.3390/e24091310

**AMA Style**

Błażkiewicz M.
Evaluation of Geometric Attractor Structure and Recurrence Analysis in Professional Dancers. *Entropy*. 2022; 24(9):1310.
https://doi.org/10.3390/e24091310

**Chicago/Turabian Style**

Błażkiewicz, Michalina.
2022. "Evaluation of Geometric Attractor Structure and Recurrence Analysis in Professional Dancers" *Entropy* 24, no. 9: 1310.
https://doi.org/10.3390/e24091310