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