# Multifractal Analysis of Movement Behavior in Association Football

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## Abstract

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## 1. Introduction

_{s}:

_{s}) versus the log (s). It follows that, if the slope of the log–log plot between the scale of observation and the descriptive statistics is equal 0 < α < 0.5, then the process has a memory, and it exhibits anti-correlations. Therefore, the successive random elements in a sequence present an increasing trend in the past that is likely to be followed by a decreasing trend in the future, and vice-versa. Conversely, if the slope is equal to 0.5 < α < 1, then the process has a memory, and it exhibits positive correlations. Therefore, the successive random elements in a sequence present an increasing trend in the past that is likely to be followed by an increasing trend in the future. If the slope is α = 0.5, then the process is indistinguishable from a random process with no memory (random walk). Finally, if the slope is 1 < α < 2, then the process is non-stationary. Most real-world phenomena yield signals that mix different patterns of correlations over time and in different time scales, most often presenting irregular behavior with interwoven periods of high and low correlation. In such cases the linear regression of log

_{2}μ(s) on log

_{2}s does not exhibit a straight-line characterizing mono-scaling behavior by a single scale exponent, conversely, additional exponents are necessary. In fact, signals with interwoven fractal subsets become more complex, hence requiring multiple scaling exponents for the full description of different parts of the signal. Consequently, in contrast to the monofractal formalism, where a global scale exponent α is defined for the entire signal, multifractal time series are defined by a set of local singularity exponents. According to Ihlen and Vereijken [19], signals present in biological systems exhibit multiscaling behavior, for example, signals of inter-stride intervals in human walking or inter-beat interval in electrocardiogram data. Moreover, almost all data points in the aforementioned signals can be represented as a singularity at time instant t

_{0}with a strength that is numerically defined by a singularity exponent α. These apparent random variations can be misinterpreted as pure noise originating from the system, however, studying the correlation structure of noise-like signals can elucidate important information about the process underpinning its generation.

## 2. Materials and Methods

#### 2.1. Participants

#### 2.2. Data Collection

^{®}software (MathWorks, Inc., Natick, MA, USA).

- Jogging: low intensity self-paced steady state running.
- HIIP: consisting of 10 s of maximum effort of linear running interspersed with 20 s active recovery.
- RC: comprised of walking, jogging, running and sprinting. To each element was assigned a number, 1 to 4, respectively, and a sequence of numbers was randomly generated using an Excel spreadsheet. The exercise intensity was changed every 5 s according to the sequence using a verbal command.
- 5 vs. 5 SSG.
- 8 vs. 8 SSG.
- 10 vs. 10 SSG.

#### 2.3. Data Processing

_{2}of the sample size (signal length). The regression type was set to “uniform”, which means that the leader coefficients were uniformly weighted in all decomposition levels. The range of statistical moments (q-moment) used varied from −5 to +5.

#### 2.4. Multifractal Discrete Wavelet Leader

#### 2.5. Interpreting the Multifractal Singularity Spectrum

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Experimental conditions: jogging; (

**b**) high intensity interval protocol (HIIP); (

**c**) running circuit; (

**d**) 5 vs. 5 small-sided games (SSG); (

**e**) 8 vs. 8 SSG; (

**f**)10 vs. 10 SSG.

**Figure 2.**Raw jogging and circuit running (RC) accelerometry signal (top row). Below, exemplification of the magnification process for scale j = 1 using different moment orders (q-moments from −3 to 3) on the wavelet leader coefficients L_x (j,k) to capture a fuller characterization of the fluctuations. Note how the negative q values amplify small fluctuation (red dashed line) whereas positive q values amplify large fluctuation (black dashed line).

Components | Conditions (Mean ± SD) | |||||
---|---|---|---|---|---|---|

Jogging | HIIP | Circuit | 5 vs. 5 | 8 vs. 8 | 10 vs. 10 | |

$h\left(0\right)$ | 0.153 ± 0.018 | 0.231 ± 0.028 | 0.325 ± 0.044 | 0.403 ± 0.052 | 0.415 ± 0.055 | 0.441 ± 0.015 |

${h}_{min}$ | 0.069 ± 0.015 | 0.060 ± 0.023 | 0.059 ± 0.020 | 0.068 ± 0.012 | 0.055 ± 0.024 | 0.062 ± 0.030 |

${h}_{max}$ | 0.284 ± 0.025 | 0.355 ± 0.029 | 0.753 ± 0.058 | 0.835 ± 0.132 | 0.936 ± 0.161 | 1.054 ± 0.101 |

Δ$h$ ${h}_{max}-{h}_{min}$ | 0.215 ± 0.020 | 0.295 ± 0.038 | 0.694 ± 0.060 | 0.767 ± 0.121 | 0.881 ± 0.144 | 0.992 ± 0.104 |

${D}_{h}\left({h}_{min}\right)$ | 0.843 ± 0.010 | 0.615 ± 0.0.82 | 0.532 ± 0.130 | 0.542 ± 0.079 | 0.451 ± 0.254 | 0.515 ± 0.072 |

${D}_{h}\left({h}_{max}\right)$ | 0.717 ± 0.037 | 0.733 ± 0.026 | 0.368 ± 0.122 | 0.345 ± 0.185 | 0.203 ± 0.278 | 0.154 ± 0.136 |

Δ${D}_{h}$ ${D}_{h}\left({h}_{max}\right)-{D}_{h}\left({h}_{min}\right)$ | −0.126 ± 0.039 | 0.118 ± 0.091 | −0.164 ± 0.176 | −0.197 ± 0.211 | −0.248 ± 0.494 | −0.361 ± 0.146 |

$L=h\left(0\right)\u2013{h}_{min}$ | 0.084 ± 0.005 | 0.171 ± 0.037 | 0.266 ± 0.054 | 0.335 ± 0.054 | 0.360 ± 0.072 | 0.380 ± 0.035 |

R =${h}_{max}-h\left(0\right)$ | 0.131 ± 0.017 | 0.124 ± 0.011 | 0.428 ± 0.035 | 0.432 ± 0.0141 | 0.521 ± 0.185 | 0.612 ± 0.0112 |

Δ$S=R-L$ | 0.047 ± 0.016 | −0.047 ± 0.041 | 0.162 ± 0.069 | 0.161 ± 0.069 | 0.162 ± 0.240 | 0.232 ± 0.129 |

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**MDPI and ACS Style**

Freitas Cruz, I.; Sampaio, J.
Multifractal Analysis of Movement Behavior in Association Football. *Symmetry* **2020**, *12*, 1287.
https://doi.org/10.3390/sym12081287

**AMA Style**

Freitas Cruz I, Sampaio J.
Multifractal Analysis of Movement Behavior in Association Football. *Symmetry*. 2020; 12(8):1287.
https://doi.org/10.3390/sym12081287

**Chicago/Turabian Style**

Freitas Cruz, Igor, and Jaime Sampaio.
2020. "Multifractal Analysis of Movement Behavior in Association Football" *Symmetry* 12, no. 8: 1287.
https://doi.org/10.3390/sym12081287