Research in football has been embracing the complex systems paradigm in order to identify different insights about key determinants of performance. The present study explored the multifractal properties of several football-related scenarios, as a candidate method to describe movement dynamics. The sample consisted of five footballers that were engaged in six different training situations (jogging, high intensity interval protocol, running circuit, 5 vs. 5, 8 vs. 8 and a 10 vs. 10 small-sided game). All kinematic measures were collected using a 100 Hz wireless and wearable inertial measurement unit (WIMUPRO©). Data were processed using a discrete wavelet leader transform in order to obtain a spectrum of singularities that could best describe the movement dynamics. The Holder exponent for each of all six conditions revealed mean values h < 0.5 indicating presence of long memory with anti-correlated behavior. A strong trend was found between the width of the multifractal spectrum and the type of task performed, with jogging showing the weakest multifractality ∆h = 0.215 ± 0.020, whereas, 10 vs. 10 small-sided game revealed the strongest ∆h = 0.992 ± 0.104. The Hausdorff dimension indicates that a maximal fluctuation rate occurs with a higher probability than that of the minimal fluctuation rate for all tasks, with the exception of the high intensity interval protocol. Moreover, the spectrum asymmetry values of jogging, running circuit, 5 vs. 5, 8 vs. 8 and 10 vs. 10 small-sided games reveal their multifractal structures are more sensitive to the local fluctuations with small magnitudes. The multifractal analysis has shown a potential to systematically elucidate the dynamics and variability structure over time for the training situations.

中文翻译：

---

协会足球运动行为的多重分形分析

足球研究一直在采用复杂系统范式，以识别关于表现关键决定因素的不同见解。本研究探索了几个与足球相关的场景的多重分形特性，作为描述运动动态的候选方法。样本由五名足球运动员组成，他们参与了六种不同的训练情况（慢跑、高强度间歇训练、跑步循环、5 对 5、8 对 8 和 10 对 10 小场比赛）。所有运动学测量均使用 100 Hz 无线和可穿戴惯性测量单元 (WIMUPRO©) 收集。使用离散小波前导变换处理数据以获得最能描述运动动力学的奇异点谱。所有六个条件中每一个的 Holder 指数显示平均值 h < 0。5 表示存在具有反相关行为的长记忆。在多重分形谱的宽度和执行的任务类型之间发现了一个强烈的趋势，慢跑显示最弱的多重分形 Δh = 0.215 ± 0.020，而 10 对 10 小边游戏显示最强的 Δh = 0.992 ± 0.104。Hausdorff 维度表明，除了高强度间隔协议之外，所有任务的最大波动率发生的概率高于最小波动率的概率。此外，慢跑、跑步电路、5 vs. 5、8 vs. 8和10 vs. 10小场比赛的频谱不对称值显示它们的多重分形结构对小幅度的局部波动更敏感。