A human is a thing that moves in space. Like all things that move in space, we can in principle use differential equations to describe their motion as a set of functions that maps time to position (and velocity, acceleration, and so on). With inanimate objects, we can reliably predict their trajectories by using differential equations that account for up to the second-order time derivative of their position, as is commonly done in analytical mechanics. With animate objects, though, and with humans, in particular, we do not know the cardinality of the set of equations that define their trajectory. We may be tempted to think, for example, that by reason of their complexity in cognition or behaviour as compared to, say, a rock, then the motion of humans requires a more complex description than the one generally used to describe the motion of physical systems. In this paper, we examine a real-world dataset on human mobility and consider the information that is added by each (computed, but denoised) additional time derivative, and find the maximum order of derivatives of the position that, for that particular dataset, cannot be expressed as a linear transformation of the previous. In this manner, we identify the dimensionality of a minimal model that correctly describes the observed trajectories. We find that every higher-order derivative after the acceleration is linearly dependent upon one of the previous time-derivatives. This measure is robust against noise and the choice for differentiation techniques that we use to compute the time-derivatives numerically as a function of the measured position. This result imposes empirical constraints on the possible sets of differential equations that can be used to describe the kinematics of a moving human.
翻译:人类是一种在空间中移动的东西。和在空间中移动的所有事物一样,我们原则上可以使用差异方程式来描述其运动,将其描述成一组功能,绘制时间定位(和速度、加速等)的复杂度或行为。如果非无动性天体,我们就可以可靠地预测其轨迹。如果使用差异方程式来计算其位置的二阶时间衍生物,就像通常在分析机理中所做的那样。如果使用动画天体,特别是人类,我们不知道用来决定其轨迹的一组方程式的基本性。例如,我们可能想用差异方程式来描述其运动的动作,因为其复杂性在认知或行为上可以绘制时间定位(以及速度、加速等等 ) 。那么,人类运动需要比通常用来描述物理系统运动的动作更复杂的描述。在这个文件中,我们查看一个真实的关于人类运动的数据集,并且考虑每个(已计算过的,但已消化的)变异性时间衍生物所增加的信息,并且找到该位置的最大序列的序列顺序,在这种变轨迹上被测量的轨迹上,这种变的轨迹是每个我们所观察到的轨迹的轨迹的轨变。在前的轨迹上,我们可以正确地描述。我们所观察到的轨迹的轨迹变。在前的轨迹变。我们用的轨迹变的轨迹变。