This paper examines several applications of principal component analysis (PCA) to physical systems. The central result demonstrates that the PCA can identify from the recorded system trajectories conserved quantities that take the form of polynomials in the system variables in an easily programmed and straightforward fashion. In particular, a data record composed of the positions and velocities generated from simulations of two-dimensional harmonic oscillator trajectories is recast into a feature record containing the values of the lowest order atomic polynomials formed from these quantities at each evaluation time. The combinations of the features that are conserved can then be obtained from the principal components of the feature record with the smallest explained variances. Additionally, two features of the application of the PCA to homogeneous periodic spin systems are identified and discussed. The first of these relates to certain characteristic behaviors of the explained variances associated with the principal components of the spin distribution that are found to be artifacts of the boundary geometry. The PCA is then employed to generate synthetic spin realizations with probability distributions in energy-magnetization space that resemble the corresponding input distributions although the associated statistical quantities are not sufficiently accurate.

本文研究了主成分分析（PCA）在物理系统中的几种应用。中心结果表明，PCA可以从记录的系统轨迹中以简单易编程的方式，以系统变量中多项式形式的保守量来识别。特别是，将由二维谐波振荡器轨迹的模拟生成的位置和速度组成的数据记录重铸到功能记录中，该功能记录包含在每个评估时间由这些量形成的最低阶原子多项式的值。然后，可以从特征记录的主成分中获得具有最小解释方差的保守特征的组合。另外，确定并讨论了PCA在均匀周期自旋系统中的两个应用特征。其中第一个涉及与自旋分布的主成分相关联的所解释方差的某些特征行为，自旋分布的主成分被发现是边界几何的伪影。然后，虽然相关的统计量不够准确，但PCA用于生成具有能量磁化空间中概率分布的合成自旋实现，该概率分布类似于相应的输入分布。