Principal component analysis for percolation with and without data preprocessing.

Abstract

In the percolation model, specific preprocessing of configuration data enables identifying phase transitions with unsupervised learning methods such as principal component analysis (PCA). However, the limitations of using original percolation configuration data and the role of preprocessing remain incompletely understood. Here, we study how PCA works in percolation. Firstly, we theoretically derive the PCA results. For the original configurations, PCA shows no insights into the percolation phase transition. The theoretical results of PCA reveal the relation between principal components and the indicators of phase transitions, i.e., order parameters and structure factors based on the preprocessing of removing unpercolating clusters. In addition, we investigate the PCA based on auxiliary Ising mapping, and the relation is also derived and validated. Secondly, we examine the intrinsic properties of the original percolation configurations and the operational principles of PCA and then reveal the challenge of PCA in percolation. The statistical independence among sites results in the absence of necessary lattice structure information, and further, the negligible threshold number of occupied (unoccupied) sites required for forming (destructing) percolating cluster results in the invisibility of the phase transition to the cluster analysis or statistical approaches. Both removing unpercolating clusters and auxiliary Ising mapping can remove this invisibility. Our work suggests that yielding discernible differences between percolating state and unpercolating state, or further introducing percolation statistical correlation among sites, gives effective data preprocessing methods.