Principal component analysis constrained by layered simple structures

The paper proposes a procedure for principal component analysis called layered principal component analysis (LPCA) to produce a simple and interpretable loading matrix. The novelty of LPCA is that a loading matrix is constrained as a sum of matrices with simple structures called layers, and the resulting simplicity of the LPCA solution is controlled by how many layers are used. LPCA is a generalization of disjoint PCA proposed as reported by Ferrara (in: Giommi (ed) Topics in theoretical and applied statistics, Springer, Cham 2016). The number of layers controls the balance of simplicity and the fit to the data, and the user can choose the desired level of simplicity between the most restrictive but simplest case with a single layer or multiple layers with better fit to the data. The optimal number of layers is specified in terms of explained variance and two information criteria. Two simulation studies were conducted to evaluate how accurately the LPCA procedure recovers the true parameter values. The results showed that LPCA was effective for parameter recovery. The paper presents three examples of LPCA applied to real data, which show the potential of LPCA for producing simple and interpretable loading matrices.