PCA and SVD with nonnegative loadings

摘要

Principal component analysis (PCA) and singular value decomposition (SVD) are widely used in multivariate statistical analysis for data reduction. The work considers exponential, logit, and multinomial parameterization of the eigenvectors’ elements that always yields nonnegative loadings of shares for variable aggregation. In contrast to regular PCA and SVD, matrix decomposition by the positive shares shows explicitly which variables and with which percent are composed into each group, so what is each variable contribution to data approximation. The least squares objective of matrix fit is reduced to Rayleigh quotient for variational description of the eigenvalues. Eigenvectors with the nonlinear parameterization can be found in Newton–Raphson optimizing procedure. Numerical examples compare the classical and nonnegative loadings results, with interpretation by the Perron–Frobenius theory for each subset of variables identified by sparse loading vectors.