Least squares formulation of robust non-negative factor analysis

Positive matrix factorization (PMF) is a recently published factor analytic technique where the left and right factor matrices (corresponding to scores and loadings) are constrained to non-negative values. The PMF model is a weighted least squares fit, weights based on the known standard deviations of the elements of the data matrix. The following aspects of PMF are discussed in this work: (1) Robust factorization (based on the Huber influence function) is achieved by iterative reweighting of individual data values. This appears especially useful if individual data values may be in error. (2) Desired rotations may be obtained automatically with the help of suitably chosen regularization terms. (3) The algorithms for PMF are discussed. A synthetic spectroscopic example is shown, demonstrating both the robust processing and the automatic rotations.