A weighted non-negative least squares algorithm for three-way ‘PARAFAC’ factor analysis

Abstract

A time-efficient algorithm PMF3 is presented for solving the three-way PARAFAC (CANDECOMP) factor analytic model. In contrast to the usual alternating least squares, the PMF3 algorithm computes changes to all three modes simultaneously. This typically leads to convergence in 40–100 iteration steps. The equations of the weighted multilinear least squares fit are given. The optional non-negativity is achieved by imposing a logarithmic penalty function. The algorithm contains a possibility for dynamical reweighting of the data during the iteration, allowing a robust analysis of outlier-containing data. The problems typical of PARAFAC models are discussed (but not solved): multiple local solutions, degenerate solutions, non-identifiable solutions. The question of how to verify the solution is discussed at length. The program PMF3 is available for 486-Pentium based PC computers.