PLS Leads to Different Algorithms for Factor Analysis and Regression

Two multivariable problems of general interest, are factor analysis and regression. This paper examines partial least squares (PLS) as a tool for both problems. For single output data sets, the familiar PLS algorithm is applicable to both problems. For multiple output problems the familiar PLS algorithm [1, 2, 3] (called fact-PLS in this paper) is appropriate for factor analysis. However fact-PLS leads to algebraically-inconistent results for regression problems. To address this issue, a new algebraically-consistent multivariable PLS algorithm, C-PLS, is developed. Unlike fact-PLS, C-PLS does not rely on iterative calculations. Another PLS approach, "one-at-a-time" PLS (OAT-PLS), is closely related to C-PLS; however OAT-PLS is also algebraically-inconsistent. A simulation study of these various PLS methods shows C-PLS to have the best estimation and prediction performance