Revisiting least squares techniques for the purposes of allocation in life cycle inventory

Abstract

In this communication we reexamine the use of various least squares techniques (namely ordinary least squares (OLS), data least squares (DLS), and total least squares (TLS)) for the purposes of allocation as proposed in an earlier article in this journal (Marvuglia et al. (Int. J. Life Cycle Assess. 15:1020-1040, 2010)). These methods are placed within the context of traditional methods of partitioning allocation. An equivalence between least squares techniques and traditional partitioning is noted and demonstrated on previously published brick production data. A short summary of the relevant least squares techniques is provided followed by a description of the problem of inventory calculation for the case of more products than processes. This is presented in terms of non-unique solutions to underdetermined systems of linear equations with intensity matrix as unknown. We provide another analysis of the Sicilian brick production case study. Upon reexamination of the brick data, a number of disparities in the published inventories for brick can be more fully explained in terms of (1) data quality, (2) additional assumptions made to extract a solution of an underdetermined system of equations, and (3) discrepancy vectors. Like other types of partitioning, inventories produced by least squares techniques correspond to one of the non-unique solutions of an underdetermined system of linear equations. Based on this insight, we advise against the use of least squares techniques as a black-box approach to the allocation problem and conclude that the recommendation of TLS on the basis of its asymptotic properties is not theoretically justified.