Multi-variate factor separation of numerical simulations

Factor separation is widely used in the analysis of numerical simulations.  It allows changes in properties of a system to be attributed to changes in multiple variables associated with that system.  There are many possible factor separation methods; here we discuss three previously-proposed methods that have been applied in the field of climate modelling: the linear factor separation, the Stein and Alpert (1993) factor separation, and the Lunt et al (2012) factor separation.  We show that, when more than two variables are being considered, none of these three methods possess all four properties of 'uniqueness', 'symmetry', 'completeness', and 'purity'.  Here, we extend each of these methods so that they do possess these properties for any number of variables, resulting in three factor separation methods -- the 'linear-sum' , the 'shared-interaction', and the 'scaled-total'.  We show that the linear-sum method and the shared-interaction method reduce to be identical in the case of four or fewer variables, and we conjecture that this holds for any number of variables.  We present the results of the factor separations in the context of studies that used the previously-proposed methods.  This reveals that only the linear-sum/shared-interaction factor separation method possesses a fifth property -- `boundedness', and as such we recommend the use of this method in applications for which these properties are desirable.   The work described here is in review in Geoscientific Model Development - see https://gmd.copernicus.org/preprints/gmd-2020-69 .