Multi-variate factorisation of numerical simulations

Factorisation (also known as 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 factorisation methods; here we discuss three previously proposed factorisations that have been applied in the field of climate modelling: the linear factorisation, the Stein and Alpert (1993) factorisation, and the Lunt et al. (2012) factorisation. 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 factorisations so that they do possess these properties for any number of variables, resulting in three factorisations - the linear-sum factorisation, the shared-interaction factorisation, and the scaled-residual factorisation. We show that the linear-sum factorisation and the shared-interaction factorisation 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 factorisations in the context of three past studies that used the previously proposed factorisations.

Automated summary

The previously proposed factorisation methods do not possess all four properties when considering more than two variables. By extending each of these methods, new factorisations are obtained that can possess these properties for any number of variables. The linear-sum factorisation and the shared-interaction factorisation are found to be identical in the case of four or fewer variables, and this similarity is conjectured to hold for any number of variables.