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

Factorisation (also known as “factor separation”) consists of attributing the total change of some property of a system to multiple components, each component being associated with a change to an internal variable of the system. Multiple tests can be carried out to inform this factorisation, with each test (or simulation in the case of numerical applications) consisting of different combinations of variables. Factorisation experiments are used in many disciplines, with early applications being in agricultural field experiments

In order to introduce and discuss previous factorisation methods, we use an example case study from the field of climate science. We turn to the Pliocene,

It is worth at this stage introducing some notation. Here, we restrict ourselves to the case where there are two possible values for each variable, denoted “0” and “1”; having more than two values increases the computational cost of a factorisation and can reduce the number of factors that can be assessed in a fixed computing budget. We name the fundamental property of the climate system that we are factorising as

The simplest factorisation that can be carried out is a linear one. For the Pliocene example with two factors, three simulations are carried out in which variables are changed consecutively, for example,

This factorisation is illustrated graphically in Fig.

Three different factorisation methods of temperature,

However, an equally valid linear factorisation would be

In contrast to the linear factorisation, the

The

In extending to

Simulations and linear factorisations in an

Averaging the edges (interpretation (i) above) would result in a factorisation:

Although this is unique, symmetric, and pure, it is not complete, because

As shown above, neither the linear nor the

Here we discuss possible extensions to the three previous factorisations discussed above, that are unique, symmetric, pure, and complete in

The linear-sum factorisation arises from a generalisation to

Each possible linear factorisation can be represented as a non-returning “path” from the vertex

This factorisation is complete (

To generalise to

As for the three-dimensional case above, let us label each vertex,

All factorisations consist of partitioning the total change,

For the linear-sum factorisation, all paths that we consider start at the origin vertex,

The

For the linear-sum factorisation, we instead carry out a weighted average, with the weight for each edge in dimension

For example, for

As stated in Sect.

For our Pliocene example for

As discussed in Sect.

This factorisation for

This is identical to the equivalent term in Eq. (

This factorisation for

In the scaled-residual factorisation, the

In

For example, for

Here we discuss three examples of papers in which the

The equivalent linear-sum/shared-interaction factorisation is given by Eq. (

The equivalent scaled-residual factorisation is given by Eq. (

In

An alternative, using the linear-sum/shared-interaction factorisation that is complete, is obtained from Eq. (

Another alternative, using the scaled-residual factorisation that is complete, is obtained from Eqs. (

Comparison of various factorisation methods.

Properties of the factorisations discussed in this paper.

The bottom row in Fig.

We also explored using a version of the scaled-residual factorisation in which the residual terms were shared, not by the absolute magnitude of the individual factors, but by their relative values, so that Eq. (

In this paper, we have reviewed three previously proposed factorisations and extended them to produce factorisations that are unique, symmetric, pure, and complete. We have presented them for three dimensions (i.e. three factors) and generalised to

The methods that we present here will be of particular use in the analysis of systems with multiple variables and are applicable beyond solely climate science.

The model fields underlying Fig.

The supplement related to this article is available online at:

DJL led the study and wrote the first draft of the paper. GAS made Fig.

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors are grateful to Harry J. Dowsett for developing the boundary conditions that underpin the Pliocene simulations discussed.

This research has been supported by NERC (grant no. NE/P01903X/1).

This paper was edited by James Annan and reviewed by Chris Brierley and one anonymous referee.