Emissions decomposition
Case study · esg-toolkit
Four causes for one change in emissions, with nothing left over.
Eighteen figures, three products across two years. Change any one of them and the four effects, the chart and the residual are computed again.
Problem
A company that made less of a dirty product and more of a clean one while its suppliers changed underneath it has four stories running at once, and changing one factor at a time leaves a remainder that belongs to none of them and can be argued in either direction.
Method
The logarithmic mean Divisia index in its additive form, LMDI-I, where a segment's emissions are total output times its share of output times its energy per unit times its emissions per unit of energy, each factor takes a share of the change weighted by the logarithmic mean of the segment's emissions in the two periods, and there is no residual term at all.
What it does
A waterfall of six bars, the base year total, the four effects in order and the target year total, and under it the check: the four effects to six decimal places, their sum, the observed difference between the two years, and the residual in exponential notation.
What it refuses to do
It refuses to hide the residual: LMDI-I leaves none analytically, so that line carries floating point noise in exponential notation on every redraw and would report a broken identity before anything else did.
It refuses to run without output, so where total output is zero or below in either period it prints that total output must be above zero in both and draws nothing.
It refuses negative inputs, reading a negative number typed into any of the eighteen fields as zero, because it would put a logarithm outside its domain and return four effects that are all NaN.
It refuses to pass off a substitution as a result: values at or below zero are replaced by 1e-12 before the logarithms are taken, which is a numerical convenience and not a finding.
It refuses to draw the chart from zero and say nothing, so the scale is computed from the values it has to show and a line under the waterfall states the range it runs over.
It refuses to explain: the four numbers say how much of the change sits with output, mix, energy intensity and the emission factor, and they name no cause.
In closing
I wanted a bridge chart whose numbers could not be moved by whoever happened to be drawing it.