Which method, and when

EMMA or mixing ratios?

They answer different questions and you almost always want both, in this order.

EMMA first. It tells you how many end-members the variability requires and which species behave as a conservative mixture. It cannot give you proportions.

Mixing ratios second. They tell you the proportions, given end-members you have already justified. They cannot tell you that your conceptual model is wrong — they will apportion any sample among any end-members you supply.

Least squares or maximum likelihood?

	mixing_ratios()	mix_ml()
Assumes	end-members known exactly	end-members uncertain
Uses	each sample independently	all samples together
Improves with more samples	no	yes
Sensitive to end-member error	strongly	weakly, once samples are many
Needs	nothing but the data	a standard deviation for every analysis

The decision rule:

• Few samples and well-characterised end-members → least squares. Below about ten samples the maximum-likelihood method has no advantage, and it can return negative end-member concentrations.

• Uncertain end-members and many mixed samples → mix_ml(). This is the situation the method was designed for, and its advantage grows with the sample count because the mixtures carry information about the sources.

The second case is the common one. End-member uncertainty is rarely analytical; it comes from spatial and temporal variability and from conceptual error. You often cannot sample a pure end-member at all.

Do I have to choose end-members by hand?

Not entirely. suggest_endmembers() ranks combinations of candidates by how many samples they enclose, and mescla.emma.archetypes optimises the same idea. Both propose; neither discovers. An end-member has to be a water that exists and that your conceptual model can name.

Reporting

Whatever you use, report: the conceptual model and why those end-members; the species retained and rejected, with reasons; the rank evidence; the fraction of samples inside the mixing hull; the assumed standard deviations and a sensitivity sweep over them; ratios with uncertainty; and the measured-versus-predicted residuals with the reactions they imply.