The meaning of this equation and the significance of the subscripts will be elaborated in later sections of this paper.
The important point which needs to be made here is that Equations 6-11 only contain ratios, and do not depend on the absolute abundances of the different argon isotopes.
It is well known that common summary statistics such as the arithmetic mean and standard deviation are unreliable in this data space.
This is because the ternary diagram occupies a narrowly restricted subspace of the realm of real numbers.
where ‘x’ stands for ‘blank’, ‘sample’ or ‘standard’.
The same formulation can be used for the interference monitors (particularly Ca) but further discussion of these will be deferred to Section 8 and Appendix A.
These restrictions cause problems because standard data reduction methods commonly assume that the data follow a Normal distribution, which requires support from -.
The zero value problem can be avoided by performing generalised linear regression of the ratios (using a logarithmic link function to ensure positive intercepts, Nelder and Wedderburn, 1972), or to cast the regression problem into a more sophisticated maximum likelihood form (Wood, 2015).
In this case, terms b, c and e in Equation 5 disappear, which leaves us with a simple three component system comprised of Ar.
Because we are only interested in the relative abundances of these three isotopes, they can be normalised to unity and plotted on a ternary diagram (Figure 1).
The easiest but by no means only way to achieve this is by forming the logratios prior to regression, yielding an [n Ar is used as a common denominator for all the ratios denoted by ‘m’ in Equation 5.
We thus obtain five time-resolved logratio matrices, one for each run in the analysis sequence. It is reasonable to expect the blank-corrected logratio signals to be correlated within each group, but uncorrelated between groups.