Incorporating analytical errors in log-ratio based compositional discriminant analysis

Uncertainties in the measurement of the geochemical composition of various sample materials are rarely included for statistical analyses of the data.
In case of log-ratio methods, incorporating errors in the analysis has even not yet been done, up to the authors' knowledge.
Many calibration procedures provide relative cell-wise errors, which can be conveniently combined to deliver error assessments for any set of log-ratios.
In this contribution we incorporate all these errors in estimates of the mean vector and covariance matrix of the data on a particular log-ratio.
Thanks to the linear/bilinear relation between mean/covariance estimates among different log-ratio representations, such error-integrating estimates are affine equivariant.
These means and covariances are the building blocks of many statistical analysis.
Here we focus on developing an error-integrating Fisher rule, but the methodology can be readily applied to other linear models with compositional variables, like regression or ANOVA.
In general, results show that the incorporation of errors produce a more conservative (and honest) assessment of the discrimination direction and separability of the subpopulations considered.
The application of using cell-wise errors and its impact on interpretation of results will be shown by case studies of geochemical composition of tea plants in relation to geological source rock.