Incorporating cell-wise errors in compositional exploratory analysis

Geochemical data sets often include values with varying uncertainties. The uncertainties are small if samples had been measured under optimal analytical conditions, but are typically high if trace elements values are close to the detection limit or if the data set includes samples of very different characteristics or with inhomogeneity. From the data itself it is not necessarily possible to distinguish geochemical valuable signals from data noise without prior knowledge about the sample set and the analytical method. Especially for a small geochemical data set not knowing or ignoring the uncertainties might lead to misinterpretations. One method to circumvent this problem is to incorporate cell-wise errors which describe the uncertainty for each value and can serve as weights in statistical analysis. However, incorporating cell-wise errors into statistical analysis of geochemical data sets is rarely applied, especially when it comes to multi-variate analysis. Geochemical data sets are mostly composed of compositional data. Accordingly, the characteristics of constrained values should be taken into account for incorporating errors.
Principal component analysis (PCA) helps to explore the variance–covariance structure with the objective of data reduction and interpretation of a multivariate setting. In combination with biplots it is a powerful tool for explorative data analysis. In our contribution we will propose a method to include cell-wise errors as weights into the PCA with the aim to use the information provided by uncertainties in an early stage of data exploration. This is another approach than the weighted PCA based on the Tucker-3 method presented by Gallo and Buccianti (2013) or the spectral map analysis (SMA) presented by Lewi (2005). We will use a geochemical data set of archaeological glasses with standard deviations for each value (based on 3 - 5 measurements) as a case study for incorporating analytical uncertainties in PCA of compositional data.