Comparing univariate and multivariate approaches for process variograms: A case study

• This paper introduces the multivariogram approach to process sampling.
• The limitations of the classical univariate approach are highlighted.
• Different multivariate variographic approaches are compared.
• Variograms of PCA highlight distinct spatial patterns through variable grouping.
• Multivariogram combined with PCA is revealed to be a powerful summarising tool.

The classical application of theory of sampling (TOS) is univariate. However, most practical situations address multi-analyte issues, in which the common belief is that one should focus a variographic study on the analyte with the most heterogeneous distribution. This paper introduces a multivariogram approach to process sampling and compares it with the classical univariate and multivariate approaches of variograms as applied to principal component analysis (PCA) scores. A case study of low-grade kaolin residue sampling for metallurgical testing is used to illustrate this methodology. A total of eight important properties are analysed, including chemical analytes, size distribution properties and pulp density. The results show that the classical univariate approach can underestimate the global sampling error if the sampling protocol is designed by using only the highest variance property. Variograms that are calculated from PCA scores highlight distinct spatial patterns through variable grouping in a reduced number of variograms. Multivariograms can be used to summarise time variations in multiple analytes and highlight the multivariate time autocorrelation aspects of these analytes. However, the resulting sampling variance is very high, and an alternative approach that applies multivariograms to PCA scores, filters noise from the data, and only keeps relevant data information, which reduces the global sampling variance, is proposed. This case study illustrates the usefulness of multivariate approaches to help multivariate variographic modelling in mineral processing and in many other fields within science and industry that deal with multi-analyte sampling issues.