Optimising multivariate variographic analysis with information from multivariate process data modelling (partial least squares)

Abstract

Whereas a classical tool from the Theory of Sampling (TOS), variographic analysis, can address practical situations with multiple variables, its application has very often been limited to one variable at a time. Recent developments have shown the benefits of using multivariate approaches for variographic characterisation of a set of variables instead of considering individual variables sequentially. Among these approaches, the multivariogram has been revealed itself to be a powerful tool when the overall time-variability of a process must be summarized in terms of a large set of properties (variables) to assess its true global variability. However, even when choosing carefully the properties of interest for the process tested to avoid unnecessary variance increase, the resulting global variance with this approach is very high. In particular, some variables which contribute to a major proportion of the global (multivariate) variability could be less important for the process performance than others having a lower variability. To address this issue, a new approach is proposed, combining the multivariogram with process modelling and multivariate data analysis methods such as Partial Least Squares (PLS) regression from chemometrics. An example from the mineral processing industry is presented, for which the process performance could be linked to key process variables (sensor data) using the PLS regression. Once introduced in the multivariogram equation, PLS model parameters (loading-weights or regression coefficients) can be used to weigh the variables according to their relevance for the process. In addition, this also permits characterisation of process performance variability with time using only the process input variables and a weighted metric according to the PLS regression model. Ultimately, this method helps to find an optimized sampling procedure in terms of frequency, sampling mode and number of increments according to the actual overall process performance. This approach has potentially many applications in the mining, feed and food, pharmaceutical or any other industry for which it can be used to reduce risks and ensure a better use and management of resources.