Multivariate control charts for grade control using principal-component analysis and time-series modelling

Abstract

Quality is one of the most important strategic issues for top management. Its importance has been increased by the increasingly stringent demands of customers in today’s globally competitive market. In general, ‘quality’ has two components: quality of design and quality of conformance. In the minerals industry grade-control procedures are used to ensure conformance with product specifications. Grade control has always been considered a challenging task for geologists and mining engineers. Various factors that determine ore quality, including the size and geometry of a deposit, geological disturbances, such as folds and faults, the texture and mineralogical composition of ore and waste and the chemical and metallurgical properties of the ore, are attributable to the nature of formation of the deposit and cannot be changed. Nevertheless, ore quality can be maintained through various means, such as accurate grade estimation, grade-control planning, minimization of ore dilution, blending and subsequent treatment. Geostatistics is often used in the quest for the desired grade control, but because grade variation is not simply localized at faces, but extends from the face to ore despatch through blending and processing, it cannot always produce the desired quality level. Erratic grade fluctuations within a deposit exacerbate the problem. In such cases the use of a statistical quality-control technique, including control charts, will contribute to the achievement of the desired grade control. The main idea behind the use of a control chart in grade control is to identify the root causes of quality variation—in particular, assignable causes on the basis of which corrective action may be taken to eradicate irregular grade fluctuations. The Shewhart X – chart is the most popular control chart used today for the monitoring and control of mean ore grade. This chart, however, can only be utilized effectively in univariate cases. Many gradecontrol operations in mining must address the control of more than one variable at a time. The usual procedure in mining is to monitor each of the variables separately using simultaneous individual Shewhart X – charts. This approach is satisfactory as long as the variables are independent, but in most cases it is observed that the variables are interrelated. Hence, simultaneous monitoring of individual variables separately will fail to recognize possible cross-correlation between the variables, and this will increase the insensitivity of the charts for the detection of out-of-control conditions. In these circumstances a single, multivariate control chart, especially the Hotelling T2 chart, will enable handling of the crosscorrelation problem. The effectiveness of the Hotelling T2 chart in multivariate cases is beyond doubt, as reflected by its widespread use in various situations.1,2,3 The underlying assumptions for the Hotelling T2 chart to function properly are that the sample observations are univariate as well as multivariate normal and statistically independent. Even though these assumptions are important and deviation will result in an adverse effect on the performance of the chart, their validity in mining situations is often questionable. The normality assumption for individual variables can sometimes be met by increasing the sample sizes according to the central limit theorem. According to this the distribution of the sample averages of n independent observations will approach normality as the number of observations in a sample increases even if the observations are not normally distributed. At this point Shewhart observed that many individual observations are non-normal, although the distribution of sample means of size four will in many cases follow the normal curve.4 Spedding and Rawlings5 observed, however, that a more extreme population distribution may require larger sample sizes to achieve a normal distribution of sample means. Even if it is possible to fulfil the normality assumption for individual variables by increasing the sample size, this does not necessarily mean that multivariate normality will be achieved where the variables are cross-correlated. Similarly, the need for sample observations to be statistically independent is a serious problem as autocorrelation between the observations becomes an inherent characteristic in mineral deposits where ore grades are spatially related. As a result, it is expected that the sample observations X(t) and X(t + 1) will be positively autocorrelated. Further, the sample means may tend to drift over time. Any one or combination of these factors does not mean that the quality characteristic is out of control—they merely represent the inherent grade variation. The application of a conventional multivariate T2 chart in such circumstances will create a false impression of out-of-control conditions, leading to unjustified searches for assignable causes. A multivariate quality-control scheme appropriate for mining applications is thus needed. A combination of principal-component analysis and time-series modelling is investigated here as a means to construct control charts in multivariate situations that can alleviate problems pertaining to their application to grade control in the minerals industry.