Monitoring the lot mean and uncertainty estimates by piecewise local modelling

Abstract

Variography is an excellent tool for monitoring the long-range trend of continuous processes. Pierre Gy has presented a method that can be used for estimating the measurement variance of a lot mean as function of sampling frequency for different sampling modes: random, stratified, and systematic sample selections. The method involves the estimation of the intercept (also called the nugget effect) of the variogram at the time point zero, and numerical integration of the variogram. The method can also be used for optimising sampling plans. At the time when variography was developed on-line analysers were not available. Samples were extracted from the process streams and analysed in laboratories. It was important to optimise the sampling plans to control the analytical costs and the reliability of the plans in estimating the estimation error. For a reliable variogram more than thirty to forty samples had to be analysed. Consequently, the results could not be used on-line. Currently process analysers are widely used to monitor continuous processes. Like in variographic estimation of the lot mean this method is based on the theory of stratified sampling. If the lot is divided into N1 strata of equal sizes (or sublots) of which n1 are sampled the variance of the lot mean aL is [EQN] Here s21 is the variance between strata mean values and s22 the within-strata variance, N2 is the size of strata as the potential number of samples and n2 the number of samples taken from the stratum. The great advance of stratified sampling is that only the within-strata variance propagates into the lot average if samples are taken from every stratum. With current process analysers measurements can be taken at short time intervals and that is used in the current method to estimate the process average and its variance continuously. Within a short range (or stratum in this case) a continuous process can be locally modelled with a line. With systematic sampling after a minimum of three measurements a line can be fitted to this range and the mean and variance of the range mean calculated. That is the first stratum. When the process progresses, the calculations are repeated for the new strata and values. It is important that the quality of the final lot can be monitored on-line, especially if lots of certain sizes and demanding quality specifications are produced. The method is tested with different kinds of simulated and real data sets. This method can be easily modified also for 2D and 3D sampling targets.