Direct Estimation of Sampling Variance From Time Series Measurements ù Comparison to Variographic Analysis

Sampling is often the most critical part in process analysis. When the mean of a continuous lot, such as a process stream within a given time interval, is estimated by calculating the mean of several samples, the uncertainty of the estimated mean depends usually on the sampling strategy. Three different strategies can be used: systematic, stratified or random sample selection. The standard deviation of the lot mean is different for these methods so that systematic sampling usually gives the smallest and random sampling the largest standard deviation. Pierre Gy developed a method where the sampling errors of one-dimensional lots for these three strategies can be estimated from a variogram, which is calculated from a variographic experiment, ie from a set of one-dimensional time series measurements with fixed sampling interval. While variogram is an excellent tool for characterising and displaying the variability of autocorrelated processes as function of time, the sampling error estimation from it is quite laborious, however. In this work an alternative method for estimating the sampling errors was developed. The process is divided into strata of equal sizes, varying the length of strata similarly to variographic analysis. For stratified sampling the standard deviation estimates are obtained by analysis of variance and for systematic sampling by fitting a linear regression line to each stratum and calculating the residual variances. By pooling the individual variance estimates the variance estimates that are corrected for autocorrelation are obtained. The new method was tested with simulated and real data sets having periodic, random and/or drift fluctuations. For stationary processes the results agreed well with the results of the GyÆs variographic method. The method can probably be generalised to estimate the sampling variance of higher-dimensional lots.