The variogram

The variogram function, V(j), is defined as Vi times the average squared difference in heterogeneity contributions between the sum of pairs of increments, as a function of j ... 

Practical interpretation of the experimental variogram is the first item to address. The variogram level and form provide valuable information on the process variation captured, as well as the quality of the sampling procedure employed. There are only three principal variogram types encountered in process sampling, but many more combinations hereof ... 

The variogram function is not defined for lag = 0 - which would correspond to extracting the exact same increment twice. Even though this is not physically possible, it is highly valuable to obtain information of the likely variance corresponding to this zero-point variability (i.e. what if it would have been possible to repeat the exact same physical sampling). The TOS identifies this variance as MPE (minimum practical error), which is also known as V(0). 

R. Heikka and P. Minkkinen, Comparison of some methods to estimate the limiting value of the variogram, vfj), for the sampling interval j=0 in sampling error estimation. Anal. Chim. Acta, 346, 277-283 (1997). 

The second order moment can be expressed either in terms of the covariance or the variogram. The covariance of the random function at points and is defined to be... 

Thus, to estimate the variogram, the drift must be known and to estimate the drift, the variogram must be known. This leads to difficulties in model identification which will be discussed later. 

We give here a simple example to demonstrate the basic idea. Suppose we are measuring our product for impurities and sample the process hourly. For illustration purposes, we use a circle to represent each sample, and the degree of shading differentiates the impurity level. We observe a pattern in the results of Figure 4.14, so that a variogram is not really needed. However, we use this example to show intuitively how the variogram is calculated and interpreted. 

Estimation of the nugget effect can provide valuable information about the process. For example, if it is substantially larger than an estimate of the material variation obtained independently, then we know that extraneous variation and/or bias is being introduced through incorrect sample collection, handling, or imacceptably large analytical variation. Extrapolation of the variogram to... 

Figure 4.15 Extrapolation of the variogram to estimate the nugget effect.

While the variogram is defined as a continuous mathematical function of time or space for continuous processes, continuity is a mathematical concept that does not exist in the physical, material world (Gy, 1998, p. 87). We must therefore be careful in its estimation. When collecting discrete data and using the formula in Appendix D, the variogram makes sense only if the following two conditions are satisfied. 

If only one or two samples are missing and not accounted for in the calculations, the variogram will not be severely affected. However, it is best to make sure that these exceptions are not ignored. 

Let us examine a few special cases to illustrate how different variograms might be interpreted. Then we will see how the variogram can improve our process understanding. 

Suppose we normally take a sample at the beginning of every 8 h work shift. Using historical data, we construct the variogram shown in Figure 4.18. 

Suppose we can reduce the analytical variation by more than half by using a more precise method. This reduces the nugget effect substantially, and we perform another variographic experiment. By overlaying all the information, the variogram looks like that in Figure 4.20. 

Suppose a plant operates on 12 h work shifts and that samples are normally taken at the beginning of each shift. For our variographic experiment, we collect a sample every hour for several days and estimate the nugget effect with a sample every minute. Combining this information results in the variogram given in Figure 4.21. 

Figure 4.23 is the variogram for the data from Figure 4.1 over a longer period. There are two ways we can interpret it. 

The variogram in Figure 4.24 shows a trend in the process variation. 

Suppose samples are taken on a fixed schedule a certain number of time units apart every hour, for example, at the beginning of each shift, or at noon every day. For the th sample, a property of interest (the critical content) is measured as a percent weight fractior of the total sample weight. This content, Cj, is recorded. After a fixed number n of samples is taken, we compute the average A of the contents c. Now we are ready to compute each point in the variogram. The jth variogram point V j) is calculated as... 

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