Extrapolation, penalty

The expression for Vy = Sy = Vres/ is true if a = Xmean however, if x is different from Xmean, an extrapolation penalty is paid that is proportional to the square of the deviation. [See Eq. (2.16).] This results in the characteristic trumpet shape observed in Figures 2.6 and 2.8. The influence of the calibration design is shown in Figure 2.8, where the corresponding individ-... 

Figure 2.8. The slopes and residuals are the same as in Figure 2.4 (50,75,100, 125, and 150% of nominal black squares), but the A -values are more densely clustered 90, 95, 100, 105, and 110% of nominal (gray squares), respectively 96, 98, 100, 102, and 104% of nominal (white squares). The following figures of merit are found for the sequence bottom, middle, top the residual standard deviations +0.00363 in all cases the coefficients of determination 0.9996, 0.9909, 0.9455 the relative confidence intervals of b +3.5%, +17.6%, 44.1%. Obviously the extrapolation penalty increases with decreasing Sx.x, and can be readily influenced by the choice of the calibration concentrations. The difference in Sxx (6250, 250 resp. 40) exerts a very large influence on the estimated confidence limits associated with a, b, Y(x), and X( y ).

Figure 3.10. The relationship between the three coefficients A, B, and C for the curves shown in Figure 3.9 the quadratic and the linear coefficients are tightly linked. The intercept suffers from higher variability because it carries the extrapolation penalty discussed in Section 2.2.5 and Figure 2.8.

In a penalty test, a property cf the system is modified to reduce the probability of the desired result. For example, to predict safety, a particular expl train interface may be tested with a standard donor and a more sensitive acceptor conversely, to predict reliability, a less sensitive acceptor material is used. If this probability is reduced sufficiently, it is possible to obtain mixed responses (that is, some fires and some no-fires) with samples of reasonable size, and to develop data from which the mean value of the penalty and its standard deviation (as well as confidence limits) can be established. These estimates can be used iri statistical extrapolation to estimate safety or reliability under the original design conditions. The term VARICOMP (VARIation of explosive COMPosition) was coined by J.N. Ayres for a method developed at the Naval Ordnance Lab, White Oak, in the 1950 s and early 1960 s (Ref 1)... 

It can never be assumed that results from one set of conditions can be extrapolated to other conditions without penalty. 

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