POD and PFP are inextricably joined
Making the detection threshold size smaller will improve the probability
of detection (POD), since more of the signals will be above the lowered
threshold (Figure 1). Unfortunately, lowering the detection threshold
size will also increase the probability of false positive (PFP), a
fact too often ignored.
Thus the "obvious" fix for an inadequate inspection - lower the decision
threshold - won't work in practice. If the threshold is lowered by
fiat, the inspection throughput will plummet as false positives
unnecessarily remove parts.
Sometimes this results in a serious communications breakdown: The engineering
staff thinks the inspection threshold is set low, where they mandated it, while
the inspections are using a higher threshold to make throughput quotas. ("Not in MY shop," I can hear
you say. Are you sure?)
How POD and PFP depend on ‚ decision:
Look closely at Figure 1. Imagine what happens when the horizontal
line representing the decision threshold (‚ decision = 200) is lowered. What happens?
- The POD curve in the inset plot moves to the left as the probability of detection increases for any given size.
- The Probability of a False Positive, the tail of the noise probability distribution plotted on the left y-axis, also
increases (gets worse).
Figure 1: How POD is related
to size through the signal vs. size (‚ vs. a) relationship.
Probability of Detection and Probability of False Positive are
unavoidably related to detection threshold. Both will increase
or decrease together as the decision threshold moves up or down, but while
an increase in POD is good, an increase in PFP is bad: Decreasing the detection threshold will improve
Probability of Detection, and make the Probability of a False
Positive worse. That is unavoidable. (There's no "free
Figure 2 graphically illustrates this by plotting the size at 90% POD
(a90) on the left vertical axis and PFP on the right vertical axis,
as functions of the decision threshold size.
Figure 2: ‚ decision†influences
PFP, and the size at 90% POD, in a trade-off.