Department of the Air Force Report, 1996

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The fading-memory recursive filter, defined by Eqs. (12) and (20), can be applied to
launch test results to estimate failure probability. For this application the values to be
filtered are the test . outcomes, with 0 representing a successful launch, and 1
representing a failure or anomalous behavior. Given a series of outcomes, the filtered
result after each launch in the series represents the estimate of failure probability at that
point. Filtered results for two filter-control constants are shown in Table 37 for a
hypothetical series of ten launches for which all but the second and fourth flights were
successful.
Table 37. Filter Application for Failure Probability
j[]
F = 0.98
F =0.90
Index
Outcome
lter factor, an
Fail. Prob.
Filter factor, an
Fail. Prob.
1
2
3
4
5
6
7
8
9
10
0
1
0
1
0
0
0
0
0
0
1.0000
0.5051
0.3401
0.2576
0.2082
0.1752
0.1517
0.1340
0.1203
0.1093
0.0
0.5051
0.3333
0.5051
0.3999
0.3299
0.2798
0.2423
0.2132
0.1899
1.0000
0.5263
0.3690
0.2908
0.2442
0.2132
0.1917
0.1756
0.1632
0.1535
0.0
0.5263
0.3321
0.5263
0.3978
0.3129
0.2529
0.2085
0.1745
0.1477
In this example, estimated failure probabilities are shown for two values of the filter
constant that force the filter to fade at two different rates. After ten launches the
estimated failure probability using F = 0.98 is 0.1899. For the faster fading-memory
filter (F =0.90), the result is 0.1477. Both estimates are less than that obtained by equal
weighting, since the two failures occurred early in the sequence. Note that after four
launches (2 successes and 2 failures) both filtered estimates exceed 0.5, since one of the
two failures occ~rred during the fourth flight.
If the l's and O's used in the example to represent failures and successes were reversed,
the same filter would provide estimates of probability of success.
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