Department of the Air Force Report, 1996

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Appendix C. Filter Characteristics
Estimating launch-vehicle failure probabilities using empirical launch data is an
uncertain process when the sample size is small and the data are obtained from an
evolving system. One approach that may be used to estimate failure probabilities is to
perform a least-squares fit to trial outcome values (0 =success, 1 =failure). For mature
launch vehicles, failure probabilities have decreased markedly from their early
experimental days. For new programs, empirical data may be scant or nonexistent.
One decision that must be made involves the type of function to- fit to the data. The
true nature of the failure-rate function may be unknown or extremely complex, or there
may be insufficient data to estimate a complex function. The easiest calculation is made
when a constant failure-rate function is assumed. However, available data appear to
indicate that failure rates decrease as a program matures, at least up to a point. If it can
be assumed that launch-vehicle failure probabilities decrease over time (i.e., as the
number of launches increases), then some non-constant function (perhaps linear or
exponential) can be chosen for the fit, or the data weighted as a function of time. In
estimating Atlas reliability, General Dynamics161 chose the latter option by adopting the
Duane model. ~s model is based on the assumption that the mean number of
launches between failures increases when causes of failure are corrected. Although this
may be the case up to- a point, eventually reliability seems to level off at a fairly
constant value. Consequently, for mature programs RTI has chosen to fit the failure-
rate function to a constant. Su<;h a fit can be based on simple least squares using a
fixed-length sliding-window filter to allow for changes in the estimated value over
time, or on a least squares fitwith unequal weighting.
If a constant function is fit to a set of data using least squares with equal weighting of
data, the solution is given by the mean:
(10)
·Consider the following example:
X1­-6
"2 = 5
"3 = 7
Then,
Recursively,
X = 6+5+7 =-18
3
3
= 6
(11)
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RTI