Post date: Apr 19, 2014 11:16:21 PM
Scott said:
Okay. The simplified code for the noncoherent system appears on the code subpage. Sorry for the confusion and delay, Mohamed. A picture of the output it creates appears at the bottom of the page.
I may still be a little confused about the inputs to be used for this case, or perhaps I lost the understanding that we worked out in those last, hurried days at UTC.
There are three inputs: probabilities for pump, sensor, and controller. (I don't actually remember whether we're doing failure probabilities or reliability probabilities. And is the latter simply one minus the former?)
When I give "kn" c-boxes to the three variables, like we used for the tank and bridge systems, I get results that look reasonable. Are they correct, Mohamed?
When I give scalar (real) values to the three variables, I seem to get correct results too.
When I give interval ranges to the three variables, the results also seem correct. The interval ranges were computed by interpreting the significant digits given in the hypothetical expert elicitations:
pump = Q(1e-3, 0.1)
sensor = Q(2e-3, 5e-2)
controller = Q(3e-3, 1/60)
where Q is the function
Q <- function(lambda,mu) return(pexp(500, lambda+mu) / (1 + mu/lambda)).
For instance, the significant digits in "1e-3" imply the interval [ 0.0005, 0.0015]. When such intervals are propagated through the Q function in Risk Calc, we get the interval inputs:
pump = [ 0.003322259, 0.02912622]
sensor = [ 0.02654867, 0.05263158]
controller = [ 0.1208633, 0.1853325]
which leads to the interval range for the system
answer = [ 0.02224406, 0.04979065].
I think this example is pretty interesting, and it illustrates how one might use expert statements even if they did not come with confessions about uncertainties. (Of course we would not display all of these digits in the inputs and outputs; they should be cropped appropriately.)
The fourth way to translate empirical inputs into parameters characterized as c-boxes is to use (hypothetical) failure times to estimate a c-box for lambda, the parameter for the exponential distribution. This c-box is a (precise) gamma distribution. The distribution that we cut at 500 hours is then a composition of this lambda c-box through the exponential shape, which turns out to be a (precise) gamma-exponential distribution. (This fourth parameterization isn't set up yet on the code subpage, but it will be easy to do using the ready code at https://sites.google.com/site/confidenceboxes/software.
We could use one or more of these four ways ("kn" c-boxes, scalars, sigdig intervals, or gamma-exponential c-boxes) to parameterize the pump problem. The first and the last offer the confidence interpretation.
Comments?
Mohamed replied:
Ok, we started the case study of non-coherent system from failure and repair rate of components (by using failure times) based on an exponential distribution but we can also started from unvailaibilities of components. I will send you a preliminary version of the first paper in the next 3 days.