Search Results for: tribus

Dick Mensing said, “Larry, you can’t give an estimate without some measure of its uncertainty!” For seismic risk analysis of nuclear power plants, we had plenty of multivariate earthquake stress data but paltry strength-at-failure data on safety-system components. So we surveyed “experts” for their opinions on strengths-at-failures distribution parameters and for the correlations between pairs of components’ strengths at failures. 

If you make estimates from population field reliability data, do the estimates have uncertainty? If all the data were population lifetimes or ages-at-failures, estimates would have no sample uncertainty, perhaps measurement error. Estimates from population field reliability data have uncertainty because typically some population members haven’t failed. If field reliability data are from renewal or replacement processes, some replacements haven’t failed and earlier renewal or replacement counts may be unknown. Regardless, estimates from population data are better than estimates from a sample, even if the population data is ships and returns counts!

What can we do without reliability function estimates? FMEA? FTA? RCA? RCM? Argue about MTBFs and availability? Weibull? Keep a low profile? Run Admirals’ tests? Look for a new, well-funded project far from the deliverable stage? 

Ask for field data; there should be enough to estimate reliability and make reliability-based decisions, even if some data are missing. Field data might even be population data!

Myron Tribus’ UCLA Statistical Thermodynamics class introduced me to entropy, -Sp(t)ln(p(t)). (p(t) is the probability of state t of a system.) Professor Tribus later advocated maximum-entropy reliability estimation, because that “…best represents the current state of knowledge about a system…” [Principle of maximum entropy – Wikipedia] Caution! This article contain statistical neurohazards.

Claude Shannon wrote that entropy (log base 2) represents information bits, “…an absolute mathematical limit on how well data from the source can be losslessly compressed onto a perfectly noiseless channel.”  [Beirlant et al.]

Maximum likelihood estimation is one way to estimate reliability from data. It maximizes the probability density function of observed data, Pp(t), e.g., for observed failures at ages t. It is equivalent to maximize -Sln(p(t). Maximum entropy reliability estimation maximizes entropy -Sp(t)ln(p(t). That’s same as maximizing the expected value, -Sp(t)ln(p(t), of the log likelihood -ln(p(t). Fine, if you have life data, ages at failures t censored or not. [Read more…]