Search Results for: entropy

Suppose installed base or cohorts in successive periods have different reliabilities due to nonstationarity? What does that do to forecasts, estimates, reliability predictions, diagnostics, spares stock levels, maintenance plans, etc.? Assuming stationarity is equivalent to assuming all installed base, cohorts, or ships have the same reliability functions. At what cost? Assuming a constant failure rate is equivalent to assuming everything has exponentially distributed time to failure or constant failure rate. At what cost?

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I needed multivariate fragility functions for seismic risk analysis of nuclear power plants. I didn’t have any test data, so Lawrence Livermore Lab paid “experts” for their opinions! I set up the questionnaires, asked for percentiles, salted the sample to check for bias, asked for percentiles of conditional fragility functions to estimate correlations, and fixed pairwise correlations to make legitimate multivariate correlation matrixes. Subjective percentiles provide more distribution information than parameter or distribution assumptions, RPNs, ABCD, high-medium-low, or RCM risk classifications.

The title was inspired by Rupert Miller’s report “What Price Kaplan Meier?” That report compares nonparametric vs. parametric reliability estimators from censored age-at-failure data. This article compares alternative, nonparametric estimators from different data: grouped, censored age-at-failure data vs. population ships and returns data required by generally accepted accounting principles. This article compares data storage and collection requirements and costs, and bias, precision, and information of nonparametric reliability estimators. 

“Component D” had some failures in its first 12 months. How many more would fail in 36-month warranty? ASQ’s Quality Progress Statistics Roundtable published the data and Weibull analysis. The data included left-censored failure counts collected at one calendar time. The Weibull analysis included actuarial failure forecasts. This article describes nonparametric alternatives to Weibull and quantifies extrapolation uncertainty. The nonparametric forecasts are larger than the Weibull forecasts. Alternative extrapolations of nonparametric failure rates from data subsets quantify uncertainty. [Read more…]

Ergodicity means that cross-section probabilities equal longitudinal lifetime probabilities. (“Ergos” is Greek for “work.” Think of “ergonomics”.) Ergodicity means that we can estimate age-specific field reliability functions from cross-section data: ships (installed base) and returns (complaints, failures, service parts’ sales, etc.). Ships and returns provide information about lifetimes. Returns are the superpositions of failures of products or their parts started at different times. What does ergodicity have to do with toilet paper? [Read more…]

Myron Tribus’ UCLA Statistical Thermodynamics class introduced me to entropy, -SUM[p(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 contains 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, PRODUCT[p(t)], e.g., for observed failures at ages t. It is equivalent to maximize -SUM[ln(p(t)]. Maximum entropy reliability estimation maximizes entropy -SUM[p(t)ln(p(t)]. That’s same as maximizing the expected value, -SUM[p(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…]