weibull-series-consequence
When does model simplification matter? Consequence analysis and adaptive model selection for Weibull series systems with masked failure data.
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Source Code
Package Registries
When Does Model Simplification Matter?
Consequence Analysis for Weibull Series Systems
When a reliability engineer simplifies a series system model by assuming all components share a common Weibull shape parameter, how much accuracy is lost – and can the data even tell the difference? This paper shows these two questions have a quantitatively aligned answer.
Key Findings
Bias and power are aligned
The common-shape model’s MTTF prediction bias and the likelihood ratio test’s detection power are aligned as functions of shape heterogeneity:
| Shape CV | MTTF Bias | LRT Power (n=500) | Recommendation |
|---|---|---|---|
| < 5% | < 0.2% | < 7% | Use reduced model confidently |
| 5–14% | 0.2–0.9% | 7–37% | Safe zone: bias negligible, test lacks power |
| 14–21% | 0.9–1.5% | 37–97% | Transition: test detects meaningful bias |
| > 27% | > 2.9% | > 99% | Use full model |
The test lacks power precisely where the consequence of using the wrong model is negligible, and has high power precisely where bias warrants switching models.
The traditional parsimony argument is wrong
A bias-variance decomposition reveals that the full model has lower MTTF variance than the reduced model even when the reduced model is correctly specified (CV = 0). At n = 100: full variance = 348, reduced variance = 417. The constraint k_1 = … = k_m forces all shape estimation error into one degree of freedom, amplifying its propagation through the nonlinear MTTF integral.
Adaptive model selection works
An LRT-based adaptive procedure achieves RMSE within 2.5% of the always-full strategy at n >= 500, while selecting the simpler model over 90% of the time when appropriate.
The LRT is well-calibrated; information criteria are not
| Criterion | Type I Error | Behavior |
|---|---|---|
| LRT (alpha = 0.05) | 4.6–6.8% | Well-calibrated |
| AIC | 8.2–12.4% | Liberal (~2x nominal) |
| BIC | 0–0.2% | Over-conservative |
Repository Structure
.
├── paper/ # LaTeX source and figures
│ ├── paper.tex # Main manuscript
│ ├── refs.bib # Bibliography
│ ├── Makefile # Build automation
│ └── image/ # Figures (PDF)
├── qrei/ # QREI submission variant
│ ├── manuscript.tex # Journal-formatted manuscript
│ └── cover-letter.tex # Cover letter
├── results/ # Simulation code and data
│ ├── consequence/ # Consequence analysis (MTTF bias, MSE)
│ │ └── figure/ # Including bias-variance decomposition
│ ├── adaptive/ # Adaptive model selection
│ ├── lrt/ # Likelihood ratio test simulations
│ │ ├── divergence/ # Type I error and power vs shape CV
│ │ ├── vary_m/ # Effect of number of components (m = 2-8)
│ │ ├── vary_p/ # Effect of masking probability (p = 0.05-0.70)
│ │ ├── vary_q/ # Effect of censoring level (q = 0.50-1.00)
│ │ └── nomasking/ # Ideal case baseline (p=0, q=1)
│ ├── 5_system_scale3/ # Scale parameter sensitivity
│ ├── 5_system_shape3/ # Shape parameter sensitivity
│ └── sim_utils.R # Shared vectorized simulation utilities
├── docs/ # GitHub Pages (paper PDF)
└── CLAUDE.md # Development guidance
Data Pipeline
R scripts (Monte Carlo) → CSV → Python analysis → PDF figures → LaTeX paper
Each experiment in results/ contains an R script that runs simulations, a CSV of raw results, and a figure/ subdirectory with a Python analysis script that produces publication-quality PDFs.
Building the Paper
cd paper
latexmk -pdf paper.tex
Or with Make:
cd paper
make # build paper.pdf
make docs # copy PDF to docs/ for GitHub Pages
Dependencies
R Packages
wei.series.md.c1.c2.c3– Weibull series system with masked dataalgebraic.mle– Maximum likelihood estimation utilitiesalgebraic.dist– Algebraic distribution functionsmd.tools– Masked data toolstidyverse,ggplot2,parallel,boot
Python
matplotlib,seaborn,pandas,numpy
LaTeX
Standard distribution with amsmath, amsthm, graphicx, natbib, hyperref
Related Work
- Towell (2023). Reliability Estimation in Series Systems with Masked Data – the C1-C2-C3 likelihood framework
wei.series.md.c1.c2.c3– R package implementing the parametric models
Citation
@article{towell2025consequence,
title={When Does Model Simplification Matter? Consequence Analysis for Weibull Series Systems},
author={Towell, Alexander},
year={2025},
note={Preprint},
url={https://github.com/queelius/weibull-series-consequence}
}
License
MIT License – see the LICENSE file for details.
Author
Alexander Towell ORCID: 0000-0001-6443-9897 lex@metafunctor.com