Monte Carlo Sampling

Monte Carlo Sampling (cross-cutting/numerics/monte-carlo-sampling)

Use when the task is estimating the distribution of an output quantity by Monte Carlo sampling: seeded pseudo-random draws over an input range, sample statistics, percentile confidence intervals, and histograms of the sampled values. The analytic GUM first order law lives in the uncertainty-propagation leaf; this leaf is the sampling alternative that needs no sensitivity coefficients.

Domain quick reference

• Monte Carlo sampling draws n pseudo-random input values from a uniform distribution on [low, high] with a fixed seed, so the draw is reproducible and the study can be audited.
• The sample mean is the arithmetic mean of the draws; the sample standard deviation uses the n - 1 denominator.
• The p-th percentile is the value below which p percent of the sample lies, computed with linear interpolation between the order statistics (percentile 50 is the median).
• The two-tailed confidence interval at level 0.95 is the percentile pair (2.5, 97.5); it brackets the central 95 percent of the sampled distribution.
• A histogram bins the draws into k equal-width intervals over the sample range; the counts sum to the sample size and the last bin is inclusive.
• Propagation applies the model function to every draw and reports the mean, standard deviation, and confidence interval of the transformed outputs; the spread of the outputs carries the effect of the input spread through the model.
• Sample statistics converge with the sample size: the error of the mean estimate shrinks like 1 / sqrt(n).

Workflow

1. Fix the input range [low, high] and the sample size n; a few thousand draws is a common starting point.
2. Draw the inputs with draw_samples(seed, n, low, high) and keep the seed for reproducibility.
3. Summarize the draws with sample_mean and sample_stddev.
4. Extract the spread with confidence_interval(draws, 0.95).
5. Inspect the shape with histogram(draws, bins) before gating the study.
6. For a model output, run propagate_samples(seed, n, low, high, func) and report the transformed statistics.

Pitfalls

• Changing the seed between runs: the study is only reproducible when the same seed is reused; record it with the results.
• Small sample sizes: a few dozen draws give noisy percentiles; the tails of the distribution need thousands of draws to stabilize.
• Reading the sample standard deviation as the population value: the n - 1 denominator is the unbiased estimate for a sample.
• Interpolating percentiles by hand: percentile uses the linear interpolation method over the sorted order statistics, not a rounding to the nearest rank.
• Histogramming a constant sample set: a zero range cannot be binned and the logic raises ValueError.
• Passing a high bound at or below the low bound, a sample size below 1, or a percentile outside [0, 100]: all raise ValueError.
• Confusing this leaf with uncertainty-propagation: the GUM first order law needs sensitivity coefficients; Monte Carlo sampling needs only the input range and the model function.

Behavior contract (gate 3)

The sampling, percentile, interval, and histogram logic is exercised by the gate 3 contract test: scripts/test_monte_carlo_sampling.py against scripts/monte_carlo_sampling_logic.py (stdlib unittest, offline). Run:

python3 scripts/test_monte_carlo_sampling.py

Compliance

• NACA Report 824 is US government work (public domain); the pack anchor per standards-map.yaml. The Monte Carlo method follows the JCGM 101 (GUM supplement 1) sampling procedure, which is generic numerical methodology, not RTCA or SAE content; summary and formulas only.
• compliance: STANDARDS-REF, gated: false.