Module point_process 

Trend tests and the power-law fit for the event times of a point process.

The data is one sequence of event times 0 < t₁ ≤ t₂ ≤ … ≤ tₙ observed on a single unit — the failures of one repairable system, the incidents of one service — and the question is whether events arrive at a constant rate (a homogeneous Poisson process, HPP) or a changing one. That is a different question from mann_kendall_test (a trend in an ordered series of values) and from weibull fitting (independent lifetimes of many units, which assumes exactly the absence of trend these tests check for).

Observation

Truncation::Time(T): the unit was watched until T ≥ tₙ, whatever happened. Truncation::Failure: watching stopped at the last event, so T = tₙ and that event carries no information about the trend — the statistics use the n − 1 events before it.

Methods

With m the number of events used (n time-truncated, n − 1 failure-truncated) and T the end of observation:

• Laplace test — U = (Σtᵢ/m − T/2) / (T·√(1/(12m))), approximately standard normal under an HPP. U > 0: events crowd towards the end (rate increasing).
• MIL-HDBK-189 test — χ² = 2·Σ ln(T/tᵢ), exactly χ² with 2m degrees of freedom under an HPP. A small value means events crowd towards the end (rate increasing).
• Power-law process (Crow-AMSAA) — intensity λβt^(β−1), expected events λt^β. MLE β̂ = n / Σ ln(T/tᵢ) (the same sum as the MIL-HDBK-189 statistic, over the same events), λ̂ = n / T^β̂. β̂ is biased upward; conditionally on n, 2nβ/β̂ ~ χ²(2m), so E[β̂] = nβ/(m − 1) and β̄ = (m − 1)/n · β̂ is unbiased — (n − 1)/n time-truncated, (n − 2)/n failure-truncated. β < 1: rate decreasing; β > 1: increasing.

References

• Ascher & Feingold (1984), Repairable Systems Reliability, ch. 5.
• Crow (1974), “Reliability analysis for complex, repairable systems”, in Reliability and Biometry, SIAM, 379–410.
• MIL-HDBK-189C (2011), Reliability Growth Management, §5.
• IEC 61710 (2013), Power law model — Goodness-of-fit tests and estimation methods.
• Rausand & Høyland (2004), System Reliability Theory, 2nd ed., §7.4.

Structs

PowerLawFit

A power-law process fit.

TrendTestResult

A trend test’s outcome.

Enums

PointProcessError

Why event times were refused.

TrendDirection

Which way the observed rate moves — the sign of the deviation from a constant rate, not a significance verdict (see p_value).

Truncation

How observation of the unit ended.

Functions

laplace_trend_test

Laplace trend test against a constant event rate.

mil_hdbk_189_test

MIL-HDBK-189 test against a constant event rate.

power_law_process_fit

Maximum-likelihood fit of a power-law process (Crow-AMSAA).