Struct ReliabilityAnalysis

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pub struct ReliabilityAnalysis { /* private fields */ }

Expand description

Reliability analysis results from a fitted Weibull distribution.

Computes reliability engineering metrics (reliability function, hazard rate, MTBF, B-life) from Weibull shape (beta) and scale (eta) parameters.

§Mathematical Background

Given a Weibull distribution with shape beta > 0 and scale eta > 0:

Reliability: R(t) = exp(-(t/eta)^beta)
Hazard rate: lambda(t) = (beta/eta) * (t/eta)^(beta-1)
MTBF: eta * Gamma(1 + 1/beta)

§Examples

use u_analytics::weibull::ReliabilityAnalysis;
let ra = ReliabilityAnalysis::new(2.0, 100.0).unwrap();
assert!((ra.reliability(0.0) - 1.0).abs() < 1e-10);
assert!(ra.hazard_rate(50.0) > 0.0);
assert!(ra.mtbf() > 0.0);
let b10 = ra.b_life(0.10).unwrap();
assert!(b10 > 0.0 && b10 < 100.0);

§References

Meeker & Escobar (1998), Statistical Methods for Reliability Data, Wiley.
Nelson, W. (1982). Applied Life Data Analysis, Wiley. §2.
Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data, 2nd ed.

Implementations§

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impl ReliabilityAnalysis

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pub fn new(shape: f64, scale: f64) -> Option<Self>

Creates a new reliability analysis from Weibull parameters.

§Arguments

shape - Shape parameter beta (must be positive and finite)
scale - Scale parameter eta (must be positive and finite)

§Returns

None if either parameter is non-positive or non-finite.

§Examples

use u_analytics::weibull::ReliabilityAnalysis;
let ra = ReliabilityAnalysis::new(2.0, 100.0);
assert!(ra.is_some());

// Invalid parameters return None
assert!(ReliabilityAnalysis::new(-1.0, 100.0).is_none());
assert!(ReliabilityAnalysis::new(2.0, 0.0).is_none());

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pub fn with_covariance(self, covariance: [[f64; 2]; 2]) -> Option<Self>

The same analysis with the covariance of the estimated (shape, scale) – row and column order shape, scale, as WeibullMleResult::covariance gives it – so the B-lives and reliabilities carry confidence intervals (Self::b_life_interval, Self::reliability_interval).

None unless the matrix is a covariance: finite, symmetric (to a relative 1e-9) and positive definite.

§Examples

use u_analytics::weibull::ReliabilityAnalysis;
let ra = ReliabilityAnalysis::new(2.0, 100.0).unwrap();
assert!(ra.clone().with_covariance([[0.04, 0.1], [0.1, 25.0]]).is_some());
assert!(ra.with_covariance([[0.04, 2.0], [2.0, 25.0]]).is_none()); // not positive definite

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pub fn covariance(&self) -> Option<[[f64; 2]; 2]>

The covariance of (shape, scale) the intervals are made from, when there is one.

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pub fn from_mle(result: &WeibullMleResult) -> Self

Creates a reliability analysis from an MLE fitting result.

§Panics

This function does not panic. MLE results always have valid parameters.

The fit’s covariance comes along, so the analysis gives confidence intervals.

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pub fn from_mrr(result: &WeibullMrrResult) -> Self

Creates a reliability analysis from an MRR fitting result.

§Panics

This function does not panic. MRR results always have valid parameters.

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pub fn shape(&self) -> f64

Returns the shape parameter (beta).

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pub fn scale(&self) -> f64

Returns the scale parameter (eta).

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pub fn reliability(&self, t: f64) -> f64

Reliability (survival) function at time t.

R(t) = exp(-(t/eta)^beta)

For t < 0, returns 1.0 (no failure before time 0).

§Examples

use u_analytics::weibull::ReliabilityAnalysis;
let ra = ReliabilityAnalysis::new(2.0, 100.0).unwrap();

// R(0) = 1.0 (full reliability at time zero)
assert!((ra.reliability(0.0) - 1.0).abs() < 1e-10);

// R(eta) = exp(-1) ≈ 0.368 for any shape parameter
let expected = (-1.0_f64).exp();
assert!((ra.reliability(100.0) - expected).abs() < 1e-10);

§Reference

Weibull (1951), Journal of Applied Mechanics 18(3), pp. 293-297.

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pub fn hazard_rate(&self, t: f64) -> f64

Failure rate (hazard function) at time t.

lambda(t) = (beta/eta) * (t/eta)^(beta-1)

beta < 1: Decreasing failure rate (infant mortality)
beta = 1: Constant failure rate (random/exponential failures)
beta > 1: Increasing failure rate (wear-out)

For t <= 0, returns 0.0.

§Examples

use u_analytics::weibull::ReliabilityAnalysis;
let ra = ReliabilityAnalysis::new(2.0, 100.0).unwrap();

// Hazard rate is positive for t > 0
assert!(ra.hazard_rate(50.0) > 0.0);

// With beta > 1, hazard rate increases over time (wear-out)
assert!(ra.hazard_rate(50.0) < ra.hazard_rate(80.0));

§Reference

Meeker & Escobar (1998), Statistical Methods for Reliability Data, Ch. 4.

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pub fn mtbf(&self) -> f64

Mean Time Between Failures (MTBF).

MTBF = eta * Gamma(1 + 1/beta)

§Examples

use u_analytics::weibull::ReliabilityAnalysis;
let ra = ReliabilityAnalysis::new(2.0, 100.0).unwrap();
let mtbf = ra.mtbf();
assert!(mtbf > 0.0);

// For beta=1 (exponential), MTBF = eta
let exp_ra = ReliabilityAnalysis::new(1.0, 50.0).unwrap();
assert!((exp_ra.mtbf() - 50.0).abs() < 1e-8);

§Reference

Johnson, Kotz & Balakrishnan (1994), Continuous Univariate Distributions, Vol. 1, Chapter 21.

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pub fn time_to_reliability(&self, p: f64) -> Option<f64>

Time at which reliability drops to a given level.

Solves R(t) = p for t:

t = eta * (-ln(p))^(1/beta)

§Arguments

p - Desired reliability level, must be in (0, 1).

§Returns

None if p is outside (0, 1).

§Reference

Abernethy (2006), The New Weibull Handbook, 5th ed.

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pub fn b_life(&self, fraction_failed: f64) -> Option<f64>

B-life: time at which a given fraction of the population has failed.

B10 life (10% failed) = b_life(0.10), which is equivalent to time_to_reliability(0.90).

§Arguments

fraction_failed - Fraction of population that has failed, must be in (0, 1).

§Returns

None if fraction_failed is outside (0, 1).

§Examples

use u_analytics::weibull::ReliabilityAnalysis;
let ra = ReliabilityAnalysis::new(2.0, 100.0).unwrap();

// B10: time when 10% of the population has failed
let b10 = ra.b_life(0.10).unwrap();
assert!(b10 > 0.0 && b10 < 100.0);

// B-lives increase with failure fraction: B5 < B10 < B50
let b5 = ra.b_life(0.05).unwrap();
let b50 = ra.b_life(0.50).unwrap();
assert!(b5 < b10 && b10 < b50);

§Reference

Abernethy (2006), The New Weibull Handbook, 5th ed., Chapter 2.

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pub fn b_life_interval( &self, fraction_failed: f64, confidence: f64, ) -> Option<(f64, WeibullInterval)>

The B-life at fraction_failed with its two-sided interval at confidence, by the delta method on ln t_p = ln η + ln(−ln(1 − p)) / β (normal on the log scale, so the limits stay positive).

Returns (estimate, interval); None without a covariance (Self::with_covariance, Self::from_mle) or when either argument is outside (0, 1).

§Examples

use u_analytics::weibull::ReliabilityAnalysis;
let ra = ReliabilityAnalysis::new(2.0, 100.0)
    .unwrap()
    .with_covariance([[0.04, 0.1], [0.1, 25.0]])
    .unwrap();
let (b10, ci) = ra.b_life_interval(0.10, 0.90).unwrap();
assert!(ci.lower < b10 && b10 < ci.upper);

§Reference

Meeker & Escobar (1998), Statistical Methods for Reliability Data, §8.4.

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pub fn reliability_interval( &self, t: f64, confidence: f64, ) -> Option<(f64, WeibullInterval)>

The reliability R(t) with its two-sided interval at confidence, by the delta method on u = ln(−ln R(t)) = β (ln t − ln η), which is unbounded – so the limits exp(−exp(u ± z·se)) stay inside (0, 1). At t ≤ 0 the model has no failures and the interval is [1, 1].

Returns (estimate, interval); None without a covariance or when confidence is outside (0, 1) or t is not finite.

§Examples

use u_analytics::weibull::ReliabilityAnalysis;
let ra = ReliabilityAnalysis::new(2.0, 100.0)
    .unwrap()
    .with_covariance([[0.04, 0.1], [0.1, 25.0]])
    .unwrap();
let (r, ci) = ra.reliability_interval(50.0, 0.95).unwrap();
assert!(0.0 < ci.lower && ci.lower < r && r < ci.upper && ci.upper < 1.0);