torch.distributions.Weibull

class Weibull extends Distribution

new Weibull(scale: number | Tensor, concentration: number | Tensor, options?: DistributionOptions)

readonlyscale(Tensor): – Scale parameter (lambda).
readonlyconcentration(Tensor): – Concentration/shape parameter (k).
readonlyarg_constraints(unknown)
readonlysupport(unknown)
readonlyhas_rsample(unknown)
readonlymean(Tensor)
readonlymode(Tensor)
readonlyvariance(Tensor)

Weibull distribution: continuous distribution for modeling failure rates and lifetimes.

Flexible distribution with adjustable shape (concentration) that can model increasing, constant, or decreasing failure rates. Essential for:

Reliability engineering (time-to-failure, product lifetime)
Survival analysis and time-to-event modeling
Extreme value statistics and risk analysis
Wind speed and weather modeling (Rayleigh as special case)
Material strength analysis (Weibull modulus)
Manufacturing quality control and failure prediction
Generalization of exponential distribution (k=1) and Rayleigh (k=2)

Parameterized by scale λ (characteristic life) and concentration k (shape). The concentration parameter controls tail behavior: k < 1 → decreasing hazard rate, k = 1 → constant (exponential), k > 1 → increasing hazard rate.

The probability density function: f(x) = (k/λ) * (x/λ)^(k-1) * exp(-(x/λ)^k) where x ≥ 0, λ > 0 (scale), k > 0 (concentration/shape)

Key properties:

Support: [0, ∞) - non-negative values only
Mean: λ * Γ(1 + 1/k) where Γ is gamma function
Median: λ * (ln(2))^(1/k)
Mode: λ * ((k-1)/k)^(1/k) for k > 1, else 0
Variance: λ² * [Γ(1 + 2/k) - Γ²(1 + 1/k)]
Hazard rate (failure rate): h(x) = (k/λ) * (x/λ)^(k-1)
- k < 1: decreasing (infant mortality)
- k = 1: constant (no memory, like exponential)
- k > 1: increasing (wear-out failures)
Skewness decreases as k increases (approaches normal)
k = 1 → exponential distribution (memoryless)
k = 2 → Rayleigh distribution (wind speed)
k → ∞ → approaches delta function (deterministic)

\begin{aligned} \text{PDF: } f(x) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1} \exp\left(-\left(\frac{x}{\lambda}\right)^k\right) \quad x \geq 0 \\ \text{CDF: } F(x) = 1 - \exp\left(-\left(\frac{x}{\lambda}\right)^k\right) \\ \text{Mean: } \mathbb{E}[X] = \lambda\Gamma\left(1 + \frac{1}{k}\right) \\ \text{Median: } m = \lambda(\ln 2)^{1/k} \\ \text{Mode: } \begin{cases} 0 & k \leq 1 \\ \lambda\left(\frac{k-1}{k}\right)^{1/k} & k > 1 \end{cases} \\ \text{Variance: } \text{Var}(X) = \lambda^2\left[\Gamma\left(1 + \frac{2}{k}\right) - \Gamma^2\left(1 + \frac{1}{k}\right)\right] \end{aligned}

Hazard rate control: Shape k directly determines failure rate behavior
Characteristic life: Scale λ is where 63.2% of population has failed
Extreme value: Weibull models extremes (min of exponentials under scaling)
Exponential limit: k → 1 recovers exponential distribution (no memory)
Rayleigh connection: k = 2 gives Rayleigh, important for wind/distance
Scale invariance: Distribution scales with λ (no shape change)

Positive support: k and λ must be strictly positive; domain is [0, ∞)
Hazard rate: Increasing hazard (k 1) means older items fail faster
Numerical issues: Very small λ or extreme k values can cause instability
Mean sensitivity: Mean grows rapidly with shape k due to Gamma function

Examples

// Standard Weibull: scale=1, concentration=2 (Rayleigh distribution)
const weibull = new torch.distributions.Weibull(1, 2);
weibull.sample();  // Similar to Rayleigh(scale=1)

// Reliability modeling: product lifetime with scale=1000 hours
const lambda = 1000;  // Characteristic lifetime (63.2% failure by this time)
const k = 1.5;  // Increasing failure rate (wear-out behavior)
const reliability = new torch.distributions.Weibull(lambda, k);
const ttf = reliability.sample([1000]);  // 1000 time-to-failure samples

// Exponential as special case: k=1 means constant failure rate
const constant_rate = new torch.distributions.Weibull(2, 1);
// Same as Exponential(rate=1/2), memoryless property

// Decreasing failure rate: k < 1 (infant mortality)
const infant_mortality = new torch.distributions.Weibull(5, 0.5);
// High early failure, improves over time

// Wind speed modeling: Rayleigh when k=2
const wind_scale = 10;  // Mean wind speed parameter
const wind_dist = new torch.distributions.Weibull(wind_scale, 2);
const wind_speeds = wind_dist.sample([100]);  // 100 wind speed samples

// Material strength analysis: Weibull modulus controls variability
const strength_scale = 1000;  // MPa
const weibull_modulus = 5;  // Higher = more consistent strength
const material = new torch.distributions.Weibull(strength_scale, weibull_modulus);
const strengths = material.sample([10000]);  // Distribution of material strengths

// Batched distributions: different failure rates for different components
const scales = torch.tensor([100, 500, 1000]);
const shapes = torch.tensor([0.8, 1.5, 2.0]);
const dists = new torch.distributions.Weibull(scales, shapes);
const samples = dists.sample();  // One sample from each component's lifetime

torch.distributions.Weibull

Examples

See Also

torch.distributions.Weibull

Examples

See Also