class

Weibull

extendsDistribution

Weibull(scale: Tensor | float, concentration: Tensor | float, validate_args: bool | None = None)

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Weibull distribution — a flexible family for lifetime and survival analysis.

Weibull(scale=λ, concentration=k) generalises both the Exponential ( $k = 1$ ) and Rayleigh ( $k = 2$ ) distributions. It is widely used for modelling time-to-failure data because the hazard rate $h(t) = (k/\lambda)(t/\lambda)^{k-1}$ can increase, be constant, or decrease depending on $k$ .

Parameters

scaleTensor | float

Scale parameter

\lambda > 0

— a characteristic life value at which the CDF is

1 - e^{-1} \approx 63.2\%

concentrationTensor | float

Shape parameter

k > 0

. Values

k < 1

give a decreasing hazard (infant mortality);

k = 1

is constant (memoryless Exponential);

k > 1

gives an increasing hazard (wear-out).

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Attributes

scaleTensor

Scale parameter

\lambda

concentrationTensor

Shape parameter

k

Notes

PDF:

p(x; \lambda, k) = \frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{k-1} \exp\!\left(-\left(\frac{x}{\lambda}\right)^k\right), \quad x > 0

Log-PDF:

\log p(x) = \log k - k\log\lambda + (k-1)\log x - (x/\lambda)^k

Moments:

Mean: $E[X] = \lambda \,\Gamma(1 + 1/k)$
Variance: $\operatorname{Var}[X] = \lambda^2 \left[\Gamma(1+2/k) - \Gamma(1+1/k)^2\right]$

Entropy:

H[X] = \gamma\left(1 - \tfrac{1}{k}\right) + \log(\lambda/k) + 1

where $\gamma \approx 0.5772$ is the Euler–Mascheroni constant.

Reparameterised sampling uses the inverse-CDF: $X = \lambda(-\log(1-U))^{1/k}$ for $U \sim \operatorname{Uniform}(0,1)$ .

Examples

>>> import lucid
>>> from lucid.distributions import Weibull
>>> # Exponential(rate=1) as a special case
>>> dist_exp = Weibull(scale=1.0, concentration=1.0)
>>> dist = Weibull(scale=2.0, concentration=1.5)
>>> samples = dist.rsample((500,))

Used by 2

Constructors

dunder

init

→None

__init__(scale: Tensor | float, concentration: Tensor | float, validate_args: bool | None = None)

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Initialise a Weibull distribution.

Parameters

scaleTensor | float

Scale parameter

\lambda > 0

concentrationTensor | float

Shape parameter

k > 0

. Values

k < 1

give a decreasing hazard;

k = 1

gives the Exponential;

k > 1

gives an increasing hazard.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Properties

prop

mean

→Tensor

mean: Tensor

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Mean of the Weibull distribution: $E[X] = \lambda \Gamma(1 + 1/k)$ .

Returns

Tensor

Mean values of shape batch_shape.

prop

mode

→Tensor

mode: Tensor

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Mode of the Weibull distribution.

\lambda \left(\frac{k - 1}{k}\right)^{1/k} \quad (k > 1), \qquad 0 \quad (k \leq 1)

For $k \leq 1$ the density decreases from the origin, so the mode is 0. The reference framework evaluates the first formula there too and answers nan or a meaningless positive number; this answers 0.

prop

support

→Constraint

support: Constraint

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Support of the Weibull distribution: $(0, \infty)$ .

Returns

Constraint

The positive constraint, as in the reference framework. At 0 the density is infinite for $k < 1$ and 0 for $k > 1$ , and the closed form is not evaluated there.

prop

variance

→Tensor

variance: Tensor

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Variance of the Weibull distribution.

$\operatorname{Var}[X] = \lambda^2 [\Gamma(1+2/k) - \Gamma(1+1/k)^2]$

Returns

Tensor

Variance values of shape batch_shape.

Instance methods

cdf

→Tensor

cdf(value: Tensor)

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Cumulative distribution function of the Weibull distribution.

F(x) = 1 - e^{-(x/\lambda)^k}, \quad x \geq 0

The Exponential(1) CDF at $(x/\lambda)^k$ ; 0 below 0.

Returns probabilities in $[0, 1]$ .

Examples

>>> Weibull(scale=1.0, concentration=1.0).cdf(lucid.tensor(1.0))
tensor(0.6321)

entropy

→Tensor

entropy()

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Entropy of the Weibull distribution.

H[X] = \gamma(1 - 1/k) + \log(\lambda/k) + 1

where $\gamma \approx 0.5772$ is the Euler-Mascheroni constant.

Returns

Tensor

Entropy values of shape batch_shape (nats).

icdf

→Tensor

icdf(value: Tensor)

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Quantile function of the Weibull distribution.

Q(p) = \lambda \left(-\ln(1 - p)\right)^{1/k}

Returns quantiles in $[0, \infty]$ .

Examples

>>> Weibull(scale=2.0, concentration=1.0).icdf(lucid.tensor(0.5))
tensor(1.386)

log_prob

→Tensor

log_prob(value: Tensor)

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Log-probability density of the Weibull distribution.

\log p(x) = \log k - k\log\lambda + (k-1)\log x - (x/\lambda)^k

Parameters

valueTensor

Positive points

x > 0

at which to evaluate.

Returns

Tensor

Log-density values of the same shape as value.

rsample

→Tensor

rsample(sample_shape: tuple[int, ...] = ())

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Draw reparameterised samples via the inverse-CDF.

Uses $X = \lambda (-\log(1-U))^{1/k}$ for $U \sim \operatorname{Uniform}(0, 1)$ . Gradients propagate through both scale and concentration.

Parameters

sample_shapetuple[int, ...]= ()

Leading shape of the output sample batch.

Returns

Tensor

Non-negative samples of shape (*sample_shape, *batch_shape).

>>> import lucid >>> from lucid.distributions import Weibull >>> # Exponential(rate=1) as a special case >>> dist_exp = Weibull(scale=1.0, concentration=1.0) >>> dist = Weibull(scale=2.0, concentration=1.5) >>> samples = dist.rsample((500,))