relife.renewal_process.RenewalProcess

relife.renewal_process.RenewalProcess

class relife.renewal_process.RenewalProcess(model: relife.model.LifetimeModel, model1: Optional[relife.model.LifetimeModel] = None)[source]

Bases: object

Renewal process.

Creates a renewal process.

Parameters

  • model (LifetimeModel) – A lifetime model representing the durations between events.

  • model1 (LifetimeModel, optional) – A lifetime model for the first renewal (delayed renewal process), by default None.

Methods

renewal_density

The renewal density.

renewal_function

The renewal function.

sample

Renewal data sampling.
renewal_function(t: numpy.ndarray, model_args: Tuple[numpy.ndarray, ...] = (), model1_args: Tuple[numpy.ndarray, ...] = ()) → numpy.ndarray[source]

The renewal function.

Parameters

  • t (1D array) – Timeline.

  • model_args (Tuple[ndarray,...], optional) – Extra arguments required by the underlying lifetime model, by default ().

  • model1_args (Tuple[ndarray,...], optional) – Extra arguments required by the lifetime model of the first renewal, by default ().

Returns

The renewal function evaluated at each point of the timeline.

Return type

ndarray

Notes

The expected total number of renewals is computed by solving the renewal equation:

\[m(t) = F_1(t) + \int_0^t m(t-x) \mathrm{d}F(x)\]

where:

  • \(m\) is the renewal function,

  • \(F\) is the cumulative distribution function of the underlying lifetime model,

  • \(F_1\) is the cumulative distribution function of the underlying lifetime model for the fist renewal in the case of a delayed renewal process.

References

1

Rausand, M., Barros, A., & Hoyland, A. (2020). System Reliability Theory: Models, Statistical Methods, and Applications. John Wiley & Sons.

renewal_density(t: numpy.ndarray, model_args: Tuple[numpy.ndarray, ...] = (), model1_args: Tuple[numpy.ndarray, ...] = ()) → numpy.ndarray[source]

The renewal density.

Parameters

  • t (1D array) – Timeline.

  • model_args (Tuple[ndarray,...], optional) – Extra arguments required by the underlying lifetime model, by default ().

  • model1_args (Tuple[ndarray,...], optional) – Extra arguments required by the lifetime model of the first renewal, by default ().

Returns

Renewal density evaluated at each point of the timeline.

Return type

ndarray

Raises

NotImplementedError – If the lifetime model is not absolutely continuous.

Notes

The renewal density is the derivative of the renewal function with respect to time. It is computed by solving the renewal equation:

\[\mu(t) = f_1(t) + \int_0^t \mu(t-x) \mathrm{d}F(x)\]

where:

  • \(\mu\) is the renewal function,

  • \(F\) is the cumulative distribution function of the underlying lifetime model,

  • \(f_1\) is the probability density function of the underlying lifetime model for the fist renewal in the case of a delayed renewal process.

References

1

Rausand, M., Barros, A., & Hoyland, A. (2020). System Reliability Theory: Models, Statistical Methods, and Applications. John Wiley & Sons.

sample(T: float, model_args: Tuple[numpy.ndarray, ...] = (), model1_args: Tuple[numpy.ndarray, ...] = (), n_samples: int = 1, random_state: Optional[int] = None) → relife.data.RenewalData[source]

Renewal data sampling.

Parameters

  • T (float) – Time at the end of the observation.

  • model_args (Tuple[ndarray,...], optional) – Extra arguments required by the underlying lifetime model, by default ().

  • model1_args (Tuple[ndarray,...], optional) – Extra arguments required by the lifetime model of the first renewal, by default ().

  • n_samples (int, optional) – Number of samples, by default 1.

  • random_state (int, optional) – Random seed, by default None.

Returns

Samples of replacement times and durations.

Return type

RenewalData

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