RenewalProcess

class relife.stochastic_process.RenewalProcess(lifetime_model, first_lifetime_model=None)

Renewal process.

Parameters:

lifetime_modelany lifetime distribution or frozen lifetime model

A lifetime model representing the durations between events.

first_lifetime_modelany lifetime distribution or frozen lifetime model, optional

A lifetime model for the first renewal (delayed renewal process). It is None by default

Attributes:

lifetime_modelany lifetime distribution or frozen lifetime model

A lifetime model representing the durations between events.

first_lifetime_modelany lifetime distribution or frozen lifetime model, optional

A lifetime model for the first renewal (delayed renewal process). It is None by default

nb_params

Number of parameters.

params

Parameters values.

params_names

Parameters names.

Methods

generate_failure_data	Generate lifetime data
renewal_density	The renewal density.
renewal_function	The renewal function.
sample	Renewal data sampling.

generate_failure_data(nb_samples, time_window, seed=None)

Generate lifetime data

This function will generate lifetime data that can be used to fit a lifetime model.

Parameters:

nb_samplesint

The size of the desired sample

time_windowtuple of two floats

Time window in which failure data are sampled

seedint, optional

Random seed, by default None.

Returns:

A dict of time, event, entry and args (covariates)

property nb_params

Number of parameters.

Returns:

int

Number of parameters.

property params

Parameters values.

Returns:

ndarray

Parameters values

Notes

If parameter values are not set, they are encoded as np.nan value.

property params_names

Parameters names.

Returns:

list of str

Parameters names

Notes

Parameters values can be requested (a.k.a. get) by their name at instance level.

renewal_density(tf, nb_steps)

The renewal density.

The renewal density corresponds to 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.

Parameters:

tffloat

The final time.

nb_stepsint

The number of steps used to discretized the time.

Returns:

tuple of two ndarrays

A tuple containing the timeline and the computed values.

References

[1]

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

renewal_function(tf, nb_steps)

The renewal function.

The renewal function gives the expected total number of renewals. It 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.

Parameters:

tffloat

The final time.

nb_stepsint

The number of steps used to discretized the time.

Returns:

tuple of two ndarrays

A tuple containing the timeline and the computed values.

References

[1]

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

sample(nb_samples, time_window, seed=None)

Renewal data sampling.

This function will sample data and encapsulate them in an object.

Parameters:

nb_samplesint

The size of the desired sample

time_windowtuple of two floats

Time window in which data are sampled

seedint, optional

Random seed, by default None.