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(size, tf, t0=0.0, seed=None)

Generate lifetime data

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

Parameters:

sizeint

The size of the desired sample

tffloat

Time at the end of the observation.

t0float, default 0

Time at the beginning of the observation.

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 of the core

Notes

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

Parameters can be by manually setting`params` through its setter, fitting the core if fit exists or by specifying all parameters values when the core object is initialized.

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.

Parameters:

tffloat

Time horizon. The renewal density will be computed up until this calendar time.

nb_stepsint

The number of steps used to compute the renewal density.

Returns:

tuple of two ndarrays

A tuple containing the timeline used to compute the renewal density and its corresponding values at each step of the timeline.

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.

renewal_function(tf, nb_steps)

The renewal function.

Parameters:

tffloat

Time horizon. The renewal function will be computed up until this calendar time.

nb_stepsint

The number of steps used to compute the renewal function.

Returns:

tuple of two ndarrays

A tuple containing the timeline used to compute the renewal function and its corresponding values at each step of the timeline.

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.

sample(size, tf, t0=0.0, seed=None)

Renewal data sampling.

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

Parameters:

sizeint

The size of the desired sample

tffloat

Time at the end of the observation.

t0float, default 0

Time at the beginning of the observation.

sizeint or tuple of 2 int

Size of the sample

seedint, optional

Random seed, by default None.