RenewalRewardProcess

class relife.stochastic_process.RenewalRewardProcess(lifetime_model, reward, discounting_rate=0.0, first_lifetime_model=None, first_reward=None)

Renewal reward process.

Parameters:

lifetime_modelany lifetime distribution or frozen lifetime model

A lifetime model representing the durations between events.

rewardReward

A reward object that answers costs or conditional costs given lifetime values

discounting_ratefloat

The discounting rate value used in the exponential discounting function

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

rewardReward

A reward object for the first renewal

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

rewardReward

A reward object that answers costs or conditional costs given lifetime values

first_rewardReward

A reward object for the first renewal. If it is not given at the initialization, it is a copy of reward.

discounting_rate

The discounting rate value

nb_params

Number of parameters.

params

Parameters values.

params_names

Parameters names.

Methods

asymptotic_expected_equivalent_annual_worth	Asymptotic expected equivalent annual worth.
asymptotic_expected_total_reward	Asymptotic expected total reward.
expected_equivalent_annual_worth	Expected equivalent annual worth.
expected_total_reward	The expected total reward.
generate_failure_data	Generate lifetime data
renewal_density	The renewal density.
renewal_function	The renewal function.
sample	Renewal data sampling.

asymptotic_expected_equivalent_annual_worth()

Asymptotic expected equivalent annual worth.

Returns:

ndarray

The assymptotic expected equivalent annual worth.

asymptotic_expected_total_reward()

Asymptotic expected total reward.

The asymptotic expected total reward is:

\[z^\infty = \lim_{t\to \infty} z(t) = \dfrac{E\left[Y e^{-\delta X}\right]}{1-E\left[e^{-\delta X}\right]}\]

where:

• \(X\) the interarrival random variable.

• \(Y\) the associated reward.

• \(D\) the exponential discount factor.

If the renewal reward process is delayed, the asymptotic expected total reward is modified as:

\[z_1^\infty = E\left[Y_1 e^{-\delta X_1}\right] + z^\infty E\left[e^{-\delta X_1}\right]\]

where:

• \(X_1\) the interarrival random variable of the first renewal.

• \(Y_1\) the associated reward of the first renewal.

Returns:

ndarray

The assymptotic expected total reward of the process.

property discounting_rate

The discounting rate value

expected_equivalent_annual_worth(tf, nb_steps)

Expected equivalent annual worth.

Gives the equivalent annual worth of the expected total reward of the process at each point of the timeline.

The equivalent annual worth at time \(t\) is equal to the expected total reward \(z\) divided by the annuity factor \(AF(t)\).

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.

expected_total_reward(tf, nb_steps)

The expected total reward.

The renewal equation solved to compute the expected reward is:

\[z(t) = \int_0^t E[Y | X = x] e^{-\delta x} \mathrm{d}F(x) + \int_0^t z(t-x) e^{-\delta x}\mathrm{d}F(x)\]

where:

• \(z\) is the expected total reward.

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

• \(X\) the interarrival random variable.

• \(Y\) the associated reward.

• \(D\) the exponential discount factor.

If the renewal reward process is delayed, the expected total reward is modified as:

\[z_1(t) = \int_0^t E[Y_1 | X_1 = x] e^{-\delta x} \mathrm{d}F_1(x) + \int_0^t z(t-x) e^{-\delta x} \mathrm{d}F_1(x)\]

where:

• \(z_1\) is the expected total reward with delay.

• \(F_1\) is the cumulative distribution function of the lifetime model for the first renewal.

• \(X_1\) the interarrival random variable of the first renewal.

• \(Y_1\) the associated reward of the first renewal.

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.

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.