RenewalRewardProcess#

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

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()[source]#

Asymptotic expected equivalent annual worth.

Returns:

ndarray: The assymptotic expected equivalent annual worth

asymptotic_expected_total_reward()[source]#

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)[source]#

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: Time horizon. The expected equivalent annual worth will be computed up until this calendar time.
nb_stepsint: The number of steps used to compute the expected equivalent annual worth.

Returns:

tuple of two ndarrays: A tuple containing the timeline used to compute the expected equivalent annual worth and its corresponding values at each step of the timeline.

expected_total_reward(tf, nb_steps)[source]#

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: Time horizon. The expected total reward will be computed up until this calendar time.
nb_stepsint: The number of steps used to compute the expected total reward.

Returns:

tuple of two ndarrays: A tuple containing the timeline used to compute the expected total reward and its corresponding values at each step of the timeline.

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)[source]#

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.

RenewalRewardProcess#

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