RenewalRewardProcess

class relife.stochastic_processes.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 lifetime_model 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 lifetime_model 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

params

params_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_number_of_events	The expected number of observed events.
expected_number_of_preventive_renewals	The expected number of preventive renewals.
expected_total_reward	The expected total reward.
generate_failure_data	Generate lifetime data
get_params	Get the parameters of this model.
get_params_names	Parameters names.
renewal_density	The renewal density.
renewal_function	The renewal function.
sample	Renewal data sampling.
set_params	Set the parameters of this model.

asymptotic_expected_equivalent_annual_worth(a0=None, ar=None)

Asymptotic expected equivalent annual worth.

Parameters:

a0float or np.ndarray, optional

Initial ages of the assets.

arfloat or np.ndarray, optional

Preventive ages of replacements.

Returns:

ndarray

The assymptotic expected equivalent annual worth.

asymptotic_expected_total_reward(a0=None, ar=None)

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.

Parameters:

a0float or np.ndarray, optional

Initial ages of the assets.

arfloat or np.ndarray, optional

Preventive ages of replacements.

Returns:

ndarray

The assymptotic expected total reward of the process.

property discounting_rate

The discounting rate value

expected_equivalent_annual_worth(tf, nb_steps, a0=None, ar=None)

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.

a0float or np.ndarray, optional

Initial ages of the assets.

arfloat or np.ndarray, optional

Preventive ages of replacements.

Returns:

tuple of two ndarrays

A tuple containing the timeline and the computed values.

expected_number_of_events(tf, nb_steps, a0=None, ar=None)

The expected number of observed events.

Here, events are assets failures, i.e. only the assets failures are counted (not the preventive replacements at ar).

The function is noted \(m_e\) and computed by solving :

\[m_e(t) = F(\text{min}(t,~a_r)) + \int_0^{t}m_e(t-x)dF_{a_r}(x)\]

where:

• \(F\) is the cumulative distribution function of the time to failure \(X\).

• \(F_{a_r}\) is the cumulative distribution of \(T = \text{min}(X,~a_r)\).

If a0 or first_lifetime_model is given, instead, we compute \(m_e^{\text{delayed}}\) by solving:

\[m_e^{\text{delayed}}(t) = F_1(\text{min}(t,~a_r)) + \int_0^{t}m_e(t-x)dF_{1_{a_r}}(x)\]

where:

• \(F_1\) is the cumulative distribution function of the first time to failure \(X_1\).

• \(F_{1_{a_r}}\) is the cumulative distribution of \(T_1 = \text{min}(X_1,~a_r)\).

Note

If ar is None, \(a_r = \infty\).

This function is complementary to expected_number_of_preventive_renewals() i.e. \(m(t) = m_e(t) + m_p(t)\).

See also renewal_function().

Parameters:

tffloat

The final time.

nb_stepsint

The number of steps used to discretized the time.

a0float or np.ndarray, optional

Initial ages of the assets.

arfloat or np.ndarray, optional

Preventive ages of replacements.

Returns:

outtuple of two ndarrays

A timeline and the corresponding values.

Notes

Preventive replacements are not considered as events. Only renewals are. Thus, there are not counted.

expected_number_of_preventive_renewals(tf, nb_steps, ar, a0=None)

The expected number of preventive renewals.

The function is noted \(m_p\) and computed by solving :

\[m_p(t) = \mathbb{1}_{t > a_r} \cdot (1 - F(a_r)) + \int_0^{t}m_p(t-x)dF_{a_r}(x)\]

where:

• \(F\) is the cumulative distribution function of the time to failure \(X\).

• \(F_{a_r}\) is the cumulative distribution of \(T = \text{min}(X,~a_r)\).

If a0 or first_lifetime_model is given, instead, we compute \(m_p^{\text{delayed}}\) by solving:

\[m_p^{\text{delayed}}(t) = \mathbb{1}_{t > a_r} \cdot (1 - F_1(a_r)) + \int_0^{t}m_p(t-x)dF_{1_{a_r}}(x)\]

where:

• \(F_1\) is the cumulative distribution function of the first time to failure \(X_1\).

• \(F_{1_{a_r}}\) is the cumulative distribution of \(T_1 = \text{min}(X_1,~a_r)\).

Note

If ar is None, \(a_r = \infty\).

This function is complementary to expected_number_of_events() i.e. \(m(t) = m_e(t) + m_p(t)\).

See also renewal_function().

Parameters:

tffloat

The final time.

nb_stepsint

The number of steps used to discretized the time.

arfloat or np.ndarray

Preventive ages of replacements.

a0float or np.ndarray, optional

Initial ages of the assets.

Returns:

outtuple of two ndarrays

A timeline and the corresponding values.

expected_total_reward(tf, nb_steps, a0=None, ar=None)

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.

a0float or np.ndarray, optional

Initial ages of the assets.

arfloat or np.ndarray, optional

Preventive ages of replacements.

Returns:

tuple of two ndarrays

A tuple containing the timeline and the computed values.

generate_failure_data(nb_samples, time_window, a0=None, ar=None, 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 data are sampled

a0float or np.ndarray, optional

Initial ages of the assets.

arfloat or np.ndarray, optional

Preventive ages of replacements.

seedint, optional

Random seed, by default None.

Returns:

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

get_params()

Get the parameters of this model.

Returns:

out1darray of number

Model parameters.

Notes

If parameter values are not set, they default to np.nan values.

get_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, a0=None, ar=None)

The renewal density.

It is the derivative \(\omega\) of the renewal function \(m\). See the renewal_function().

\[\omega(t) = m'(t) = f_1(t) + \int_0^t \omega(t-x) \mathrm{d}F(x)\]

where:

• \(F\) is the cumulative distribution function of the time to failure \(X\).

• \(f_1\) is the probability density function of the first time to failure \(X_1\).

If ar is given, \(F\) becomes \(F_{a_r}\) defined by \(T = \text{min}(X,~a_r) \sim F_{a_r}\). The same applies for \(X_1\). \(F_1\) becomes \(F_{1_{a_r}}\) defined by \(T_1 = \text{min}(X_1,~a_r) \sim F_{1_{a_r}}\).

If a0 is given, \(F_1\) becomes \(\mathbb{P}(X \leq t |~ X > a_0)\).

Parameters:

tffloat

The final time.

nb_stepsint

The number of steps used to discretized the time.

a0float or np.ndarray, optional

Initial ages of the assets.

arfloat or np.ndarray, optional

Preventive ages of replacements.

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, a0=None, ar=None)

The renewal function.

It gives the expected total number of renewals \(m\). It is computed by solving the renewal equation:

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

where:

• \(F\) is the cumulative distribution function of the time to failure \(X\).

• \(F_1\) is the cumulative distribution function of the first time to failure \(X_1\).

If ar is given, \(F\) becomes \(F_{a_r}\) defined by \(T = \text{min}(X,~a_r) \sim F_{a_r}\). The same applies for \(X_1\). \(F_1\) becomes \(F_{1_{a_r}}\) defined by \(T_1 = \text{min}(X_1,~a_r) \sim F_{a_r}\).

If a0 is given, \(F_1\) becomes \(\mathbb{P}(X \leq t |~ X > a_0)\).

Parameters:

tffloat

The final time.

nb_stepsint

The number of steps used to discretized the time.

a0float or np.ndarray, optional

Initial ages of the assets.

arfloat or np.ndarray, optional

Preventive ages of replacements.

Returns:

outtuple of two ndarrays

A timeline and the corresponding 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, a0=None, ar=None, 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

a0float or np.ndarray, optional

Initial ages of the assets.

arfloat or np.ndarray, optional

Preventive ages of replacements.

seedint, optional

Random seed, by default None.

set_params(new_params)

Set the parameters of this model.

Parameters:

new_paramsarray-like of floats

Model parameters.

Notes

set_params definition expects an array-like of floats. At runtime, complex parameters might be setted temporarily to approximate fitted parameters covariance. This is contradictory to the given typing. At the moment, we don’t see a better solution and we believe that this is actually a limitation of what be expressed in the static typesystem.