RenewalProcess#

class relife.stochastic_processes.RenewalProcess(lifetime_model, first_lifetime_model=None)[source]#

Renewal process.

Parameters:

lifetime_modelParametricLifetimeModel: Lifetime model representing durations between events.
first_lifetime_modelParametricLifetimeModel, optional: Lifetime model for the first renewal in a delayed renewal process. Defaults to lifetime_model.

Methods

`expected_number_of_events`	The expected number of observed events.
`expected_number_of_preventive_renewals`	The expected number of preventive renewals.
`get_params`	Get the parameters of this model.
`is_fitted`	Whether fitting results are set.
`is_parametrized`	Whether at least one parameter value is set.
`renewal_density`	The renewal density.
`renewal_function`	The renewal function.
`set_params`	Set the parameters of this model.

expected_number_of_events(tf, nb_steps, *, a0=None, ar=None)[source]#

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 discretize the time.
arfloat or 1d array: Preventive ages of replacements.
a0float or 1d array, optional: Initial ages of the assets.

Returns:

outtuple of np.ndarray: Timeline and corresponding values.

get_params()#

Get the parameters of this model.

Returns:

out1darray of floats: Model parameters.

Notes

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

is_fitted()#: Whether fitting results are set.

is_parametrized()#: Whether at least one parameter value is set.

renewal_density(tf, nb_steps, *, a0=None, ar=None)[source]#

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 discretize the time.
a0float or 1d array, optional: Initial ages of the assets.
arfloat or 1d array, optional: Preventive ages of replacements.

Returns:

tuple of np.ndarray: Timeline and corresponding 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)[source]#

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 discretize the time.
a0float or 1d array, optional: Initial ages of the assets.
arfloat or 1d array, optional: Preventive ages of replacements.

Returns:

outtuple of np.ndarray: Timeline and corresponding values.

References

[1]

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

set_params(new_params)#

Set the parameters of this model.

Parameters:

new_params1d array-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 can be expressed in the static typesystem.