pyuncertainnumber.propagation.mixed_up¶

Functions¶

`interval_monte_carlo`(→ pyuncertainnumber.pba.pbox_abc.Pbox)	Interval Monte Carlo for propagation of pbox
`slicing`(→ pyuncertainnumber.pba.pbox_abc.Pbox)	classic slicing algoritm for rigorous propagation of pbox
`double_monte_carlo`(...)	Double-loop Monte Carlo or nested Monte Carlo for mixed uncertainty propagation
`bi_imc`(x, y, func[, dependency, n_sam])	Bivariate interval monte carlo for convenience

Module Contents¶

pyuncertainnumber.propagation.mixed_up.interval_monte_carlo(vars: list[pyuncertainnumber.pba.intervals.Interval | pyuncertainnumber.pba.distributions.Distribution | pyuncertainnumber.pba.pbox_abc.Pbox], func: callable, interval_strategy, n_sam: int, dependency: pyuncertainnumber.pba.dependency.Dependency = None, random_state=None, side_effects=False, **kwargs) → pyuncertainnumber.pba.pbox_abc.Pbox¶

Interval Monte Carlo for propagation of pbox

Parameters:

vars (list) – a list of constructs
func (callable) – response function. By default, iterable signature is expected.
interval_strategy (str) – strategy for interval discretisation, options include {‘direct’, ‘endpoints’, ‘subinterval’}
n_sam (int) – number of samples for each input
dependency – dependency structure (e.g. vine copula or archimedean copula
random_state – random seed for reproducibility
side_effects (bool) – whether return auxiliary outputs (side effects) during propagation If true, the alpha-cut samples in the uniform space will be returned as well. otherwise, the default is False and only the p-box is returned.

Note

When choosing interval_strategy, “direct” requires function signature to take a list of inputs, whereas “subinterval” and “endpoints” require the function to have a vectorised signature.

Returns:: Pbox

Example

>>> from pyuncertainnumber import pba
>>> def foo(x): return x[0] ** 3 + x[1] + x[2]
>>> a = pba.normal([2, 3], [1])
>>> b = pba.normal([10, 14], [1])
>>> c = pba.normal([4, 5], [1])
>>> corre_matrix = np.array([[1, 0.5, 0.3], [0.5, 1, 0.4], [0.3, 0.4, 1]])
>>> de = pba.Dependency(family='gaussian', corr=corre_matrix)
>>> mix = interval_monte_carlo(vars=[a,b,c],
>>> ...       func=foo,
>>> ...       n_sam=20,
>>> ...       dependency=de,
>>> ...       interval_strategy='direct')

pyuncertainnumber.propagation.mixed_up.slicing(vars: list[pyuncertainnumber.pba.distributions.Distribution | pyuncertainnumber.pba.intervals.Interval | pyuncertainnumber.pba.pbox_abc.Pbox], func, interval_strategy, n_slices, outer_discretisation=True, dependency=None, **kwargs) → pyuncertainnumber.pba.pbox_abc.Pbox¶

classic slicing algoritm for rigorous propagation of pbox

Parameters:

vars (list) – list of constructs
func (callable) – response function
interval_strategy (str) – strategy for interval discretisation, options include {‘direct’, ‘endpoints’, ‘subinterval’}
n_slices – number of slices for each input
outer_discretisation (bool) – whether to use outer discretisation for pbox. By default is True for rigorous propagation; however, alpha-cut style interval are also supported.
dependency – dependency structure (e.g. vine copula or archimedean copula).

Tip

Merely independence assumption is supported by now. Other dependency structures are at beta developement now.

Note

Returns:: Pbox

Example

>>> from pyuncertainnumber import pba
>>> def foo(x): return x[0] ** 3 + x[1] + x[2]
>>> a = pba.normal([2, 3], [1])
>>> b = pba.normal([10, 14], [1])
>>> c = pba.normal([4, 5], [1])
>>> mix = slicing(vars=[a,b,c],
>>> ...       func=foo,
>>> ...       n_slices=20,
>>> ...       interval_strategy='direct')

pyuncertainnumber.propagation.mixed_up.double_monte_carlo(joint_distribution: pyuncertainnumber.pba.distributions.Distribution | pyuncertainnumber.pba.distributions.JointDistribution, epistemic_vars: list[pyuncertainnumber.pba.intervals.Interval], n_a: int, n_e: int, func: callable, side_effects=False, parallel=False) → tuple[pyuncertainnumber.pba.pbox_abc.Pbox, list, numpy.ndarray]¶

Double-loop Monte Carlo or nested Monte Carlo for mixed uncertainty propagation

Parameters:

joint_distribution (Distribution or JointDistribution) – an aleatoric sampler based on joint distribution of aleatory variables (or marginal one in 1d case). A sampler is basically anything (univariate or multivariate) that has the sample interface whereby it can sample a given number of samples.
epistemic_vars (list) – a list epistemic variables in the form of Interval
n_a (int) – number of aleatory samples
n_e (int) – number of epistemic samples
parallel (Boolean) – parallel processing. Only use it for heavy computation (black-box) due to overhead

Hint

consider a function mapping f(X) -> y

\(X\) in \(R^5\) with `n_a=1000`will suggest f(1000, 5)
resulting sample array: with n_e=2, the response \(y\) : (n_ep+2, n_a) e.g. (4, 1000)

side_effects (bool): whether return auxiliary outputs (side effects) during propagation: If true, the alpha-cut samples in the uniform space will be returned as well. otherwise, the default is False and only the p-box is returned.

Returns:

a p-box enveloping all the CDFs from the epistemic samples
- a list of ECDFs for each epistemic sample
- numpy array of shape (n_e+2, n_a) as a collection of CDFs for the response
- the epistemic samples used

Otherwise, just the p-box.

Return type:

If side_effects is True, a tuple containing the following items

Note

The result array can be interpreted as a collection of CDFs for the response function evaluated at the aleatory samples for each epistemic sample. One can further envelope these CDFs into a Pbox or UncertainNumber object.

Example

>>> from pyuncertainnumber import pba
>>> # vectorised function signature with matrix input (2D np.ndarray)
>>> def foo_vec(x):
...     return x[:, 0] ** 3 + x[:, 1] + x[:, 2] + x[:, 3]

>>> dist_a = pba.Distribution('gaussian', (5, 1))
>>> dist_b = pba.Distribution('uniform', (2, 3))
>>> c = pba.Dependency('gaussian', params=0.8)
>>> joint_dist = pba.JointDistribution(copula=c, marginals=[dist_a, dist_b])

>>> xe1 = pba.I(1, 2)
>>> xe2 = pba.I(3, 4)

>>> t = double_monte_carlo(
...     joint_distribution=joint_dist,
...     epistemic_vars=[xe1, xe2],
...     n_a=20,
...     n_e=3,
...     func=foo_vec
... )

pyuncertainnumber.propagation.mixed_up.bi_imc(x, y, func, dependency=None, n_sam=100)¶

Bivariate interval monte carlo for convenience

Parameters:

x (Pbox) – Pbox
y (Pbox) – Pbox
func – callable which takes vector-type of inputs
dependency – dependency structure (regular copula)