Probability Boxes#

class pba.pbox.Pbox#

A probability distribution is a mathematical function that gives the probabilities of occurrence for diﬀerent possible values of a variable. Probability boxes (p-boxes) represent interval bounds on probability distributions. The simplest kind of p-box can be expressed mathematically as

\[\mathcal{F}(x) = [\underline{F}(x),\overline{F}(x)], \ \underline{F}(x)\geq \overline{F}(x)\ \forall x \in \mathbb{R}\]

where \(\underline{F}(x)\) is the function that defines the left bound of the p-box and \(\overline{F}(x)\) defines the right bound of the p-box. In PBA the left and right bounds are each stored as a NumPy array containing the percent point function (the inverse of the cumulative distribution function) for steps evenly spaced values between 0 and 1. P-boxes can be defined using all the probability distributions that are available through SciPy’s statistics library,

Naturally, precise probability distributions can be defined in PBA by defining a p-box with precise inputs. This means that within probability bounds analysis probability distributions are considered a special case of a p-box with zero width. Resultantly, all methodology that applies to p-boxes can also be applied to probability distributions.

Distribution-free p-boxes can also be generated when the underlying distribution is unknown but parameters such as the mean, variance or minimum/maximum bounds are known. Such p-boxes make no assumption about the shape of the distribution and instead return bounds expressing all possible distributions that are valid given the known information. Such p-boxes can be constructed making use of Chebyshev, Markov and Cantelli inequalities from probability theory.

Attention

It is usually better to define p-boxes using distributions or non-parametric methods (see ). This constructor is provided for completeness and for the construction of p-boxes with precise inputs.

Parameters:: left – Left bound of the p-box. Can be a list, NumPy array, Interval or numeric type. If left is None, the left bound is set to -inf.

__init__(left=None, right=None, steps=None, shape=None, mean_left=None, mean_right=None, var_left=None, var_right=None, interpolation='linear')#

Attention

It is usually better to define p-boxes using distributions or non-parametric methods (see ). This constructor is provided for completeness and for the construction of p-boxes with precise inputs.

Parameters:: left – Left bound of the p-box. Can be a list, NumPy array, Interval or numeric type. If left is None, the left bound is set to -inf.

add(other: Pbox | Interval | float | int, method='f') → Pbox#: Adds to Pbox to other using the defined dependency method

env(other)#

Computes the envelope of two Pboxes.

Parameters: - other: Pbox or numeric value

The other Pbox or numeric value to compute the envelope with.

Returns: - Pbox

The envelope Pbox.

Raises: - ArithmeticError: If both Pboxes have different number of steps.

get_x()#: returns the x values for plotting

get_y()#: returns the y values for plotting

hi()#: Returns the right-most value in the interval

imp(other)#: Returns the imposition of self with other

lo()#: Returns the left-most value in the interval

mean() → Interval#: Returns the mean of the pbox

median() → Interval#: Returns the median of the distribution

min(other, method='f')#

Returns a new Pbox object that represents the element-wise minimum of two Pboxes.

Parameters:

other: Another Pbox object or a numeric value.
method: Calculation method to determine the minimum. Can be one of ‘f’, ‘p’, ‘o’, ‘i’.

Returns:

Pbox

pow(other: Pbox | Interval | float | int, method='f') → Pbox#

Raises a p-box to the power of other using the defined dependency method

Parameters:

other – Pbox, Interval or numeric type
method –

Returns:

Pbox

Return type:

Pbox

straddles(N, endpoints=True) → bool#

Parameters#

Nnumeric: Number to check
endpointsbool: Whether to include the endpoints within the check

Returns#

True: If \(\mathrm{left} \leq N \leq \mathrm{right}\) (Assuming endpoints=True)
False: Otherwise

straddles_zero(endpoints=True) → bool#: Checks whether \(0\) is within the p-box

truncate(a, b, method='f')#: Equivalent to self.min(a,method).max(b,method)