Prescribing probability functions

class equadratures.weight.Weight(weight_function, support=None, pdf=False, mean=None, variance=None)

The class offers a template to input bespoke weight (probability density) functions. The resulting weight function can be given to Parameter to create a bespoke analytical or data-driven parameter.

Parameters

• weight_function (Callable,numpy.ndarray) – A callable function or an array of data representing the weights.

• support (list, optional) – Lower and upper bounds of the weight respectively. Values such as -inf or inf are not acceptable.

• pdf (bool, optional) – If set to True, then the weight_function is assumed to be normalised to integrate to unity. Otherwise, the integration constant is computed and used to normalise weight_function.

• mean (float, optional) – User-defined mean for distribution. When provided, the code does not compute the mean of the weight_function over its support.

• variance (float, optional) – User-defined variance for distribution. When provided, the code does not compute the variance of the weight_function over its support.

Example

Analytical weight functions

>>> # exp(-x)/sqrt(x)
>>> pdf_1 = Weight(lambda x: np.exp(-x)/ np.sqrt(x), [0.00001, -np.log(1e-10)],
>>>        pdf=False)
>>>
>>> # A triangular distribution
>>> a = 3.
>>> b = 6.
>>> c = 4.
>>> mean = (a + b + c)/3.
>>> var = (a**2 + b**2 + c**2 - a*b - a*c - b*c)/18.
>>> pdf_2 = Weight(lambda x : 2*(x-a)/((b-a)*(c-a)) if (a <= x < c)
>>>                         else( 2/(b-a) if (x == c)
>>>                         else( 2*(b-x)/((b-a)*(b-c)))),
>>>             support=[a, b], pdf=True)
>>>
>>> # Passing to Parameter
>>> param = Parameter(distribution='analytical', weight_function=pdf_2, order=2)

Data driven weight functions

>>> # Constructing a kde based on given data, using Rilverman's rule for bandwidth selection
>>> pdf_2 = Weight( stats.gaussian_kde(data, bw_method='silverman'),
>>>        support=[-3, 3.2])
>>>
>>> # Passing to Parameter
>>> param = Parameter(distribution='analytical', weight_function=pdf, order=2)

get_pdf(points=None)

Returns the pdf associated with the distribution.

Parameters

points (numpy.ndarray, optional) – Array of points to evaluate pdf at.

Returns

Array with shape ( len(points),1 ) containing the probability distribution.

Return type

numpy.ndarray