StandardDistributionPolynomialFactory¶

class
StandardDistributionPolynomialFactory
(*args)¶ Build orthonormal or orthogonal univariate polynomial families.
 Parameters
 arg
Distribution
orOrthonormalizationAlgorithm
Either a
Distribution
implementing the probability measure according to which the polynomial family is orthonormal or anOrthonormalizationAlgorithm
.In the first case, the implementation will switch to the suitable specific orthonormal univariate polynomial family if any (see the notes below), or it will default to the
AdaptiveStieltjesAlgorithm
to build an orthonormal univariate polynomial family.
 arg
Notes
Use this functionality with caution:
The polynomials exist if and only if the distribution admits finite moments of all orders. Even if some algorithms manage to compute something, it will be plain numerical noise.
Even if the polynomials exist, they form an Hilbertian basis wrt the dot product induced by the distribution if and only if the distribution is determinate, ie is characterized by its moments. For example, the
LogNormal
distribution has orthonormal polynomials of arbitrary degree but the projection onto the functional space generated by these polynomials (seeFunctionalChaosAlgorithm
) may converge to a function that differs significantly from the function being projected.
OpenTURNS implements the following specific orthonormal 1 univariate polynomial families together with their associated standard distributions:
Standard distribution
Polynomial
Aside, OpenTURNS also implements generic algorithms for building orthonormal univariate polynomial families with respect to any arbitrary probability measure (implemented as a
Distribution
). OpenTURNS implements the followingOrthonormalizationAlgorithm
’s:AdaptiveStieltjesAlgorithm
(default),
 1
A polynomial family is said to be orthonormal with respect to the probability measure if and only if:
where denotes Kronecker’s delta.
Examples
>>> import openturns as ot
Build the specific orthonormal polynomial factory associated to the normal distribution (Hermite):
>>> polynomial_factory = ot.StandardDistributionPolynomialFactory(ot.Normal()) >>> for i in range(3): ... print(polynomial_factory.build(i)) 1 X 0.707107 + 0.707107 * X^2
Build an orthonormal polynomial factory for the Weibull distribution with the default orthonormalization algorithm:
>>> polynomial_factory = ot.StandardDistributionPolynomialFactory(ot.Weibull()) >>> for i in range(3): ... print(polynomial_factory.build(i)) 1 1 + X 1  2 * X + 0.5 * X^2
Build an orthonormal polynomial factory for the lognormal distribution with Chebychev’s othonormalization algorithm:
>>> algorithm = ot.AdaptiveStieltjesAlgorithm(ot.Weibull()) >>> polynomial_factory = ot.StandardDistributionPolynomialFactory(algorithm) >>> for i in range(3): ... print(polynomial_factory.build(i)) 1 1 + X 1  2 * X + 0.5 * X^2
Methods
build
(self, degree)Build the th order orthogonal univariate polynomial.
buildCoefficients
(self, degree)Build the th order orthogonal univariate polynomial coefficients.
Build the recurrence coefficients.
getClassName
(self)Accessor to the object’s name.
getId
(self)Accessor to the object’s id.
getMeasure
(self)Accessor to the associated probability measure.
getName
(self)Accessor to the object’s name.
getNodesAndWeights
(self, n)Build the th order quadrature scheme.
getRecurrenceCoefficients
(self, n)Accessor to the recurrence coefficients of the th order.
getRoots
(self, n)Accessor to the recurrence coefficients of the th order.
getShadowedId
(self)Accessor to the object’s shadowed id.
getVisibility
(self)Accessor to the object’s visibility state.
hasName
(self)Test if the object is named.
hasVisibleName
(self)Test if the object has a distinguishable name.
setName
(self, name)Accessor to the object’s name.
setShadowedId
(self, id)Accessor to the object’s shadowed id.
setVisibility
(self, visible)Accessor to the object’s visibility state.

__init__
(self, *args)¶ Initialize self. See help(type(self)) for accurate signature.

build
(self, degree)¶ Build the th order orthogonal univariate polynomial.
 Parameters
 kint,
Polynomial order.
 Returns
 polynomial
OrthogonalUniVariatePolynomial
Requested orthogonal univariate polynomial.
 polynomial
Examples
>>> import openturns as ot >>> polynomial_factory = ot.HermiteFactory() >>> print(polynomial_factory.build(2)) 0.707107 + 0.707107 * X^2

buildCoefficients
(self, degree)¶ Build the th order orthogonal univariate polynomial coefficients.
 Parameters
 kint,
Polynomial order.
 Returns
 coefficients
Point
Coefficients of the requested orthogonal univariate polynomial.
 coefficients
Examples
>>> import openturns as ot >>> polynomial_factory = ot.HermiteFactory() >>> print(polynomial_factory.buildCoefficients(2)) [0.707107,0,0.707107]

buildRecurrenceCoefficientsCollection
(self, degree)¶ Build the recurrence coefficients.
Build the recurrence coefficients of the orthogonal univariate polynomial family up to the th order.
 Parameters
 kint,
Polynomial order.
 Returns
 recurrence_coefficientslist of
Point
All the tecurrence coefficients up to the requested order.
 recurrence_coefficientslist of
Examples
>>> import openturns as ot >>> polynomial_factory = ot.HermiteFactory() >>> print(polynomial_factory.buildRecurrenceCoefficientsCollection(2)) [[1,0,0],[0.707107,0,0.707107]]

getClassName
(self)¶ Accessor to the object’s name.
 Returns
 class_namestr
The object class name (object.__class__.__name__).

getId
(self)¶ Accessor to the object’s id.
 Returns
 idint
Internal unique identifier.

getMeasure
(self)¶ Accessor to the associated probability measure.
 Returns
 measure
Distribution
The associated probability measure (according to which the polynomials are orthogonal).
 measure
Notes
Two polynomials P and Q are orthogonal with respect to the probability measure if and only if their dot product:
where and .
Examples
>>> import openturns as ot >>> polynomial_factory = ot.HermiteFactory() >>> print(polynomial_factory.getMeasure()) Normal(mu = 0, sigma = 1)

getName
(self)¶ Accessor to the object’s name.
 Returns
 namestr
The name of the object.

getNodesAndWeights
(self, n)¶ Build the th order quadrature scheme.
Associated with the orthogonal univariate polynomials family.
 Parameters
 kint,
Polynomial order.
 Returns
Examples
>>> import openturns as ot >>> polynomial_factory = ot.HermiteFactory() >>> nodes, weights = polynomial_factory.getNodesAndWeights(3) >>> print(nodes) [1.73205,...,1.73205] >>> print(weights) [0.166667,0.666667,0.166667]

getRecurrenceCoefficients
(self, n)¶ Accessor to the recurrence coefficients of the th order.
Of the orthogonal univariate polynomial.
 Parameters
 kint,
Polynomial order.
 Returns
 recurrence_coefficients
Point
The recurrence coefficients of the th order orthogonal univariate polynomial.
 recurrence_coefficients
Notes
Any sequence of orthogonal polynomials has a recurrence formula relating any three consecutive polynomials as follows:
Examples
>>> import openturns as ot >>> polynomial_factory = ot.HermiteFactory() >>> print(polynomial_factory.getRecurrenceCoefficients(3)) [0.5,0,0.866025]

getRoots
(self, n)¶ Accessor to the recurrence coefficients of the th order.
Of the orthogonal univariate polynomial.
 Parameters
 kint,
Polynomial order.
 Returns
 roots
Point
The roots of the th order orthogonal univariate polynomial.
 roots
Examples
>>> import openturns as ot >>> polynomial_factory = ot.HermiteFactory() >>> print(polynomial_factory.getRoots(3)) [1.73205,...,1.73205]

getShadowedId
(self)¶ Accessor to the object’s shadowed id.
 Returns
 idint
Internal unique identifier.

getVisibility
(self)¶ Accessor to the object’s visibility state.
 Returns
 visiblebool
Visibility flag.

hasName
(self)¶ Test if the object is named.
 Returns
 hasNamebool
True if the name is not empty.

hasVisibleName
(self)¶ Test if the object has a distinguishable name.
 Returns
 hasVisibleNamebool
True if the name is not empty and not the default one.

setName
(self, name)¶ Accessor to the object’s name.
 Parameters
 namestr
The name of the object.

setShadowedId
(self, id)¶ Accessor to the object’s shadowed id.
 Parameters
 idint
Internal unique identifier.

setVisibility
(self, visible)¶ Accessor to the object’s visibility state.
 Parameters
 visiblebool
Visibility flag.