Distributions

FVectorDistribution{TSpace<:VectorSpaceFiber, T}

An abstract distribution for vector bundle fiber-valued distributions (values from a fiber of a vector bundle at point x from the given manifold). For example used for tangent vector-valued distributions.

FVectorSupport(space::AbstractManifold, VectorSpaceFiber)

Value support for vector bundle fiber-valued distributions (values from a fiber of a vector bundle at a point from the given manifold). For example used for tangent vector-valued distributions.

FVectorvariate

Structure that subtypes VariateForm, indicating that a single sample is a vector from a fiber of a vector bundle.

MPointDistribution{TM<:AbstractManifold}

An abstract distribution for points on manifold of type TM.

MPointSupport(M::AbstractManifold)

Value support for manifold-valued distributions (values from given AbstractManifold M).

MPointvariate

Structure that subtypes VariateForm, indicating that a single sample is a point on a manifold.

support(d::FVectorDistribution)

Get the object of type FVectorSupport for the distribution d.

uniform_distribution(M::Grassmann{ℝ}, p)

Uniform distribution on given (real-valued) Grassmann M. Specifically, this is the normalized Haar measure on M. Generated points will be of similar type as p.

The implementation is based on Section 2.5.1 in [Chi03]; see also Theorem 2.2.2(iii) in [Chi03].

uniform_distribution(M::ProjectiveSpace{ℝ}, p)

Uniform distribution on given ProjectiveSpace M. Generated points will be of similar type as p.

uniform_distribution(M::Sphere{ℝ}, p)

Uniform distribution on given Sphere M. Generated points will be of similar type as p.

uniform_distribution(M::Stiefel{ℝ}, p)

Uniform distribution on given (real-valued) Stiefel M. Specifically, this is the normalized Haar and Hausdorff measure on M. Generated points will be of similar type as p.

The implementation is based on Section 2.5.1 in [Chi03]; see also Theorem 2.2.1(iii) in [Chi03].

NormalRotationDistribution(M::Rotations, d::Distribution, x::TResult)

Distribution that returns a random point on the manifold Rotations M. Random point is generated using base distribution d and the type of the result is adjusted to TResult.

See normal_rotation_distribution for details.

PowerFVectorDistribution([type::VectorSpaceFiber], [x], distr)

Generates a random vector at a point from vector space (a fiber of a tangent bundle) of type type using the power distribution of distr.

Vector space type and point can be automatically inferred from distribution distr.

PowerPointDistribution(M::AbstractPowerManifold, distribution)

Power distribution on manifold M, based on distribution.

ProductFVectorDistribution([type::VectorSpaceFiber], [x], distrs...)

Generates a random vector at point x from vector space (a fiber of a tangent bundle) of type type using the product distribution of given distributions.

Vector space type and x can be automatically inferred from distributions distrs.

ProductPointDistribution(M::ProductManifold, distributions)

Product distribution on manifold M, combined from distributions.

ProjectedFVectorDistribution(type::VectorSpaceFiber, p, d, project!)

Generates a random vector from ambient space of manifold type.manifold at point p and projects it to vector space of type type using function project!, see project for documentation. Generated arrays are of type TResult.

ProjectedPointDistribution(M::AbstractManifold, d, proj!, p)

Generates a random point in ambient space of M and projects it to M using function proj!. Generated arrays are of type TResult, which can be specified by providing the p argument.

normal_tvector_distribution(M::Euclidean, p, σ)

Normal distribution in ambient space with standard deviation σ projected to tangent space at p.

projected_distribution(M::AbstractManifold, d, [p=rand(d)])

Wrap the standard distribution d into a manifold-valued distribution. Generated points will be of similar type to p. By default, the type is not changed.