GenDirectionalStats.jl

This package contains Gen probability distributions for plane rotations, rotations in 3D space, and directions in 3D space. The package also contains several Gen involutive MCMC moves on 3D rotations.

Probability distributions

Probability distributions on plane rotations

Plane rotations are represented as elements of the following concrete type, which is defined in Rotations.jl:

• GenDirectionalStats.Rot2 = Rotations.RotMatrix{2,Float64,4}

The supported probability distributions on plane rotations are:

• GenDirectionalStats.uniform_rot2(): Uniform distribution on plane rotations.

• GenDirectionalStats.von_mises_rot2(location::Rot2, concentration::Real): The unimodal distribution von Mises distribution on plane rotations with a mode at location and given concentration.

The reference measure for random choices of this type is the uniform measure on the circle S^1. The circle has a natural identification with the group of plane rotations SO(2), and under this identification the uniform measure coincides with the Haar measure. The uniform measure is uniquely determined up to a scaling constant; this package chooses the scaling constant so that the measure of the space of all plane rotations is 2 * pi. The density functions of all probability distribution(s) are defined relative to this measure. For example, the probability density of the uniform distribution uniform_rot2 at any point is 1/(2 * pi).

The plot below shows samples from these distributions (where I is the identity rotation). Rotations are visualized as 2D coordinate frames (x-axis is red, y-axis is green). The modes for the von Mises distributions (location) are shown as the bold coordinate frame, a single sample from each distribution is shown as a light coordinate frame, and 250 samples are represented by the endpoints of their two coordinate axis unit vectors.

Probability distributions on 3D directions

Directions in 3D space are represented as elements of the following concrete type:

• GenDirectionalStats.UnitVector3

The supported probability distributions on 3D directions are:

• GenDirectionalStats.uniform_3d_direction(): Uniform distribution on 3D directions.

• GenDirectionalStats.vmf_3d_direction(location::UnitVector3, concentration::Real): A unimodal von Mises Fisher distribution on the space of unit 3-vectors with a mode at location and given concentration.

The reference measure for random choices of this type is the uniform measure on the two-dimensional sphere S^2. The uniform measure is uniquely determined up to a scaling constant; this package chooses the scaling constant so that the measure of the whole space S^2 is 4 * pi (so the total measure equals the surface area). The density functions of all probability distribution(s) are defined relative to this Haar measure. For example, the probability density of the uniform distribution uniform_3d_direction is 1/(4 * pi).S

The plot below shows samples from these distributions. The blue unit vectors are the modes of the von Mises Fisher distributions, and 250 samples of unit 3-vectors from each distribution are shown as their end-points (points on the unit sphere).

Probability distributions on 3D rotations

Rotations in 3D space, i.e. elements of the group SO(3) are represented as elements of the following concrete type, which is defined in Rotations.jl.

• GenDirectionalStats.Rot3 = Rotations.UnitQuaternion{Float64}

The supported probability distributions on 3D rotations are:

• GenDirectionalStats.uniform_rot3(): Uniform distribution on 3D rotations.

• GenDirectionalStats.vmf_rot3(location::Rot3, concentration::Real): A unimodal distribution on 3D rotations that is based on the von Mises Fisher distribution on the space of unit 3-vectors with a mode at location and given concentration. The distribution is a mixture of

• GenDirectionalStats.uniform_vmf_rot3(location::Rot3, concentration::Real, prob_outlier::Real):

The reference measure for random choices of this type is the Haar measure on SO(3). This measure coincides with the pushforward of the uniform measure on S^3 by the double cover S^3 -> SO(3) induced by the group action of quaternions as rotations. This package defines the measure of all of SO(3) as pi * pi (this value is chosen to be half of the surface area of the 3-sphere S^3, motivated by the aforementioned double cover). The density functions of all probability distribution(s) are defined relative to this Haar measure. For example, the probability density of the uniform distribution uniform_rot3 at any point is 1/(pi * pi).

The plot below shows samples from these distributions (where I is the identity rotation). 3D rotations are visualized as 3D coordinate frames centered at the origin. The mode of each von Mises Fisher distribution is bold, one sample is shown non-bold, and 250 other samples are shown just via the endpoints of their coordinate axes.

Hopf fibration

The following functions convert back and forth between 3D rotations (Rot3) and pairs of plane rotations (Rot2) and 3D directions (UnitVector3):

• GenDirectionalStats.from_direction_and_plane_rotation(direction::UnitVector3, plane_rotation::Rot2)::Rot3

• GenDirectionalStats.to_direction_and_plane_rotation(rot3::Rot3)::Tuple{UnitVector3,Rot2}

The 3D direction gives the z-axis of the rotated coordinate frame, and the plane rotation gives the rotation around the z-axis. The 3D rotation corresponding to "zero plane rotation" is the rotation that carries the old z-axis to the new z-axis along a minor arc of great circle (i.e., the shortest possible path). This parametrization of 3D rotations can be understood in terms of the Hopf fibration S^3 -> S^2: choosing the shortest possible path of z-axis amounts to choosing a distinguished element (up to sign) of the fiber over each element of S^2. These distinguished elements allow us to locally identify SO(3) with the Cartesian product S^2 x S^1, except at the singularities at {[0, 0, -1]} x S^1 (anti-parallel z-axis). Note that this parametrization is not the same as axis-angle parametrization. Radon-Nikodym derivatives relating the measures involved are given by the constants:

• GenDirectionalStats.TO_DIRECTION_AND_PLANE_ROTATION_JACOBIAN_CORRECTION

• GenDirectionalStats.FROM_DIRECTION_AND_PLANE_ROTATION_JACOBIAN_CORRECTION

MCMC kernels

You can implement random walk or data-driven Metropolis-Hastings kernels using the unimodal probability distributions for each data type.

Specialized reversible jump MCMC kernels on 3D rotations

This package also includes specialized reversible jump moves on 3D rotations. Each move takes a trace, the address of the 3D rotation in the trace to move (must have type GenDirectionalStats.Rot3), and possible other arguments.

Rotate by a uniformly chosen angle around a given axis.

This is useful for exploring rotations of objects that have some sort of cylindrical self-similarity (e.g. coffee mugs).

trace, = GenDirectionalStats.uniform_angle_fixed_axis_mh(trace, addr, axis)

Rotate by a random small angle around a random axis

This is useful for constructing random walks on the space of rotations.

trace, = GenDirectionalStats.small_angle_random_axis(trace, addr, width)

Rotate by a random small angle around a given axis

trace, = GenDirectionalStats.small_angle_fixed_axis_mh(trace, addr, axis, width)

Flip 180 degrees around a given axis

trace, = GenDirectionalStats.flip_around_fixed_axis_mh(trace, addr, axis)

Installing

From the Julia package manager REPL, run:

add https://github.com/probcomp/GenDirectionalStats.jl