Some Computation on Sphere (Updating)

Motivation

Recently, in research, we encounter quite a few statistical problems on sphere. For example,

Head direction tuning
3d direction of object
3d direction of body parts
Some 3d tuning

There are many standard statistical operations on Euclidean space, like getting mean, standard deviation and generate uniform distribution, fitting a model etc. We can perform these operation without thinking.

But these things become a problem on the sphere, because of the intrinsic Geometry and Topology of the sphere.

Note: Simulation_and_Visualization_of_Spherical_Distributions an 2018 review by some UCSB statisticians is a great guide to the topic.

Generate Uniform Distribution

Problem: Generate a uniform distribution on the sphere.

Normalized Gaussian distribution will do the work, due to the intrinsic rotation symmetry. However, a uniform distribution does not guarantee to generate no clusters.

Generate Uniform Mesh

This problem differs from the former one, because we want the nearby points to have nearly equal distance from each other. Random sample from uniform distribution may well give us clustered points if the sample number is not very high.

https://en.wikipedia.org/wiki/Geodesic_polyhedron

geodesic icosphere

Define the Mean of Points

Problem: Given a set of points on a sphere, compute their mean.

Note: that circularity happens in the parameter space (intrinsic coordinate of the manifold), but not in the embedded space! So many work could be done easier by resorting to the extrinsic coordinates.

Check this repo for different method to generate points on a sphere and compare.

https://github.com/gradywright/spherepts

Fit a Distribution of Sphere

Problem: Given a set of sample points $z_i$ on the sphere, fit a distribution defined on sphere as the model of data.

Fit a Function Defined on Sphere

Problem: Given a set of sample points $z_i$ and their function value $y_i$, estimate a function defined on sphere $f(z;\theta):S^2\mapsto \R$, such that $\arg\min_\theta|f(z_i;\theta)-y_i|$

Problem Variant: If we know the noise scheme (like that of neuron firing rate or photon counts), then the problem could be formulated as a statistical estimation problem

\[y_i =f(z_i;\theta)+\epsilon\\ \arg\min_\theta\ \mathcal \log L(\theta;y_i,z_i)=\sum_i \log p(y_i,z_i,\theta)\]

Solution:

The problem is intrisically the same to the curve fitting problem on Euclidean space, which could be solved by optimizing the loss function in the $\theta$ space by gradient or non-gradient optimization methods. The only tricky part is the parametrization of $f$.

For example, the function defining Kent distribution is

\[f(\mathbf {x} )={\frac {A}{c(\kappa ,\beta )}}\exp\{\kappa {\boldsymbol {\gamma }}_{1}^{T}\cdot \mathbf {x} +\beta [({\boldsymbol {\gamma }}_{2}^{T}\cdot \mathbf {x} )^{2}-({\boldsymbol {\gamma }}_{3}^{T}\cdot \mathbf {x} )^{2}]\}\]

The parameters are an amplitude parameter $A$, and 2 parameters characterizing the shape and peakness of the function $\kappa,\beta$. And 3 unit-vectors $\gamma_1,\gamma_2,\gamma_3$ forming a set of orthonormal basis ($[\gamma_1,\gamma_2,\gamma_3]\in SO(3)$ which is a 3 parameter matrix group), thus there are 6 parameters for this function. And we can fit this function just as normal with matlab fitting routines.

ft = fittype( @(theta, phi, psi, kappa, beta, A, x, y) KentFunc(theta, phi, psi, kappa, beta, A, x, y), ...
        'independent', {'x', 'y'},'dependent',{'z'});
Parameter = fit([theta_data(:), phi_data(:)], score_data, ft, ...
                'StartPoint', [0, 0, pi/2, 0.1, 0.1, 0.1], ...
                'Lower', [-pi, -pi/2,  0, -Inf,   0,   0], ...
                'Upper', [ pi,  pi/2,  pi,  Inf, Inf, Inf],...)%, ...
                    'Robust', 'LAR' );
V = coeffvalues(Parameter);
CI = confint(Parameter);

Visualize Function on Sphere

Refer to this note for plotting in matlab

https://math.boisestate.edu/~wright/montestigliano/PlottingOnTheSphere.pdf

Numerical Integration on Sphere

Problem: Given a set of points and their function evaluations on a sphere ${p_i\in S^2,f(p_i)}$ or given a function handle, that could be evaluated at any points on a sphere. Compute the integration of the function on sphere. $$

\[## Numerical Differentiation on Sphere\]