libbingham Matlab Tutorial

Contents

What is a Bingham Distribution?

The Bingham distribution is an antipodally-symmetric probability distribution on a unit hypersphere. Its probability density function (PDF) is

$$ f(\vec{x} ; \Lambda, V) = \frac{1}{F} \exp \{ \sum_{i=1}^d \lambda_i (\vec{v_i}^T \vec{x})^2 \} $$

where $\vec{x}$ is a unit vector on the surface of the sphere $\bf{S}^d \subset \bf{R}^{d+1}$, $F$ is a normalization constant, $\Lambda$ is a vector of non-positive ($\leq 0$) concentration parameters, and the columns of the $(d+1) \times d$ matrix $V$ are orthogonal unit vectors.

Note that a large-magnitude $\lambda_i$ indicates that the distribution is highly peaked along the direction $\vec{v_i}$, while a small-magnitude $\lambda_i$ indicates that the distribution is spread out along $\vec{v_i}$.

The Bingham distribution is derived from a zero-mean Gaussian on $\bf{R}^{d+1}$, conditioned to lie on the surface of the unit hypersphere $\bf{S}^d$. Thus, the exponent of the Bingham PDF is the same as the exponent of a zero-mean Gaussian distribution (in principal components form, with one of the eigenvalues of the covariance matrix set to infinity).

The Bingham distribution is the maximum entropy distribution on the hypersphere which matches the sample inertia matrix $E[\vec{x} \vec{x}^T]$. Therefore, it may be better suited to representing random process noise on the hypersphere than some other distributions, such as (projected) tangent-space Gaussians. Binghams are also quite flexible, since a concentration parameter, $\lambda_i$, of zero indicates that the distribution is completely uniform in the direction of $\vec{v_i}$. They are therefore very useful in tracking problems where there is high, anisotropic noise.

Matlab Support in libbingham

Starting with version 0.3.0, all of the core functions in libbingham are supported in Matlab. But, as in the C library, most functions only support Bingham distributions up to dimension 4. (That is, up to $\bf{S}^3$.)

Creating a Bingham Distribution in Matlab

Bingham distributions are represented as a Matlab struct, with fields d, V, Z, and F and dF (which are computed by libbingham). To create a new Bingham distribution, create a new struct with dimension d, orthogonal direction matrix V, and concentration parameters Z. For example, the uniform Bingham distribution on the 3-D sphere $\bf{S}^2$ is:

B = struct();
B.d = 3;
B.Z = [0,0];
B.V = [0,0; 1,0; 0,1];

To look up the normalization constant and its partial derivatives with respect to Z, use:

[B.F B.dF] = bingham_F(B.Z);

Fitting

Given a matrix X with unit vectors in the rows, you can compute the maximum likelihood Bingham distribution given X with bingham_fit(). For example:

% create n 4-D unit vectors
n = 10;
X = randn(n,4);
X = X./repmat(sqrt(sum(X.^2,2)), [1,4]);

% Fit a Bingham distribution to X
B = bingham_fit(X);

Sampling

To sample n unit vectors from a Bingham distribution, use:

Y = bingham_sample(B,n);

Computing the PDF

To compute the PDF of a unit vector x under a Bingham B, use:

f = bingham_pdf(x,B);

Computing the Mode

mu = bingham_mode(B);

Computing the Entropy

h = bingham_entropy(B);

Computing the Scatter Matrix

To compute the scatter matrix, $E[\vec{x} \vec{x}^T]$, use:

S = bingham_scatter(B);

Multiplying two Binghams

Two multiply two Binghams, B1 and B2, use:

B = bingham_mult(B1,B2);

Special Functions for the Quaternion Bingham Distribution

B2 = bingham_pre_rotate_3d(B,q);
B2 = bingham_post_rotate_3d(q,B);
B2 = bingham_invert_3d(B);