PopulationVMFHypercircle

class svvamp.PopulationVMFHypercircle(V, C, vmf_concentration, vmf_probability=None, vmf_pole=None)

Population drawn with Von Mises-Fisher distributions on the C-2-sphere

Parameters:

• V – Integer. Number of voters.
• C – Integer. Number of candidates.
• vmf_concentration – 1d array. Let us note k its size (number of ‘groups’). vmf_concentration[i] is the VMF concentration of group i.
• vmf_probability – 1d array of size k. vmf_probability[i] is the probability, for a voter, to be in group i (up to normalization). If None, then groups have equal probabilities.
• vmf_pole – 2d array of size (k, C). vmf_pole[i, :] is the pole of the VMF distribution for group i.

Returns:

A Population object.

We work on the C-2-sphere: vectors of \(\mathbb{R}^C\) with Euclidean norm equal to 1 and that are orthogonal to [1, ..., 1]. It is a representation of the classical Von Neumann-Morgenstern utility space. Cf. working paper Durand et al. ‘Geometry on the Utility Sphere’.

Before all computations, the poles are projected onto the hyperplane and normalized. So, the only source of concentration for group i is vmf_concentration[i], not the norm of vmf_pole[i]. If pole is None, then each pole is drawn independently and uniformly on the C-2-sphere.

For each voter c, we draw a group i at random, according to vmf_probability (normalized beforehand if necessary). Then, v‘s utility vector is drawn according to a Von Mises-Fisher distribution of pole vmf_pole[i, :] and concentration vmf_concentration[i], using Ulrich’s method modified by Wood.

Once group i is chosen, then up to a normalization constant, the density of probability for a unit vector x is exp(vmf_concentration[i] vmf.pole[i, :].x), where vmf.pole[i, :].x is a dot product.

References:

Ulrich (1984) - Computer Generation of Distributions on the m-Sphere

Wood (1994) - Simulation of the von Mises Fisher distribution