class svvamp.GeneratorProfileVMFHypersphere(n_v, n_c, vmf_concentration, vmf_probability=None, vmf_pole=None, stretching=1, sort_voters=False)[source]#

Profile generator using the Von Mises-Fisher distribution on the n_c - 1-sphere

Parameters:

  • n_v (int) – Number of voters.

  • n_c (int) – Number of candidates.

  • stretching (number) – Number between 0 and numpy.inf (both included).

  • vmf_concentration (float or list or ndarray) – 1d array. Denote k its size (number of ‘groups’). vmf_concentration[i] is the VMF concentration of group i.

  • vmf_probability (float or list or ndarray or None) – 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 the groups have equal probabilities.

  • vmf_pole (list or ndarray) – 2d array of size (k, n_c). vmf_pole[i, :] is the pole of the VMF distribution for group i.

  • sort_voters (bool) – This argument is passed to Profile.

Notes

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.

Then, v’s utility vector is sent onto the spheroid that is the image of the sphere by a dilatation of factor stretching along the direction [1, …, 1]. For example, if stretching = 1, we stay on the unit sphere of \(\mathbb{R}^n_c\). Cf. Durand et al. ‘Geometry on the Utility Sphere’.

N.B.: if stretching != 1, it amounts to move the poles. For example, if the pole [1, 0, 0, 0] is given and stretching = 0, then the actual pole will be [0.75, - 0.25, - 0.25, - 0.25] (up to a multiplicative constant).

poles are normalized before being used. So, the only source of concentration for group i is vmf_concentration[i], not the norm of vmf_pole[i]. If vmf_pole is None, then each pole is drawn independently and uniformly on the sphere.

Examples

Typical usage:

>>> initialize_random_seeds()
>>> generator = GeneratorProfileVMFHypersphere(n_v=5, n_c=3, vmf_concentration=10, sort_voters=True)
>>> profile = generator()
>>> profile.preferences_ut
array([[ 0.95891992, -0.12067454,  0.25672991],
       [ 0.86494714,  0.06265355,  0.49793673],
       [ 0.94128919, -0.27215259,  0.19976892],
       [ 0.80831535, -0.13959226,  0.57196179],
       [ 0.60411981, -0.07180745,  0.79365165]])

You can specify a pole:

>>> initialize_random_seeds()
>>> generator = GeneratorProfileVMFHypersphere(n_v=5, n_c=3, vmf_concentration=10, vmf_pole=[.7, .7, 0],
...                                            sort_voters=True)
>>> profile = generator()
>>> profile.preferences_ut
array([[ 0.75351327,  0.5734909 , -0.32144353],
       [ 0.5984744 ,  0.64868975, -0.47013828],
       [ 0.38344905,  0.91929245, -0.08870296],
       [ 0.6432514 ,  0.75827724, -0.1060342 ],
       [ 0.25385897,  0.94748174, -0.19450957]])

With several poles:

>>> initialize_random_seeds()
>>> generator = GeneratorProfileVMFHypersphere(n_v=5, n_c=3, vmf_concentration=[np.inf, np.inf],
...                                            vmf_probability=[.9, .1], sort_voters=True)
>>> profile = generator()
>>> profile.preferences_ut
array([[ 2.2408932 ,  1.86755799, -0.97727788],
       [ 1.76405235,  0.40015721,  0.97873798],
       [ 1.76405235,  0.40015721,  0.97873798],
       [ 1.76405235,  0.40015721,  0.97873798],
       [ 1.76405235,  0.40015721,  0.97873798]])

If the probabilities are not explicitly given, poles have equal probabilities:

>>> initialize_random_seeds()
>>> generator = GeneratorProfileVMFHypersphere(n_v=5, n_c=3, vmf_concentration=[np.inf, np.inf],
...                                            sort_voters=True)
>>> profile = generator()
>>> profile.preferences_ut
array([[ 2.2408932 ,  1.86755799, -0.97727788],
       [ 2.2408932 ,  1.86755799, -0.97727788],
       [ 2.2408932 ,  1.86755799, -0.97727788],
       [ 1.76405235,  0.40015721,  0.97873798],
       [ 1.76405235,  0.40015721,  0.97873798]])

With some stretching:

>>> initialize_random_seeds()
>>> generator = GeneratorProfileVMFHypersphere(n_v=5, n_c=3, vmf_concentration=0, stretching=100,
...                                            sort_voters=True)
>>> profile = generator()
>>> profile.preferences_ut
array([[ 0.14719473,  0.14577802,  0.14320266],
       [ 0.17534594,  0.17348776,  0.17273085],
       [-0.3104039 , -0.3047001 , -0.31555119],
       [ 0.25555345,  0.25232602,  0.25907932],
       [ 0.59993219,  0.59426747,  0.60493974]])

>>> initialize_random_seeds()
>>> generator = GeneratorProfileVMFHypersphere(n_v=5, n_c=3, vmf_concentration=0, stretching=.01,
...                                            sort_voters=True)
>>> profile = generator()
>>> profile.preferences_ut
array([[ 0.18174638,  0.04007574, -0.21746036],
       [ 0.15084759, -0.03497065, -0.11066129],
       [-0.02165249,  0.54872752, -0.53638159],
       [-0.00739108, -0.33013437,  0.34519503],
       [ 0.02790245, -0.53856937,  0.52865831]])

References

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

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