S2Fun.discrepancy

kernel discrepancy between a discrete sample and a spherical density

Description

discrepancy measures how well a list of directions \(v_j\) with weights \(c_j\) represents the density function \(f\), by the squared distance

\[ J(v,c) = \| \mu - f \|_{\psi}^2 = \sum_{n=1}^N \frac{4\pi A_n}{2n+1} \sum_{k=-n}^{n} | \hat{\mu}_n^{k} - \hat{f}_n^{k} |^2, \qquad \mu = \lambda \, \sum_{j=1}^M c_j \, \delta_{v_j}, \quad \lambda = \int_{S^2} f(v) \,dv, \]

i.e. the functional optimalSample minimizes. Here \(A_n\) are the Legendre coefficients of the restricted distance kernel and \(N\) is the bandwidth - only the harmonic degrees up to \(N\) are compared, so state the one the sample is meant for.

Degree 0 is left out. The restricted distance kernel is only conditionally positive definite, its coefficient of degree 0 being negative, and the weights are normalized to sum up to 1, which makes that degree vanish anyway.

Scaling \(f\) multiplies the result by \(\lambda^2\), hence discrepancies are comparable only between samples of the same function.

Give the same bandwidth the sample was optimized for. Above it nothing was optimized, and since a sample gains its accuracy up to bandwidth partly at the expense of the higher degrees, scoring beyond it ranks two samples by the part neither of them minimized.

Syntax

res = discrepancy(sF,v)
res = discrepancy(sF,v,'weights',c)
res = discrepancy(sF,v,'bandwidth',64)

Input

Output

Options

See also

S2Fun.optimalSample S2Fun.discreteSample S2RestrictedDistanceKernel