Comparing distributions on S2

Consider two independent finite samples {x1i} i=1k and {x 2i} i=1k of size k from distributions F1 and F2 on the unit 2-sphere, S2. We consider following sample covariances at point q ∈ M (an observation point)

∑k -→ -→ ˆΣ (q) = 1- (qxi)(qxi)′(1 - ---π---)2, s = 1,2. s k s s -→i i=1 2||qxs||
Let Σs(q) and Ω ∈ ℝn2×n2 be the mean and covariance of Z = (-→ qX)(-→ qX)′, for X ~ F s. Thus, we assume that the mean and covarince of the tensor valued trandom variable Z exist. As an application of the Central Limit Theorem we obtained the asymptotic
√-- ˆ ˆ -1 -1 k (Σ1(q)Σ 2 (q) - In) ⇝ Nn ×n(0,2Ω ⊗ Σ (q)),
provided that Σ1(q) = Σ2(q) = Σ(q). Further application of delta method gives us
√-- -1 ′ T -1 ′ ξk(q;h) := k (h(ˆΣ1(q)ˆΣ 2 (q))- h (In )) ⇝ N (0,2[h(In)] (Ω ⊗Σ (q))[h (In)]),
(1)

for similarity invariant h which gradient h′(.) ∈ ℝn2 is continuous and does not vanish at In. For example, ξk(q; tr) = √ -- k(tr(ˆ Σ1(q) ˆ Σ2-1(q)) - n) and ξk(q; det) = √ -- k(det(ˆ Σ1(q) ˆ Σ2-1(q)) - 1).

We apply bootstrapping method to utilize (1). For k′ < k, let ym,m = 1,...,M be instances of statistic ξk(q; h) based on subsamples of size k′ of the initial k-samples. Then, according to (1), ξ = ¯y ∕s.e.(y) goes to N(0, 1) in distribution as k →∞.

The following applet simulates distributions from different families and calculates ξ statistics. Xi-pval is the p-value of ξ statistics. We choose the observation point q randomly on S2. The user can choose the sample size k and whether the two samples are equally distributed or not, H0 hypothesis. Sub-sample size is fixed to k′ = k∕2 and M = k∕4. Possible choices for function h are the trace, determinant and h(A) = tr(log(A)). Shown are the two k-samples in red and blue and the observation point in green. Drag the mouse over to change the view point.

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