198.
where xi = (cos ϕi , sin ϕi )t and yi = (cos ψi , sin ψi )t . For a time seriesϕt (t = 1, 2, ...) of circular data, this deﬁnition can be carried over toautocorrelations n i,j=1;i=j sin(ϕi − ϕj ) sin(ϕi+k − ϕj+k ) r(k) = n 2 (7.17) i,j=1;i=j sin (ϕi − ϕj )or det(n−1 n−k t i=1 xi xi+k ) rϕ (k) = n−k (7.18) det(n−1 t i=1 xi xi )7.2.3 Probability distributionsA probability distribution for circular data is a distribution F on the in-terval [0, 2π). The sample statistics deﬁned in Section 7.1 are estimates ofthe corresponding population counterparts in Table 7.2. Most frequently used distributions are the uniform, cardioid, wrapped,von Mises, and mixture distributions.Uniform distribution U ([0, 2π)): u F (u) = P (0 ≤ ϕ ≤ u) = 1{0 ≤ u < 2π}, 2π 1 f (ϕ) = F (ϕ) = 1{0 ≤ u < 2π}. 2πIn this case, µp = ρp = 0, the mean direction µϕ is not deﬁned, and thecircular standard deviation σ and dispersion δ are inﬁnite. This expressesthe fact that there is no preference for any direction and variability istherefore maximal.Cardioid (or Cosine) distribution C(µ, ρ): ρ u F (u) = [ sin(u − µ) + ]1{0 ≤ u < 2π} π 2πand 1
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