x θ

x = (cosθ, senθ)

z = e^{iθ} = cosθ + isenθ

x_{1}, …, x_{n}

θ_{i} i = 1, …, n

θ̄ θ_{1}, …, θ_{n} x_{1}+ …+ x_{n} x̄ x_{1}, …, x_{n} x_{j}

(cosθ_{j}, senθ_{j}) (C̄, S̄)

C̄ = \frac{1}{n} ∑_{j=1}^{n} cosθ_{j} (1)

S̄ = \frac{1}{n} ∑_{j=1}^{n} senθ_{j} (2)

θ̄ ???? x_{1}+ …+ x_{n}

C̄ = R̄cosθ̄ (3) S̄ = R̄senθ̄ (4)

R̄ > 0 R̄

R̄ = \sqrt{C̄^{2} + S̄^{2}} (5)

θ̄ R̄ = 0 R̄ > 0 θ̄

θ̄ = \begin{cases}tan^{−1}(S̄/C̄) & C̄ ≥ 0 \\ tan^{−1}(S̄/C̄) + π & C̄ < 0 \end{cases}

θ̄ (θ_{1}+ …+ θ_{n})/n

θ̄ (θ_{1}, …, θ_{n})

ϕ [ϕ, ϕ+ π) ϕ

ϕ+ π R̄ ?? x̄ R̄ = \sqrt{C̄^{2} + S̄^{2}} x_{1}, …, x_{n} 0 ≤

R̄ ≤ 1 θ_{1}, …, θ_{n}

R̄ = 1 θ_{1}, …, θ_{n}

R̄ R̄ R̄ x_{1}+ …+ x_{n} R = nR̄ R̄

v = \sqrt{−2logR̄} (6)

θ ξ

θ− ξ, 2π−

(θ−

ξ)) = π− |π−

|θ− ξ||

θ_{1}, …, θ_{n} α 0(α) =

\frac{1}{n} ∑_{i=1}^{n}(π− |π− |θ_{i}− α||) d_{0} θ̃ d_{0}(θ̃)

θ F (x) = P r(0 < θ ≤ x), 0 ≤ x ≤ 2∗π

F (x+2π)−F (x) = 1, −∞ < x < ∞ (7)

??

π α ≤ β ≤ α+ 2π

P r(α < θ ≤ β) = F (β)−F (α) = ∫_{α}^{β} dF (x)

(8)

lim_{x→∞}F (x) = ∞lim_{x→−∞}F (x) = −∞

F (0) = 0F (2π) = 1

?? F (β)−

F (α) ∫_{α}^{β} f(x)dx = F (β)− F (α), −∞ <

α ≤ β < ∞ f(θ) ≥

0 (−∞, ∞)

∫_{0}^{2π} f(θ)dθ =

1 f(θ) =

f(θ+ 2π) (−∞, ∞)

?? \frac{2πr}{m}

?? p_{r} =

\frac{1}{m}

P r(θ = v + \frac{2πr}{m}) = p_{r}r = 0, 1, ⋯, m − 1 (9)

p_{r} ≥ 0, ∑_{r=0}^{m−1} p_{r} = 1 (10)

θ) = \frac{1}{2π}

α ≤ β ≤ α+ 2π α < θ ≤

β) = \frac{β−α}{2π}

?? θ = μ

θ = μ

θ = μ+ π e^{2κ} κ

g(θ; μ, κ) = \frac{1}{2πI_{0}(κ)}e^{κcos(θ−μ)} (11)

F (θ; 0, κ) = \frac{1}{I_{0}(κ)} ∫_{0}^{θ} e^{κcosu}du

I_{0}

I_{0}(κ) = \frac{1}{2π} ∫_{0}^{2π} e^{κcosθ}dθ, I_{0}(κ) = ∑_{r=0}^{∞} \frac{1}{(r!)^{2}} (\frac{κ}{2})^{2r}

μ kappaappa

mises.epsRepresentacindelafuncindedensidaddeDistribucinvonMisesM(0,κ κ

R̄S_{n}^{2} κ ?? ?? ?? κ M(π, κ) κ ?? ?? ??

S_{n}^{2}(θ) = \frac{1}{n} ∑_{i=1}^{n} d^{2}(θ_{i}, θ) (12)

θ = min_{y}S_{n}^{2}(y)

d^{2}(θ_{i}, θj) = 2(1 − (cos(θ_{i} − θ_{j}))) (13) d^{2}(θ_{i}, θj) = π − |π − |θ_{i} − θ_{j}|| (14)

R̄ = \sqrt{C̄^{2} + S̄^{2}}

\sqrt{−2log(R̄)} κ S_{n}^{2} κ R̄ κ R̄S_{n}^{2} κ