Editor's Notes
- \cos(\bm{x}, \bm{y}) = \frac{\bm{x}\cdot\bm{y}}{|\bm{x}||\bm{y}|}
\cos(\bm{x}, \bm{y}) = \bm{x}\cdot\bm{y}
(\bm{x}-\bm{y})^2 = |\bm{x}|^2+2\bm{x}\cdot\bm{y}+|\bm{y}|^2
(\bm{x}-\bm{y})^2 = 2\bm{x}\cdot\bm{y} + 2
- \mathcal{M}(\bm{x} | \bm{\mu}, \kappa)
= \frac{\kappa^{M/2-1}}{ (2\pi)^{M/2} I_{M/2-1}(\kappa)}\exp(\kappa \bm{\mu}^{\top} \bm{x} )
I_{\alpha}(\kappa) = \frac{2^{-\alpha}\kappa^{-\alpha}}{\sqrt{\pi} \Gamma(\alpha + (1/2))}\int_0^{\phi}d\phi \sin^{2\alpha}\phi e^{\kappa \cos \phi}
- L(\bm{\mu}, \kappa | \mathcal{D}) = \ln \prod_{n=1}^{N}c_{M}(\kappa) e^{\kappa \bm{\mu}^{\top} \bm{x}^{(n)}}
= \sum_{n=1}^{N} \bigg( \ln c_{M}(\kappa) + \kappa \bm{\mu}^{\top} \bm{x}^{(n)} \bigg)
c_{M}(\kappa) = \frac{\kappa^{M/2-1}}{ (2\pi)^{M/2} I_{M/2-1}(\kappa)}
- L(\bm{\mu}, \kappa | \mathcal{D}) = \sum_{n=1}^{N} \bigg( \ln c_{M}(\kappa) + \kappa \bm{\mu}^{\top} \bm{x}^{(n)} \bigg)
\bm{\mu}^{\top}\bm{\mu} = 1
0 = \frac{\partial}{\partial \bm{\mu}} \bigg\{ L(\bm{\mu}, \kappa | \mathcal{D}) - \lambda \bm{\mu}^{\top} \bm{\mu} \bigg\} = \kappa \sum_{n=1}^{N} \bm{x}^{(n)} - 2 \lambda \bm{\mu}
- 0 = \kappa \sum_{n=1}^{N} \bm{x}^{(n)} - 2 \lambda \bm{\mu}
\hat{\bm{\mu}}=\frac{\bm{m}}{\sqrt{\bm{m}^{\top} \bm{m}}}
\bm{m} = \frac{1}{N}\sum_{n=1}^{N}\bm{x}^{(n)}
- 0 = \kappa \sum_{n=1}^{N} \bm{x}^{(n)} - 2 \lambda \bm{\mu}
\bm{m} = \frac{1}{N}\sum_{n=1}^{N}\bm{x}^{(n)}
\frac{\kappa \bm{m}}{2 \bm{\mu}} = \lambda
\bm{\mu}^{\top}\bm{\mu} = 1
\frac{\kappa^2 \bm{m}^{\top}\bm{m}}{2^2 } = \lambda^2
\frac{\kappa \sqrt{\bm{m}^{\top}\bm{m}}}{2 } = \lambda
\hat{\bm{\mu}}=\frac{\bm{m}}{\sqrt{\bm{m}^{\top} \bm{m}}}
- a(\bm{x}') = 1 - \hat{\bm{\mu}}^{\top}\bm{x}'
\hat{\bm{\mu}}^{\top}\bm{x}'
- q(z) = \int_{-\infty}^{\infty}d\bm{x}\delta(z-f(x_{1},...,x_{M}))p(x_{1},...,x_{M})
p(a) =\int_{S_M}d\bm{x}\delta(a-(1-\hat{\bm{\mu}}^{\top}\bm{x}))c_{M}(\kappa)\exp(\kappa \hat{\bm{\mu}}^{\top} \bm{x})
p(a) \propto \int_{0}^{\pi}d\theta_1 \sin^{M-2}\theta_1 \delta(a-(1-\cos \theta_1))\exp(\kappa \cos \theta_1)
q(z) = \int_{-\infty}^{\infty}dx\delta(x-b)f(x)=f(b)
p(a) \propto (2a-a^2)^{(M-3)/2}\exp(\kappa (1-a))
-
p(a) \propto (2a-a^2)^{(M-3)/2}\exp(\kappa (1-a))
p(a) \propto a^{(M-1)/2-1}\exp(-\kappa a)
- \bm{x'} \sim \mathcal{M}(\bm{\mu},\kappa)
1-\bm{\mu}^{\top}\bm{x'} \sim \chi ^2\bigg(M-1,\frac{1}{2\kappa}\bigg)
- \langle a \rangle = \int_{0}^{\infty} da \ a \chi^2(a|m,s) = ms
\langle a^2 \rangle = \int_{0}^{\infty} da \ a^2 \chi^2(a|m,s) = m(m+2)s^2
\langle a \rangle \approx \frac{1}{N}\sum_{n=1}^{N} a^{(n)}
\langle a^2 \rangle \approx \frac{1}{N}\sum_{n=1}^{N} (a^{(n)})^2
\hat{m}_{\mbox{mo}} = \frac{2\langle a \rangle ^2}{\langle a^2 \rangle - {\langle a \rangle}^2}
\hat{s}_{\mbox{mo}} = \frac{\langle a^2 \rangle - {\langle a \rangle}^2}{2\langle a \rangle ^2}
