# 威沙特分佈

参数 ${\displaystyle n>0\!}$ 自由度 (實數)${\displaystyle \mathbf {V} >0\,}$ 尺度矩陣 (正定) ${\displaystyle \mathbf {W} \!}$是正定的 ${\displaystyle {\frac {\left|\mathbf {W} \right|^{\frac {n-p-1}{2}}}{2^{\frac {np}{2}}\left|{\mathbf {V} }\right|^{\frac {n}{2}}\Gamma _{p}({\frac {n}{2}})}}\exp \left(-{\frac {1}{2}}{\rm {Tr}}({\mathbf {V} }^{-1}\mathbf {W} )\right)}$ ${\displaystyle n\mathbf {V} }$ ${\displaystyle (n-p-1)\mathbf {V} {\text{ for }}n\geq p+1}$ ${\displaystyle \Theta \mapsto \left|{\mathbf {I} }-2i\,{\mathbf {\Theta } }{\mathbf {V} }\right|^{-n/2}}$

## 定義

${\displaystyle X_{(i)}{=}(x_{i}^{1},\dots ,x_{i}^{p})^{T}\sim N_{p}(0,V),}$

${\displaystyle S=X^{T}X=\sum _{i=1}^{n}X_{(i)}X_{(i)}^{T},\,\!}$

${\displaystyle \mathbf {S} }$有該機率分佈通常記為

${\displaystyle \mathbf {S} \sim W_{p}(\mathbf {V} ,n).}$

## 機率密度函數

${\displaystyle f_{\mathbf {W} }(w)={\frac {\left|w\right|^{(n-p-1)/2}\exp \left[-{\rm {trace}}({\mathbf {V} }^{-1}w/2)\right]}{2^{np/2}\left|{\mathbf {V} }\right|^{n/2}\Gamma _{p}(n/2)}}}$

${\displaystyle \Gamma _{p}(n/2)=\pi ^{p(p-1)/4}\Pi _{j=1}^{p}\Gamma \left[(n+1-j)/2\right].}$

## 特徵函數

${\displaystyle \Theta \mapsto \left|{\mathbf {I} }-2i\,{\mathbf {\Theta } }{\mathbf {V} }\right|^{-n/2}.}$

${\displaystyle \Theta \mapsto {\mathcal {E}}\left\{\mathrm {exp} \left[i\cdot \mathrm {trace} ({\mathbf {W} }{\mathbf {\Theta } })\right]\right\}=\left|{\mathbf {I} }-2i{\mathbf {\Theta } }{\mathbf {V} }\right|^{-n/2}}$

（這裡的${\displaystyle \Theta }$${\displaystyle {\mathbf {I} }}$ 皆為與${\displaystyle {\mathbf {V} }}$維度相同的矩陣。（${\displaystyle {\mathbf {I} }}$單位矩陣，而${\displaystyle i}$為－1的平方根）.[3]

## 理論架構

${\displaystyle \scriptstyle {\mathbf {W} }}$為一自由度為m，共變異矩陣為${\displaystyle \scriptstyle {\mathbf {V} }}$的威沙特分佈，記為—${\displaystyle \scriptstyle {\mathbf {W} }\sim {\mathbf {W} }_{p}({\mathbf {V} },m)}$—其中${\displaystyle \scriptstyle {\mathbf {C} }}$為一${\displaystyle q\times p}$q秩矩陣，則[4]

${\displaystyle {\mathbf {C} }{\mathbf {W} }{\mathbf {C} '}\sim {\mathbf {W} }_{q}\left({\mathbf {C} }{\mathbf {V} }{\mathbf {C} '},m\right).}$

### 推論1

${\displaystyle {\mathbf {z} }}$為一非負${\displaystyle p\times 1}$常數向量，則[4] ${\displaystyle {\mathbf {z} '}{\mathbf {W} }{\mathbf {z} }\sim \sigma _{z}^{2}\chi _{m}^{2}}$.

### 推論2

${\displaystyle {\mathbf {z} '}=(0,\ldots ,0,1,0,\ldots ,0)}$ 的情形下（亦即第j個元素為1其他為0），推論1可導出

${\displaystyle w_{jj}\sim \sigma _{jj}\chi _{m}^{2}}$

## 分佈抽樣

1. 生成一隨機${\displaystyle p\times p}$三角矩陣 ${\displaystyle {\textbf {A}}}$使得：
• ${\displaystyle a_{ii}=(\chi _{n-i+1}^{2})^{1/2}}$，意即 ${\displaystyle a_{ii}}$為一${\displaystyle \chi _{n-i+1}^{2}}$卡方分佈隨機樣本的平方根。
• ${\displaystyle a_{ij}}$其中${\displaystyle j，為一${\displaystyle N_{1}(0,1)}$常態分佈的隨機樣本。[8]
2. 計算${\displaystyle {\textbf {V}}={\textbf {L}}{\textbf {L}}^{T}}$Cholesky分解
3. 計算${\displaystyle {\textbf {X}}={\textbf {L}}{\textbf {A}}{\textbf {A}}^{T}{\textbf {L}}^{T}}$。此時，${\displaystyle {\textbf {X}}}$ 為一${\displaystyle W_{p}({\textbf {V}},n)}$的隨機樣本。

${\displaystyle {\textbf {V}}={\textbf {I}}}$，則因${\displaystyle {\textbf {V}}={\textbf {I}}{\textbf {I}}^{T}}$，可以直接以${\displaystyle {\textbf {X}}={\textbf {A}}{\textbf {A}}^{T}}$進行抽樣。

## 參考資料

1. ^ Wishart, J. The generalised product moment distribution in samples from a normal multivariate population. Biometrika. 1928, 20A (1–2): 32–52. JFM 54.0565.02. JSTOR 2331939. doi:10.1093/biomet/20A.1-2.32.
2. ^ Uhlig, H. On Singular Wishart and Singular Multivariate Beta Distributions. The Annals of Statistics. 1994, 22: 395–405. doi:10.1214/aos/1176325375.
3. ^ Anderson, T. W. An Introduction to Multivariate Statistical Analysis 3rd. Hoboken, N. J.: Wiley Interscience. 2003: 259. ISBN 0-471-36091-0.
4. Rao, C. R. Linear Statistical Inference and its Applications. Wiley. 1965: 535.
5. ^ Seber, George A. F. Multivariate Observations. Wiley. 2004. ISBN 978-0471691211.
6. ^ Chatfield, C.; Collins, A. J. Introduction to Multivariate Analysis. London: Chapman and Hall. 1980: 103–108. ISBN 0-412-16030-7.
7. ^ Smith, W. B.; Hocking, R. R. Algorithm AS 53: Wishart Variate Generator. Journal of the Royal Statistical Society, Series C. 1972, 21 (3): 341–345. JSTOR 2346290.
8. ^ Anderson, T. W. An Introduction to Multivariate Statistical Analysis 3rd. Hoboken, N. J.: Wiley Interscience. 2003: 257. ISBN 0-471-36091-0.