查看“威沙特分佈”的源代码

{{Link style|time=2015-12-11T09:57:38+00:00}}
{{機率分佈
|name       =威沙特
|type       =密度
|pdf_image  =
|cdf_image  =
|parameters =<math> n > 0\!</math> 自由度 ([[實數]])<br /><math>\mathbf{V} > 0\,</math> 尺度矩陣 ([[正定矩陣|正定]])
|support    =<math>\mathbf{W}\!</math>是正定的
|pdf        =<math>\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)</math>
|cdf        =
|mean       =<math>n \mathbf{V}</math>
|median     =
|mode       =<math>(n-p-1)\mathbf{V}\text{ for }n \geq p+1</math>
|variance   =
|skewness   =
|kurtosis   =
|entropy    =
|mgf        =
|char       =<math>\Theta \mapsto \left|{\mathbf I} - 2i\,{\mathbf\Theta}{\mathbf V}\right|^{-n/2}</math>
}}

以[[統計學家]][[约翰·威沙特]]為名的'''威沙特分佈'''是[[統計學]]上的一種[[半正定]]矩陣隨機分佈。<ref>{{cite journal |first=J. |last=Wishart |authorlink=John Wishart (statistician) |title=The generalised product moment distribution in samples from a normal multivariate population |journal=[[Biometrika]] |volume=20A |issue=1–2 |pages=32–52 |year=1928 |doi=10.1093/biomet/20A.1-2.32 |jfm=54.0565.02 |jstor=2331939}}</ref>這個分佈在[[多變量分析]]的[[协方差矩阵|共變異矩陣]]估計上相當重要。

== 定義 ==

假設''X''為一''n'' × ''p''矩陣，其各行（row）來自同一均值向量為<math>\mathbf 0</math>的<math>p</math>維[[多變量常態分佈]]且彼此[[統計獨立性|獨立]]。
:<math>X_{(i)}{=}(x_i^1,\dots,x_i^p)^T\sim N_p(0,V),</math>

則威沙特分佈為<math>p\times p</math>[[:en:Scatter matrix|散異矩陣]]
:<math>S=X^T X = \sum_{i = 1}^{n} X_{(i)} X_{(i)}^T, \,\!</math>
的[[機率分佈]]。

<math>\mathbf S</math>有該機率分佈通常記為
:<math>\mathbf S\sim W_p(\mathbf V,n).</math>

其中正[[整數]]<math>n</math>為[[自由度]]。有時亦記號為<math>W(\mathbf V, p, n)</math>。若<math> p = 1</math>且<math>\mathbf V=1</math>則該分佈退化為一自由度為<math>n</math>的單變量[[卡方分佈]]。

== 常見應用 ==

威沙特分佈常用於多變量的[[概似比檢定]]，亦用於[[隨機矩陣]]的頻譜理論中。

== 機率密度函數 ==

威沙特分佈具有下述的[[機率密度函數]]：

令'<math>\mathbf W</math>為一<math>p\times p</math>正定對稱隨機變數矩陣。令<math>\mathbf V</math>為一特定正定<math>p\times p</math>矩陣。

如此，若<math>n > p</math>，則<math>\mathbf W</math>服從於一具自由度''n''的威沙特分佈且有機率度函數<math>f_W</math>

:<math>
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)
}
</math>

其中<math>\Gamma_p(\cdot)</math>為[[多變量Gamma分佈]]，其定義為

:<math>
\Gamma_p(n/2)=
\pi^{p(p-1)/4}\Pi_{j=1}^p
\Gamma\left[ (n+1-j)/2\right].
</math>

上述定義可推廣至任一實數<math>n> p -1</math><ref name="Uhlig1994">{{Cite journal | doi = 10.1214/aos/1176325375| title = On Singular Wishart and Singular Multivariate Beta Distributions| journal = The Annals of Statistics| volume = 22| pages = 395–405| year = 1994| last1 = Uhlig | first1 = H. }}</ref>

== 特徵函數 ==
威沙特分佈的[[特徵函數]]為
:<math>
\Theta \mapsto \left|{\mathbf I} - 2i\,{\mathbf\Theta}{\mathbf V}\right|^{-n/2}.
</math>
也就是說
:<math>\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}
</math>

其中<math>{\mathcal E}(\cdot)</math>為期望值

（這裡的<math>\Theta</math>及<math>{\mathbf I}</math> 皆為與<math>{\mathbf V}</math>維度相同的矩陣。（<math>{\mathbf I}</math> 為[[單位矩陣]]，而<math>i</math>為－1的[[平方根]]）.<ref>{{cite book | last = Anderson | first = T. W. | authorlink = T. W. Anderson | title = An Introduction to Multivariate Statistical Analysis | publisher = [[Wiley Interscience]] | edition = 3rd | location = Hoboken, N. J. | year = 2003 | page = 259 | isbn = 0-471-36091-0 }}</ref>

== 理論架構 ==

若<math>\scriptstyle {\mathbf W}</math>為一自由度為''m''，共變異矩陣為<math>\scriptstyle {\mathbf V}</math>的威沙特分佈，記為—<math>\scriptstyle {\mathbf W}\sim{\mathbf W}_p({\mathbf V},m)</math>—其中<math>\scriptstyle{\mathbf C}</math>為一<math>q\times p</math>的''q''秩矩陣，則<ref name="rao">{{cite book |last=Rao |first=C. R. |title=Linear Statistical Inference and its Applications |location= |publisher=Wiley |year=1965 |page=535 }}</ref>
:<math>
{\mathbf C}{\mathbf W}{\mathbf C'}
\sim
{\mathbf W}_q\left({\mathbf C}{\mathbf V}{\mathbf C'},m\right).
</math>

=== 推論1 ===
若<math>{\mathbf z}</math>為一非負<math>p\times 1</math>常數向量，則<ref name="rao"/>
<math>{\mathbf z'}{\mathbf W}{\mathbf z}\sim\sigma_z^2\chi_m^2</math>.

則在此情形下，<math>\chi_m^2</math>為一[[卡方分佈]]且<math>\sigma_z^2={\mathbf z'}{\mathbf V}{\mathbf z}</math>（因<math>{\mathbf V}</math>為正定，所以<math>\sigma_z^2</math>為一正常數）。

=== 推論2 ===
在<math>{\mathbf z'}=(0,\ldots,0,1,0,\ldots,0)</math> 的情形下（亦即第j個元素為1其他為0），推論1可導出
:<math>w_{jj}\sim\sigma_{jj}\chi^2_m</math>
為矩陣的每一個對對角元素的邊際分佈。

統計學家[[:en:George Seber|George Seber]]曾論證威沙特分佈並非多變量卡方分佈，這是因為非對角元素的邊際分佈並非卡方分佈，Seber傾向於將''某某多變量''分佈此一遣詞用於所有元素的邊際分佈皆相同的情形。<ref>{{cite book | last = Seber | first = George A. F. | title = Multivariate Observations | publisher = [[John Wiley & Sons|Wiley]] | year = 2004 | isbn = 978-0471691211 }}</ref>

== 多變量常態分佈的估計 ==

由於威沙特分佈可視為一多變量常態分佈其[[协方差矩阵|共變異矩陣]]的[[最大概似估計量]]（MLE）的的分佈，其衍自MLE的計算可為令人驚喜地簡約而優雅。<ref>{{cite book |first=C. |last=Chatfield |first2=A. J. |last2=Collins |year=1980 |title=Introduction to Multivariate Analysis |location=London |publisher=Chapman and Hall |pages=103–108 |isbn=0-412-16030-7 }}</ref> 基於[[頻譜理論]]，可將一純量視為一<math>1\times 1</math>矩陣的跡（trace）。請參考[[共變異矩陣的估計]]。

== 分佈抽樣 ==

以下的演算法取材自 Smith & Hocking (1972)。<ref>{{cite journal |title=Algorithm AS 53: Wishart Variate Generator |first1= W. B. |last1=Smith |first2= R. R. |last2=Hocking |journal=[[Journal of the Royal Statistical Society, Series C]] |volume=21 |issue=3 |year=1972 |pages=341–345 |jstor=2346290}}</ref>一個來自自由度為''n''及共變異矩陣為<math>\mathbf V</math>的威沙特分佈的<math>p\times p</math>（其中<math>n \geq p</math>）隨機樣本可以如下方式抽樣而得：

# 生成一隨機<math>p\times p</math>下[[三角矩陣]] <math>{\textbf A}</math>使得：
#* <math>a_{ii}=(\chi^2_{n-i+1})^{1/2}</math>，意即 <math>a_{ii}</math>為一<math>\chi^2_{n-i+1}</math>卡方分佈隨機樣本的平方根。 
#* <math>a_{ij}</math>其中<math>j<i</math>，為一<math>N_1(0,1)</math>[[常態分佈]]的隨機樣本。<ref>{{cite book | last = Anderson | first = T. W. | authorlink = T. W. Anderson | title = An Introduction to Multivariate Statistical Analysis | publisher = [[Wiley Interscience]] | edition = 3rd | location = Hoboken, N. J. | year = 2003 | page = 257 | isbn = 0-471-36091-0 }}</ref>
# 計算<math>{\textbf V} = {\textbf L}{\textbf L}^T</math>的[[Cholesky分解]]。
# 計算<math>{\textbf X} = {\textbf L}{\textbf A}{\textbf A}^T{\textbf L}^T</math>。此時，<math>{\textbf X}</math> 為一<math>W_p({\textbf V},n)</math>的隨機樣本。

若<math>{\textbf V}={\textbf I}</math>，則因<math>{\textbf V}={\textbf I}{\textbf I}^T</math>，可以直接以<math>{\textbf X} = {\textbf A}{\textbf A}^T</math>進行抽樣。

== 參考條目 ==

* [[共變異矩陣的估計]]
* [[Hotelling的T平方分佈]]
* [[逆威沙特分佈]]

== 參考資料 ==
{{Reflist}}
{{Refbegin}}
*{{cite book |last=Gelman |first=Andrew |date=2003 |title=Bayesian Data Analysis |url=http://www.stat.columbia.edu/~gelman/book/ |publisher=Chapman & Hall |page=582 |isbn=158488388X |access-date=3 June 2015 |location=Boca Raton, Fla. |edition=2nd}}
*{{cite journal| last=Zanella| first=A.|author2=Chiani, M. |author3=Win, M.Z. |title=On the marginal distribution of the eigenvalues of wishart matrices| journal=IEEE Transactions on Communications|date=April 2009| volume=57| issue=4| pages=1050–1060 | doi=10.1109/TCOMM.2009.04.070143}}
*{{cite book |first=C. M. |last=Bishop |title=Pattern Recognition and Machine Learning |location= |publisher=Springer |year=2006 |page=693}}
*{{cite journal | last1 = Pearson | first1 = Karl | author1-link = Karl Pearson | last2 = Jeffery | first2 = G. B. | author2-link = George Barker Jeffery | last3 = Elderton | first3 = Ethel M. | author3-link = Ethel M. Elderton | title = On the Distribution of the First Product Moment-Coefficient, in Samples Drawn from an Indefinitely Large Normal Population | journal = Biometrika | volume = 21 | issue = | pages = 164–201 | publisher = Biometrika Trust | date = December 1929  | jstor = 2332556 | doi = 10.2307/2332556}}
*{{cite journal | last = Craig | first = Cecil C. | title = On the Frequency Function of xy | journal = Ann. Math. Statist. | volume = 7 | issue = | pages = 1–15 | year = 1936 | url = http://projecteuclid.org/euclid.aoms/1177732541 | doi = 10.1214/aoms/1177732541}}
*{{cite journal |doi=10.1214/aop/1176990455 |last=Peddada and Richards |first1=Shyamal Das |last2=Richards |first2=Donald St. P. |title=Proof of a Conjecture of M. L. Eaton on the Characteristic Function of the Wishart Distribution, |journal=[[Annals of Probability]] |volume=19 |issue=2 |pages=868–874 |year=1991 }}
*{{cite journal |doi=10.1007/BF01078179 |first=S.G. |last=Gindikin |title=Invariant generalized functions in homogeneous domains, |journal=[[Funct. Anal. Appl.]] |volume=9 |issue=1 |pages=50–52 |year=1975}}
*{{cite journal |first=Paul S. |last=Dwyer |title=Some Applications of Matrix Derivatives in Multivariate Analysis |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assoc.]] |year=1967 |volume=62 |issue=318 |pages=607–625 |jstor=2283988 }}
{{Refend}}

{{概率分布类型列表|威沙特分佈}}

[[Category:连续分布]]
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