协方差据/h1>
测验据/h4>
相关......据/h4>
- 可能性据/span>>据/span>
这据strong>协方差据/strong>概括对多个随机变量的差异概念。代替测量单个随机变量的波动,协方差衡量彼此两个变量的波动。据/p>
定义据/h2>
回忆,据一种href="//www.parkandroid.com/wiki/variance-definition/" class="wiki_link" title="方差" target="_blank">方差据/一种>是单个随机变量的平均值的平均平方偏差据span class="katex">
X据/span>:据span class="katex-display">
var.据/span>(据/span>X据/span>)据/span>=据/span>E.据/span>[据/span>(据/span>X据/span>-据/span>E.据/span>[据/span>X据/span>]据/span>)据/span>2据/span>]据/span>。据/span>协方差采用类似的功能形式。据/p>
这据strong>协方差据/strong>
COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>随机变量据span class="katex">
X据/span>和据span class="katex">
y据/span>被定义为据span class="katex-display">
COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>=据/span>E.据/span>[据/span>(据/span>X据/span>-据/span>E.据/span>[据/span>X据/span>]据/span>)据/span>(据/span>y据/span>-据/span>E.据/span>[据/span>y据/span>]据/span>)据/span>]据/span>。据/span>
现在,代替测量单个变量的波动,协方差措施据em>两个变量如何一起波动据/em>。对于强大的协方差,都是据span class="katex">
X据/span>-据/span>E.据/span>[据/span>X据/span>]据/span>和据span class="katex">
y据/span>-据/span>E.据/span>[据/span>y据/span>]据/span>必须在同一时间或换句话说,更改在一起。据p>
计算协方差据/h2>
通过服用找到协方差通常更简单据span class="katex-display">
COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>=据/span>E.据/span>[据/span>X据/span>y据/span>-据/span>E.据/span>[据/span>X据/span>]据/span>y据/span>-据/span>X据/span>E.据/span>[据/span>y据/span>]据/span>+据/span>E.据/span>[据/span>X据/span>]据/span>E.据/span>[据/span>y据/span>]据/span>]据/span>=据/span>E.据/span>[据/span>X据/span>y据/span>]据/span>-据/span>E.据/span>[据/span>X据/span>]据/span>E.据/span>[据/span>y据/span>]据/span>。据/span>换句话说,计算协方差,可以等同地找到据span class="katex">
E.据/span>[据/span>X据/span>y据/span>]据/span>除了手段据span class="katex">
X据/span>和据span class="katex">
y据/span>)。据/p>
让据span class="katex">
X据/span>和据span class="katex">
y据/span>是随机变量据/p>
找到据span class="katex">
COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>。据/p>
同样,人们可以在差异方面找到表达式:据span class="katex-display">
var.据/span>(据/span>X据/span>+据/span>y据/span>)据/span>=据/span>E.据/span>[据/span>(据/span>X据/span>+据/span>y据/span>-据/span>E.据/span>[据/span>X据/span>]据/span>-据/span>E.据/span>[据/span>y据/span>]据/span>)据/span>2据/span>]据/span>=据/span>E.据/span>[据/span>(据/span>X据/span>-据/span>E.据/span>[据/span>X据/span>]据/span>)据/span>2据/span>]据/span>+据/span>E.据/span>[据/span>(据/span>y据/span>-据/span>E.据/span>[据/span>y据/span>]据/span>)据/span>2据/span>]据/span>+据/span>2据/span>E.据/span>[据/span>(据/span>X据/span>-据/span>E.据/span>[据/span>X据/span>]据/span>)据/span>(据/span>y据/span>-据/span>E.据/span>[据/span>y据/span>]据/span>)据/span>]据/span>=据/span>var.据/span>(据/span>X据/span>)据/span>+据/span>var.据/span>(据/span>y据/span>)据/span>+据/span>2据/span>COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>。据/span> 这一结果的广义陈述如下。据/p>
总和的方差。据/strong>给定随机变量据span class="katex">
X据/span>一世据/span>,每个都有有限的方差,据/p>
var.据/span>(据/span>一世据/span>σ.据/span>X据/span>一世据/span>)据/span>=据/span>一世据/span>σ.据/span>var.据/span>(据/span>X据/span>一世据/span>)据/span>+据/span>2据/span>一世据/span>据据/span>j据/span>σ.据/span>COV.据/span>(据/span>X据/span>一世据/span>那据/span>X据/span>j据/span>)据/span>。据/span>
协方差 - 属性据/h2>
协方差继承了与之相同的属性据一种href="//www.parkandroid.com/wiki/inner-product/" class="wiki_link" title="内部产品" target="_blank">内部产品据/一种>从线性代数。证据涉及直接的代数,并作为读者作为练习。据/p>
给予常数据span class="katex"> 一种据/span>和随机变量据span class="katex"> X据/span>那据span class="katex"> y据/span>, 和据span class="katex"> Z.据/span>,以下属性持有:据/p>
- COV.据/span>(据/span>X据/span>那据/span>X据/span>)据/span>=据/span>var.据/span>(据/span>X据/span>)据/span>≥据/span>0.据/span>
- COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>=据/span>COV.据/span>(据/span>y据/span>那据/span>X据/span>)据/span>
- COV.据/span>(据/span>一种据/span>X据/span>那据/span>y据/span>)据/span>=据/span>一种据/span>COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>
- COV.据/span>(据/span>X据/span>那据/span>一种据/span>)据/span>=据/span>0.据/span>
- COV.据/span>(据/span>X据/span>+据/span>y据/span>那据/span>Z.据/span>)据/span>=据/span>COV.据/span>(据/span>X据/span>那据/span>Z.据/span>)据/span>+据/span>COV.据/span>(据/span>y据/span>那据/span>Z.据/span>)据/span>。据/li>
鉴于知识据span class="katex">
COV.据/span>(据/span>W.据/span>那据/span>y据/span>)据/span>那据span class="katex">
COV.据/span>(据/span>W.据/span>那据/span>Z.据/span>)据/span>那据span class="katex">
COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>, 和据span class="katex">
COV.据/span>(据/span>X据/span>那据/span>Z.据/span>)据/span>,以下哪项必须计算?据/p>
一世。据span class="katex">
COV.据/span>(据/span>W.据/span>+据/span>X据/span>那据/span>y据/span>+据/span>Z.据/span>)据/span> II。据span class="katex">
COV.据/span>(据/span>y据/span>+据/span>Z.据/span>那据/span>W.据/span>+据/span>X据/span>)据/span> III。据span class="katex">
COV.据/span>(据/span>W.据/span>那据/span>X据/span>+据/span>y据/span>+据/span>Z.据/span>)据/span> IV。据span class="katex">
COV.据/span>(据/span>W.据/span>那据/span>X据/span>+据/span>y据/span>+据/span>Z.据/span>)据/span>,如果知道这一点据span class="katex">
W.据/span>和据span class="katex">
X据/span>是独立的据/p>
让据span class="katex"> X据/span>和据span class="katex"> y据/span>是随机变量据span class="katex"> var.据/span>(据/span>X据/span>)据/span>=据/span>σ.据/span>2据/span>和据span class="katex"> y据/span>=据/span>一种据/span>X据/span>, 在哪里据span class="katex"> σ.据/span>和据span class="katex"> 一种据/span>是常数。决定据span class="katex"> COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>。据/p>
内部产品特性产量据/p>
COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>=据/span>COV.据/span>(据/span>X据/span>那据/span>一种据/span>X据/span>)据/span>=据/span>COV.据/span>(据/span>一种据/span>X据/span>那据/span>X据/span>)据/span>=据/span>一种据/span>COV.据/span>(据/span>X据/span>那据/span>X据/span>)据/span>=据/span>一种据/span>σ.据/span>2据/span>。据/span>
结果,据一种href="//www.parkandroid.com/wiki/cauchy-schwarz-inequality/" class="wiki_link" title="Cauchy-Schwarz不平等" target="_blank">Cauchy-Schwarz不平等据/一种>持有共聚航。据/p>
cauchy - schwarz不等式。据/strong>给定随机变量据span class="katex"> X据/span>和据span class="katex"> y据/span>那据/p>
[据/span>COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>]据/span>2据/span>≤.据/span>var.据/span>(据/span>X据/span>)据/span>var.据/span>(据/span>y据/span>)据/span>。据/span>
协方差的一个关键属性是,独立随机变量具有零协方差。据/p>
独立变量的协方差。据/strong>如果据span class="katex"> X据/span>和据span class="katex"> y据/span>是独立的随机变量,然后据span class="katex"> COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>=据/span>0.据/span>。据/span>
如果据span class="katex"> X据/span>和据span class="katex"> y据/span>是独立的,然后据span class="katex"> E.据/span>[据/span>X据/span>y据/span>]据/span>=据/span>E.据/span>[据/span>X据/span>]据/span>E.据/span>[据/span>y据/span>]据/span>因此据span class="katex"> COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>=据/span>0.据/span>。(回想起那个据span class="katex"> E.据/span>[据/span>X据/span>y据/span>]据/span>=据/span>E.据/span>[据/span>X据/span>]据/span>E.据/span>[据/span>y据/span>]据/span>这是事实的简单后果据span class="katex"> P.据/span>(据/span>X据/span>|据/span>y据/span>)据/span>=据/span>P.据/span>(据/span>X据/span>)据/span>。)据/p>
具有零协方差的依赖变量。据/strong>但是,交谈不完全是真实的。作为一个简单的例子,假设据span class="katex"> X据/span>是标准的正常随机变量,是据span class="katex"> y据/span>=据/span>X据/span>2据/span>。注意到知识据span class="katex"> X据/span>完全决定据span class="katex"> y据/span>在这种情况下据span class="katex"> X据/span>和据span class="katex"> y据/span>非常明确依赖。然而,通过对称性来保持这种情况据span class="katex-display"> COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>=据/span>E.据/span>[据/span>X据/span>y据/span>]据/span>-据/span>E.据/span>[据/span>X据/span>]据/span>E.据/span>[据/span>y据/span>]据/span>=据/span>0.据/span>。据/span>
简单的推论如下。据/p>
独立变量总和的方差。据/strong>给定独立随机变量据span class="katex"> X据/span>一世据/span>,每个都有有限的方差,据/p>
var.据/span>(据/span>一世据/span>σ.据/span>X据/span>一世据/span>)据/span>=据/span>一世据/span>σ.据/span>var.据/span>(据/span>X据/span>一世据/span>)据/span>。据/span>
自从此以来据span class="katex"> X据/span>一世据/span>是独立的,这一定是这样的据span class="katex"> COV.据/span>(据/span>X据/span>一世据/span>那据/span>X据/span>j据/span>)据/span>=据/span>0.据/span>对所有人据span class="katex"> 一世据/span>据/span>=据/span>j据/span>结果直接从总和定理的方差遵循。据/p>
协方差矩阵据/h2>
处理大量随机变量时据span class="katex">
X据/span>一世据/span>考虑一个有意义的是有意义的据strong>协方差矩阵据/strong>谁的据span class="katex">
m据/span>那据/span>N据/span>当然是据span class="katex">
COV.据/span>(据/span>X据/span>m据/span>那据/span>X据/span>N据/span>)据/span>。据/p>
自据span class="katex">
COV.据/span>(据/span>X据/span>那据/span>y据/span>)据/span>=据/span>COV.据/span>(据/span>y据/span>那据/span>X据/span>)据/span>,协方差矩阵是对称的。据/p>
参考文献据/h2>
[1]莫里斯H.据em>概率和统计数据据/em>。第二版。Addison-Wesley,1985年。据/p>