Covariance 协方差与协方差矩阵

协方差听起来像是跟方差相对应的概念，但是在我看来，正是“方差”这个概念可能一直或多或少的误导我对协方差的理解。在我看来，协方差只是在计算形式上长得很像方差，或者说是从方差演变而来，但实际意义还是有不小的差距。

1 方差

方差可以代表数据的分散程度，方差越小，数据点越集中于均值，方差越大，数据点距离均值的分散程度越大。首先回顾以下方差的定义。给定随机变量和对应的概率密度函数，定义为

具体对于连续型随机变量，方差计算方式如下：

而离散型性随机变量的计算方式则为

计算机接触到的都是离散变量，如果假设所有样本是相互独立的，方差可以通过以下方式计算：

其中为样本均值，即

通过对方差的定义式 (1) 进行展开，可以得到方差另外一种计算方法：

2 协方差

协方差定义在两个随机变量上。随机变量和的协方差计算方式如下：

其中为样本的数量，分别为随机变量和的均值。

可以看到协方差的形式跟方差有点像。但协方差代表的是随机变量和线性相关程度的大小。协方差可以为负数，协方差为负数时，绝对值越大代表和负线性相关的程度越大；当协方差为正数时，绝对值越大代表和正线性相关的程度越大。

将协方差进行归一化，就得到了相关系数

当或时，代表和成线性关系，即当，有；当，有。

协方差与方差的关系
两个随机变量的和的协方差为

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以此可见方差与协方差的关系。

3 协方差矩阵

为随机变量的协方差矩阵通常被记为%22%20aria-hidden%3D%22true%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-3A3%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E#card=math&code=%5CSigma&id=vje6i)，是一个矩阵。其中

其中为 随机变量的方差，为随机变量之间的协方差。

通过上面的定义可以看到协方差矩阵是个对称阵，故协方差矩阵可以进行特征分解。

害可以线性代数的语言来描述协方差矩阵。求的协方差可以分为两步，对于个样本中每一个样本，求，然后全部个样本上求和。记变量是一个列向量，记矩阵

其中为样本均值，而。

进而协方差矩阵的计算方式为

光这么看协方差矩阵感觉不知道它具体应该怎么用，需要结合案例。实际上，可以把协方差矩阵看作一个线性变换矩阵。协方差表征了两个随机变量之间的线性关系，而协方差矩阵则代表了两两随机变量之间的线性关系。如果将高维随机变量某一样本点通过协方差矩阵进行线性变换，得到，此时就服从于一个新的分布。

大多数情况下我们知道的分布，希望反推的分布，则可以计算进而得到的分布。具体示例见多元高斯分布。