python - 为什么 numpy cov 对角元素和 var 函数有不同的值？

Question

In [127]: x = np.arange(2)

In [128]: np.cov(x,x)
Out[128]:
array([[ 0.5,  0.5],
       [ 0.5,  0.5]])

In [129]: x.var()
Out[129]: 0.25

为什么会这样？我认为协方差矩阵对角线元素应该是序列的方差。

Answer 1

在 numpy 中，cov 默认为 1 的“delta degree of freedom”，而 var 默认为 0 的 ddof。从注释到 numpy。变种

Notes
-----
The variance is the average of the squared deviations from the mean,
i.e.,  ``var = mean(abs(x - x.mean())**2)``.

The mean is normally calculated as ``x.sum() / N``, where ``N = len(x)``.
If, however, `ddof` is specified, the divisor ``N - ddof`` is used
instead.  In standard statistical practice, ``ddof=1`` provides an
unbiased estimator of the variance of a hypothetical infinite population.
``ddof=0`` provides a maximum likelihood estimate of the variance for
normally distributed variables.

所以你可以通过以下方式让他们同意:

In [69]: cov(x,x)#defaulting to ddof=1
Out[69]: 
array([[ 0.5,  0.5],
       [ 0.5,  0.5]])

In [70]: x.var(ddof=1)
Out[70]: 0.5

In [71]: cov(x,x,ddof=0)
Out[71]: 
array([[ 0.25,  0.25],
       [ 0.25,  0.25]])

In [72]: x.var()#defaulting to ddof=0
Out[72]: 0.25