python - numpy 内部如何实现协方差？

Question

这是协方差矩阵的定义。http://en.wikipedia.org/wiki/Covariance_matrix#Definition

矩阵中的每个元素，除了主对角线，（如果我没记错的话）简化为 E(x_{i} * x_{j}) - mean(i)*mean(j) 其中 i 和 j 是行协方差矩阵的编号和列编号。

从 numpy 文档中，

x = np.array([[0, 2], [1, 1], [2, 0]]).T
x
array([[0, 1, 2], [2, 1, 0]])    
np.cov(x)
array([[ 1., -1.],
   [-1.,  1.]])

第一行即 [0, 1, 2] 对应于 X_{0}，第二行即 [2, 1, 0] 对应于 X_{1} X_{0}*X_{1} 的期望值是如何计算的，因为随机变量的分布是未知的？

谢谢。

Answer 1

只需检查代码。
cov在\site-packages\numpy\lib\function_base.py

def cov(m, y=None, rowvar=1, bias=0, ddof=None):
    """
    Estimate a covariance matrix, given data.

    Covariance indicates the level to which two variables vary together.
    If we examine N-dimensional samples, :math:`X = [x_1, x_2, ... x_N]^T`,
    then the covariance matrix element :math:`C_{ij}` is the covariance of
    :math:`x_i` and :math:`x_j`. The element :math:`C_{ii}` is the variance
    of :math:`x_i`.

    Parameters
    ----------
    m : array_like
        A 1-D or 2-D array containing multiple variables and observations.
        Each row of `m` represents a variable, and each column a single
        observation of all those variables. Also see `rowvar` below.

...

    if y is not None:
        y = array(y, copy=False, ndmin=2, dtype=float)
        X = concatenate((X,y), axis)

    X -= X.mean(axis=1-axis)[tup]
    if rowvar:
        N = X.shape[1]
    else:
        N = X.shape[0]

    if ddof is None:
        if bias == 0:
            ddof = 1
        else:
            ddof = 0
    fact = float(N - ddof)

    if not rowvar:
        return (dot(X.T, X.conj()) / fact).squeeze()
    else:
        return (dot(X, X.T.conj()) / fact).squeeze()