numpy.cov¶

numpy.cov(m, y=None, rowvar=1, bias=0, ddof=None)[source]¶

Estimate a covariance matrix, given data.

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

Parameters:

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. y : array_like, optional An additional set of variables and observations. y has the same form as that of m. rowvar : int, optional If rowvar is non-zero (default), then each row represents a variable, with observations in the columns. Otherwise, the relationship is transposed: each column represents a variable, while the rows contain observations. bias : int, optional Default normalization is by `(N - 1)`, where `N` is the number of observations given (unbiased estimate). If bias is 1, then normalization is by `N`. These values can be overridden by using the keyword `ddof` in numpy versions >= 1.5. ddof : int, optional New in version 1.5. If not `None` normalization is by `(N - ddof)`, where `N` is the number of observations; this overrides the value implied by `bias`. The default value is `None`.
Returns:	out : ndarray The covariance matrix of the variables.

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.

y : array_like, optional

An additional set of variables and observations. y has the same form as that of m.

rowvar : int, optional

If rowvar is non-zero (default), then each row represents a variable, with observations in the columns. Otherwise, the relationship is transposed: each column represents a variable, while the rows contain observations.

bias : int, optional

Default normalization is by (N - 1), where N is the number of observations given (unbiased estimate). If bias is 1, then normalization is by N. These values can be overridden by using the keyword ddof in numpy versions >= 1.5.

ddof : int, optional

New in version 1.5.

If not None normalization is by (N - ddof), where N is the number of observations; this overrides the value implied by bias. The default value is None.

Returns:

out : ndarray

The covariance matrix of the variables.

See also

corrcoef: Normalized covariance matrix

Examples

Consider two variables, $x_0$ and $x_1$ , which correlate perfectly, but in opposite directions:

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

Note how $x_0$ increases while $x_1$ decreases. The covariance matrix shows this clearly:

>>> np.cov(x)
array([[ 1., -1.],
       [-1.,  1.]])

Note that element $C_{0,1}$ , which shows the correlation between $x_0$ and $x_1$ , is negative.

Further, note how x and y are combined:

>>> x = [-2.1, -1,  4.3]
>>> y = [3,  1.1,  0.12]
>>> X = np.vstack((x,y))
>>> print np.cov(X)
[[ 11.71        -4.286     ]
 [ -4.286        2.14413333]]
>>> print np.cov(x, y)
[[ 11.71        -4.286     ]
 [ -4.286        2.14413333]]
>>> print np.cov(x)
11.71

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