numpy.cov¶

numpy.cov(m, y=None, rowvar=True, bias=False, ddof=None, fweights=None, aweights=None)[source]¶

Estimate a covariance matrix, given data and weights.

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 $x_i$ .

See the notes for an outline of the algorithm.

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 : bool, optional If rowvar is True (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 : bool, optional Default normalization (False) is by `(N - 1)`, where `N` is the number of observations given (unbiased estimate). If bias is True, then normalization is by `N`. These values can be overridden by using the keyword `ddof` in numpy versions >= 1.5. ddof : int, optional If not `None` the default value implied by bias is overridden. Note that `ddof=1` will return the unbiased estimate, even if both fweights and aweights are specified, and `ddof=0` will return the simple average. See the notes for the details. The default value is `None`. New in version 1.5. fweights : array_like, int, optional 1-D array of integer frequency weights; the number of times each observation vector should be repeated. New in version 1.10. aweights : array_like, optional 1-D array of observation vector weights. These relative weights are typically large for observations considered “important” and smaller for observations considered less “important”. If `ddof=0` the array of weights can be used to assign probabilities to observation vectors. New in version 1.10.
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 : bool, optional: If rowvar is True (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 : bool, optional: Default normalization (False) is by (N - 1), where N is the number of observations given (unbiased estimate). If bias is True, then normalization is by N. These values can be overridden by using the keyword ddof in numpy versions >= 1.5.
ddof : int, optional: If not None the default value implied by bias is overridden. Note that ddof=1 will return the unbiased estimate, even if both fweights and aweights are specified, and ddof=0 will return the simple average. See the notes for the details. The default value is None.

New in version 1.5.
fweights : array_like, int, optional: 1-D array of integer frequency weights; the number of times each observation vector should be repeated.

New in version 1.10.
aweights : array_like, optional: 1-D array of observation vector weights. These relative weights are typically large for observations considered “important” and smaller for observations considered less “important”. If ddof=0 the array of weights can be used to assign probabilities to observation vectors.

New in version 1.10.

Returns:

out : ndarray: The covariance matrix of the variables.

See also

corrcoef: Normalized covariance matrix

Notes

Assume that the observations are in the columns of the observation array m and let f = fweights and a = aweights for brevity. The steps to compute the weighted covariance are as follows:

>>> m = np.arange(10, dtype=np.float64)
>>> f = np.arange(10) * 2
>>> a = np.arange(10) ** 2.
>>> ddof = 9 # N - 1
>>> w = f * a
>>> v1 = np.sum(w)
>>> v2 = np.sum(w * a)
>>> m -= np.sum(m * w, axis=None, keepdims=True) / v1
>>> cov = np.dot(m * w, m.T) * v1 / (v1**2 - ddof * v2)

Note that when a == 1, the normalization factor v1 / (v1**2 - ddof * v2) goes over to 1 / (np.sum(f) - ddof) as it should.

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.stack((x, y), axis=0)
>>> np.cov(X)
array([[11.71      , -4.286     ], # may vary
       [-4.286     ,  2.144133]])
>>> np.cov(x, y)
array([[11.71      , -4.286     ], # may vary
       [-4.286     ,  2.144133]])
>>> np.cov(x)
array(11.71)

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