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 Ndimensional 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:  m : array_like
A 1D or 2D 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)
, whereN
is the number of observations given (unbiased estimate). If bias is True, then normalization is byN
. These values can be overridden by using the keywordddof
in numpy versions >= 1.5. ddof : int, optional
If not
None
the default value implied by bias is overridden. Note thatddof=1
will return the unbiased estimate, even if both fweights and aweights are specified, andddof=0
will return the simple average. See the notes for the details. The default value isNone
.New in version 1.5.
 fweights : array_like, int, optional
1D array of integer frequency weights; the number of times each observation vector should be repeated.
New in version 1.10.
 aweights : array_like, optional
1D 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
anda = aweights
for brevity. The steps to compute the weighted covariance are as follows:>>> w = f * a >>> v1 = np.sum(w) >>> v2 = np.sum(w * a) >>> m = np.sum(m * w, axis=1, keepdims=True) / v1 >>> cov = np.dot(m * w, m.T) * v1 / (v1**2  ddof * v2)
Note that when
a == 1
, the normalization factorv1 / (v1**2  ddof * v2)
goes over to1 / (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) >>> 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