numpy.cov   numpy.cov(m, y=None, rowvar=True, bias=False, ddof=None, fweights=None, aweights=None, *, dtype=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 
 
marray_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.  
yarray_like, optional


An additional set of variables and observations. y has the same form as that of m.  
rowvarbool, 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.  
biasbool, 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.  
ddofint, 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.   
fweightsarray_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.   
aweightsarray_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.   
dtypedata-type, optional


Data-type of the result. By default, the return data-type will have at least numpy.float64 precision.  New in version 1.20.     Returns 
 
outndarray


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 = 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)