cov(m, y=None, rowvar=True, bias=False, ddof=None, fweights=None, aweights=None)¶
Estimate a covariance matrix, given data and weights.
Covariance indicates the level to which two variables vary together. If we examine N-dimensional samples, , then the covariance matrix element is the covariance of and . The element is the variance of .
See the notes for an outline of the algorithm.
- 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
Nis 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
ddofin numpy versions >= 1.5.
- ddof : int, optional
Nonethe default value implied by bias is overridden. Note that
ddof=1will return the unbiased estimate, even if both fweights and aweights are specified, and
ddof=0will return the simple average. See the notes for the details. The default value is
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=0the array of weights can be used to assign probabilities to observation vectors.
New in version 1.10.
- out : ndarray
The covariance matrix of the variables.
- Normalized covariance matrix
Assume that the observations are in the columns of the observation array m and let
f = fweightsand
a = aweightsfor 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.
Consider two variables, and , 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 increases while decreases. The covariance matrix shows this clearly:
>>> np.cov(x) array([[ 1., -1.], [-1., 1.]])
Note that element , which shows the correlation between and , 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)