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.
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
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.
- ddofint, 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.
- 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=0the array of weights can be used to assign probabilities to observation vectors.
New in version 1.10.
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)