numpy.cov()
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,
, 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.
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 seerowvar
below.y : array_like, optional
An additional set of variables and observations.
y
has the same form as that ofm
.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). Ifbias
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 bybias
is overridden. Note thatddof=1
will return the unbiased estimate, even if bothfweights
andaweights
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
1-D array of integer freguency 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 letf = 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,
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
andy
are combined:>>> x = [-2.1, -1, 4.3] >>> y = [3, 1.1, 0.12] >>> X = np.vstack((x,y)) >>> 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
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https://docs.scipy.org/doc/numpy-1.11.0/reference/generated/numpy.cov.html