cov(m, y=None, rowvar=1, bias=0, ddof=None)
This docstring was copied from numpy.cov.
Some inconsistencies with the Dask version may exist.
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.
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) # doctest: +SKIP >>> f = np.arange(10) * 2 # doctest: +SKIP >>> a = np.arange(10) ** 2. # doctest: +SKIP >>> ddof = 1 # doctest: +SKIP >>> w = f * a # doctest: +SKIP >>> v1 = np.sum(w) # doctest: +SKIP >>> v2 = np.sum(w * a) # doctest: +SKIP >>> m -= np.sum(m * w, axis=None, keepdims=True) / v1 # doctest: +SKIP >>> cov = np.dot(m * w, m.T) * v1 / (v1**2 - ddof * v2) # doctest: +SKIP
Note that when a == 1
, the normalization factor v1 / (v1**2 - ddof * v2)
goes over to 1 / (np.sum(f) - ddof)
as it should.
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 :None:None:`rowvar` below.
An additional set of variables and observations. y has the same form as that of m.
If :None:None:`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.
Default normalization (False) is by (N - 1)
, where N
is the number of observations given (unbiased estimate). If :None:None:`bias` is True, then normalization is by N
. These values can be overridden by using the keyword ddof
in numpy versions >= 1.5.
If not None
the default value implied by :None:None:`bias` is overridden. Note that ddof=1
will return the unbiased estimate, even if both :None:None:`fweights` and :None:None:`aweights` are specified, and ddof=0
will return the simple average. See the notes for the details. The default value is None
.
1-D array of integer frequency weights; the number of times each observation vector should be repeated.
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.
Data-type of the result. By default, the return data-type will have at least numpy.float64
precision.
The covariance matrix of the variables.
Estimate a covariance matrix, given data and weights.
corrcoef
Normalized covariance matrix
Consider two variables, $x_0$ and $x_1$ , which correlate perfectly, but in opposite directions:
This example is valid syntax, but we were not able to check execution>>> x = np.array([[0, 2], [1, 1], [2, 0]]).T # doctest: +SKIP
... x # doctest: +SKIP array([[0, 1, 2], [2, 1, 0]])
Note how $x_0$ increases while $x_1$ decreases. The covariance matrix shows this clearly:
This example is valid syntax, but we were not able to check execution>>> np.cov(x) # doctest: +SKIP 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 :None:None:`x` and y are combined:
>>> x = [-2.1, -1, 4.3] # doctest: +SKIPThis example is valid syntax, but we were not able to check execution
... y = [3, 1.1, 0.12] # doctest: +SKIP
... X = np.stack((x, y), axis=0) # doctest: +SKIP
... np.cov(X) # doctest: +SKIP array([[11.71 , -4.286 ], # may vary [-4.286 , 2.144133]])
>>> np.cov(x, y) # doctest: +SKIP array([[11.71 , -4.286 ], # may vary [-4.286 , 2.144133]])This example is valid syntax, but we were not able to check execution
>>> np.cov(x) # doctest: +SKIP array(11.71)See :
The following pages refer to to this document either explicitly or contain code examples using this.
dask.array.routines.corrcoef
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