#### Previous topic

numpy.correlate

#### Next topic

numpy.histogram

# numpy.cov

`numpy.``cov`(m, y=None, rowvar=1, bias=0, 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.

  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 see rowvar below. y : array_like, optional An additional set of variables and observations. y has the same form as that of m. rowvar : int, optional If rowvar is non-zero (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 : int, optional Default normalization is by `(N - 1)`, where `N` corresponds to the number of observations given (unbiased estimate). If bias is 1, then normalization is by `N`. These values can be overridden by using the keyword `ddof` in numpy versions >= 1.5. ddof : int, 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. 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 let `f = fweights` and `a = 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 factor `v1 / (v1**2 - ddof * v2)` goes over to `1 / (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 and y 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