# numpy.cov

numpy.cov(m, y=None, rowvar=True, bias=False, ddof=None, fweights=None, aweights=None, *, dtype=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, \(X = [x_1, x_2, ... x_N]^T\), 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.

  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 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 `N` is 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 `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 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=0` the array of weights can be used to assign probabilities to observation vectors.

      New in version 1.10.

    dtype data-type, optional

      Data-type of the result. By default, the return data-type will have at least `numpy.float64` precision.

      New in version 1.20.

  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:

      >>> m = np.arange(10, dtype=np.float64)
      >>> f = np.arange(10) * 2
      >>> a = np.arange(10) ** 2.
      >>> ddof = 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.

  Examples

  Consider two variables, \(x_0\) and \(x_1\), 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 \(x_0\) increases while \(x_1\) decreases. The covariance matrix shows this clearly:

      >>> np.cov(x)
      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 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)