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# numpy.random.multivariate_normal

`numpy.random.``multivariate_normal`(mean, cov[, size])

  Draw random samples from a multivariate normal distribution.

  The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. Such a distribution is specified by its mean and covariance matrix. These parameters are analogous to the mean (average or “center”) and variance (standard deviation, or “width,” squared) of the one-dimensional normal distribution.

  Parameters : | mean : 1-D array_like, of length N Mean of the N-dimensional distribution. cov : 2-D array_like, of shape (N, N) Covariance matrix of the distribution. Must be symmetric and positive semi-definite for “physically meaningful” results. size : int or tuple of ints, optional Given a shape of, for example, `(m,n,k)`, `m*n*k` samples are generated, and packed in an m-by-n-by-k arrangement. Because each sample is N-dimensional, the output shape is `(m,n,k,N)`. If no shape is specified, a single (N-D) sample is returned.
  Returns : | out : ndarray The drawn samples, of shape size, if that was provided. If not, the shape is `(N,)`. In other words, each entry `out[i,j,...,:]` is an N-dimensional value drawn from the distribution.

  Notes

  The mean is a coordinate in N-dimensional space, which represents the location where samples are most likely to be generated. This is analogous to the peak of the bell curve for the one-dimensional or univariate normal distribution.

  Covariance indicates the level to which two variables vary together. From the multivariate normal distribution, we draw N-dimensional samples, . The covariance matrix element is the covariance of and . The element is the variance of (i.e. its “spread”).

  Instead of specifying the full covariance matrix, popular approximations include:

  > - Spherical covariance (cov is a multiple of the identity matrix)
  > - Diagonal covariance (cov has non-negative elements, and only on the diagonal)

  This geometrical property can be seen in two dimensions by plotting generated data-points:

      >>> mean = [0,0]
      >>> cov = [[1,0],[0,100]] # diagonal covariance, points lie on x or y-axis

      >>> import matplotlib.pyplot as plt
      >>> x,y = np.random.multivariate_normal(mean,cov,5000).T
      >>> plt.plot(x,y,'x'); plt.axis('equal'); plt.show()

  Note that the covariance matrix must be non-negative definite.

  References

  Papoulis, A., Probability, Random Variables, and Stochastic Processes, 3rd ed., New York: McGraw-Hill, 1991.

  Duda, R. O., Hart, P. E., and Stork, D. G., Pattern Classification, 2nd ed., New York: Wiley, 2001.

  Examples

      >>> mean = (1,2)
      >>> cov = [[1,0],[1,0]]
      >>> x = np.random.multivariate_normal(mean,cov,(3,3))
      >>> x.shape
      (3, 3, 2)

  The following is probably true, given that 0.6 is roughly twice the standard deviation:

      >>> print list( (x[0,0,:] - mean) < 0.6 )
      [True, True]