numpy.random.multivariate_normal¶

numpy.random.
multivariate_normal
(mean, cov[, size, check_valid, tol])¶ Draw random samples from a multivariate normal distribution.
The multivariate normal, multinormal or Gaussian distribution is a generalization of the onedimensional 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 onedimensional normal distribution.
Parameters:  mean : 1D array_like, of length N
Mean of the Ndimensional distribution.
 cov : 2D array_like, of shape (N, N)
Covariance matrix of the distribution. It must be symmetric and positivesemidefinite for proper sampling.
 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 mbynbyk arrangement. Because each sample is Ndimensional, the output shape is(m,n,k,N)
. If no shape is specified, a single (ND) sample is returned. check_valid : { ‘warn’, ‘raise’, ‘ignore’ }, optional
Behavior when the covariance matrix is not positive semidefinite.
 tol : float, optional
Tolerance when checking the singular values in covariance matrix.
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 Ndimensional value drawn from the distribution.
Notes
The mean is a coordinate in Ndimensional 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 onedimensional or univariate normal distribution.
Covariance indicates the level to which two variables vary together. From the multivariate normal distribution, we draw Ndimensional 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 nonnegative elements, and only on the diagonal)
This geometrical property can be seen in two dimensions by plotting generated datapoints:
>>> mean = [0, 0] >>> cov = [[1, 0], [0, 100]] # diagonal covariance
Diagonal covariance means that points are oriented along x or yaxis:
>>> 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 positive semidefinite (a.k.a. nonnegativedefinite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed.
References
[1] Papoulis, A., “Probability, Random Variables, and Stochastic Processes,” 3rd ed., New York: McGrawHill, 1991. [2] 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], [0, 1]] >>> 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:
>>> list((x[0,0,:]  mean) < 0.6) [True, True]