multivariate_normal(mean, cov[, shape])

Draw 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, which are analogous to the mean (average or "centre") and variance (standard deviation squared or "width") of the one-dimensional normal distribution.

Parameters

mean : (N,) ndarray

Mean of the N-dimensional distribution.

cov : (N, N) ndarray

Covariance matrix of the distribution.

shape : 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 sample is returned.

Returns

out : ndarray

The drawn samples, arranged according to shape. If the shape given is (m, n, ...), then the shape of out is (m, n, ... , N).

In other words, each entry out[i, j, ... , :] is an N-dimensional value drawn from the distribution.

See also

normal scipy.stats.norm : Provides random variates, as well as probability density

function, cumulative density function, etc.

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, X = [x_1, x_2, ... , x_N]. 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 (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
>>> x, y = np.random.multivariate_normal(mean, cov, 5000).T

>>> import matplotlib.pyplot as plt
>>> plt.plot(x, y, 'x'); plt.axis('equal'); plt.show()

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

References

[1]	A. Papoulis, "Probability, Random Variables, and Stochastic Processes," 3rd ed., McGraw-Hill Companies, 1991

[2]	R.O. Duda, P.E. Hart, and D.G. Stork, "Pattern Classification," 2nd ed., 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:

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