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:

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. It must be symmetric and positive-semidefinite for proper sampling. size : int or tuple of ints, optional Given a shape of, for example, `(m,n,k)`, `mnk` 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.

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. It must be symmetric and positive-semidefinite 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 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, $X = [x_1, x_2, ... x_N]$ . 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

Diagonal covariance means that points are oriented along 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 positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed.

References

[R241]

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

[R242]

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]

Links:

https://docs.scipy.org/doc/numpy-1.11.0/reference/generated/numpy.random.multivariate_normal.html

doc_NumPy

2017-01-10 18:18:06

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