multivariate_normal(mean, cov, size=None, check_valid='warn', tol=1e-8)

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.

.. note:: New code should use the ``multivariate_normal`` method of a ``default_rng()`` instance instead; see `random-quick-start`.

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)``, ``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. 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. cov is cast to double before the check.

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.

See Also -------- Generator.multivariate_normal: which should be used for new code.

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, :math:`X = `x_1, x_2, ... x_N`

`. The covariance matrix element :math:`C_*j*` is the covariance of :math:`x_i` and :math:`x_j`. The element :math:`C_*i*` is the variance of :math:`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 ---------- .. `1`

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

# random