flopscope.numpy.random.RandomState.multivariate

fnp.random.RandomState.multivariate_normal(self, mean, cov, size=None, check_valid='warn', tol=1e-08)

Draw random samples from a multivariate normal distribution.

Adapted from NumPy docs np.random.RandomState.multivariate_normal

Arearandom

Typecounted

NumPy Refnp.random.RandomState.multivariate_normal

Cost

\text{numel}(\text{output})

Flopscope Context

Legacy multivariate normal; cost = numel(output).

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 Generator instance instead; please see the 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.

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 = flops.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

footnote

1

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

footnote

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 = flops.random.multivariate_normal(mean, cov, (3, 3))
>>> x.shape
(3, 3, 2)

Here we generate 800 samples from the bivariate normal distribution with mean [0, 0] and covariance matrix [[6, -3], [-3, 3.5]]. The expected variances of the first and second components of the sample are 6 and 3.5, respectively, and the expected correlation coefficient is -3/sqrt(6*3.5) ≈ -0.65465.

>>> cov = flops.array([[6, -3], [-3, 3.5]])
>>> pts = flops.random.multivariate_normal([0, 0], cov, size=800)

Check that the mean, covariance, and correlation coefficient of the sample are close to the expected values:

>>> pts.mean(axis=0)
array([ 0.0326911 , -0.01280782])  # may vary
>>> flops.cov(pts.T)
array([[ 5.96202397, -2.85602287],
       [-2.85602287,  3.47613949]])  # may vary
>>> flops.corrcoef(pts.T)[0, 1]
-0.6273591314603949  # may vary

We can visualize this data with a scatter plot. The orientation of the point cloud illustrates the negative correlation of the components of this sample.

>>> import matplotlib.pyplot as plt
>>> plt.plot(pts[:, 0], pts[:, 1], '.', alpha=0.5)
>>> plt.axis('equal')
>>> plt.grid()
>>> plt.show()

flopscope.numpy.random.RandomState.multivariate_normal

Parameters

Returns

See also

Notes

References

Examples