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 : | 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. Must be symmetric and
positive semi-definite for “physically meaningful” results.
size : 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,
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is the covariance of
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and
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The element
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is the variance of
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(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
>>> 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 non-negative definite.
References
Papoulis, A., Probability, Random Variables, and Stochastic Processes,
3rd ed., New York: McGraw-Hill, 1991.
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],[1,0]]
>>> 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]