multivariate_normal(mean, cov[, shape])
Draw samples from a multivariate normal distribution.
The multivariate normal, multinormal or Gaussian distribution is a generalisation 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.
- 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.
- 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.
normal scipy.stats.norm : Provides random variates, as well as probability density function, cumulative density function, etc.
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 = [x1, x2, ...xN]. The covariance matrix element Cij is the covariance of xi and xj. The element Cii is the variance of xi (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'); pyplot.show()
Note that the covariance matrix must be non-negative definite.
>>> 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]