scipy.stats.matrix_normal#
- scipy.stats.matrix_normal = <scipy.stats._multivariate.matrix_normal_gen object>[source]#
- A matrix normal random variable. - The mean keyword specifies the mean. The rowcov keyword specifies the among-row covariance matrix. The ‘colcov’ keyword specifies the among-column covariance matrix. - Parameters:
- meanarray_like, optional
- Mean of the distribution (default: None) 
- rowcovarray_like, optional
- Among-row covariance matrix of the distribution (default: - 1)
- colcovarray_like, optional
- Among-column covariance matrix of the distribution (default: - 1)
- seed{None, int, np.random.RandomState, np.random.Generator}, optional
- Used for drawing random variates. If seed is None, the RandomState singleton is used. If seed is an int, a new - RandomStateinstance is used, seeded with seed. If seed is already a- RandomStateor- Generatorinstance, then that object is used. Default is None.
 
 - Notes - If mean is set to None then a matrix of zeros is used for the mean. The dimensions of this matrix are inferred from the shape of rowcov and colcov, if these are provided, or set to - 1if ambiguous.- rowcov and colcov can be two-dimensional array_likes specifying the covariance matrices directly. Alternatively, a one-dimensional array will be be interpreted as the entries of a diagonal matrix, and a scalar or zero-dimensional array will be interpreted as this value times the identity matrix. - The covariance matrices specified by rowcov and colcov must be (symmetric) positive definite. If the samples in X are \(m \times n\), then rowcov must be \(m \times m\) and colcov must be \(n \times n\). mean must be the same shape as X. - The probability density function for - matrix_normalis\[f(X) = (2 \pi)^{-\frac{mn}{2}}|U|^{-\frac{n}{2}} |V|^{-\frac{m}{2}} \exp\left( -\frac{1}{2} \mathrm{Tr}\left[ U^{-1} (X-M) V^{-1} (X-M)^T \right] \right),\]- where \(M\) is the mean, \(U\) the among-row covariance matrix, \(V\) the among-column covariance matrix. - The allow_singular behaviour of the - multivariate_normaldistribution is not currently supported. Covariance matrices must be full rank.- The - matrix_normaldistribution is closely related to the- multivariate_normaldistribution. Specifically, \(\mathrm{Vec}(X)\) (the vector formed by concatenating the columns of \(X\)) has a multivariate normal distribution with mean \(\mathrm{Vec}(M)\) and covariance \(V \otimes U\) (where \(\otimes\) is the Kronecker product). Sampling and pdf evaluation are \(\mathcal{O}(m^3 + n^3 + m^2 n + m n^2)\) for the matrix normal, but \(\mathcal{O}(m^3 n^3)\) for the equivalent multivariate normal, making this equivalent form algorithmically inefficient.- Added in version 0.17.0. - Examples - >>> import numpy as np >>> from scipy.stats import matrix_normal - >>> M = np.arange(6).reshape(3,2); M array([[0, 1], [2, 3], [4, 5]]) >>> U = np.diag([1,2,3]); U array([[1, 0, 0], [0, 2, 0], [0, 0, 3]]) >>> V = 0.3*np.identity(2); V array([[ 0.3, 0. ], [ 0. , 0.3]]) >>> X = M + 0.1; X array([[ 0.1, 1.1], [ 2.1, 3.1], [ 4.1, 5.1]]) >>> matrix_normal.pdf(X, mean=M, rowcov=U, colcov=V) 0.023410202050005054 - >>> # Equivalent multivariate normal >>> from scipy.stats import multivariate_normal >>> vectorised_X = X.T.flatten() >>> equiv_mean = M.T.flatten() >>> equiv_cov = np.kron(V,U) >>> multivariate_normal.pdf(vectorised_X, mean=equiv_mean, cov=equiv_cov) 0.023410202050005054 - Alternatively, the object may be called (as a function) to fix the mean and covariance parameters, returning a “frozen” matrix normal random variable: - >>> rv = matrix_normal(mean=None, rowcov=1, colcov=1) >>> # Frozen object with the same methods but holding the given >>> # mean and covariance fixed. - Methods - pdf(X, mean=None, rowcov=1, colcov=1) - Probability density function. - logpdf(X, mean=None, rowcov=1, colcov=1) - Log of the probability density function. - rvs(mean=None, rowcov=1, colcov=1, size=1, random_state=None) - Draw random samples. - entropy(rowcol=1, colcov=1) - Differential entropy.