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Cholesky factor of the covariance matrix of vector-variate random samples.

This function can be use to fit a multivariate normal to data.

import tensorflow_probability as tfp; tfp = tfp.substrates.numpy
tfd = tfp.distributions

# Assume data.shape = (1000, 2).  1000 samples of a random variable in R^2.
observed_data = read_data_samples(...)

# The mean is easy
mu = tf.reduce_mean(observed_data, axis=0)

# Get the scale matrix
L = tfp.stats.cholesky_covariance(observed_data)

# Make the best fit multivariate normal (under maximum likelihood condition).
mvn = tfd.MultivariateNormalTriL(loc=mu, scale_tril=L)

# Plot contours of the pdf.
xs, ys = tf.meshgrid(
    tf.linspace(-5., 5., 50), tf.linspace(-5., 5., 50), indexing='ij')
xy = tf.stack((tf.reshape(xs, [-1]), tf.reshape(ys, [-1])), axis=-1)
pdf = tf.reshape(mvn.prob(xy), (50, 50))
CS = plt.contour(xs, ys, pdf, 10)
plt.clabel(CS, inline=1, fontsize=10)

Why does this work? Given vector-variate random variables X = (X1, ..., Xd), one may obtain the sample covariance matrix in R^{d x d} (see tfp.stats.covariance).

The Cholesky factor of this matrix is analogous to standard deviation for scalar random variables: Suppose X has covariance matrix C, with Cholesky factorization C = L L^T Then multiplying a vector of iid random variables which have unit variance by L produces a vector with covariance L L^T, which is the same as X.

observed_data = read_data_samples(...)
L = tfp.stats.cholesky_covariance(observed_data, sample_axis=0)

# Make fake_data with the same covariance as observed_data.
uncorrelated_normal = tf.random.stateless_normal(shape=(500, 10))
fake_data = tf.linalg.matvec(L, uncorrelated_normal)

x Numeric Tensor. The rightmost dimension of x indexes events. E.g. dimensions of a random vector.
sample_axis Scalar or vector Tensor designating axis holding samples. Default value: 0 (leftmost dimension). Cannot be the rightmost dimension (since this indexes events).
keepdims Boolean. Whether to keep the sample axis as singletons.
name Python str name prefixed to Ops created by this function. Default value: None (i.e., 'covariance').

chol Tensor of same dtype as x. The last two dimensions hold lower triangular matrices (the Cholesky factors).