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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
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_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).