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Windowed estimates of variance.
tfp.substrates.jax.stats.windowed_variance( x, low_indices=None, high_indices=None, axis=0, name=None )
Computes variances among data in the Tensor
x along the given windows:
result[i] = variance(x[low_indices[i]:high_indices[i]+1])
accurately and efficiently. To wit, if K is the size of
N is the size of
axis, the computation takes O(K + N) work, O(log(N))
depth (the length of the longest series of operations that are
performed sequentially), and only uses O(1) TensorFlow kernel
invocations. The underlying algorithm is an adaptation of the
streaming reduction for accurate variance computations given in .
This function can be useful for assessing the behavior over time of trailing-window estimators from some iterative process, such as the last half of an MCMC chain.
x has shape
Bx + [N] + E, where the
Bx component has
high_indices broadcast to shape
[M]. Then each element of
must be between 0 and N+1, and the shape of the output will be
Bx + [M] + E. Batch shape in the indices is not currently supported.
The default windows are
[0, 1), [1, 2), [1, 3), [2, 4), [2, 5), ...
This corresponds to analyzing
x as though it were streaming, for
example successive states of an MCMC sampler, and we were interested
in the variance of the last half of the data at each point.
: Philippe Pebay. Formulas for Robust, One-Pass Parallel Computation of Covariances and Arbitrary-Order Statistical Moments. Technical Report SAND2008-6212, 2008. https://prod-ng.sandia.gov/techlib-noauth/access-control.cgi/2008/086212.pdf