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# tfp.stats.cumulative_variance

Cumulative estimates of variance.

Given `N` samples of a scalar-valued random variable `X`, we can compute cumulative variance estimates

result[i] = variance(x[0:i+1])

in O(N) work and O(log(N)) depth (the length of the longest series of operations that are performed sequentially), with O(1) TF kernel invocations. This implementation also arranges to do so in a numerically accurate manner, i.e., without incurring a subtraction of floating-point numbers of size quadratic in the data `x`. The underlying algorithm is from .

`x` A numeric `Tensor` holding samples.
`sample_axis` Scalar `Tensor` designating the axis holding samples. Other axes are treated in batch. Default value: `0` (leftmost dimension).
`name` Python `str` name prefixed to Ops created by this function. Default value: `None` (i.e., `'cumulative_variance'`).

`cum_var` A `Tensor` of same shape and dtype as `x` giving cumulative variance estimates. The zeroth element is the variance of a size-1 set of samples, so 0.

: 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

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