Welford Online Statistics

📐 Definition

Given samples $x_1,\dots,x_n$ , maintain mean $\mu_n$ and accumulated second moment $M2_n$ via

\delta = x_n - \mu_{n-1},\qquad \mu_n = \mu_{n-1} + \frac{\delta}{n},\qquad M2_n = M2_{n-1} + \delta (x_n - \mu_n)

initialized with $\mu_0 = 0$ , $M2_0 = 0$ , $n=0$ . The variance is $\sigma_n^{2} = M2_n / n$ (population) or $M2_n/(n-1)$ (unbiased, $n>1$ ).

Domain and Codomain

Applies to real or complex data streams with finite second moments. For complex data with variance defined as $\sigma^2=\mathbb{E}|X-\mathbb{E}X|^2$ , a common update is

M2_n = M2_{n-1} + \delta\,\overline{(x_n-\mu_n)}

which keeps $M2_n$ real and nonnegative.

⚙️ Key Properties

Single-pass and numerically stable; $M2_n$ accumulates squared deviations without storing history. Order-invariant: results depend only on the multiset of samples, not their order.

For constant data, variance remains zero. As $n \to \infty$ for stationary processes with finite variance, estimates converge almost surely to true moments.

🎯 Special Cases and Limits

Constant stream yields zero variance.
Suitable for long streams where two-pass variance would be numerically fragile or memory-intensive.

Exponential moving averages provide biased but rapidly adapting moments; basic variance formulas require two passes or stored history.

Usage in Oakfield

Oakfield uses Welford-style online accumulation in its runtime field statistics engine:

SimFieldStatsAccumulator updates mean and M2 in a single pass over field data (numerically stable, streaming-friendly).
The resulting means/variances (plus magnitude moments) feed UI diagnostics and downstream metrics without storing full histories.

Historical Foundations

📜 One-Pass Variance

Welford’s update is a classic numerically stable one-pass method for mean and variance computation and is widely used in streaming statistics.

🌍 Modern Perspective

It is a standard building block for online diagnostics and adaptive methods that must track variance without storing history.

📚 References

Welford, “Note on a Method for Calculating Corrected Sums of Squares and Products” (1962)
Chan, Golub, LeVeque, “Algorithms for Computing the Sample Variance” (1983)