Numerically stable, scalable formulas for parallel and online computation of higher-order multivariate central moments with arbitrary weights

Abstract

Formulas for incremental or parallel computation of second order central moments have long been known, and recent extensions of these formulas to univariate and multivariate moments of arbitrary order have been developed. Such formulas are of key importance in scenarios where incremental results are required and in parallel and distributed systems where communication costs are high. We survey these recent results, and improve them with arbitrary-order, numerically stable one-pass formulas which we further extend with weighted and compound variants. We also develop a generalized correction factor for standard two-pass algorithms that enables Dedicated to the Memory of Dr Timothy J. Baker (1948Baker ( -2006 the maintenance of accuracy over nearly the full representable range of the input, avoiding the need for extended-precision arithmetic. We then empirically examine algorithm correctness for pairwise update formulas up to order four as well as condition number and relative error bounds for eight different central moment formulas, each up to degree six, to address the trade-offs between numerical accuracy and speed of the various algorithms. Finally, we demonstrate the use of the most elaborate among the above mentioned formulas, with the utilization of the compound moments for a practical large-scale scientific application.

An Empirical Study of Moment Estimators for Quantile Approximation

“…Early works such as those of Youngs and Cramer [31] and Chan et al [5] discuss the accumulation of rounding errors in the sample variance calculation and propose single-pass update formulas with significantly improved conditioning over naïve methods. More recently, Pébay et al [26] have generalised these formulas to higherorder central moments, providing both an incremental update formula for processing elements one 3:9 at a time and a parallel update formula for merging partial sample moment estimates. The latter enables us to compute the power sum…”

Series arc fault detection in photovoltaic system by small‐signal impedance and noise monitoring

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Algorithm for error-free determination of the variance of all contiguous subsequences and fixed-length contiguous subsequences for a sequence of industrial measurement data

“…Peaby, Terriberry, Kolla and Bennett published very valuable results that combine both of these research directions. In article Pébay et al (2016) they defined general recursive formulas for determining higher order moments. Their approach can be the basis for many parallel and numerically stable algorithms.…”