Numerically stable, scalable formulas for parallel and online computation of higherorder multivariate central moments with arbitrary weights
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 arbitraryorder, numerically stable onepass formulas which we further extend with weighted and compound variants. We also develop a generalized correction factor for standard twopass algorithms that enables the maintenance of accuracy over nearly the full representable range of the input, avoiding the need for extendedprecision 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 tradeoffs 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 largescale scientific application.
 Authors:

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 Sandia National Lab. (SNLCA), Livermore, CA (United States)
 The Xiph.Org Foundation, Arlington, VA (United States)
 Publication Date:
 Report Number(s):
 SAND20159451J
Journal ID: ISSN 09434062; 607859
 Grant/Contract Number:
 AC0494AL85000
 Type:
 Accepted Manuscript
 Journal Name:
 Computational Statistics
 Additional Journal Information:
 Journal Volume: 31; Journal Issue: 4; Journal ID: ISSN 09434062
 Research Org:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org:
 USDOE National Nuclear Security Administration (NNSA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; Descriptive Statistics; Statistical Moments; Parallel Computing; Large Data Analysis
 OSTI Identifier:
 1427275
Pebay, Philippe, Terriberry, Timothy B., Kolla, Hemanth, and Bennett, Janine. Numerically stable, scalable formulas for parallel and online computation of higherorder multivariate central moments with arbitrary weights. United States: N. p.,
Web. doi:10.1007/s001800150637z.
Pebay, Philippe, Terriberry, Timothy B., Kolla, Hemanth, & Bennett, Janine. Numerically stable, scalable formulas for parallel and online computation of higherorder multivariate central moments with arbitrary weights. United States. doi:10.1007/s001800150637z.
Pebay, Philippe, Terriberry, Timothy B., Kolla, Hemanth, and Bennett, Janine. 2016.
"Numerically stable, scalable formulas for parallel and online computation of higherorder multivariate central moments with arbitrary weights". United States.
doi:10.1007/s001800150637z. https://www.osti.gov/servlets/purl/1427275.
@article{osti_1427275,
title = {Numerically stable, scalable formulas for parallel and online computation of higherorder multivariate central moments with arbitrary weights},
author = {Pebay, Philippe and Terriberry, Timothy B. and Kolla, Hemanth and Bennett, Janine},
abstractNote = {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 arbitraryorder, numerically stable onepass formulas which we further extend with weighted and compound variants. We also develop a generalized correction factor for standard twopass algorithms that enables the maintenance of accuracy over nearly the full representable range of the input, avoiding the need for extendedprecision 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 tradeoffs 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 largescale scientific application.},
doi = {10.1007/s001800150637z},
journal = {Computational Statistics},
number = 4,
volume = 31,
place = {United States},
year = {2016},
month = {3}
}