{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T19:13:39Z","timestamp":1770578019799,"version":"3.49.0"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,3,19]],"date-time":"2021-03-19T00:00:00Z","timestamp":1616112000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,3,19]],"date-time":"2021-03-19T00:00:00Z","timestamp":1616112000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Stat"],"published-print":{"date-parts":[[2021,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The article presents an algorithm for fast and error-free determination of statistics such as the arithmetic mean and variance of all contiguous subsequences and fixed-length contiguous subsequences for a\u00a0sequence of industrial measurement data. Additionally, it shows that both floating-point and integer representation can be used to perform this kind of statistical calculations. The author proves a\u00a0theorem on the number of bits of precision that an arithmetic type must have to guarantee error-free determination of the arithmetic mean and variance. The article also presents the extension of Welford\u2019s formula for determining variance for the sliding window method\u2014determining the variance of fixed-length contiguous subsequences. The section dedicated to implementation tests shows the running times of individual algorithms depending on the arithmetic type used. The research shows that the use of integers in calculations makes the determination of the aforementioned statistics much faster.<\/jats:p>","DOI":"10.1007\/s00180-021-01096-1","type":"journal-article","created":{"date-parts":[[2021,3,19]],"date-time":"2021-03-19T13:04:49Z","timestamp":1616159089000},"page":"2813-2840","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Algorithm for error-free determination of the variance of all contiguous subsequences and fixed-length contiguous subsequences for a sequence of industrial measurement data"],"prefix":"10.1007","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6629-0029","authenticated-orcid":false,"given":"Andrzej","family":"Chmielowiec","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,3,19]]},"reference":[{"issue":"1","key":"1096_CR1","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1080\/00401706.1959.10489850","volume":"1","author":"G Box","year":"1959","unstructured":"Box G, Hunter J (1959) Condensed calculations for evolutionary operation programs. Technometrics 1(1):77\u201395","journal-title":"Technometrics"},{"key":"1096_CR2","doi-asserted-by":"crossref","unstructured":"Chalapathy R, Chawla S (2019) Deep learning for anomaly detection: a survey. CoRR. abs\/1901.03407","DOI":"10.1145\/3394486.3406704"},{"key":"1096_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-51461-6_3","author":"T Chan","year":"1982","unstructured":"Chan T, Golub G, LeVeque R (1982) Updating formulae and a pairwise algorithm for computing sample variances. COMPSTAT. https:\/\/doi.org\/10.1007\/978-3-642-51461-6_3","journal-title":"COMPSTAT"},{"issue":"3","key":"1096_CR4","doi-asserted-by":"publisher","first-page":"242","DOI":"10.2307\/2683386","volume":"37","author":"T Chan","year":"1983","unstructured":"Chan T, Golub G, LeVeque R (1983) Algorithms for computing the sample variance: analysis and recommendations. Am Stat 37(3):242\u2013247. https:\/\/doi.org\/10.2307\/2683386","journal-title":"Am Stat"},{"issue":"9","key":"1096_CR5","doi-asserted-by":"publisher","first-page":"526","DOI":"10.1145\/359146.359152","volume":"22","author":"T Chan","year":"1979","unstructured":"Chan T, Lewis J (1979) Computing standard deviations: accuracy. Commun ACM 22(9):526\u2013531. https:\/\/doi.org\/10.1145\/359146.359152","journal-title":"Commun ACM"},{"issue":"8","key":"1096_CR6","doi-asserted-by":"publisher","first-page":"458","DOI":"10.1145\/360933.360981","volume":"18","author":"I Cotton","year":"1975","unstructured":"Cotton I (1975) Remark on stably updating mean and standard deviation of data. Commun ACM 18(8):458. https:\/\/doi.org\/10.1145\/360933.360981","journal-title":"Commun ACM"},{"key":"1096_CR7","volume-title":"Out of the crisis","author":"W Deming","year":"1986","unstructured":"Deming W (1986) Out of the crisis. MIT Press, Cambridge"},{"key":"1096_CR8","doi-asserted-by":"publisher","first-page":"340","DOI":"10.1007\/s10489-015-0713-7","volume":"44","author":"J Ding","year":"2016","unstructured":"Ding J, Liu Y, Zhang L, Wang J, Liu Y (2016) An anomaly detection approach for multiple monitoring data series based on latent correlation probabilistic model. Appl Intell 44:340\u2013361. https:\/\/doi.org\/10.1007\/s10489-015-0713-7","journal-title":"Appl Intell"},{"key":"1096_CR9","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/s00357-015-9171-5","volume":"32","author":"K Evans","year":"2015","unstructured":"Evans K, Love T, Thurston S (2015) Outlier identification in model-based cluster analysis. J Classif 32:63\u201384. https:\/\/doi.org\/10.1007\/s00357-015-9171-5","journal-title":"J Classif"},{"key":"1096_CR10","doi-asserted-by":"crossref","unstructured":"Gouin F, Ancourt C, Guettier C (2016) Threewise: a local variance algorithm for gpu. In: 2016 IEEE International Conference on CSE, IEEE International Conference on EUC, International Symposium DCABES, pp 257\u2013262. https:\/\/doi.org\/10.1109\/CSE-EUC-DCABES.2016.194","DOI":"10.1109\/CSE-EUC-DCABES.2016.194"},{"issue":"1","key":"1096_CR11","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1145\/360569.360662","volume":"18","author":"R Hanson","year":"1975","unstructured":"Hanson R (1975) Stably updating mean and standard deviation of data. Commun ACM 18(1):57\u201358. https:\/\/doi.org\/10.1145\/360569.360662","journal-title":"Commun ACM"},{"key":"1096_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-3994-4","volume-title":"Identification of outliers","author":"D Hawkins","year":"1980","unstructured":"Hawkins D (1980) Identification of outliers. Springer, Dordrecht"},{"key":"1096_CR13","doi-asserted-by":"crossref","unstructured":"Hyndman R, Wang E, Laptev N (2015) Large-scale unusual time series detection. In: 2015 IEEE interna-tional conference on data mining workshop (ICDMW), pp 1616\u20131619. https:\/\/doi.org\/10.1109\/ICDMW.2015.104","DOI":"10.1109\/ICDMW.2015.104"},{"key":"1096_CR14","doi-asserted-by":"crossref","unstructured":"IEEE: Ieee std 754-2008. Tech. rep., IEEE (2008). https:\/\/doi.org\/10.1109\/ieeestd.2008.4610935","DOI":"10.1109\/IEEESTD.2008.4610935"},{"key":"1096_CR15","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1007\/BF02418571","volume":"30","author":"J Jensen","year":"1906","unstructured":"Jensen J (1906) Sur les fonctions convexes et les in\u00e9galit\u00e9s entre les valeurs moyennes. Acta Math 30:175\u2013193. https:\/\/doi.org\/10.1007\/BF02418571","journal-title":"Acta Math"},{"issue":"1","key":"1096_CR16","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1145\/363707.363723","volume":"8","author":"W Kahan","year":"1965","unstructured":"Kahan W (1965) Pracniques: further remarks on reducing truncation errors. Commun ACM 8(1):40. https:\/\/doi.org\/10.1145\/363707.363723","journal-title":"Commun ACM"},{"issue":"2","key":"1096_CR17","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1007\/s10115-004-0172-7","volume":"8","author":"E Keogh","year":"2005","unstructured":"Keogh E, Lin J (2005) Clustering of time-series subsequences is meaningless: implications for previousand future research. Knowl Inf Syst 8(2):154\u2013177. https:\/\/doi.org\/10.1007\/s10115-004-0172-7","journal-title":"Knowl Inf Syst"},{"key":"1096_CR18","doi-asserted-by":"crossref","unstructured":"Keogh E, Lin J, Fu A (2005) Hot sax: efficiently finding the most unusual time series subsequence. In: Fifth IEEE international conference on data mining (ICDM\u201905), p 8. https:\/\/doi.org\/10.1109\/ICDM.2005.79","DOI":"10.1109\/ICDM.2005.79"},{"issue":"1","key":"1096_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10115-006-0034-6","volume":"11","author":"E Keogh","year":"2006","unstructured":"Keogh E, Lin J, Lee S, Herle H (2006) Finding the most unusual time series subsequence: algorithms and applications. Knowl Inf Syst 11(1):1\u201327. https:\/\/doi.org\/10.1007\/s10115-006-0034-6","journal-title":"Knowl Inf Syst"},{"key":"1096_CR20","unstructured":"Knuth D (1981) The art of computer programming, vol II: seminumerical algorithms, 2nd edn. Addison-Wesley"},{"key":"1096_CR21","doi-asserted-by":"publisher","first-page":"3228","DOI":"10.1016\/j.ins.2006.11.007","volume":"177","author":"V Kreinovich","year":"2007","unstructured":"Kreinovich V, Nguyen H, Wu B (2007) On-line algorithms for computing mean and variance of interval data, and their use in intelligent systems. Inf Sci 177:3228\u20133238. https:\/\/doi.org\/10.1016\/j.ins.2006.11.007","journal-title":"Inf Sci"},{"key":"1096_CR22","doi-asserted-by":"crossref","unstructured":"Laptev N, Amizadeh S, Flint I (2015) Generic and scalable framework for automated time-series anomaly detection. In: KDD\u201915: Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp 1939\u20131947. https:\/\/doi.org\/10.1145\/2783258.2788611","DOI":"10.1145\/2783258.2788611"},{"issue":"348","key":"1096_CR23","doi-asserted-by":"publisher","first-page":"859","DOI":"10.2307\/2286154","volume":"69","author":"R Ling","year":"1974","unstructured":"Ling R (1974) Comparison of several algorithms for computing sample means and variances. J Am Stat Assoc 69(348):859\u2013866. https:\/\/doi.org\/10.2307\/2286154","journal-title":"J Am Stat Assoc"},{"issue":"7","key":"1096_CR24","doi-asserted-by":"publisher","first-page":"496","DOI":"10.1145\/365719.365958","volume":"9","author":"P Neely","year":"1966","unstructured":"Neely P (1966) Comparison of several algorithms for computation of means, standard deviations and correlation coefficients. Commun ACM 9(7):496\u2013499. https:\/\/doi.org\/10.1145\/365719.365958","journal-title":"Commun ACM"},{"key":"1096_CR25","doi-asserted-by":"crossref","unstructured":"P\u00e9bay P (2008) Formulas for robust, one-pass parallel computation of covariances and arbitrary-order statistical moments. Tech. Rep. SAND2008-6212, Sandia National Laboratories. https:\/\/doi.org\/10.2172\/1028931","DOI":"10.2172\/1028931"},{"issue":"4","key":"1096_CR26","doi-asserted-by":"publisher","first-page":"1305","DOI":"10.1007\/s00180-015-0637-z","volume":"31","author":"P P\u00e9bay","year":"2016","unstructured":"P\u00e9bay P, Terriberry T, Kolla H, Bennett J (2016) Numerically stable, scalable formulas for parallel and online computation of higher-order multivariate central moments with arbitrary weights. Comput Stat 31(4):1305\u20131325. https:\/\/doi.org\/10.1007\/s00180-015-0637-z","journal-title":"Comput Stat"},{"issue":"3","key":"1096_CR27","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1145\/363162.363205","volume":"10","author":"B Rodden","year":"1967","unstructured":"Rodden B (1967) Error-free methods for statistical computations. Commun ACM 10(3):179\u2013180. https:\/\/doi.org\/10.1145\/363162.363205","journal-title":"Commun ACM"},{"key":"1096_CR28","doi-asserted-by":"crossref","unstructured":"Schubert E, Gertz M (2018) Numerically stable parallel computation of (co-)variance. In: SSDBM\u201918. https:\/\/doi.org\/10.1145\/3221269.3223036","DOI":"10.1145\/3221269.3223036"},{"issue":"12","key":"1096_CR29","doi-asserted-by":"publisher","first-page":"1976","DOI":"10.14778\/2824032.2824115","volume":"8","author":"E Schubert","year":"2015","unstructured":"Schubert E, Koos A, Emrich T, Z\u00fcfle A, Schmid K, Zimek A (2015) A framework for clustering uncertain data. Proc VLDB Endowment 8(12):1976\u20131979. https:\/\/doi.org\/10.14778\/2824032.2824115","journal-title":"Proc VLDB Endowment"},{"key":"1096_CR30","doi-asserted-by":"crossref","unstructured":"Selvamuthu D, Das D (2018) Introduction to statistical methods, design of experiments and statistical quality control. Springer Nature Singapore Pte Ltd., Singapore","DOI":"10.1007\/978-981-13-1736-1"},{"key":"1096_CR31","unstructured":"Senin P, Lin J, Wang X, Oates T, Gandhi S, Boedihardjo A, Chen C, Frankenstein S (2015) Time seriesanomaly discovery with grammar-based compression. In: Proceedings of the 18th international conference on extending database technology, EDBT 2015, pp 481\u2013492. https:\/\/doi.org\/10.5441\/002\/edbt.2015.42"},{"key":"1096_CR32","volume-title":"Economic control of quality manufactured product","author":"W Shewart","year":"1931","unstructured":"Shewart W (1931) Economic control of quality manufactured product. D. Van Nostrand, New York"},{"key":"1096_CR33","unstructured":"Terriberry T (2008) Computing higher-order moments online. http:\/\/people.xiph.org\/tterribe\/notes\/homs.html"},{"issue":"3","key":"1096_CR34","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1145\/362929.362961","volume":"11","author":"A Van Reeken","year":"1968","unstructured":"Van Reeken A (1968) Letters to the editor: dealing with Neely\u2019s algorithms. Commun ACM 11(3):149\u2013150. https:\/\/doi.org\/10.1145\/362929.362961","journal-title":"Commun ACM"},{"key":"1096_CR35","doi-asserted-by":"crossref","unstructured":"Villaverde K, Xiang G (2008) Estimating variance under interval and fuzzy uncertainty: Parallel algorithms. In: IEEE International Conference on Fuzzy Systems, pp 1030\u20131033. https:\/\/doi.org\/10.1109\/FUZZY.2008.4630496","DOI":"10.1109\/FUZZY.2008.4630496"},{"key":"1096_CR36","doi-asserted-by":"publisher","first-page":"1806","DOI":"10.1007\/s10618-018-0569-7","volume":"32","author":"X Wang","year":"2018","unstructured":"Wang X, Lin J, Patel NMB (2018) Exact variable-length anomaly detection algorithmfor univariate and multivariate time series. Data Mind Kowl Discov 32:1806\u20131844. https:\/\/doi.org\/10.1007\/s10618-018-0569-7","journal-title":"Data Mind Kowl Discov"},{"issue":"3","key":"1096_CR37","doi-asserted-by":"publisher","first-page":"419","DOI":"10.2307\/1266577","volume":"4","author":"B Welford","year":"1962","unstructured":"Welford B (1962) Note on a method for calculating corrected sums of squares and products. Technometrics 4(3):419\u2013420. https:\/\/doi.org\/10.2307\/1266577","journal-title":"Technometrics"},{"issue":"9","key":"1096_CR38","doi-asserted-by":"publisher","first-page":"532","DOI":"10.1145\/359146.359153","volume":"22","author":"D West","year":"1979","unstructured":"West D (1979) Updating mean and variance estimates: an improved method. Commun ACM 22(9):532\u2013535. https:\/\/doi.org\/10.1145\/359146.359153","journal-title":"Commun ACM"},{"issue":"3","key":"1096_CR39","doi-asserted-by":"publisher","first-page":"657","DOI":"10.1080\/00401706.1971.10488826","volume":"13","author":"E Youngs","year":"1971","unstructured":"Youngs E, Cramer E (1971) Some results relevant to choice of sum and sum-of-product algorithms. Technometrics 13(3):657\u2013665. https:\/\/doi.org\/10.1080\/00401706.1971.10488826","journal-title":"Technometrics"},{"issue":"2","key":"1096_CR40","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1109\/SURV.2010.021510.00088","volume":"12","author":"Y Zhang","year":"2010","unstructured":"Zhang Y, Meratnia NPH (2010) Outlier detection techniques for wireless sensor networks: a survey. IEEE Commun Surv Tutor 12(2):159\u2013170. https:\/\/doi.org\/10.1109\/SURV.2010.021510.00088","journal-title":"IEEE Commun Surv Tutor"}],"container-title":["Computational Statistics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-021-01096-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00180-021-01096-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-021-01096-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,18]],"date-time":"2021-11-18T07:03:42Z","timestamp":1637219022000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00180-021-01096-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,19]]},"references-count":40,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,12]]}},"alternative-id":["1096"],"URL":"https:\/\/doi.org\/10.1007\/s00180-021-01096-1","relation":{},"ISSN":["0943-4062","1613-9658"],"issn-type":[{"value":"0943-4062","type":"print"},{"value":"1613-9658","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,19]]},"assertion":[{"value":"29 October 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 March 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 March 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}