{"title":"算法1034:一种计算Qn鲁棒统计量的加速算法,并对常数进行校正","authors":"Thierry Fahmy","doi":"https://dl.acm.org/doi/10.1145/3576920","DOIUrl":null,"url":null,"abstract":"<p>The robust scale estimator <i>Q<sub>n</sub></i> developed by Croux and Rousseeuw [3], for the computation of which they provided a deterministic algorithm, has proven to be very useful in several domains including in quality management and time series analysis. It has interesting mathematical (50% breakdown, 82% Asymptotic Relative Efficiency) and computing (<i>O(nlogn)</i> time, <i>O</i>(<i>n</i>) space) properties. While working on a faster algorithm to compute <i>Q<sub>n</sub></i>, we have discovered an error in the computation of the <i>d</i> constant, and as a consequence in the <i>d<sub>n</sub></i> constants that are used to scale the statistic for consistency with the variance of a normal sample. These errors have been reproduced in several articles including in the International Standard Organisation 13,528 [12] document. In this article, we fix the errors and present a new approach, which includes a new algorithm, allowing computations to run 1.3 to 4.5 times faster when <i>n</i> grows from 10 to 100,000.</p>","PeriodicalId":50935,"journal":{"name":"ACM Transactions on Mathematical Software","volume":"72 ","pages":""},"PeriodicalIF":2.7000,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Algorithm 1034: An Accelerated Algorithm to Compute the Qn Robust Statistic, with Corrections to Constants\",\"authors\":\"Thierry Fahmy\",\"doi\":\"https://dl.acm.org/doi/10.1145/3576920\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The robust scale estimator <i>Q<sub>n</sub></i> developed by Croux and Rousseeuw [3], for the computation of which they provided a deterministic algorithm, has proven to be very useful in several domains including in quality management and time series analysis. It has interesting mathematical (50% breakdown, 82% Asymptotic Relative Efficiency) and computing (<i>O(nlogn)</i> time, <i>O</i>(<i>n</i>) space) properties. While working on a faster algorithm to compute <i>Q<sub>n</sub></i>, we have discovered an error in the computation of the <i>d</i> constant, and as a consequence in the <i>d<sub>n</sub></i> constants that are used to scale the statistic for consistency with the variance of a normal sample. These errors have been reproduced in several articles including in the International Standard Organisation 13,528 [12] document. In this article, we fix the errors and present a new approach, which includes a new algorithm, allowing computations to run 1.3 to 4.5 times faster when <i>n</i> grows from 10 to 100,000.</p>\",\"PeriodicalId\":50935,\"journal\":{\"name\":\"ACM Transactions on Mathematical Software\",\"volume\":\"72 \",\"pages\":\"\"},\"PeriodicalIF\":2.7000,\"publicationDate\":\"2023-03-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACM Transactions on Mathematical Software\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/https://dl.acm.org/doi/10.1145/3576920\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"COMPUTER SCIENCE, SOFTWARE ENGINEERING\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Transactions on Mathematical Software","FirstCategoryId":"94","ListUrlMain":"https://doi.org/https://dl.acm.org/doi/10.1145/3576920","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
Algorithm 1034: An Accelerated Algorithm to Compute the Qn Robust Statistic, with Corrections to Constants
The robust scale estimator Qn developed by Croux and Rousseeuw [3], for the computation of which they provided a deterministic algorithm, has proven to be very useful in several domains including in quality management and time series analysis. It has interesting mathematical (50% breakdown, 82% Asymptotic Relative Efficiency) and computing (O(nlogn) time, O(n) space) properties. While working on a faster algorithm to compute Qn, we have discovered an error in the computation of the d constant, and as a consequence in the dn constants that are used to scale the statistic for consistency with the variance of a normal sample. These errors have been reproduced in several articles including in the International Standard Organisation 13,528 [12] document. In this article, we fix the errors and present a new approach, which includes a new algorithm, allowing computations to run 1.3 to 4.5 times faster when n grows from 10 to 100,000.
期刊介绍:
As a scientific journal, ACM Transactions on Mathematical Software (TOMS) documents the theoretical underpinnings of numeric, symbolic, algebraic, and geometric computing applications. It focuses on analysis and construction of algorithms and programs, and the interaction of programs and architecture. Algorithms documented in TOMS are available as the Collected Algorithms of the ACM at calgo.acm.org.