{"title":"Optimal Piecewise Polynomial Approximation for Minimum Computing Cost by Using Constrained Least Squares.","authors":"Jieun Song, Bumjoo Lee","doi":"10.3390/s24123991","DOIUrl":null,"url":null,"abstract":"<p><p>In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed.</p>","PeriodicalId":21698,"journal":{"name":"Sensors","volume":null,"pages":null},"PeriodicalIF":3.4000,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11207727/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Sensors","FirstCategoryId":"103","ListUrlMain":"https://doi.org/10.3390/s24123991","RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"CHEMISTRY, ANALYTICAL","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed.
本文提出了一种最佳近似算法,利用受限最小二乘法将非线性函数和/或离散数据简化为分段多项式。在对时间敏感的应用或资源有限的嵌入式系统中,近似函数的运行时间与其精度同样重要。所提出的算法以最小的计算成本搜索最优分片多项式(OPP),同时确保误差低于指定阈值。这是通过使用具有最佳阶数和区间数的平滑分片多项式来实现的。计算成本只取决于多项式的复杂性,即运行时函数调用的阶数和区间数。在以往的研究中,用户必须决定一个或所有阶次和区间数。相比之下,OPP 近似算法可以同时决定这两个因素。为了获得最佳近似值,我们计算了所有可能的片断多项式组合的计算成本,并按升序列出了特定目标 CPU 的离线计算成本。针对给定的样本点,通过约束最小二乘法和随机选择法对每种组合进行了优化。然后,检查近似误差是否低于预定值。当误差允许时,该组合被选为最优近似值,否则将检查下一个组合。为了验证性能,对几个有代表性的函数进行了检查和分析。
期刊介绍:
Sensors (ISSN 1424-8220) provides an advanced forum for the science and technology of sensors and biosensors. It publishes reviews (including comprehensive reviews on the complete sensors products), regular research papers and short notes. Our aim is to encourage scientists to publish their experimental and theoretical results in as much detail as possible. There is no restriction on the length of the papers. The full experimental details must be provided so that the results can be reproduced.