{"title":"Degree selection methods for curve estimation via Bernstein polynomials","authors":"","doi":"10.1007/s00180-024-01473-6","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>Bernstein Polynomial (BP) bases can uniformly approximate any continuous function based on observed noisy samples. However, a persistent challenge is the data-driven selection of a suitable degree for the BPs. In the absence of noise, asymptotic theory suggests that a larger degree leads to better approximation. However, in the presence of noise, which reduces bias, a larger degree also results in larger variances due to high-dimensional parameter estimation. Thus, a balance in the classic bias-variance trade-off is essential. The main objective of this work is to determine the minimum possible degree of the approximating BPs using probabilistic methods that are robust to various shapes of an unknown continuous function. Beyond offering theoretical guidance, the paper includes numerical illustrations to address the issue of determining a suitable degree for BPs in approximating arbitrary continuous functions.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00180-024-01473-6","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
Bernstein Polynomial (BP) bases can uniformly approximate any continuous function based on observed noisy samples. However, a persistent challenge is the data-driven selection of a suitable degree for the BPs. In the absence of noise, asymptotic theory suggests that a larger degree leads to better approximation. However, in the presence of noise, which reduces bias, a larger degree also results in larger variances due to high-dimensional parameter estimation. Thus, a balance in the classic bias-variance trade-off is essential. The main objective of this work is to determine the minimum possible degree of the approximating BPs using probabilistic methods that are robust to various shapes of an unknown continuous function. Beyond offering theoretical guidance, the paper includes numerical illustrations to address the issue of determining a suitable degree for BPs in approximating arbitrary continuous functions.
摘要 伯恩斯坦多项式（BP）基可以根据观测到的噪声样本均匀地近似任何连续函数。然而，一个长期存在的难题是如何根据数据为 BP 选择合适的阶数。在没有噪声的情况下，渐近理论表明，阶数越大，逼近效果越好。然而，在有噪声的情况下，噪声会减少偏差，但由于高维参数估计，较大的度数也会导致较大的方差。因此，传统的偏差-方差权衡中的平衡至关重要。这项工作的主要目的是利用概率方法确定近似 BP 的最小可能度，这种方法对未知连续函数的各种形状都具有鲁棒性。除了提供理论指导外，本文还通过数值说明来解决在逼近任意连续函数时如何确定 BP 的合适度这一问题。
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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