Jiyang Wang, Wanfeng Liang, Lijie Li, Yue Wu, Xiaoyan Ma
{"title":"A new robust covariance matrix estimation for high-dimensional microbiome data","authors":"Jiyang Wang, Wanfeng Liang, Lijie Li, Yue Wu, Xiaoyan Ma","doi":"10.1111/anzs.12415","DOIUrl":null,"url":null,"abstract":"<div>\n \n <p>Microbiome data typically lie in a high-dimensional simplex. One of the key questions in metagenomic analysis is to exploit the covariance structure for this kind of data. In this paper, a framework called approximate-estimate-threshold (AET) is developed for the robust basis covariance estimation for high-dimensional microbiome data. To be specific, we first construct a proxy matrix <span></span><math>\n <semantics>\n <mrow>\n <mi>Γ</mi>\n </mrow>\n <annotation>$$ \\boldsymbol{\\Gamma} $$</annotation>\n </semantics></math>, which is almost indistinguishable from the real basis covariance matrix <span></span><math>\n <semantics>\n <mrow>\n <mi>∑</mi>\n </mrow>\n <annotation>$$ \\boldsymbol{\\Sigma} $$</annotation>\n </semantics></math>. Then, any estimator <span></span><math>\n <semantics>\n <mrow>\n <mover>\n <mrow>\n <mi>Γ</mi>\n </mrow>\n <mo>^</mo>\n </mover>\n </mrow>\n <annotation>$$ \\hat{\\boldsymbol{\\Gamma}} $$</annotation>\n </semantics></math> satisfying some conditions can be used to estimate <span></span><math>\n <semantics>\n <mrow>\n <mi>Γ</mi>\n </mrow>\n <annotation>$$ \\boldsymbol{\\Gamma} $$</annotation>\n </semantics></math>. Finally, we impose a thresholding step on <span></span><math>\n <semantics>\n <mrow>\n <mover>\n <mrow>\n <mi>Γ</mi>\n </mrow>\n <mo>^</mo>\n </mover>\n </mrow>\n <annotation>$$ \\hat{\\boldsymbol{\\Gamma}} $$</annotation>\n </semantics></math> to obtain the final estimator <span></span><math>\n <semantics>\n <mrow>\n <mover>\n <mrow>\n <mi>∑</mi>\n </mrow>\n <mo>^</mo>\n </mover>\n </mrow>\n <annotation>$$ \\hat{\\boldsymbol{\\Sigma}} $$</annotation>\n </semantics></math>. In particular, this paper applies a Huber-type estimator <span></span><math>\n <semantics>\n <mrow>\n <mover>\n <mrow>\n <mi>Γ</mi>\n </mrow>\n <mo>^</mo>\n </mover>\n </mrow>\n <annotation>$$ \\hat{\\boldsymbol{\\Gamma}} $$</annotation>\n </semantics></math>, and achieves robustness by only requiring the boundedness of 2+<span></span><math>\n <semantics>\n <mrow>\n <mi>ϵ</mi>\n </mrow>\n <annotation>$$ \\epsilon $$</annotation>\n </semantics></math> moments for some <span></span><math>\n <semantics>\n <mrow>\n <mi>ϵ</mi>\n <mo>∈</mo>\n <mo>(</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mn>2</mn>\n <mo>]</mo>\n </mrow>\n <annotation>$$ \\epsilon \\in \\left(0,2\\right] $$</annotation>\n </semantics></math>. We derive the convergence rate of <span></span><math>\n <semantics>\n <mrow>\n <mover>\n <mrow>\n <mi>∑</mi>\n </mrow>\n <mo>^</mo>\n </mover>\n </mrow>\n <annotation>$$ \\hat{\\boldsymbol{\\Sigma}} $$</annotation>\n </semantics></math> under the spectral norm, and provide theoretical guarantees on support recovery. Extensive simulations and a real example are used to illustrate the empirical performance of our method.</p>\n </div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1111/anzs.12415","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Microbiome data typically lie in a high-dimensional simplex. One of the key questions in metagenomic analysis is to exploit the covariance structure for this kind of data. In this paper, a framework called approximate-estimate-threshold (AET) is developed for the robust basis covariance estimation for high-dimensional microbiome data. To be specific, we first construct a proxy matrix , which is almost indistinguishable from the real basis covariance matrix . Then, any estimator satisfying some conditions can be used to estimate . Finally, we impose a thresholding step on to obtain the final estimator . In particular, this paper applies a Huber-type estimator , and achieves robustness by only requiring the boundedness of 2+ moments for some . We derive the convergence rate of under the spectral norm, and provide theoretical guarantees on support recovery. Extensive simulations and a real example are used to illustrate the empirical performance of our method.