Journal of the Royal Statistical Society Series B-Statistical Methodology最新文献

筛选
英文 中文
Root cause discovery via permutations and Cholesky decomposition. 通过排列和乔列斯基分解发现根本原因。
IF 3.8 1区 数学
Journal of the Royal Statistical Society Series B-Statistical Methodology Pub Date : 2025-10-16 eCollection Date: 2026-07-01 DOI: 10.1093/jrsssb/qkaf066
Jinzhou Li, Benjamin B Chu, Ines F Scheller, Julien Gagneur, Marloes H Maathuis
{"title":"Root cause discovery via permutations and Cholesky decomposition.","authors":"Jinzhou Li, Benjamin B Chu, Ines F Scheller, Julien Gagneur, Marloes H Maathuis","doi":"10.1093/jrsssb/qkaf066","DOIUrl":"10.1093/jrsssb/qkaf066","url":null,"abstract":"<p><p>This work is motivated by the following problem: Can we identify the disease-causing gene in a patient affected by a monogenic disorder? This problem is an instance of root cause discovery. Specifically, we aim to identify the intervened variable in one interventional sample using a set of observational samples as reference. We consider a linear structural equation model where the causal ordering is unknown. We begin by examining a simple method that uses squared z-scores and characterize the conditions under which this method succeeds and fails, showing it generally cannot identify the root cause. We then prove, without additional assumptions, that the root cause is identifiable even if the causal ordering is not. Two key ingredients of this identifiability result are the use of permutations and the Cholesky decomposition, which allow us to exploit an invariant property across different permutations to discover the root cause. Furthermore, we characterize permutations that yield the correct root cause and, based on this, propose a valid method for root cause discovery. We also adapt this approach to high-dimensional settings. Finally, we evaluate our methods through simulations and apply the high-dimensional method to discover disease-causing genes in the gene expression dataset that motivates this work.</p>","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"88 3","pages":"779-798"},"PeriodicalIF":3.8,"publicationDate":"2025-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13370348/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148457338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Inference of dependency knowledge graph for Electronic Health Records. 电子病历依赖知识图谱的推理。
IF 3.8 1区 数学
Journal of the Royal Statistical Society Series B-Statistical Methodology Pub Date : 2025-09-29 eCollection Date: 2026-04-01 DOI: 10.1093/jrsssb/qkaf061
Zhiwei Xu, Ziming Gan, Doudou Zhou, Shuting Shen, Junwei Lu, Tianxi Cai
{"title":"Inference of dependency knowledge graph for Electronic Health Records.","authors":"Zhiwei Xu, Ziming Gan, Doudou Zhou, Shuting Shen, Junwei Lu, Tianxi Cai","doi":"10.1093/jrsssb/qkaf061","DOIUrl":"10.1093/jrsssb/qkaf061","url":null,"abstract":"<p><p>The effective analysis of high-dimensional Electronic Health Record (EHR) data, with substantial potential for healthcare research, presents notable methodological challenges. Employing predictive modeling guided by a knowledge graph (KG), which enables efficient feature selection, can enhance both statistical efficiency and interpretability. While various methods have emerged for constructing KGs, existing techniques often lack statistical certainty concerning the presence of links between entities, especially in scenarios where the utilization of patient-level EHR data is limited due to privacy concerns. In this paper, we propose the first inferential framework for deriving a sparse KG with statistical guarantee based on a dynamic log-linear topic model. Within this model, the KG embeddings are estimated by performing singular value decomposition on the empirical pointwise mutual information matrix, offering a scalable solution. We then establish entrywise asymptotic normality for the KG low-rank estimator, enabling the recovery of sparse graph edges with controlled type I error. Our work uniquely addresses the under-explored domain of statistical inference about non-linear statistics under the low-rank temporal dependent models, a critical gap in existing research. We validate our approach through extensive simulation studies and then apply the method to real-world EHR data in constructing clinical KGs and generating clinical feature embeddings.</p>","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"88 2","pages":"637-656"},"PeriodicalIF":3.8,"publicationDate":"2025-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13070795/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147693145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Robust Detection of Watermarks for Large Language Models Under Human Edits. 人工编辑下大型语言模型水印的鲁棒检测。
IF 3.8 1区 数学
Journal of the Royal Statistical Society Series B-Statistical Methodology Pub Date : 2025-09-22 DOI: 10.1093/jrsssb/qkaf056
Xiang Li, Feng Ruan, Huiyuan Wang, Qi Long, Weijie J Su
{"title":"Robust Detection of Watermarks for Large Language Models Under Human Edits.","authors":"Xiang Li, Feng Ruan, Huiyuan Wang, Qi Long, Weijie J Su","doi":"10.1093/jrsssb/qkaf056","DOIUrl":"10.1093/jrsssb/qkaf056","url":null,"abstract":"<p><p>Watermarking has offered an effective approach to distinguishing text generated by large language models (LLMs) from human-written text. However, the pervasive presence of human edits on LLM-generated text dilutes watermark signals, thereby significantly degrading detection performance of existing methods. In this paper, by modeling human edits through mixture model detection, we introduce a new method in the form of a truncated goodness-of-fit test for detecting watermarked text under human edits, which we refer to as Tr-GoF. We prove that the Tr-GoF test achieves optimality in robust detection of the Gumbel-max watermark in a certain asymptotic regime of substantial text modifications and vanishing watermark signals. Importantly, Tr-GoF achieves this optimality <i>adaptively</i> as it does not require precise knowledge of human edit levels or probabilistic specifications of the LLMs, in contrast to the optimal but impractical (Neyman-Pearson) likelihood ratio test. Moreover, we establish that the Tr-GoF test attains the highest detection efficiency rate in a certain regime of moderate text modifications. In stark contrast, we show that sum-based detection rules, as employed by existing methods, fail to achieve optimal robustness in both regimes because the additive nature of their statistics is less resilient to edit-induced noise. Finally, we demonstrate the competitive and sometimes superior empirical performance of the Tr-GoF test on both synthetic data and open-source LLMs in the OPT and LLaMA families.</p>","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":" ","pages":""},"PeriodicalIF":3.8,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12851586/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146088005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Covariate-assisted bounds on causal effects with instrumental variables. 工具变量因果效应的协变量辅助界。
IF 3.6 1区 数学
Journal of the Royal Statistical Society Series B-Statistical Methodology Pub Date : 2025-05-27 eCollection Date: 2025-11-01 DOI: 10.1093/jrsssb/qkaf028
Alexander W Levis, Matteo Bonvini, Zhenghao Zeng, Luke Keele, Edward H Kennedy
{"title":"Covariate-assisted bounds on causal effects with instrumental variables.","authors":"Alexander W Levis, Matteo Bonvini, Zhenghao Zeng, Luke Keele, Edward H Kennedy","doi":"10.1093/jrsssb/qkaf028","DOIUrl":"10.1093/jrsssb/qkaf028","url":null,"abstract":"<p><p>When an exposure of interest is confounded by unmeasured factors, an instrumental variable (IV) can be used to identify and estimate certain causal contrasts. Identification of the marginal average treatment effect (ATE) from IVs relies on strong untestable structural assumptions. When one is unwilling to assert such structure, IVs can nonetheless be used to construct bounds on the ATE. Famously, Alexander Balke and Judea Pearl proved tight bounds on the ATE for a binary outcome, in a randomized trial with noncompliance and no covariate information. We demonstrate how these bounds remain useful in observational settings with baseline confounders of the IV, as well as randomized trials with measured baseline covariates. The resulting bounds on the ATE are nonsmooth functionals, and thus standard nonparametric efficiency theory is not immediately applicable. To remedy this, we propose (1) under a novel margin condition, influence function-based estimators of the bounds that can attain parametric convergence rates when the nuisance functions are modelled flexibly, and (2) estimators of smooth approximations of these bounds. We propose extensions to continuous outcomes, explore finite sample properties in simulations, and illustrate the proposed estimators in an observational study targeting the effect of higher education on wages.</p>","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"87 5","pages":"1508-1527"},"PeriodicalIF":3.6,"publicationDate":"2025-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12602419/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145507800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A statistical view of column subset selection. 列子集选择的统计视图。
IF 3.8 1区 数学
Journal of the Royal Statistical Society Series B-Statistical Methodology Pub Date : 2025-05-16 DOI: 10.1093/jrsssb/qkaf023
Anav Sood, Trevor Hastie
{"title":"A statistical view of column subset selection.","authors":"Anav Sood, Trevor Hastie","doi":"10.1093/jrsssb/qkaf023","DOIUrl":"10.1093/jrsssb/qkaf023","url":null,"abstract":"<p><p>We consider the problem of selecting a small subset of representative variables from a large dataset. In the computer science literature, this dimensionality reduction problem is typically formalized as column subset selection (CSS). Meanwhile, the typical statistical formalization is to find an information-maximizing set of principal variables. This paper shows that these two approaches are equivalent, and moreover, both can be viewed as maximum-likelihood estimation within a certain semi-parametric model. Within this model, we establish suitable conditions under which the CSS estimate is consistent in high dimensions, specifically in the proportional asymptotic regime where the number of variables over the sample size converges to a constant. Using these connections, we show how to efficiently (1) perform CSS using only summary statistics from the original dataset; (2) perform CSS in the presence of missing and/or censored data; and (3) select the subset size for CSS in a hypothesis testing framework.</p>","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":" ","pages":""},"PeriodicalIF":3.8,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12288642/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144734981","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Product Centred Dirichlet Processes for Bayesian Multiview Clustering. 贝叶斯多视图聚类的产品中心Dirichlet过程。
IF 3.8 1区 数学
Journal of the Royal Statistical Society Series B-Statistical Methodology Pub Date : 2025-04-30 DOI: 10.1093/jrsssb/qkaf021
Alexander Dombowsky, David B Dunson
{"title":"Product Centred Dirichlet Processes for Bayesian Multiview Clustering.","authors":"Alexander Dombowsky, David B Dunson","doi":"10.1093/jrsssb/qkaf021","DOIUrl":"10.1093/jrsssb/qkaf021","url":null,"abstract":"<p><p>While there is an immense literature on Bayesian methods for clustering, the multiview case has received little attention. This problem focuses on obtaining distinct but statistically dependent clusterings in a common set of entities for different data types. For example, clustering patients into subgroups with subgroup membership varying according to the domain of the patient variables. A challenge is how to model the across-view dependence between the partitions of patients into subgroups. The complexities of the partition space make standard methods to model dependence, such as correlation, infeasible. In this article, we propose CLustering with Independence Centring (CLIC), a clustering prior that uses a single parameter to explicitly model dependence between clusterings across views. CLIC is induced by the product centred Dirichlet process (PCDP), a novel hierarchical prior that bridges between independent and equivalent partitions. We show appealing theoretic properties, provide a finite approximation and prove its accuracy, present a marginal Gibbs sampler for posterior computation, and derive closed form expressions for the marginal and joint partition distributions for the CLIC model. On synthetic data and in an application to epidemiology, CLIC accurately characterizes view-specific partitions while providing inference on the dependence level.</p>","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":" ","pages":""},"PeriodicalIF":3.8,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12392789/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144976818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Augmented balancing weights as linear regression. 增广平衡权作为线性回归。
IF 3.8 1区 数学
Journal of the Royal Statistical Society Series B-Statistical Methodology Pub Date : 2025-04-24 eCollection Date: 2026-07-01 DOI: 10.1093/jrsssb/qkaf019
David Bruns-Smith, Oliver Dukes, Avi Feller, Elizabeth L Ogburn
{"title":"Augmented balancing weights as linear regression.","authors":"David Bruns-Smith, Oliver Dukes, Avi Feller, Elizabeth L Ogburn","doi":"10.1093/jrsssb/qkaf019","DOIUrl":"10.1093/jrsssb/qkaf019","url":null,"abstract":"<p><p>We provide a novel characterization of augmented balancing weights, also known as automatic debiased machine learning. These popular <i>doubly robust</i> estimators combine outcome modelling with balancing weights-weights that achieve covariate balance directly instead of estimating and inverting the propensity score. When the outcome and weighting models are both linear in some (possibly infinite) basis, we show that the augmented estimator is equivalent to a single linear model with coefficients that combine those of the original outcome model with those from unpenalized ordinary least-squares (OLS). Under certain choices of regularization parameters, the augmented estimator in fact collapses to the OLS estimator alone. We then extend these results to specific outcome and weighting models. We first show that the augmented estimator that uses (kernel) ridge regression for both outcome and weighting models is equivalent to a single, undersmoothed (kernel) ridge regression-implying a novel analysis of undersmoothing. When the weighting model is instead lasso-penalized, we demonstrate a familiar 'double selection' property. Our framework opens the black box on this increasingly popular class of estimators, bridges the gap between existing results on the semiparametric efficiency of undersmoothed and doubly robust estimators, and provides new insights into the performance of augmented balancing weights.</p>","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"88 3","pages":"699-723"},"PeriodicalIF":3.8,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13370344/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148473844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Two-phase rejective sampling and its asymptotic properties. 两相拒绝抽样及其渐近性质。
IF 3.6 1区 数学
Journal of the Royal Statistical Society Series B-Statistical Methodology Pub Date : 2025-02-10 DOI: 10.1093/jrsssb/qkaf002
Shu Yang, Peng Ding
{"title":"Two-phase rejective sampling and its asymptotic properties.","authors":"Shu Yang, Peng Ding","doi":"10.1093/jrsssb/qkaf002","DOIUrl":"10.1093/jrsssb/qkaf002","url":null,"abstract":"<p><p>Rejective sampling improves design and estimation efficiency of single-phase sampling when auxiliary information in a finite population is available. When such auxiliary information is unavailable, we propose to use two-phase rejective sampling (TPRS), which involves measuring auxiliary variables for the sample of units in the first phase, followed by the implementation of rejective sampling for the outcome in the second phase. We explore the asymptotic design properties of double expansion and regression estimators under TPRS. We show that TPRS enhances the efficiency of the double-expansion estimator, rendering it comparable to a regression estimator. We further refine the design to accommodate varying importance of covariates and extend it to multi-phase sampling. We start with the theory for the population mean and then extend the theory to parameters defined by general estimating equations. Our asymptotic results for TPRS immediately cover the existing single-phase rejective sampling, under which the asymptotic theory has not been fully established.</p>","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":" ","pages":""},"PeriodicalIF":3.6,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12355938/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144876438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Probabilistic Richardson extrapolation. 概率Richardson外推法。
IF 3.1 1区 数学
Journal of the Royal Statistical Society Series B-Statistical Methodology Pub Date : 2024-12-26 eCollection Date: 2025-04-01 DOI: 10.1093/jrsssb/qkae098
Chris J Oates, Toni Karvonen, Aretha L Teckentrup, Marina Strocchi, Steven A Niederer
{"title":"Probabilistic Richardson extrapolation.","authors":"Chris J Oates, Toni Karvonen, Aretha L Teckentrup, Marina Strocchi, Steven A Niederer","doi":"10.1093/jrsssb/qkae098","DOIUrl":"https://doi.org/10.1093/jrsssb/qkae098","url":null,"abstract":"<p><p>For over a century, extrapolation methods have provided a powerful tool to improve the convergence order of a numerical method. However, these tools are not well-suited to modern computer codes, where multiple continua are discretized and convergence orders are not easily analysed. To address this challenge, we present a probabilistic perspective on Richardson extrapolation, a point of view that unifies classical extrapolation methods with modern multi-fidelity modelling, and handles uncertain convergence orders by allowing these to be statistically estimated. The approach is developed using Gaussian processes, leading to <i>Gauss-Richardson Extrapolation</i>. Conditions are established under which extrapolation using the conditional mean achieves a polynomial (or even an exponential) speed-up compared to the original numerical method. Further, the probabilistic formulation unlocks the possibility of experimental design, casting the selection of fidelities as a continuous optimization problem, which can then be (approximately) solved. A case study involving a computational cardiac model demonstrates that practical gains in accuracy can be achieved using the GRE method.</p>","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"87 2","pages":"457-479"},"PeriodicalIF":3.1,"publicationDate":"2024-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11985099/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144058583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Robust evaluation of longitudinal surrogate markers with censored data. 用删节数据对纵向替代标记进行稳健评估。
IF 3.6 1区 数学
Journal of the Royal Statistical Society Series B-Statistical Methodology Pub Date : 2024-12-26 eCollection Date: 2025-07-01 DOI: 10.1093/jrsssb/qkae119
Denis Agniel, Layla Parast
{"title":"Robust evaluation of longitudinal surrogate markers with censored data.","authors":"Denis Agniel, Layla Parast","doi":"10.1093/jrsssb/qkae119","DOIUrl":"10.1093/jrsssb/qkae119","url":null,"abstract":"<p><p>The development of statistical methods to evaluate surrogate markers is an active area of research. In many clinical settings, the surrogate marker is not simply a single measurement but is instead a longitudinal trajectory of measurements over time, e.g. fasting plasma glucose measured every 6 months for 3 years. In general, available methods developed for the single-surrogate setting cannot accommodate a longitudinal surrogate marker. Furthermore, many of the methods have not been developed for use with primary outcomes that are time-to-event outcomes and/or subject to censoring. In this paper, we propose robust methods to evaluate a longitudinal surrogate marker in a censored time-to-event outcome setting. Specifically, we propose a method to define and estimate the proportion of the treatment effect on a censored primary outcome that is explained by the treatment effect on a longitudinal surrogate marker measured up to time <math><msub><mi>t</mi> <mn>0</mn></msub> </math> . We accommodate both potential censoring of the primary outcome and of the surrogate marker. A simulation study demonstrates a good finite-sample performance of our proposed methods. We illustrate our procedures by examining repeated measures of fasting plasma glucose, a surrogate marker for diabetes diagnosis, using data from the diabetes prevention programme.</p>","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"87 3","pages":"891-907"},"PeriodicalIF":3.6,"publicationDate":"2024-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12256123/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144638525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
相关产品
×
本文献相关产品
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:604180095
Book学术官方微信
小红书