Biostatistics最新文献

筛选
英文 中文
Unlocking the power of time-since-infection models: data augmentation for improved instantaneous reproduction number estimation. 释放感染后时间模型的威力:通过数据扩增改进瞬时繁殖数估算。
IF 1.8 3区 数学
Biostatistics Pub Date : 2024-12-31 DOI: 10.1093/biostatistics/kxae054
Jiasheng Shi, Yizhao Zhou, Jing Huang
{"title":"Unlocking the power of time-since-infection models: data augmentation for improved instantaneous reproduction number estimation.","authors":"Jiasheng Shi, Yizhao Zhou, Jing Huang","doi":"10.1093/biostatistics/kxae054","DOIUrl":"10.1093/biostatistics/kxae054","url":null,"abstract":"<p><p>The time-since-infection (TSI) models, which use disease surveillance data to model infectious diseases, have become increasingly popular due to their flexibility and capacity to address complex disease control questions. However, a notable limitation of TSI models is their primary reliance on incidence data. Even when hospitalization data are available, existing TSI models have not been crafted to improve the estimation of disease transmission or to estimate hospitalization-related parameters-metrics crucial for understanding a pandemic and planning hospital resources. Moreover, their dependence on reported infection data makes them vulnerable to variations in data quality. In this study, we advance TSI models by integrating hospitalization data, marking a significant step forward in modeling with TSI models. We introduce hospitalization propensity parameters to jointly model incidence and hospitalization data. We use a composite likelihood function to accommodate complex data structure and a Monte Carlo expectation-maximization algorithm to estimate model parameters. We analyze COVID-19 data to estimate disease transmission, assess risk factor impacts, and calculate hospitalization propensity. Our model improves the accuracy of estimating the instantaneous reproduction number in TSI models, particularly when hospitalization data is of higher quality than incidence data. It enables the estimation of key infectious disease parameters without relying on contact tracing data and provides a foundation for integrating TSI models with other infectious disease models.</p>","PeriodicalId":55357,"journal":{"name":"Biostatistics","volume":"26 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11878408/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143558889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Penalized likelihood optimization for censored missing value imputation in proteomics.
IF 1.8 3区 数学
Biostatistics Pub Date : 2024-12-31 DOI: 10.1093/biostatistics/kxaf006
Lucas Etourneau, Laura Fancello, Samuel Wieczorek, Nelle Varoquaux, Thomas Burger
{"title":"Penalized likelihood optimization for censored missing value imputation in proteomics.","authors":"Lucas Etourneau, Laura Fancello, Samuel Wieczorek, Nelle Varoquaux, Thomas Burger","doi":"10.1093/biostatistics/kxaf006","DOIUrl":"https://doi.org/10.1093/biostatistics/kxaf006","url":null,"abstract":"<p><p>Label-free bottom-up proteomics using mass spectrometry and liquid chromatography has long been established as one of the most popular high-throughput analysis workflows for proteome characterization. However, it produces data hindered by complex and heterogeneous missing values, which imputation has long remained problematic. To cope with this, we introduce Pirat, an algorithm that harnesses this challenge using an original likelihood maximization strategy. Notably, it models the instrument limit by learning a global censoring mechanism from the data available. Moreover, it estimates the covariance matrix between enzymatic cleavage products (ie peptides or precursor ions), while offering a natural way to integrate complementary transcriptomic information when multi-omic assays are available. Our benchmarking on several datasets covering a variety of experimental designs (number of samples, acquisition mode, missingness patterns, etc.) and using a variety of metrics (differential analysis ground truth or imputation errors) shows that Pirat outperforms all pre-existing imputation methods. Beyond the interest of Pirat as an imputation tool, these results pinpoint the need for a paradigm change in proteomics imputation, as most pre-existing strategies could be boosted by incorporating similar models to account for the instrument censorship or for the correlation structures, either grounded to the analytical pipeline or arising from a multi-omic approach.</p>","PeriodicalId":55357,"journal":{"name":"Biostatistics","volume":"26 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143694529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
DifferentialRegulation: a Bayesian hierarchical approach to identify differentially regulated genes. DifferentialRegulation:一种贝叶斯分层方法,用于识别差异调控基因。
IF 1.8 3区 数学
Biostatistics Pub Date : 2024-10-01 DOI: 10.1093/biostatistics/kxae017
Simone Tiberi, Joël Meili, Peiying Cai, Charlotte Soneson, Dongze He, Hirak Sarkar, Alejandra Avalos-Pacheco, Rob Patro, Mark D Robinson
{"title":"DifferentialRegulation: a Bayesian hierarchical approach to identify differentially regulated genes.","authors":"Simone Tiberi, Joël Meili, Peiying Cai, Charlotte Soneson, Dongze He, Hirak Sarkar, Alejandra Avalos-Pacheco, Rob Patro, Mark D Robinson","doi":"10.1093/biostatistics/kxae017","DOIUrl":"10.1093/biostatistics/kxae017","url":null,"abstract":"<p><p>Although transcriptomics data is typically used to analyze mature spliced mRNA, recent attention has focused on jointly investigating spliced and unspliced (or precursor-) mRNA, which can be used to study gene regulation and changes in gene expression production. Nonetheless, most methods for spliced/unspliced inference (such as RNA velocity tools) focus on individual samples, and rarely allow comparisons between groups of samples (e.g. healthy vs. diseased). Furthermore, this kind of inference is challenging, because spliced and unspliced mRNA abundance is characterized by a high degree of quantification uncertainty, due to the prevalence of multi-mapping reads, ie reads compatible with multiple transcripts (or genes), and/or with both their spliced and unspliced versions. Here, we present DifferentialRegulation, a Bayesian hierarchical method to discover changes between experimental conditions with respect to the relative abundance of unspliced mRNA (over the total mRNA). We model the quantification uncertainty via a latent variable approach, where reads are allocated to their gene/transcript of origin, and to the respective splice version. We designed several benchmarks where our approach shows good performance, in terms of sensitivity and error control, vs. state-of-the-art competitors. Importantly, our tool is flexible, and works with both bulk and single-cell RNA-sequencing data. DifferentialRegulation is distributed as a Bioconductor R package.</p>","PeriodicalId":55357,"journal":{"name":"Biostatistics","volume":" ","pages":"1079-1093"},"PeriodicalIF":1.8,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11639160/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141421995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Projection-based two-sample inference for sparsely observed multivariate functional data. 基于投影的稀疏观测多变量函数数据的双样本推断。
IF 1.8 3区 数学
Biostatistics Pub Date : 2024-10-01 DOI: 10.1093/biostatistics/kxae004
Salil Koner, Sheng Luo
{"title":"Projection-based two-sample inference for sparsely observed multivariate functional data.","authors":"Salil Koner, Sheng Luo","doi":"10.1093/biostatistics/kxae004","DOIUrl":"10.1093/biostatistics/kxae004","url":null,"abstract":"<p><p>Modern longitudinal studies collect multiple outcomes as the primary endpoints to understand the complex dynamics of the diseases. Oftentimes, especially in clinical trials, the joint variation among the multidimensional responses plays a significant role in assessing the differential characteristics between two or more groups, rather than drawing inferences based on a single outcome. We develop a projection-based two-sample significance test to identify the population-level difference between the multivariate profiles observed under a sparse longitudinal design. The methodology is built upon widely adopted multivariate functional principal component analysis to reduce the dimension of the infinite-dimensional multi-modal functions while preserving the dynamic correlation between the components. The test applies to a wide class of (non-stationary) covariance structures of the response, and it detects a significant group difference based on a single p-value, thereby overcoming the issue of adjusting for multiple p-values that arise due to comparing the means in each of components separately. Finite-sample numerical studies demonstrate that the test maintains the type-I error, and is powerful to detect significant group differences, compared to the state-of-the-art testing procedures. The test is carried out on two significant longitudinal studies for Alzheimer's disease and Parkinson's disease (PD) patients, namely, TOMMORROW study of individuals at high risk of mild cognitive impairment to detect differences in the cognitive test scores between the pioglitazone and the placebo groups, and Azillect study to assess the efficacy of rasagiline as a potential treatment to slow down the progression of PD.</p>","PeriodicalId":55357,"journal":{"name":"Biostatistics","volume":" ","pages":"1156-1177"},"PeriodicalIF":1.8,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11639128/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139984624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Functional support vector machine. 功能支持向量机
IF 1.8 3区 数学
Biostatistics Pub Date : 2024-10-01 DOI: 10.1093/biostatistics/kxae007
Shanghong Xie, R Todd Ogden
{"title":"Functional support vector machine.","authors":"Shanghong Xie, R Todd Ogden","doi":"10.1093/biostatistics/kxae007","DOIUrl":"10.1093/biostatistics/kxae007","url":null,"abstract":"<p><p>Linear and generalized linear scalar-on-function modeling have been commonly used to understand the relationship between a scalar response variable (e.g. continuous, binary outcomes) and functional predictors. Such techniques are sensitive to model misspecification when the relationship between the response variable and the functional predictors is complex. On the other hand, support vector machines (SVMs) are among the most robust prediction models but do not take account of the high correlations between repeated measurements and cannot be used for irregular data. In this work, we propose a novel method to integrate functional principal component analysis with SVM techniques for classification and regression to account for the continuous nature of functional data and the nonlinear relationship between the scalar response variable and the functional predictors. We demonstrate the performance of our method through extensive simulation experiments and two real data applications: the classification of alcoholics using electroencephalography signals and the prediction of glucobrassicin concentration using near-infrared reflectance spectroscopy. Our methods especially have more advantages when the measurement errors in functional predictors are relatively large.</p>","PeriodicalId":55357,"journal":{"name":"Biostatistics","volume":" ","pages":"1178-1194"},"PeriodicalIF":1.8,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11639177/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140112299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Correction to: Exponential family measurement error models for single-cell CRISPR screens. 更正:单细胞 CRISPR 筛选的指数族测量误差模型。
IF 1.8 3区 数学
Biostatistics Pub Date : 2024-10-01 DOI: 10.1093/biostatistics/kxae022
{"title":"Correction to: Exponential family measurement error models for single-cell CRISPR screens.","authors":"","doi":"10.1093/biostatistics/kxae022","DOIUrl":"10.1093/biostatistics/kxae022","url":null,"abstract":"","PeriodicalId":55357,"journal":{"name":"Biostatistics","volume":" ","pages":"1273"},"PeriodicalIF":1.8,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141319004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bayesian semiparametric model for sequential treatment decisions with informative timing. 具有信息时间的序列治疗决策的贝叶斯半参数模型。
IF 1.8 3区 数学
Biostatistics Pub Date : 2024-10-01 DOI: 10.1093/biostatistics/kxad035
Arman Oganisian, Kelly D Getz, Todd A Alonzo, Richard Aplenc, Jason A Roy
{"title":"Bayesian semiparametric model for sequential treatment decisions with informative timing.","authors":"Arman Oganisian, Kelly D Getz, Todd A Alonzo, Richard Aplenc, Jason A Roy","doi":"10.1093/biostatistics/kxad035","DOIUrl":"10.1093/biostatistics/kxad035","url":null,"abstract":"<p><p>We develop a Bayesian semiparametric model for the impact of dynamic treatment rules on survival among patients diagnosed with pediatric acute myeloid leukemia (AML). The data consist of a subset of patients enrolled in a phase III clinical trial in which patients move through a sequence of four treatment courses. At each course, they undergo treatment that may or may not include anthracyclines (ACT). While ACT is known to be effective at treating AML, it is also cardiotoxic and can lead to early death for some patients. Our task is to estimate the potential survival probability under hypothetical dynamic ACT treatment strategies, but there are several impediments. First, since ACT is not randomized, its effect on survival is confounded over time. Second, subjects initiate the next course depending on when they recover from the previous course, making timing potentially informative of subsequent treatment and survival. Third, patients may die or drop out before ever completing the full treatment sequence. We develop a generative Bayesian semiparametric model based on Gamma Process priors to address these complexities. At each treatment course, the model captures subjects' transition to subsequent treatment or death in continuous time. G-computation is used to compute a posterior over potential survival probability that is adjusted for time-varying confounding. Using our approach, we estimate the efficacy of hypothetical treatment rules that dynamically modify ACT based on evolving cardiac function.</p>","PeriodicalId":55357,"journal":{"name":"Biostatistics","volume":" ","pages":"947-961"},"PeriodicalIF":1.8,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11471958/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139479547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Mendelian randomization analysis using multiple biomarkers of an underlying common exposure. 利用潜在共同暴露的多种生物标志物进行孟德尔随机分析。
IF 1.8 3区 数学
Biostatistics Pub Date : 2024-10-01 DOI: 10.1093/biostatistics/kxae006
Jin Jin, Guanghao Qi, Zhi Yu, Nilanjan Chatterjee
{"title":"Mendelian randomization analysis using multiple biomarkers of an underlying common exposure.","authors":"Jin Jin, Guanghao Qi, Zhi Yu, Nilanjan Chatterjee","doi":"10.1093/biostatistics/kxae006","DOIUrl":"10.1093/biostatistics/kxae006","url":null,"abstract":"<p><p>Mendelian randomization (MR) analysis is increasingly popular for testing the causal effect of exposures on disease outcomes using data from genome-wide association studies. In some settings, the underlying exposure, such as systematic inflammation, may not be directly observable, but measurements can be available on multiple biomarkers or other types of traits that are co-regulated by the exposure. We propose a method for MR analysis on latent exposures (MRLE), which tests the significance for, and the direction of, the effect of a latent exposure by leveraging information from multiple related traits. The method is developed by constructing a set of estimating functions based on the second-order moments of GWAS summary association statistics for the observable traits, under a structural equation model where genetic variants are assumed to have indirect effects through the latent exposure and potentially direct effects on the traits. Simulation studies show that MRLE has well-controlled type I error rates and enhanced power compared to single-trait MR tests under various types of pleiotropy. Applications of MRLE using genetic association statistics across five inflammatory biomarkers (CRP, IL-6, IL-8, TNF-α, and MCP-1) provide evidence for potential causal effects of inflammation on increasing the risk of coronary artery disease, colorectal cancer, and rheumatoid arthritis, while standard MR analysis for individual biomarkers fails to detect consistent evidence for such effects.</p>","PeriodicalId":55357,"journal":{"name":"Biostatistics","volume":" ","pages":"1015-1033"},"PeriodicalIF":1.8,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11879930/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140066298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 Bayesian approach for investigating the pharmacogenetics of combination antiretroviral therapy in people with HIV. 研究艾滋病病毒感染者抗逆转录病毒联合疗法药物遗传学的贝叶斯方法。
IF 1.8 3区 数学
Biostatistics Pub Date : 2024-10-01 DOI: 10.1093/biostatistics/kxae001
Wei Jin, Yang Ni, Amanda B Spence, Leah H Rubin, Yanxun Xu
{"title":"A Bayesian approach for investigating the pharmacogenetics of combination antiretroviral therapy in people with HIV.","authors":"Wei Jin, Yang Ni, Amanda B Spence, Leah H Rubin, Yanxun Xu","doi":"10.1093/biostatistics/kxae001","DOIUrl":"10.1093/biostatistics/kxae001","url":null,"abstract":"<p><p>Combination antiretroviral therapy (ART) with at least three different drugs has become the standard of care for people with HIV (PWH) due to its exceptional effectiveness in viral suppression. However, many ART drugs have been reported to associate with neuropsychiatric adverse effects including depression, especially when certain genetic polymorphisms exist. Pharmacogenetics is an important consideration for administering combination ART as it may influence drug efficacy and increase risk for neuropsychiatric conditions. Large-scale longitudinal HIV databases provide researchers opportunities to investigate the pharmacogenetics of combination ART in a data-driven manner. However, with more than 30 FDA-approved ART drugs, the interplay between the large number of possible ART drug combinations and genetic polymorphisms imposes statistical modeling challenges. We develop a Bayesian approach to examine the longitudinal effects of combination ART and their interactions with genetic polymorphisms on depressive symptoms in PWH. The proposed method utilizes a Gaussian process with a composite kernel function to capture the longitudinal combination ART effects by directly incorporating individuals' treatment histories, and a Bayesian classification and regression tree to account for individual heterogeneity. Through both simulation studies and an application to a dataset from the Women's Interagency HIV Study, we demonstrate the clinical utility of the proposed approach in investigating the pharmacogenetics of combination ART and assisting physicians to make effective individualized treatment decisions that can improve health outcomes for PWH.</p>","PeriodicalId":55357,"journal":{"name":"Biostatistics","volume":" ","pages":"1034-1048"},"PeriodicalIF":1.8,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139747854","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fast matrix completion in epigenetic methylation studies with informative covariates. 在带有信息协变量的表观遗传甲基化研究中快速完成矩阵。
IF 1.8 3区 数学
Biostatistics Pub Date : 2024-10-01 DOI: 10.1093/biostatistics/kxae016
Mélina Ribaud, Aurélie Labbe, Khaled Fouda, Karim Oualkacha
{"title":"Fast matrix completion in epigenetic methylation studies with informative covariates.","authors":"Mélina Ribaud, Aurélie Labbe, Khaled Fouda, Karim Oualkacha","doi":"10.1093/biostatistics/kxae016","DOIUrl":"10.1093/biostatistics/kxae016","url":null,"abstract":"<p><p>DNA methylation is an important epigenetic mark that modulates gene expression through the inhibition of transcriptional proteins binding to DNA. As in many other omics experiments, the issue of missing values is an important one, and appropriate imputation techniques are important in avoiding an unnecessary sample size reduction as well as to optimally leverage the information collected. We consider the case where relatively few samples are processed via an expensive high-density whole genome bisulfite sequencing (WGBS) strategy and a larger number of samples is processed using more affordable low-density, array-based technologies. In such cases, one can impute the low-coverage (array-based) methylation data using the high-density information provided by the WGBS samples. In this paper, we propose an efficient Linear Model of Coregionalisation with informative Covariates (LMCC) to predict missing values based on observed values and covariates. Our model assumes that at each site, the methylation vector of all samples is linked to the set of fixed factors (covariates) and a set of latent factors. Furthermore, we exploit the functional nature of the data and the spatial correlation across sites by assuming some Gaussian processes on the fixed and latent coefficient vectors, respectively. Our simulations show that the use of covariates can significantly improve the accuracy of imputed values, especially in cases where missing data contain some relevant information about the explanatory variable. We also showed that our proposed model is particularly efficient when the number of columns is much greater than the number of rows-which is usually the case in methylation data analysis. Finally, we apply and compare our proposed method with alternative approaches on two real methylation datasets, showing how covariates such as cell type, tissue type or age can enhance the accuracy of imputed values.</p>","PeriodicalId":55357,"journal":{"name":"Biostatistics","volume":" ","pages":"1062-1078"},"PeriodicalIF":1.8,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11471954/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141293984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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:481959085
Book学术官方微信