Pub Date : 2025-10-29DOI: 10.1146/annurev-statistics-042424-114518
Šárka Hudecová
Assessing the performance of an estimated model for count time series is critical for subsequent statistical inference and distributional forecasting. This article reviews the most commonly used count time series models and focuses on evaluating their goodness of fit. Various formal statistical tests are presented, along with useful graphical diagnostic tools. The methods are illustrated on a real data example.
{"title":"Structure Assessment in Count Time Series","authors":"Šárka Hudecová","doi":"10.1146/annurev-statistics-042424-114518","DOIUrl":"https://doi.org/10.1146/annurev-statistics-042424-114518","url":null,"abstract":"Assessing the performance of an estimated model for count time series is critical for subsequent statistical inference and distributional forecasting. This article reviews the most commonly used count time series models and focuses on evaluating their goodness of fit. Various formal statistical tests are presented, along with useful graphical diagnostic tools. The methods are illustrated on a real data example.","PeriodicalId":48855,"journal":{"name":"Annual Review of Statistics and Its Application","volume":"113 1","pages":""},"PeriodicalIF":7.9,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145397498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-14DOI: 10.1146/annurev-statistics-042424-063308
Stevenson Bolivar, Shuo-Chieh Huang, Rong Chen
This article provides a comprehensive overview of statistical methods developed for the analysis of tensor time series data, which have become increasingly prevalent across various fields such as economics, finance, biology, engineering, and the social sciences. The review focuses on three primary approaches: autoregressive modeling, factor modeling, and segmentation approaches. These methods leverage the inherent tensor structure to offer advantages such as dimension reduction, enhanced interpretability, and computational efficiency. The review focuses on model settings and their potential interpretations, discussing various estimation techniques for these models and their associated theoretical properties. In addition, we outline various applications using these models and discuss potential directions for future developments.
{"title":"Analysis of Tensor Time Series","authors":"Stevenson Bolivar, Shuo-Chieh Huang, Rong Chen","doi":"10.1146/annurev-statistics-042424-063308","DOIUrl":"https://doi.org/10.1146/annurev-statistics-042424-063308","url":null,"abstract":"This article provides a comprehensive overview of statistical methods developed for the analysis of tensor time series data, which have become increasingly prevalent across various fields such as economics, finance, biology, engineering, and the social sciences. The review focuses on three primary approaches: autoregressive modeling, factor modeling, and segmentation approaches. These methods leverage the inherent tensor structure to offer advantages such as dimension reduction, enhanced interpretability, and computational efficiency. The review focuses on model settings and their potential interpretations, discussing various estimation techniques for these models and their associated theoretical properties. In addition, we outline various applications using these models and discuss potential directions for future developments.","PeriodicalId":48855,"journal":{"name":"Annual Review of Statistics and Its Application","volume":"27 1","pages":""},"PeriodicalIF":7.9,"publicationDate":"2025-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145289224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-14DOI: 10.1146/annurev-statistics-042424-103337
Guadalupe Gómez Melis, Ramon Oller, Klaus Langohr
Survival analysis is essential for modeling time-to-event data across various fields, including medicine, engineering, and the social sciences. A major challenge in this field is handling censored data, particularly partly interval-censored data, where event times are either precisely recorded or only known to fall within a specific interval. Proper statistical modeling of such data is crucial for drawing valid conclusions and making accurate predictions. This article reviews regression models for analyzing interval-censored responses and their implementation in R. Following an introduction to the nonparametric maximum likelihood estimator, we focus on four major regression models: the accelerated failure time model, the proportional hazards model, the proportional odds model, and the generalized odds-rate model. For each, we review the state of the art, outline its methodology, discuss implementation strategies, and illustrate practical applications using real-world data. The article concludes with a discussion of current challenges, alternative modeling approaches, and potential directions for future research.
{"title":"Regression Models with Interval-Censored Variables","authors":"Guadalupe Gómez Melis, Ramon Oller, Klaus Langohr","doi":"10.1146/annurev-statistics-042424-103337","DOIUrl":"https://doi.org/10.1146/annurev-statistics-042424-103337","url":null,"abstract":"Survival analysis is essential for modeling time-to-event data across various fields, including medicine, engineering, and the social sciences. A major challenge in this field is handling censored data, particularly partly interval-censored data, where event times are either precisely recorded or only known to fall within a specific interval. Proper statistical modeling of such data is crucial for drawing valid conclusions and making accurate predictions. This article reviews regression models for analyzing interval-censored responses and their implementation in R. Following an introduction to the nonparametric maximum likelihood estimator, we focus on four major regression models: the accelerated failure time model, the proportional hazards model, the proportional odds model, and the generalized odds-rate model. For each, we review the state of the art, outline its methodology, discuss implementation strategies, and illustrate practical applications using real-world data. The article concludes with a discussion of current challenges, alternative modeling approaches, and potential directions for future research.","PeriodicalId":48855,"journal":{"name":"Annual Review of Statistics and Its Application","volume":"1 1","pages":""},"PeriodicalIF":7.9,"publicationDate":"2025-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145289381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-01DOI: 10.1146/annurev-statistics-042424-113016
Gen Li, Eric F. Lock
With advancements in technology and the decreasing cost of data acquisition, high-throughput omics data have become increasingly prevalent in biomedical research. These data are often collected across multiple omics modalities at different molecular levels, offering a comprehensive perspective on underlying biological mechanisms. However, the multimodal nature of multiomics data presents unique and complex challenges for statistical analysis. In this article, we provide a comprehensive review of recent advancements in statistical methods for multiomics data integration. We discuss key topics in unsupervised learning (including dimension reduction, clustering, and network analysis), supervised learning (including regression, classification, and mediation analysis), and other areas. Finally, we highlight unresolved challenges and propose promising directions for future research to further advance the field.
{"title":"Integrative Analysis of Multimodal Omics Data","authors":"Gen Li, Eric F. Lock","doi":"10.1146/annurev-statistics-042424-113016","DOIUrl":"https://doi.org/10.1146/annurev-statistics-042424-113016","url":null,"abstract":"With advancements in technology and the decreasing cost of data acquisition, high-throughput omics data have become increasingly prevalent in biomedical research. These data are often collected across multiple omics modalities at different molecular levels, offering a comprehensive perspective on underlying biological mechanisms. However, the multimodal nature of multiomics data presents unique and complex challenges for statistical analysis. In this article, we provide a comprehensive review of recent advancements in statistical methods for multiomics data integration. We discuss key topics in unsupervised learning (including dimension reduction, clustering, and network analysis), supervised learning (including regression, classification, and mediation analysis), and other areas. Finally, we highlight unresolved challenges and propose promising directions for future research to further advance the field.","PeriodicalId":48855,"journal":{"name":"Annual Review of Statistics and Its Application","volume":"114 1","pages":""},"PeriodicalIF":7.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145203439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-10DOI: 10.1146/annurev-statistics-041124-044143
Jialiang Li, Jingli Wang, Yuetao Yu
We review recent advances in change-point detection methods across three important fields of statistics: (a) We first present a subgroup identification method based on a multi-threshold change plane model where the subgroup boundaries are defined by a high-dimensional hyperplane in the covariate space. Subjects grouped into different regions may receive more individualized treatments in medical research studies and achieve improved health outcomes. (b) We then consider the estimation of discontinuity for functional process data. Many longitudinal or functional responses may exhibit abrupt jumps, and our methodology effectively accommodates such complicated nonsmooth features. (c) Finally, we explore change-point estimation within dynamic networks using a recently proposed network autoregressive model. This framework demonstrates that community structures in networks can shift similarly to changes observed in time series data. These reviews highlight the wide-ranging applications of change-point detection methodologies in modern data analysis.
{"title":"Change-Point Detection and Its Modern Applications","authors":"Jialiang Li, Jingli Wang, Yuetao Yu","doi":"10.1146/annurev-statistics-041124-044143","DOIUrl":"https://doi.org/10.1146/annurev-statistics-041124-044143","url":null,"abstract":"We review recent advances in change-point detection methods across three important fields of statistics: (<jats:italic>a</jats:italic>) We first present a subgroup identification method based on a multi-threshold change plane model where the subgroup boundaries are defined by a high-dimensional hyperplane in the covariate space. Subjects grouped into different regions may receive more individualized treatments in medical research studies and achieve improved health outcomes. (<jats:italic>b</jats:italic>) We then consider the estimation of discontinuity for functional process data. Many longitudinal or functional responses may exhibit abrupt jumps, and our methodology effectively accommodates such complicated nonsmooth features. (<jats:italic>c</jats:italic>) Finally, we explore change-point estimation within dynamic networks using a recently proposed network autoregressive model. This framework demonstrates that community structures in networks can shift similarly to changes observed in time series data. These reviews highlight the wide-ranging applications of change-point detection methodologies in modern data analysis.","PeriodicalId":48855,"journal":{"name":"Annual Review of Statistics and Its Application","volume":"43 1","pages":""},"PeriodicalIF":7.9,"publicationDate":"2025-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145043395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-10DOI: 10.1146/annurev-statistics-042424-061122
Hamish William Patten, Zineb Bhaby
This article examines the role of statistics in the humanitarian sector, with a particular focus on disasters caused by natural hazards. It begins by outlining current applications, including primary data collection, anticipatory action frameworks, Earth observation, mobile positioning data, and artificial intelligence. It then highlights key challenges such as gaps and biases in disaster impact and response data, difficulties in communicating statistical findings clearly, inequities in aid allocation, and the widespread outsourcing of statistics-related work. In exploring future applications, the article discusses the potential of impact-based early warning models, dynamic population data, and artificial intelligence to enhance communication and decision-making. Throughout, emphasis is placed on the need for interoperable systems as well as ethical and inclusive data practices. In doing so, the article presents statistics as both a diagnostic and strategic tool for strengthening the effectiveness, fairness, and responsiveness of humanitarian action in disaster contexts.
{"title":"Disasters, Statistics, and the Humanitarian Sector","authors":"Hamish William Patten, Zineb Bhaby","doi":"10.1146/annurev-statistics-042424-061122","DOIUrl":"https://doi.org/10.1146/annurev-statistics-042424-061122","url":null,"abstract":"This article examines the role of statistics in the humanitarian sector, with a particular focus on disasters caused by natural hazards. It begins by outlining current applications, including primary data collection, anticipatory action frameworks, Earth observation, mobile positioning data, and artificial intelligence. It then highlights key challenges such as gaps and biases in disaster impact and response data, difficulties in communicating statistical findings clearly, inequities in aid allocation, and the widespread outsourcing of statistics-related work. In exploring future applications, the article discusses the potential of impact-based early warning models, dynamic population data, and artificial intelligence to enhance communication and decision-making. Throughout, emphasis is placed on the need for interoperable systems as well as ethical and inclusive data practices. In doing so, the article presents statistics as both a diagnostic and strategic tool for strengthening the effectiveness, fairness, and responsiveness of humanitarian action in disaster contexts.","PeriodicalId":48855,"journal":{"name":"Annual Review of Statistics and Its Application","volume":"85 1","pages":""},"PeriodicalIF":7.9,"publicationDate":"2025-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145043391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-28DOI: 10.1146/annurev-statistics-042324-123749
Ben Van Calster, Maarten van Smeden, Wouter van Amsterdam, Maarten Coemans, Laure Wynants, Ewout W. Steyerberg
The current status of applied clinical prediction modeling is poor. Many models are developed with suboptimal methods and are not evaluated, and hence have little impact on clinical care. We review 12 challenges—provocatively labeled enemies—that jeopardize the creation of prediction models that make it to clinical practice to improve treatment decisions and clinical outcomes for individual patients. The challenges cover four areas: context, data, design and analysis, and scientific culture. We provide negative examples and recommendations for improvement, but also highlight positive examples and developments. Greater awareness of the complexities surrounding clinical prediction modeling is needed among researchers, funding agencies, health professionals as end users, and all of us as potential patients. To improve the utility of prediction models for healthcare and society, we need fewer but better models as well as more resources for model validation, impact assessment, and implementation.
{"title":"The Enemies of Reliable and Useful Clinical Prediction Models: A Review of Statistical and Scientific Challenges","authors":"Ben Van Calster, Maarten van Smeden, Wouter van Amsterdam, Maarten Coemans, Laure Wynants, Ewout W. Steyerberg","doi":"10.1146/annurev-statistics-042324-123749","DOIUrl":"https://doi.org/10.1146/annurev-statistics-042324-123749","url":null,"abstract":"The current status of applied clinical prediction modeling is poor. Many models are developed with suboptimal methods and are not evaluated, and hence have little impact on clinical care. We review 12 challenges—provocatively labeled enemies—that jeopardize the creation of prediction models that make it to clinical practice to improve treatment decisions and clinical outcomes for individual patients. The challenges cover four areas: context, data, design and analysis, and scientific culture. We provide negative examples and recommendations for improvement, but also highlight positive examples and developments. Greater awareness of the complexities surrounding clinical prediction modeling is needed among researchers, funding agencies, health professionals as end users, and all of us as potential patients. To improve the utility of prediction models for healthcare and society, we need fewer but better models as well as more resources for model validation, impact assessment, and implementation.","PeriodicalId":48855,"journal":{"name":"Annual Review of Statistics and Its Application","volume":"20 1","pages":""},"PeriodicalIF":7.9,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144915659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-27DOI: 10.1146/annurev-statistics-042424-052920
Alan E. Gelfand, Erin M. Schliep
With increased data collection, the need to fuse data sources has emerged as an important and rapidly growing research activity in the statistical community. In considering spatial and spatio-temporal datasets to examine complex environmental and ecological processes of interest, we often have multiple sources that are jointly informative about features of interest of the processes. Model-based data fusion aims to leverage information from these sources to improve inference and prediction. In the spatial statistics setting, these data could be geostatistical; areal; or point patterns with varying spatial resolutions, supports, and domains. Given two or more sources, we explore stochastic modeling to implement a suitable fusion with full inference and uncertainty quantification. We illustrate these ideas using three environmental and ecological examples: precipitation, marine mammal abundance, and joint species distributions.
{"title":"Model-Based Spatial Data Fusion","authors":"Alan E. Gelfand, Erin M. Schliep","doi":"10.1146/annurev-statistics-042424-052920","DOIUrl":"https://doi.org/10.1146/annurev-statistics-042424-052920","url":null,"abstract":"With increased data collection, the need to fuse data sources has emerged as an important and rapidly growing research activity in the statistical community. In considering spatial and spatio-temporal datasets to examine complex environmental and ecological processes of interest, we often have multiple sources that are jointly informative about features of interest of the processes. Model-based data fusion aims to leverage information from these sources to improve inference and prediction. In the spatial statistics setting, these data could be geostatistical; areal; or point patterns with varying spatial resolutions, supports, and domains. Given two or more sources, we explore stochastic modeling to implement a suitable fusion with full inference and uncertainty quantification. We illustrate these ideas using three environmental and ecological examples: precipitation, marine mammal abundance, and joint species distributions.","PeriodicalId":48855,"journal":{"name":"Annual Review of Statistics and Its Application","volume":"10 1","pages":""},"PeriodicalIF":7.9,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144910905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-04DOI: 10.1146/annurev-statistics-040522-015431
Aurélien Bibaut, Nathan Kallus
Adaptive experiments such as multi-armed bandits adapt the treatment-allocation policy and/or the decision to stop the experiment to the data observed so far. This has the potential to improve outcomes for study participants within the experiment, to improve the chance of identifying the best treatments after the experiment, and to avoid wasting data. As an experiment (and not just a continually optimizing system), it is still desirable to draw statistical inferences with frequentist guarantees. The concentration inequalities and union bounds that generally underlie adaptive experimentation algorithms can yield overly conservative inferences, but at the same time, the asymptotic normality we would usually appeal to in nonadaptive settings can be imperiled by adaptivity. In this article we aim to explain why, how, and when adaptivity is in fact an issue for inference and, when it is, to understand the various ways to fix it: reweighting to stabilize variances and recover asymptotic normality, using always-valid inference based on joint normality of an asymptotic limiting sequence, and characterizing and inverting the nonnormal distributions induced by adaptivity.
{"title":"Demystifying Inference After Adaptive Experiments","authors":"Aurélien Bibaut, Nathan Kallus","doi":"10.1146/annurev-statistics-040522-015431","DOIUrl":"https://doi.org/10.1146/annurev-statistics-040522-015431","url":null,"abstract":"Adaptive experiments such as multi-armed bandits adapt the treatment-allocation policy and/or the decision to stop the experiment to the data observed so far. This has the potential to improve outcomes for study participants within the experiment, to improve the chance of identifying the best treatments after the experiment, and to avoid wasting data. As an experiment (and not just a continually optimizing system), it is still desirable to draw statistical inferences with frequentist guarantees. The concentration inequalities and union bounds that generally underlie adaptive experimentation algorithms can yield overly conservative inferences, but at the same time, the asymptotic normality we would usually appeal to in nonadaptive settings can be imperiled by adaptivity. In this article we aim to explain why, how, and when adaptivity is in fact an issue for inference and, when it is, to understand the various ways to fix it: reweighting to stabilize variances and recover asymptotic normality, using always-valid inference based on joint normality of an asymptotic limiting sequence, and characterizing and inverting the nonnormal distributions induced by adaptivity.","PeriodicalId":48855,"journal":{"name":"Annual Review of Statistics and Its Application","volume":"26 1","pages":"407-423"},"PeriodicalIF":7.9,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144565895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-21DOI: 10.1146/annurev-statistics-040522-013920
Namjoon Suh, Guang Cheng
In this article, we review the literature on statistical theories of neural networks from three perspectives: approximation, training dynamics, and generative models. In the first part, results on excess risks for neural networks are reviewed in the nonparametric framework of regression. These results rely on explicit constructions of neural networks, leading to fast convergence rates of excess risks. Nonetheless, their underlying analysis only applies to the global minimizer in the highly nonconvex landscape of deep neural networks. This motivates us to review the training dynamics of neural networks in the second part. Specifically, we review articles that attempt to answer the question of how a neural network trained via gradient-based methods finds a solution that can generalize well on unseen data. In particular, two well-known paradigms are reviewed: the neural tangent kernel and mean-field paradigms. Last, we review the most recent theoretical advancements in generative models, including generative adversarial networks, diffusion models, and in-context learning in large language models from two of the same perspectives, approximation and training dynamics.
{"title":"A Survey on Statistical Theory of Deep Learning: Approximation, Training Dynamics, and Generative Models","authors":"Namjoon Suh, Guang Cheng","doi":"10.1146/annurev-statistics-040522-013920","DOIUrl":"https://doi.org/10.1146/annurev-statistics-040522-013920","url":null,"abstract":"In this article, we review the literature on statistical theories of neural networks from three perspectives: approximation, training dynamics, and generative models. In the first part, results on excess risks for neural networks are reviewed in the nonparametric framework of regression. These results rely on explicit constructions of neural networks, leading to fast convergence rates of excess risks. Nonetheless, their underlying analysis only applies to the global minimizer in the highly nonconvex landscape of deep neural networks. This motivates us to review the training dynamics of neural networks in the second part. Specifically, we review articles that attempt to answer the question of how a neural network trained via gradient-based methods finds a solution that can generalize well on unseen data. In particular, two well-known paradigms are reviewed: the neural tangent kernel and mean-field paradigms. Last, we review the most recent theoretical advancements in generative models, including generative adversarial networks, diffusion models, and in-context learning in large language models from two of the same perspectives, approximation and training dynamics.","PeriodicalId":48855,"journal":{"name":"Annual Review of Statistics and Its Application","volume":"111 1","pages":""},"PeriodicalIF":7.9,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142684813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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