In a recent article, J. Peyhardi gives a number of novel results related to quasi Pólya thinning which encompass a number of important mixture relationships between univariate discrete distributions. In this note, I explore the duals of the general results on convolution thinning given in Peyhardi's Theorem 1 in order to obtain new relationships and to gain new insights into old relationships. Some consequences—for integer‐valued autoregressive processes—and analogues—in the continuous case—are noted.
{"title":"Duals of convolution thinned relationships","authors":"M. C. Jones","doi":"10.1111/stan.12337","DOIUrl":"https://doi.org/10.1111/stan.12337","url":null,"abstract":"In a recent article, J. Peyhardi gives a number of novel results related to quasi Pólya thinning which encompass a number of important mixture relationships between univariate discrete distributions. In this note, I explore the duals of the general results on convolution thinning given in Peyhardi's Theorem 1 in order to obtain new relationships and to gain new insights into old relationships. Some consequences—for integer‐valued autoregressive processes—and analogues—in the continuous case—are noted.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140197893","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}
{"title":"Estimation and convergence rates in the distributional single index model","authors":"Fadoua Balabdaoui, Alexander Henzi, Lukas Looser","doi":"10.1111/stan.12336","DOIUrl":"https://doi.org/10.1111/stan.12336","url":null,"abstract":"The distributional single index model is a semiparametric regression model in which the conditional distribution functions <mjx-container aria-label=\"upper P left parenthesis upper Y less than or equals y vertical bar upper X equals x right parenthesis equals upper F 0 left parenthesis theta 0 left parenthesis x right parenthesis comma y right parenthesis\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"36,34\" data-semantic-content=\"10\" data-semantic- data-semantic-role=\"equality\" data-semantic-speech=\"upper P left parenthesis upper Y less than or equals y vertical bar upper X equals x right parenthesis equals upper F 0 left parenthesis theta 0 left parenthesis x right parenthesis comma y right parenthesis\" data-semantic-type=\"relseq\"><mjx-mrow data-semantic-children=\"0,27\" data-semantic-content=\"35,0\" data-semantic- data-semantic-parent=\"37\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"36\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"36\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"26\" data-semantic-content=\"1,9\" data-semantic- data-semantic-parent=\"36\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"27\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"24,5,25\" data-semantic-content=\"5\" data-semantic- data-semantic-parent=\"27\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mjx-mrow data-semantic-children=\"2,4\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"26\" data-semantic-role=\"inequality\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"24\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,≤\" data-semantic-parent=\"24\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"24\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"punctuated\" data-semanti","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140172068","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}
We analyze nonparametric estimators for the distribution function of the incubation time in the singly and doubly interval censoring model. The classical approach is to use parametric families like Weibull, log‐normal or gamma distributions in the estimation procedure. We propose nonparametric estimates for functions of the observations, which stay closer to the data than the classical parametric methods. We also give explicit limit distributions for discrete versions of the models and apply this to compute confidence intervals. The methods complement the analysis of the continuous model in Groeneboom (2021, 2023). R scripts for computation of the estimates are provided in Groeneboom (2020).
{"title":"Estimation of the incubation time distribution in the singly and doubly interval censored model","authors":"Piet Groeneboom","doi":"10.1111/stan.12335","DOIUrl":"https://doi.org/10.1111/stan.12335","url":null,"abstract":"We analyze nonparametric estimators for the distribution function of the incubation time in the singly and doubly interval censoring model. The classical approach is to use parametric families like Weibull, log‐normal or gamma distributions in the estimation procedure. We propose nonparametric estimates for functions of the observations, which stay closer to the data than the classical parametric methods. We also give explicit limit distributions for discrete versions of the models and apply this to compute confidence intervals. The methods complement the analysis of the continuous model in Groeneboom (2021, 2023). <jats:styled-content>R</jats:styled-content> scripts for computation of the estimates are provided in Groeneboom (2020).","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139952885","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}
Self-exciting point processes allow modeling the temporal location of an event of interest, considering the history provided by previously observed events. This family of point processes is commonly used in several areas such as criminology, economics, or seismology, among others. The standard formulation of the self-exciting process implies assuming that the underlying stochastic process is dependent on its previous history over the entire period under analysis. In this paper, we consider the possibility of modeling a point pattern through a point process whose structure is not necessarily of self-exciting type at every instant or temporal interval. Specifically, we propose a mixture point process model that allows the point process to be either self-exciting or homogeneous Poisson, depending on the instant within the study period. The performance of this model is evaluated both through a simulation study and a case study. The results indicate that the model is able to detect the presence of instants in time, referred to as change points, where the nature of the process varies.
{"title":"Point process modeling through a mixture of homogeneous and self-exciting processes","authors":"Álvaro Briz-Redón, Jorge Mateu","doi":"10.1111/stan.12334","DOIUrl":"https://doi.org/10.1111/stan.12334","url":null,"abstract":"Self-exciting point processes allow modeling the temporal location of an event of interest, considering the history provided by previously observed events. This family of point processes is commonly used in several areas such as criminology, economics, or seismology, among others. The standard formulation of the self-exciting process implies assuming that the underlying stochastic process is dependent on its previous history over the entire period under analysis. In this paper, we consider the possibility of modeling a point pattern through a point process whose structure is not necessarily of self-exciting type at every instant or temporal interval. Specifically, we propose a mixture point process model that allows the point process to be either self-exciting or homogeneous Poisson, depending on the instant within the study period. The performance of this model is evaluated both through a simulation study and a case study. The results indicate that the model is able to detect the presence of instants in time, referred to as change points, where the nature of the process varies.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139483210","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}
We consider inference about the parameter that determines the distribution of the data. In frequentist inference a very important and useful idea is that data reduction to a sufficient statistic does not lose any information about this parameter. We recall two justifications for this idea in frequentist inference. We then examine the extent to which these justifications carry over to conditional frequentist inference inference, which consists of carrying out frequentist inference conditional on an ancillary statistic. This examination shows that, in the context of conditional frequentist inference, first reducing data to a sufficient statistic is not always justified, so we should first condition on an ancillary statistic. Finally, we describe two types of practically-important statistical models that illustrate this finding.
{"title":"The concept of sufficiency in conditional frequentist inference","authors":"Paul Kabaila, A. H. Welsh","doi":"10.1111/stan.12333","DOIUrl":"https://doi.org/10.1111/stan.12333","url":null,"abstract":"We consider inference about the parameter that determines the distribution of the data. In frequentist inference a very important and useful idea is that data reduction to a sufficient statistic does not lose any information about this parameter. We recall two justifications for this idea in frequentist inference. We then examine the extent to which these justifications carry over to conditional frequentist inference inference, which consists of carrying out frequentist inference conditional on an ancillary statistic. This examination shows that, in the context of conditional frequentist inference, first reducing data to a sufficient statistic is not always justified, so we should first condition on an ancillary statistic. Finally, we describe two types of practically-important statistical models that illustrate this finding.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138510084","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}
Collapsibility is a practical and useful technique for dimension reduction in multidimensional contingency tables. In this paper, we consider marginal log‐linear models for studying collapsibility and related aspects in such tables. These models generalize ordinary log‐linear and multivariate logistic models, besides several others. First, we obtain some characteristic properties of marginal log‐linear parameters. Then we define collapsibility and strict collapsibility of these parameters in a general sense. Several necessary and sufficient conditions for collapsibility and strict collapsibility are derived based on simple functions of only the cell probabilities, which are easily verifiable. These include results for an arbitrary set of marginal log‐linear parameters having some common effects. The connections of strict collapsibility to various forms of independence of the variables are explored. We analyze some real‐life datasets to illustrate the above results on collapsibility and strict collapsibility. Finally, we obtain a result relating parameters with the same effect, but different margins for an arbitrary table, and demonstrate smoothness of marginal log‐linear models under collapsibility conditions. This article is protected by copyright. All rights reserved.
{"title":"Marginal Log‐linear Parameters and their Collapsibility for Categorical Data","authors":"S. Ghosh, P. Vellaisamy","doi":"10.1111/stan.12332","DOIUrl":"https://doi.org/10.1111/stan.12332","url":null,"abstract":"Collapsibility is a practical and useful technique for dimension reduction in multidimensional contingency tables. In this paper, we consider marginal log‐linear models for studying collapsibility and related aspects in such tables. These models generalize ordinary log‐linear and multivariate logistic models, besides several others. First, we obtain some characteristic properties of marginal log‐linear parameters. Then we define collapsibility and strict collapsibility of these parameters in a general sense. Several necessary and sufficient conditions for collapsibility and strict collapsibility are derived based on simple functions of only the cell probabilities, which are easily verifiable. These include results for an arbitrary set of marginal log‐linear parameters having some common effects. The connections of strict collapsibility to various forms of independence of the variables are explored. We analyze some real‐life datasets to illustrate the above results on collapsibility and strict collapsibility. Finally, we obtain a result relating parameters with the same effect, but different margins for an arbitrary table, and demonstrate smoothness of marginal log‐linear models under collapsibility conditions. This article is protected by copyright. All rights reserved.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134993188","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}
This note develops a generalized ‐variate hazard function for censored data in survival analysis. It introduces a generalized recursive formula, extending the bivariate and trivariate cases introduced by Clayton and Cuzick (1985, Journal of the Royal Statistical Society: Series A (General) , 148(2):82–108) and Kittani (1995, Journal of Mathematical Sciences , 67–74), respectively. The newly developed function is explicitly specified by association parameters and marginal hazard functions.
本文发展了生存分析中截尾数据的广义变量风险函数。它引入了一个广义的递归公式，扩展了Clayton和Cuzick (1985, Journal of Royal Statistical Society: Series a (General)， 148(2): 82-108)和Kittani (1995, Journal of Mathematical Sciences, 67-74)分别介绍的二元和三变量情况。该函数由关联参数和边际风险函数明确表示。
{"title":"Generalized K‐Variate Proportional Hazard Function For Censored Survival Data","authors":"Hilmi Fadel Kittani","doi":"10.1111/stan.12327","DOIUrl":"https://doi.org/10.1111/stan.12327","url":null,"abstract":"This note develops a generalized ‐variate hazard function for censored data in survival analysis. It introduces a generalized recursive formula, extending the bivariate and trivariate cases introduced by Clayton and Cuzick (1985, Journal of the Royal Statistical Society: Series A (General) , 148(2):82–108) and Kittani (1995, Journal of Mathematical Sciences , 67–74), respectively. The newly developed function is explicitly specified by association parameters and marginal hazard functions.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135088197","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}
Abstract In this paper we study how the expectations of power means behave asymptotically as some relevant parameter approaches infinity and how to approximate them accurately for general non‐negative continuous probability distributions. We derive approximation formulae for such expectations as distribution mean increases, and apply them to some commonly used distributions in statistics and financial mathematics. By numerical computations we demonstrate the accuracy of the proposed formulae which behave well even for smaller sample sizes. Furthermore, analysis of behaviour depending on sample size contributes to interesting connections with the power mean of probability distribution. This article is protected by copyright. All rights reserved.
{"title":"Asymptotic approximations of expectations of power means","authors":"Tomislav Buri, Lenka Mihokovi","doi":"10.1111/stan.12331","DOIUrl":"https://doi.org/10.1111/stan.12331","url":null,"abstract":"Abstract In this paper we study how the expectations of power means behave asymptotically as some relevant parameter approaches infinity and how to approximate them accurately for general non‐negative continuous probability distributions. We derive approximation formulae for such expectations as distribution mean increases, and apply them to some commonly used distributions in statistics and financial mathematics. By numerical computations we demonstrate the accuracy of the proposed formulae which behave well even for smaller sample sizes. Furthermore, analysis of behaviour depending on sample size contributes to interesting connections with the power mean of probability distribution. This article is protected by copyright. All rights reserved.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135342476","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}
An Intrinsic Gaussian Markov Random Field (IGMRF) can be used to induce conditional dependence in Bayesian hierarchical models. IGMRFs have both a precision matrix, which defines the neighborhood structure of the model, and a precision, or scaling, parameter. Previous studies have shown the importance of selecting the prior for this scaling parameter appropriately for different types of IGMRF, as it can have a substantial impact on posterior estimates. Here, we focus on cases in one and two dimensions, where tuning of the prior is achieved by mapping it to the marginal SD of an IGMRF of corresponding dimensionality. We compare the effects of scaling various IGMRFs, including an application to real two‐dimensional blood pressure data using MCMC methods.
{"title":"Scaling priors for Intrinsic Gaussian Markov Random Fields applied to blood pressure data","authors":"Maria‐Zafeiria Spyropoulou, James Bentham","doi":"10.1111/stan.12330","DOIUrl":"https://doi.org/10.1111/stan.12330","url":null,"abstract":"An Intrinsic Gaussian Markov Random Field (IGMRF) can be used to induce conditional dependence in Bayesian hierarchical models. IGMRFs have both a precision matrix, which defines the neighborhood structure of the model, and a precision, or scaling, parameter. Previous studies have shown the importance of selecting the prior for this scaling parameter appropriately for different types of IGMRF, as it can have a substantial impact on posterior estimates. Here, we focus on cases in one and two dimensions, where tuning of the prior is achieved by mapping it to the marginal SD of an IGMRF of corresponding dimensionality. We compare the effects of scaling various IGMRFs, including an application to real two‐dimensional blood pressure data using MCMC methods.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135585243","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}
One of the main objectives of the time series analysis is forecasting, so both Machine Learning methods and statistical methods have been proposed in the literature. In this study, we compare the forecasting performance of some of these approaches. In addition to traditional forecasting methods, which are the Naive and Seasonal Naive Methods, S/ARIMA, Exponential Smoothing, TBATS, Bayesian Exponential Smoothing Models with Trend Modifications and STL Decomposition, the forecasts are also obtained using seven different machine learning methods, which are Random Forest, Support Vector Regression, XGBoosting, BNN, RNN, LSTM, and FFNN, and the hybridization of both statistical time series and machine learning methods. The data set is selected proportionally from various time domains in M4 Competition data set. Thereby, we aim to create a forecasting guide by considering different preprocessing approaches, methods, and data sets having various time domains. After the experiment, the performance and impact of all methods are discussed. Therefore, most of the best models are mainly selected from machine learning methods for forecasting. Moreover, the forecasting performance of the model is affected by both the time frequency and forecast horizon. Lastly, the study suggests that the hybrid approach is not always the best model for forecasting. Hence, this study provides guidelines to understand which method will perform better at different time series frequencies.
{"title":"Forecasting Performance of Machine Learning, Time Series and Hybrid Methods for Low and High Frequency Time Series","authors":"Ozancan Ozdemir, Ceylan Yozgatlıgil","doi":"10.1111/stan.12326","DOIUrl":"https://doi.org/10.1111/stan.12326","url":null,"abstract":"One of the main objectives of the time series analysis is forecasting, so both Machine Learning methods and statistical methods have been proposed in the literature. In this study, we compare the forecasting performance of some of these approaches. In addition to traditional forecasting methods, which are the Naive and Seasonal Naive Methods, S/ARIMA, Exponential Smoothing, TBATS, Bayesian Exponential Smoothing Models with Trend Modifications and STL Decomposition, the forecasts are also obtained using seven different machine learning methods, which are Random Forest, Support Vector Regression, XGBoosting, BNN, RNN, LSTM, and FFNN, and the hybridization of both statistical time series and machine learning methods. The data set is selected proportionally from various time domains in M4 Competition data set. Thereby, we aim to create a forecasting guide by considering different preprocessing approaches, methods, and data sets having various time domains. After the experiment, the performance and impact of all methods are discussed. Therefore, most of the best models are mainly selected from machine learning methods for forecasting. Moreover, the forecasting performance of the model is affected by both the time frequency and forecast horizon. Lastly, the study suggests that the hybrid approach is not always the best model for forecasting. Hence, this study provides guidelines to understand which method will perform better at different time series frequencies.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775586","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}
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