首页 > 最新文献

Journal of Applied Probability最新文献

英文 中文
The dutch draw: constructing a universal baseline for binary classification problems 荷兰平局:构建二元分类问题的通用基线
IF 1 4区 数学 Q3 STATISTICS & PROBABILITY Pub Date : 2024-09-19 DOI: 10.1017/jpr.2024.52
Etienne van de Bijl, Jan Klein, Joris Pries, Sandjai Bhulai, Mark Hoogendoorn, Rob van der Mei

Novel prediction methods should always be compared to a baseline to determine their performance. Without this frame of reference, the performance score of a model is basically meaningless. What does it mean when a model achieves an $F_1$ of 0.8 on a test set? A proper baseline is, therefore, required to evaluate the ‘goodness’ of a performance score. Comparing results with the latest state-of-the-art model is usually insightful. However, being state-of-the-art is dynamic, as newer models are continuously developed. Contrary to an advanced model, it is also possible to use a simple dummy classifier. However, the latter model could be beaten too easily, making the comparison less valuable. Furthermore, most existing baselines are stochastic and need to be computed repeatedly to get a reliable expected performance, which could be computationally expensive. We present a universal baseline method for all binary classification models, named the Dutch Draw (DD). This approach weighs simple classifiers and determines the best classifier to use as a baseline. Theoretically, we derive the DD baseline for many commonly used evaluation measures and show that in most situations it reduces to (almost) always predicting either zero or one. Summarizing, the DD baseline is general, as it is applicable to any binary classification problem; simple, as it can be quickly determined without training or parameter tuning; and informative, as insightful conclusions can be drawn from the results. The DD baseline serves two purposes. First, it is a robust and universal baseline that enables comparisons across research papers. Second, it provides a sanity check during the prediction model’s development process. When a model does not outperform the DD baseline, it is a major warning sign.

新的预测方法应始终与基线进行比较,以确定其性能。如果没有这个参照基准,模型的性能得分基本上毫无意义。当一个模型在测试集上的 $F_1$ 达到 0.8 时,这意味着什么?因此,评估性能分数的 "好坏 "需要一个适当的基准。将结果与最新的最先进模型进行比较通常很有意义。然而,最新模型是动态的,因为不断有新的模型被开发出来。与先进模型相反,也可以使用简单的虚拟分类器。但是,后一种模型很容易被打败,从而降低了比较的价值。此外,现有的基线大多是随机的,需要反复计算才能获得可靠的预期性能,这可能会导致计算成本高昂。我们提出了一种适用于所有二元分类模型的通用基线方法,名为 "荷兰平局"(Dutch Draw,DD)。这种方法可对简单分类器进行权衡,并确定用作基线的最佳分类器。从理论上讲,我们为许多常用的评估指标推导出了 DD 基线,并表明在大多数情况下,它(几乎)总是预测 0 或 1。概括地说,DD 基线是通用的,因为它适用于任何二元分类问题;简单的,因为它无需训练或参数调整即可快速确定;信息丰富的,因为可以从结果中得出深刻的结论。DD 基线有两个作用。首先,它是一个稳健、通用的基线,可以对不同的研究论文进行比较。其次,它在预测模型的开发过程中提供了合理性检查。当一个模型的性能没有超过 DD 基线时,这是一个重要的警示信号。
{"title":"The dutch draw: constructing a universal baseline for binary classification problems","authors":"Etienne van de Bijl, Jan Klein, Joris Pries, Sandjai Bhulai, Mark Hoogendoorn, Rob van der Mei","doi":"10.1017/jpr.2024.52","DOIUrl":"https://doi.org/10.1017/jpr.2024.52","url":null,"abstract":"<p>Novel prediction methods should always be compared to a baseline to determine their performance. Without this frame of reference, the performance score of a model is basically meaningless. What does it mean when a model achieves an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918134025706-0265:S0021900224000524:S0021900224000524_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$F_1$</span></span></img></span></span> of 0.8 on a test set? A proper baseline is, therefore, required to evaluate the ‘goodness’ of a performance score. Comparing results with the latest state-of-the-art model is usually insightful. However, being state-of-the-art is dynamic, as newer models are continuously developed. Contrary to an advanced model, it is also possible to use a simple dummy classifier. However, the latter model could be beaten too easily, making the comparison less valuable. Furthermore, most existing baselines are stochastic and need to be computed repeatedly to get a reliable expected performance, which could be computationally expensive. We present a universal baseline method for all <span>binary classification</span> models, named the <span>Dutch Draw</span> (DD). This approach weighs simple classifiers and determines the best classifier to use as a baseline. Theoretically, we derive the DD baseline for many commonly used evaluation measures and show that in most situations it reduces to (almost) always predicting either zero or one. Summarizing, the DD baseline is <span>general</span>, as it is applicable to any binary classification problem; <span>simple</span>, as it can be quickly determined without training or parameter tuning; and <span>informative</span>, as insightful conclusions can be drawn from the results. The DD baseline serves two purposes. First, it is a robust and universal baseline that enables comparisons across research papers. Second, it provides a sanity check during the prediction model’s development process. When a model does not outperform the DD baseline, it is a major warning sign.</p>","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Transience of continuous-time conservative random walks 连续时间保守随机游走的短暂性
IF 1 4区 数学 Q3 STATISTICS & PROBABILITY Pub Date : 2024-09-18 DOI: 10.1017/jpr.2024.46
Satyaki Bhattacharya, Stanislav Volkov
We consider two continuous-time generalizations of conservative random walks introduced in Englander and Volkov (2022), an orthogonal and a spherically symmetrical one; the latter model is also known as random flights. For both models, we show the transience of the walks when $dge 2$ and that the rate of direction changing follows a power law $t^{-alpha}$ , $0<alphale 1$ , or the law $(!ln t)^{-beta}$ where $beta>2$ .
我们考虑了英格兰德和沃尔科夫(2022)引入的保守随机漫步的两种连续时间广义模型,一种是正交模型,另一种是球对称模型;后一种模型也被称为随机飞行。对于这两种模型,我们都证明了当$dge 2$时随机游走的瞬时性,并且方向改变的速率遵循幂律$t^{-alpha}$ ,$0<alphale 1$,或律$(!ln t)^{-beta}$ (其中$beta>2$)。
{"title":"Transience of continuous-time conservative random walks","authors":"Satyaki Bhattacharya, Stanislav Volkov","doi":"10.1017/jpr.2024.46","DOIUrl":"https://doi.org/10.1017/jpr.2024.46","url":null,"abstract":"We consider two continuous-time generalizations of conservative random walks introduced in Englander and Volkov (2022), an orthogonal and a spherically symmetrical one; the latter model is also known as <jats:italic>random flights</jats:italic>. For both models, we show the transience of the walks when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000469_inline1.png\"/> <jats:tex-math> $dge 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and that the rate of direction changing follows a power law <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000469_inline2.png\"/> <jats:tex-math> $t^{-alpha}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000469_inline3.png\"/> <jats:tex-math> $0&lt;alphale 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, or the law <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000469_inline4.png\"/> <jats:tex-math> $(!ln t)^{-beta}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000469_inline5.png\"/> <jats:tex-math> $beta&gt;2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Efficiency of reversible MCMC methods: elementary derivations and applications to composite methods 可逆 MCMC 方法的效率:基本推导及对复合方法的应用
IF 1 4区 数学 Q3 STATISTICS & PROBABILITY Pub Date : 2024-09-18 DOI: 10.1017/jpr.2024.48
Radford M. Neal, Jeffrey S. Rosenthal
We review criteria for comparing the efficiency of Markov chain Monte Carlo (MCMC) methods with respect to the asymptotic variance of estimates of expectations of functions of state, and show how such criteria can justify ways of combining improvements to MCMC methods. We say that a chain on a finite state space with transition matrix P efficiency-dominates one with transition matrix Q if for every function of state it has lower (or equal) asymptotic variance. We give elementary proofs of some previous results regarding efficiency dominance, leading to a self-contained demonstration that a reversible chain with transition matrix P efficiency-dominates a reversible chain with transition matrix Q if and only if none of the eigenvalues of $Q-P$ are negative. This allows us to conclude that modifying a reversible MCMC method to improve its efficiency will also improve the efficiency of a method that randomly chooses either this or some other reversible method, and to conclude that improving the efficiency of a reversible update for one component of state (as in Gibbs sampling) will improve the overall efficiency of a reversible method that combines this and other updates. It also explains how antithetic MCMC can be more efficient than independent and identically distributed sampling. We also establish conditions that can guarantee that a method is not efficiency-dominated by any other method.
我们回顾了比较马尔可夫链蒙特卡罗(MCMC)方法在状态函数期望估计值渐近方差方面的效率的标准,并说明了这些标准如何证明将改进 MCMC 方法结合起来的方法是合理的。如果对每个状态函数而言,具有过渡矩阵 P 的有限状态空间链的渐近方差较小(或相等),我们就认为该链的效率优于具有过渡矩阵 Q 的有限状态空间链。我们给出了以前关于效率优势的一些结果的基本证明,从而得出一个自足的论证:当且仅当 $Q-P$ 的特征值都不是负数时,具有过渡矩阵 P 的可逆链效率优势于具有过渡矩阵 Q 的可逆链。这让我们得出结论:修改一种可逆 MCMC 方法以提高其效率,也会提高随机选择这种或其他可逆方法的方法的效率;还让我们得出结论:提高状态的一个组成部分的可逆更新(如吉布斯采样)的效率,会提高结合这种更新和其他更新的可逆方法的整体效率。这也解释了反可逆 MCMC 如何比独立同分布采样更高效。我们还建立了一些条件,以保证一种方法的效率不会被任何其他方法所支配。
{"title":"Efficiency of reversible MCMC methods: elementary derivations and applications to composite methods","authors":"Radford M. Neal, Jeffrey S. Rosenthal","doi":"10.1017/jpr.2024.48","DOIUrl":"https://doi.org/10.1017/jpr.2024.48","url":null,"abstract":"We review criteria for comparing the efficiency of Markov chain Monte Carlo (MCMC) methods with respect to the asymptotic variance of estimates of expectations of functions of state, and show how such criteria can justify ways of combining improvements to MCMC methods. We say that a chain on a finite state space with transition matrix <jats:italic>P</jats:italic> efficiency-dominates one with transition matrix <jats:italic>Q</jats:italic> if for every function of state it has lower (or equal) asymptotic variance. We give elementary proofs of some previous results regarding efficiency dominance, leading to a self-contained demonstration that a reversible chain with transition matrix <jats:italic>P</jats:italic> efficiency-dominates a reversible chain with transition matrix <jats:italic>Q</jats:italic> if and only if none of the eigenvalues of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000482_inline1.png\"/> <jats:tex-math> $Q-P$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are negative. This allows us to conclude that modifying a reversible MCMC method to improve its efficiency will also improve the efficiency of a method that randomly chooses either this or some other reversible method, and to conclude that improving the efficiency of a reversible update for one component of state (as in Gibbs sampling) will improve the overall efficiency of a reversible method that combines this and other updates. It also explains how antithetic MCMC can be more efficient than independent and identically distributed sampling. We also establish conditions that can guarantee that a method is not efficiency-dominated by any other method.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
A non-homogeneous alternating renewal process model for interval censoring 区间剔除的非均质交替更新过程模型
IF 1 4区 数学 Q3 STATISTICS & PROBABILITY Pub Date : 2024-09-16 DOI: 10.1017/jpr.2024.54
M. N. M. van Lieshout, R. L. Markwitz

Previous approaches to modelling interval-censored data have often relied on assumptions of homogeneity in the sense that the censoring mechanism, the underlying distribution of occurrence times, or both, are assumed to be time-invariant. In this work, we introduce a model which allows for non-homogeneous behaviour in both cases. In particular, we outline a censoring mechanism based on a non-homogeneous alternating renewal process in which interval generation is assumed to be time-dependent, and we propose a Markov point process model for the underlying occurrence time distribution. We prove the existence of this process and derive the conditional distribution of the occurrence times given the intervals. We provide a framework within which the process can be accurately modelled, and subsequently compare our model to the homogeneous approach through a number of illustrative examples.

以往的区间删失数据建模方法通常依赖于同质性假设,即假设删失机制、发生时间的基本分布或两者都是时间不变的。在这项工作中,我们引入了一个模型,允许在这两种情况下的非均质性行为。特别是,我们概述了一种基于非均质交替更新过程的删减机制,其中假设区间的生成与时间有关,我们还为基本的发生时间分布提出了一个马尔可夫点过程模型。我们证明了这一过程的存在性,并推导出了给定区间的发生时间的条件分布。我们提供了一个框架,在此框架内可以对该过程进行精确建模,随后通过一些示例将我们的模型与同质方法进行比较。
{"title":"A non-homogeneous alternating renewal process model for interval censoring","authors":"M. N. M. van Lieshout, R. L. Markwitz","doi":"10.1017/jpr.2024.54","DOIUrl":"https://doi.org/10.1017/jpr.2024.54","url":null,"abstract":"<p>Previous approaches to modelling interval-censored data have often relied on assumptions of homogeneity in the sense that the censoring mechanism, the underlying distribution of occurrence times, or both, are assumed to be time-invariant. In this work, we introduce a model which allows for non-homogeneous behaviour in both cases. In particular, we outline a censoring mechanism based on a non-homogeneous alternating renewal process in which interval generation is assumed to be time-dependent, and we propose a Markov point process model for the underlying occurrence time distribution. We prove the existence of this process and derive the conditional distribution of the occurrence times given the intervals. We provide a framework within which the process can be accurately modelled, and subsequently compare our model to the homogeneous approach through a number of illustrative examples.</p>","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
An algorithm to construct coherent systems using signatures 利用签名构建连贯系统的算法
IF 1 4区 数学 Q3 STATISTICS & PROBABILITY Pub Date : 2024-09-16 DOI: 10.1017/jpr.2024.60
T. V. Rao, Sameen Naqvi

The system signature is a useful tool for studying coherent systems. For a given coherent system, various methods have been proposed in the literature to compute its signature. However, when any system signature is given, the literature does not address how to construct the corresponding coherent system(s). In this article we propose an algorithm to address this research gap. This algorithm enables the validation of whether a provided probability vector qualifies as a signature. If it does, the algorithm proceeds to generate the corresponding coherent system(s). To illustrate the applicability of this algorithm, we consider all three and four-dimensional probability vectors, verify if they are signatures, and finally obtain 5 and 20 coherent systems, respectively, which coincides with the literature (Shaked and Suarez-Llorens 2003).

系统签名是研究相干系统的有用工具。对于给定的相干系统,文献中提出了各种计算其特征的方法。然而,当给出任何系统签名时,文献并没有讨论如何构建相应的相干系统。在本文中,我们提出了一种算法来解决这一研究空白。该算法可以验证所提供的概率向量是否符合签名条件。如果符合,算法就会继续生成相应的相干系统。为了说明该算法的适用性,我们考虑了所有三维和四维概率向量,验证它们是否是签名,最后分别得到 5 个和 20 个相干系统,这与文献(Shaked 和 Suarez-Llorens 2003 年)不谋而合。
{"title":"An algorithm to construct coherent systems using signatures","authors":"T. V. Rao, Sameen Naqvi","doi":"10.1017/jpr.2024.60","DOIUrl":"https://doi.org/10.1017/jpr.2024.60","url":null,"abstract":"<p>The system signature is a useful tool for studying coherent systems. For a given coherent system, various methods have been proposed in the literature to compute its signature. However, when any system signature is given, the literature does not address how to construct the corresponding coherent system(s). In this article we propose an algorithm to address this research gap. This algorithm enables the validation of whether a provided probability vector qualifies as a signature. If it does, the algorithm proceeds to generate the corresponding coherent system(s). To illustrate the applicability of this algorithm, we consider all three and four-dimensional probability vectors, verify if they are signatures, and finally obtain 5 and 20 coherent systems, respectively, which coincides with the literature (Shaked and Suarez-Llorens 2003).</p>","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Quenched worst-case scenario for root deletion in targeted cutting of random recursive trees 随机递归树定向切割中根删除的淬火最坏情况假设
IF 1 4区 数学 Q3 STATISTICS & PROBABILITY Pub Date : 2024-09-03 DOI: 10.1017/jpr.2024.40
Laura Eslava, Sergio I. López, Marco L. Ortiz
We propose a method for cutting down a random recursive tree that focuses on its higher-degree vertices. Enumerate the vertices of a random recursive tree of size <jats:italic>n</jats:italic> according to the decreasing order of their degrees; namely, let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000408_inline1.png"/> <jats:tex-math>$(v^{(i)})_{i=1}^{n}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> be such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000408_inline2.png"/> <jats:tex-math>$deg(v^{(1)}) geq cdots geq deg (v^{(n)})$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The targeted vertex-cutting process is performed by sequentially removing vertices <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000408_inline3.png"/> <jats:tex-math>$v^{(1)}, v^{(2)}, ldots, v^{(n)}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> and keeping only the subtree containing the root after each removal. The algorithm ends when the root is picked to be removed. The total number of steps for this procedure, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000408_inline4.png"/> <jats:tex-math>$K_n$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, is upper bounded by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000408_inline5.png"/> <jats:tex-math>$Z_{geq D}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which denotes the number of vertices that have degree at least as large as the degree of the root. We prove that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000408_inline6.png"/> <jats:tex-math>$ln Z_{geq D}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> grows as <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000408_inline7.png"/> <jats:tex-math>$ln n$</jats:tex-math> </jats:alternatives> </jats:inline-formula> asymptotically and obtain its limiting behavior in probability. Moreover, we obtain that the <jats:italic>k</jats:italic>th moment of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000408_inline8.png"/> <jats:tex-math>$ln Z_{geq D}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> is proportional to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic
我们提出了一种专注于高阶顶点的随机递归树切割方法。根据随机递归树的度数递减顺序,列举大小为 n 的随机递归树的顶点;即,让 $(v^{(i)})_{i=1}^{n}$ 是这样的: $deg(v^{(1)})cdots geq deg (v^{(n)})$ 。目标顶点切割过程是通过依次删除顶点 $v^{(1)}、v^{(2)}、ldots、v^{(n)}$ 来完成的,每次删除后只保留包含根的子树。当选中要删除的根时,算法结束。这个过程的总步数 $K_n$ 的上界是 $Z_{geq D}$,它表示阶数至少和根阶数一样大的顶点的数量。我们证明 $ln Z_{geq D}$ 会随着 $ln n$ 的增长而渐进增长,并得到它在概率上的极限行为。此外,我们还得到 $ln Z_{geq D}$ 的第 k 矩与 $(!ln n)^k$ 成正比。因此,我们得到 $K_n$ 的一阶增长上限为 $n^{1-ln 2}$,这大大小于如果顶点是均匀随机选择时所需的删除次数。
{"title":"Quenched worst-case scenario for root deletion in targeted cutting of random recursive trees","authors":"Laura Eslava, Sergio I. López, Marco L. Ortiz","doi":"10.1017/jpr.2024.40","DOIUrl":"https://doi.org/10.1017/jpr.2024.40","url":null,"abstract":"We propose a method for cutting down a random recursive tree that focuses on its higher-degree vertices. Enumerate the vertices of a random recursive tree of size &lt;jats:italic&gt;n&lt;/jats:italic&gt; according to the decreasing order of their degrees; namely, let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline1.png\"/&gt; &lt;jats:tex-math&gt;$(v^{(i)})_{i=1}^{n}$&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be such that &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline2.png\"/&gt; &lt;jats:tex-math&gt;$deg(v^{(1)}) geq cdots geq deg (v^{(n)})$&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. The targeted vertex-cutting process is performed by sequentially removing vertices &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline3.png\"/&gt; &lt;jats:tex-math&gt;$v^{(1)}, v^{(2)}, ldots, v^{(n)}$&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and keeping only the subtree containing the root after each removal. The algorithm ends when the root is picked to be removed. The total number of steps for this procedure, &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline4.png\"/&gt; &lt;jats:tex-math&gt;$K_n$&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, is upper bounded by &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline5.png\"/&gt; &lt;jats:tex-math&gt;$Z_{geq D}$&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, which denotes the number of vertices that have degree at least as large as the degree of the root. We prove that &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline6.png\"/&gt; &lt;jats:tex-math&gt;$ln Z_{geq D}$&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; grows as &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline7.png\"/&gt; &lt;jats:tex-math&gt;$ln n$&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; asymptotically and obtain its limiting behavior in probability. Moreover, we obtain that the &lt;jats:italic&gt;k&lt;/jats:italic&gt;th moment of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline8.png\"/&gt; &lt;jats:tex-math&gt;$ln Z_{geq D}$&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is proportional to &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Stochastic gradient descent for barycenters in Wasserstein space 瓦瑟尔斯坦空间双曲中心的随机梯度下降
IF 1 4区 数学 Q3 STATISTICS & PROBABILITY Pub Date : 2024-09-03 DOI: 10.1017/jpr.2024.39
Julio Backhoff, Joaquin Fontbona, Gonzalo Rios, Felipe Tobar
We present and study a novel algorithm for the computation of 2-Wasserstein population barycenters of absolutely continuous probability measures on Euclidean space. The proposed method can be seen as a stochastic gradient descent procedure in the 2-Wasserstein space, as well as a manifestation of a law of large numbers therein. The algorithm aims to find a Karcher mean or critical point in this setting, and can be implemented ‘online’, sequentially using independent and identically distributed random measures sampled from the population law. We provide natural sufficient conditions for this algorithm to almost surely converge in the Wasserstein space towards the population barycenter, and we introduce a novel, general condition which ensures uniqueness of Karcher means and, moreover, allows us to obtain explicit, parametric convergence rates for the expected optimality gap. We also study the mini-batch version of this algorithm, and discuss examples of families of population laws to which our method and results can be applied. This work expands and deepens ideas and results introduced in an early version of Backhoff-Veraguas et al. (2022), in which a statistical application (and numerical implementation) of this method is developed in the context of Bayesian learning.
我们提出并研究了一种新算法,用于计算欧几里得空间上绝对连续概率量的 2-Wasserstein 人口原点。所提出的方法可以看作是 2-Wasserstein 空间中的随机梯度下降过程,也是其中大数定律的一种表现形式。该算法旨在找到卡彻均值或临界点,并且可以 "在线 "实施,连续使用从群体规律中采样的独立且同分布的随机度量。我们为该算法提供了自然充分条件,使其在瓦瑟斯坦空间中几乎肯定收敛于群体原点,我们还引入了一个新颖的一般条件,确保卡彻均值的唯一性,此外,我们还能为预期最优性差距获得明确的参数收敛率。我们还研究了这一算法的迷你批处理版本,并讨论了我们的方法和结果可以应用的种群定律族的例子。这项工作拓展并深化了 Backhoff-Veraguas 等人(2022 年)早期版本中提出的观点和结果,其中在贝叶斯学习的背景下开发了该方法的统计应用(和数值实现)。
{"title":"Stochastic gradient descent for barycenters in Wasserstein space","authors":"Julio Backhoff, Joaquin Fontbona, Gonzalo Rios, Felipe Tobar","doi":"10.1017/jpr.2024.39","DOIUrl":"https://doi.org/10.1017/jpr.2024.39","url":null,"abstract":"We present and study a novel algorithm for the computation of 2-Wasserstein population barycenters of absolutely continuous probability measures on Euclidean space. The proposed method can be seen as a stochastic gradient descent procedure in the 2-Wasserstein space, as well as a manifestation of a law of large numbers therein. The algorithm aims to find a Karcher mean or critical point in this setting, and can be implemented ‘online’, sequentially using independent and identically distributed random measures sampled from the population law. We provide natural sufficient conditions for this algorithm to almost surely converge in the Wasserstein space towards the population barycenter, and we introduce a novel, general condition which ensures uniqueness of Karcher means and, moreover, allows us to obtain explicit, parametric convergence rates for the expected optimality gap. We also study the mini-batch version of this algorithm, and discuss examples of families of population laws to which our method and results can be applied. This work expands and deepens ideas and results introduced in an early version of Backhoff-Veraguas <jats:italic>et al.</jats:italic> (2022), in which a statistical application (and numerical implementation) of this method is developed in the context of Bayesian learning.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Uncertainty quantification and confidence intervals for naive rare-event estimators 天真罕见事件估计器的不确定性量化和置信区间
IF 1 4区 数学 Q3 STATISTICS & PROBABILITY Pub Date : 2024-09-02 DOI: 10.1017/jpr.2024.43
Yuanlu Bai, Henry Lam
We consider the estimation of rare-event probabilities using sample proportions output by naive Monte Carlo or collected data. Unlike using variance reduction techniques, this naive estimator does not have an a priori relative efficiency guarantee. On the other hand, due to the recent surge of sophisticated rare-event problems arising in safety evaluations of intelligent systems, efficiency-guaranteed variance reduction may face implementation challenges which, coupled with the availability of computation or data collection power, motivate the use of such a naive estimator. In this paper we study the uncertainty quantification, namely the construction, coverage validity, and tightness of confidence intervals, for rare-event probabilities using only sample proportions. In addition to the known normality, Wilson, and exact intervals, we investigate and compare them with two new intervals derived from Chernoff’s inequality and the Berry–Esseen theorem. Moreover, we generalize our results to the natural situation where sampling stops by reaching a target number of rare-event hits. Our findings show that the normality and Wilson intervals are not always valid, but they are close to the newly developed valid intervals in terms of half-width. In contrast, the exact interval is conservative, but safely guarantees the attainment of the nominal confidence level. Our new intervals, while being more conservative than the exact interval, provide useful insights into understanding the tightness of the considered intervals.
我们考虑使用蒙特卡洛模拟或收集的数据输出的样本比例来估计罕见事件概率。与使用方差缩小技术不同的是,这种天真的估计方法没有先验的相对效率保证。另一方面,由于最近在智能系统安全评估中出现了大量复杂的罕见事件问题,有效率保证的方差缩小可能面临实施上的挑战,再加上计算或数据收集能力的可用性,促使我们使用这种天真估计器。在本文中,我们仅使用样本比例来研究罕见事件概率的不确定性量化,即置信区间的构建、覆盖有效性和紧密性。除了已知的正态性区间、威尔逊区间和精确区间外,我们还研究并比较了从切尔诺夫不等式和贝里-埃森定理推导出的两个新区间。此外,我们还将结果推广到自然情况下,即在达到目标罕见事件命中数时停止采样。我们的研究结果表明,正态区间和威尔逊区间并不总是有效的,但就半宽度而言,它们接近于新开发的有效区间。相比之下,精确区间比较保守,但能安全地保证达到名义置信水平。我们的新区间虽然比精确区间更保守,但为理解所考虑区间的紧密性提供了有用的见解。
{"title":"Uncertainty quantification and confidence intervals for naive rare-event estimators","authors":"Yuanlu Bai, Henry Lam","doi":"10.1017/jpr.2024.43","DOIUrl":"https://doi.org/10.1017/jpr.2024.43","url":null,"abstract":"We consider the estimation of rare-event probabilities using sample proportions output by naive Monte Carlo or collected data. Unlike using variance reduction techniques, this naive estimator does not have an a priori relative efficiency guarantee. On the other hand, due to the recent surge of sophisticated rare-event problems arising in safety evaluations of intelligent systems, efficiency-guaranteed variance reduction may face implementation challenges which, coupled with the availability of computation or data collection power, motivate the use of such a naive estimator. In this paper we study the uncertainty quantification, namely the construction, coverage validity, and tightness of confidence intervals, for rare-event probabilities using only sample proportions. In addition to the known normality, Wilson, and exact intervals, we investigate and compare them with two new intervals derived from Chernoff’s inequality and the Berry–Esseen theorem. Moreover, we generalize our results to the natural situation where sampling stops by reaching a target number of rare-event hits. Our findings show that the normality and Wilson intervals are not always valid, but they are close to the newly developed valid intervals in terms of half-width. In contrast, the exact interval is conservative, but safely guarantees the attainment of the nominal confidence level. Our new intervals, while being more conservative than the exact interval, provide useful insights into understanding the tightness of the considered intervals.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Limit theorems of occupation times of normalized binary contact path processes on lattices 网格上归一化二元接触路径过程占用时间的极限定理
IF 1 4区 数学 Q3 STATISTICS & PROBABILITY Pub Date : 2024-08-27 DOI: 10.1017/jpr.2024.41
Xiaofeng Xue
The binary contact path process (BCPP) introduced in Griffeath (1983) describes the spread of an epidemic on a graph and is an auxiliary model in the study of improving upper bounds of the critical value of the contact process. In this paper, we are concerned with limit theorems of the occupation time of a normalized version of the BCPP (NBCPP) on a lattice. We first show that the law of large numbers of the occupation time process is driven by the identity function when the dimension of the lattice is at least 3 and the infection rate of the model is sufficiently large conditioned on the initial state of the NBCPP being distributed with a particular invariant distribution. Then we show that the centered occupation time process of the NBCPP converges in finite-dimensional distributions to a Brownian motion when the dimension of the lattice and the infection rate of the model are sufficiently large and the initial state of the NBCPP is distributed with the aforementioned invariant distribution.
格里菲斯(Griffeath,1983 年)提出的二元接触路径过程(BCPP)描述了图上流行病的传播,是研究提高接触过程临界值上限的辅助模型。在本文中,我们关注的是归一化 BCPP(NBCPP)在网格上的占领时间的极限定理。我们首先证明,当晶格的维数至少为 3,且模型的感染率足够大时,NBCPP 的初始状态以特定的不变分布为条件,占领时间过程的大数定律由特征函数驱动。然后我们证明,当网格维数和模型感染率足够大,且 NBCPP 的初始状态分布为上述不变分布时,NBCPP 的居中占领时间过程在有限维分布中收敛于布朗运动。
{"title":"Limit theorems of occupation times of normalized binary contact path processes on lattices","authors":"Xiaofeng Xue","doi":"10.1017/jpr.2024.41","DOIUrl":"https://doi.org/10.1017/jpr.2024.41","url":null,"abstract":"The binary contact path process (BCPP) introduced in Griffeath (1983) describes the spread of an epidemic on a graph and is an auxiliary model in the study of improving upper bounds of the critical value of the contact process. In this paper, we are concerned with limit theorems of the occupation time of a normalized version of the BCPP (NBCPP) on a lattice. We first show that the law of large numbers of the occupation time process is driven by the identity function when the dimension of the lattice is at least 3 and the infection rate of the model is sufficiently large conditioned on the initial state of the NBCPP being distributed with a particular invariant distribution. Then we show that the centered occupation time process of the NBCPP converges in finite-dimensional distributions to a Brownian motion when the dimension of the lattice and the infection rate of the model are sufficiently large and the initial state of the NBCPP is distributed with the aforementioned invariant distribution.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Maximizing the probability of visiting a set infinitely often for a Markov decision process with Borel state and action spaces 最大化具有 Borel 状态和行动空间的马尔可夫决策过程无限次访问集合的概率
IF 1 4区 数学 Q3 STATISTICS & PROBABILITY Pub Date : 2024-08-22 DOI: 10.1017/jpr.2024.25
François Dufour, Tomás Prieto-Rumeau
We consider a Markov control model with Borel state space, metric compact action space, and transitions assumed to have a density function with respect to some probability measure satisfying some continuity conditions. We study the optimization problem of maximizing the probability of visiting some subset of the state space infinitely often, and we show that there exists an optimal stationary Markov policy for this problem. We endow the set of stationary Markov policies and the family of strategic probability measures with adequate topologies (namely, the narrow topology for Young measures and the $ws^infty$ -topology, respectively) to obtain compactness and continuity properties, which allow us to obtain our main results.
我们考虑了一个马尔可夫控制模型,该模型具有 Borel 状态空间、度量紧凑的行动空间,并假定过渡具有关于满足某些连续性条件的概率度量的密度函数。我们研究了最大化无限次访问状态空间某个子集的概率的优化问题,并证明了该问题存在最优的静态马尔可夫策略。我们为静态马尔可夫策略集和策略概率度量族赋予了适当的拓扑(即分别为杨度量的窄拓扑和 $ws^infty$ -拓扑),从而获得了紧凑性和连续性特性,这使我们能够得到我们的主要结果。
{"title":"Maximizing the probability of visiting a set infinitely often for a Markov decision process with Borel state and action spaces","authors":"François Dufour, Tomás Prieto-Rumeau","doi":"10.1017/jpr.2024.25","DOIUrl":"https://doi.org/10.1017/jpr.2024.25","url":null,"abstract":"We consider a Markov control model with Borel state space, metric compact action space, and transitions assumed to have a density function with respect to some probability measure satisfying some continuity conditions. We study the optimization problem of maximizing the probability of visiting some subset of the state space infinitely often, and we show that there exists an optimal stationary Markov policy for this problem. We endow the set of stationary Markov policies and the family of strategic probability measures with adequate topologies (namely, the narrow topology for Young measures and the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000251_inline1.png\"/> <jats:tex-math> $ws^infty$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-topology, respectively) to obtain compactness and continuity properties, which allow us to obtain our main results.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
期刊
Journal of Applied Probability
全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
×
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
现在去查看 取消
×
提示
确定
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 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 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1