树结构回归与积分微分

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摘要翻译：
本文提供了关于回归的有根二叉树结构规则的几乎肯定的限制行为的问题的答案。例子表明戈登和奥尔申在1984年提出的问题有否定的答案。对于这些回归函数及其相关的二叉树结构近似序列的例子，对于除第一类集合中的回归函数之外的所有回归函数，几乎肯定一致性在全概率事件上显著失败。一个结果是，二叉树结构规则（如CART）几乎肯定的一致性要求的条件超出了以下要求：（1）回归函数在${\mathcal{L}}^1$，(2)欧几里得特征空间的划分为边平行于坐标轴的多边形，(3)划分的网格几乎肯定地变得任意精细，(4)每个多边形的经验学习样本内容“足够大”。本文中的材料包括达德利在讨论中提出的一个问题的解决方案。主要结果有一个推论，即某些贝叶斯-风险一致性分类规则缺乏几乎肯定的一致性。
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英文标题：
《Tree-structured regression and the differentiation of integrals》
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作者：
Richard A. Olshen
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Statistics Theory        统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Statistics        统计学
二级分类：Statistics Theory        统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
  This paper provides answers to questions regarding the almost sure limiting behavior of rooted, binary tree-structured rules for regression. Examples show that questions raised by Gordon and Olshen in 1984 have negative answers. For these examples of regression functions and sequences of their associated binary tree-structured approximations, for all regression functions except those in a set of the first category, almost sure consistency fails dramatically on events of full probability. One consequence is that almost sure consistency of binary tree-structured rules such as CART requires conditions beyond requiring that (1) the regression function be in ${\mathcal {L}}^1$, (2) partitions of a Euclidean feature space be into polytopes with sides parallel to coordinate axes, (3) the mesh of the partitions becomes arbitrarily fine almost surely and (4) the empirical learning sample content of each polytope be ``large enough.'' The material in this paper includes the solution to a problem raised by Dudley in discussions. The main results have a corollary regarding the lack of almost sure consistency of certain Bayes-risk consistent rules for classification.
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PDF链接：
https://arxiv.org/pdf/708.0385

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关键词：Multivariate Consistency Experiments asymptotics Foundations regression structured binary 缺乏 函数