机器学习代写|决策树作业代写decision tree代考| Selecting Splits

statistics-lab™ 为您的留学生涯保驾护航 在代写决策树decision tree方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写决策树decision tree代写方面经验极为丰富，各种代写决策树decision tree相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

机器学习代写|决策树作业代写decision tree代考|Selecting Splits

A major issue in top-down induction of decision trees is which attribute(s) to choose for splitting a node in subsets. For the case of axis-parallel decision trees (also known as univariate), the problem is to choose the attribute that better discriminates the input data. A decision rule based on such an attribute is thus generated, and the input data is filtered according to the outcomes of this rule. For oblique decision trees (also known as multivariate), the goal is to find a combination of attributes with good discriminatory power. Either way, both strategies are concerned with ranking attributes quantitatively.

We have divided the work in univariate criteria in the following categories: (i) information theory-based criteria; (ii) distance-based criteria; (iii) other classification criteria; and (iv) regression criteria. These categories are sometimes fuzzy and do not constitute a taxonomy by any means. Many of the criteria presented in a given category can be shown to be approximations of criteria in other categories.

机器学习代写|决策树作业代写decision tree代考|Information Theory-Based Criteria

Examples of this category are criteria based, directly or indirectly, on Shannon’s entropy [104]. Entropy is known to be a unique function which satisfies the four axioms of uncertainty. It represents the average amount of information when coding each class into a codeword with ideal length according to its probability. Some interesting facts regarding entropy are:

• For a fixed number of classes, entropy increases as the probability distribution of classes becomes more uniform;
• If the probability distribution of classes is uniform, entropy increases logarithmically as the number of classes in a sample increases;
• If a partition induced on a set $\mathbf{X}$ by an attribute $a_{j}$ is a refinement of a partition induced by $a_{i}$, then the entropy of the partition induced by $a_{j}$ is never higher than the entropy of the partition induced by $a_{i}$ (and it is only equal if the class distribution is kept identical after partitioning). This means that progressively refining a set in sub-partitions will continuously decrease the entropy value, regardless of the class distribution achieved after partitioning a set.

The first splitting criterion that arose based on entropy is the global mutual information (GMI) $[41,102,108]$, given by:
$$G M I\left(a_{i}, \mathbf{X}, y\right)=\frac{1}{N_{x}} \sum_{l=1}^{k} \sum_{j=1}^{\left|a_{i}\right|} N_{v j \cap \cap_{i}} \log {e} \frac{N{v_{j} \cap \cap_{y}} N_{x}}{N_{v_{j}, \bullet} N_{\mathbf{\bullet}, u}}$$
Ching et al. [22] propose the use of GMI as a tool for supervised discretization. They name it class-attribute mutual information, though the criterion is exactly the same. GMI is bounded by zero (when $a_{i}$ and $y$ are completely independent) and its maximum value is $\max \left(\log {2}\left|a{i}\right|, \log {2} k\right.$ ) (when there is a maximum correlation between $a{i}$ and $y$ ). Ching et al. [22] reckon this measure is biased towards attributes with many distinct values, and thus propose the following normalization called classattribute interdependence redundancy (CAIR):
$$\operatorname{CAIR}\left(a_{i}, \mathbf{X}, y\right)=\frac{G M I}{-\sum_{j=1}^{\left|a_{i}\right|} \sum_{l=1}^{k} p_{v_{j}} \cap \cap_{y} \log {2} p{v_{j} \cap \mathrm{y}{t}}}$$ which is actually dividing GMI by the joint entropy of $a{i}$ and $y$. Clearly CAIR $\left(a_{i}, \mathbf{X}, y\right) \geq 0$, since both GMI and the joint entropy are greater (or equal) than zero. In fact, $0 \leq \operatorname{CAIR}\left(a_{i}, \mathbf{X}, y\right) \leq 1$, with $\operatorname{CAIR}\left(a_{i}, \mathbf{X}, y\right)=0$ when $a_{i}$ and $y$ are totally independent and $\operatorname{CAIR}\left(a_{i}, \mathbf{X}, y\right)=1$ when they are totally dependent. The term redundancy in CAIR comes from the fact that one may discretize a continuous attribute in intervals in such a way that the class-attribute interdependence is kept intact (i.e., redundant values are combined in an interval). In the decision tree partitioning context, we must look for an attribute that maximizes CAIR (or similarly, that maximizes GMI).

机器学习代写|决策树作业代写decision tree代考|Distance-Based Criteria

Criteria in this category evaluate separability, divergency or discrimination between classes. They measure the distance between class probability distributions.

A popular distance criterion which is also from the class of impurity-based criteria is the Gini index $[12,39,88]$. It is given by:
$$\phi^{G i n i}(y, \mathbf{X})=1-\sum_{l=1}^{k} p_{\bullet}, y^{2}$$

Breiman et al. [12] also acknowledge Gini’s bias towards attributes with many values. They propose the twoing binary criterion for solving this matter. It belongs to the class of binary criteria, which requires attributes to have their domain split into two mutually exclusive subdomains, allowing binary splits only. For every binary criteria, the process of dividing attribute $a_{i}$ values into two subdomains, $d_{1}$ and $d_{2}$, is exhaustive ${ }^{1}$ and the division that maximizes its value is selected for attribute $a_{i}$. In other words, a binary criterion $\beta$ is tested over all possible subdomains in order to provide the optimal binary split, $\beta^{}$ : $$\beta^{}=\max {d{1}, d_{2}} \beta\left(a_{i}, d_{1}, d_{2}, \mathbf{X}, y\right)$$
s.t.
\begin{aligned} &d_{1} \cup d_{2}=\operatorname{dom}\left(a_{i}\right) \ &d_{1} \cap d_{2}=\emptyset \end{aligned}
Now that we have defined binary criteria, the twoing binary criterion is given by:
$$\beta^{\text {twoing }}\left(a_{i}, d_{1}, d_{2}, \mathbf{X}, y\right)=0.25 \times p_{d_{1}, \bullet} \times p_{d_{2}, \bullet} \times\left(\sum_{l=1}^{k} a b s\left(p_{y_{i} \mid d_{1}}-p_{y_{i} \mid d_{2}}\right)\right)^{2}$$
where $a b s(.)$ returns the absolute value.
Friedman [38] and Rounds [99] propose a binary criterion based on the Kolmogorov-Smirnoff (KS) distance for handling binary-class problems:
$$\beta^{K S}\left(a_{i}, d_{1}, d_{2}, \mathbf{X}, y\right)=a b s\left(p_{d_{1} \mid y_{1}}-p_{d_{1} \mid y_{2}}\right)$$
Haskell and Noui-Mehidi [45] propose extending $\beta^{K S}$ for handling multi-class problems. Utgoff and Clouse [120] also propose a multi-class extension to $\beta^{K S}$, as well as missing data treatment, and they present empirical results which show their criterion is similar in accuracy to Quinlan’s gain ratio, but produces smaller-sized trees.

机器学习代写|决策树作业代写decision tree代考|Information Theory-Based Criteria

• 对于固定数量的类，熵随着类的概率分布变得更加均匀而增加；
• 如果类的概率分布是均匀的，则熵会随着样本中类数的增加而对数增加；
• 如果在集合上引起分区X按属性一种j是由以下引起的分区的细化一种一世，然后由下式诱导的分区的熵一种j永远不会高于由一种一世（并且只有在分区后类分布保持相同时才相等）。这意味着逐步细化子分区中的集合将不断降低熵值，而不管划分集合后实现的类分布如何。

G米一世(一种一世,X,是)=1ñX∑l=1ķ∑j=1|一种一世|ñ在j∩∩一世日志⁡和ñ在j∩∩是ñXñ在j,∙ñ∙,在

机器学习代写|决策树作业代写decision tree代考|Distance-Based Criteria

φG一世n一世(是,X)=1−∑l=1ķp∙,是2

d1∪d2=dom⁡(一种一世) d1∩d2=∅

b合二为一 (一种一世,d1,d2,X,是)=0.25×pd1,∙×pd2,∙×(∑l=1ķ一种bs(p是一世∣d1−p是一世∣d2))2

Friedman [38] 和 Rounds [99] 提出了一种基于 Kolmogorov-Smirnoff (KS) 距离的二元标准，用于处理二元类问题：
bķ小号(一种一世,d1,d2,X,是)=一种bs(pd1∣是1−pd1∣是2)
Haskell 和 Noui-Mehidi [45] 建议扩展bķ小号用于处理多类问题。Utgoff 和 Clouse [120] 还提出了一个多类扩展bķ小号，以及缺失数据处理，他们提供的经验结果表明，他们的标准在准确性上与 Quinlan 的增益率相似，但会产生更小的树。

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。