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

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• 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。

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