### 机器学习代写|决策树作业代写decision tree代考|Decision-Tree Induction

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Origins

Automatically generating rules in the form of decision trees has been object of study of most research fields in which data exploration techniques have been developed [78]. Disciplines like engineering (pattern recognition), statistics, decision theory, and more recently artificial intelligence (machine learning) have a large number of studies dedicated to the generation and application of decision trees.

In statistics, we can trace the origins of decision trees to research that proposed building binary segmentation trees for understanding the relationship between target and input attributes. Some examples are AID [107], MAID [40], THAID [76], and CHAID [55]. The application that motivated these studies is survey data analysis. In engineering (pattern recognition), research on decision trees was motivated by the need to interpret images from remote sensing satellites in the 70 s [46]. Decision trees, and induction methods in general, arose in machine learning to avoid the knowledge acquisition bottleneck for expert systems [78].

Specifically regarding top-down induction of decision trees (by far the most popular approach of decision-tree induction), Hunt’s Concept Learning System (CLS) [49] can be regarded as the pioneering work for inducing decision trees. Systems that directly descend from Hunt’s CLS are ID3 [91], ACLS [87], and Assistant [57].

## 机器学习代写|决策树作业代写decision tree代考|Basic Concepts

Decision trees are an efficient nonparametric method that can be applied either to classification or to regression tasks. They are hierarchical data structures for supervised learning whereby the input space is split into local regions in order to predict the dependent variable [2].

A decision tree can be seen as a graph $G=(V, E)$ consisting of a finite, nonempty set of nodes (vertices) $V$ and a set of edges $E$. Such a graph has to satisfy the following properties [101]:

• The edges must be ordered pairs $(v, w)$ of vertices, i.e., the graph must be directed;
• There can be no cycles within the graph, i.e., the graph must be acyclic;
• There is exactly one node, called the root, which no edges enter;
• Every node, except for the root, has exactly one entering edge;
• There is a unique path-a sequence of edges of the form $\left(v_{1}, v_{2}\right),\left(v_{2}, v_{3}\right), \ldots$, $\left(v_{n-1}, v_{n}\right)$-from the root to each node;
• When there is a path from node $v$ to $w, v \neq w, v$ is a proper ancestor of $w$ and $w$ is a proper descendant of $v$. A node with no proper descendant is called a leaf (or a terminal). All others are called internal nodes (except for the root).

Root and internal nodes hold a test over a given data set attribute (or a set of attributes), and the edges correspond to the possible outcomes of the test. Leaf nodes can either hold class labels (classification), continuous values (regression), (non-) linear models (regression), or even models produced by other machine learning algorithms. For predicting the dependent variable value of a certain instance, one has to navigate through the decision tree. Starting from the root, one has to follow the edges according to the results of the tests over the attributes. When reaching a leaf node, the information it contains is responsible for the prediction outcome. For instance, a traditional decision tree for classification holds class labels in its leaves.
Decision trees can be regarded as a disjunction of conjunctions of constraints on the attribute values of instances [74]. Each path from the root to a leaf is actually a conjunction of attribute tests, and the tree itself allows the choice of different paths, that is, a disjunction of these conjunctions.

Other important definitions regarding decision trees are the concepts of depth and breadth. The average number of layers (levels) from the root node to the terminal nodes is referred to as the average depth of the tree. The average number of internal nodes in each level of the tree is referred to as the average breadth of the tree. Both depth and breadth are indicators of tree complexity, that is, the higher their values are, the more complex the corresponding decision tree is.

In Fig. 2.1, an example of a general decision tree for classification is presented. Circles denote the root and internal nodes whilst squares denote the leaf nodes. In

this particular example, the decision tree is designed for classification and thus the leaf nodes hold class labels.

There are many decision trees that can be grown from the same data. Induction of an optimal decision tree from data is considered to be a hard task. For instance, Hyafil and Rivest [50] have shown that constructing a minimal binary tree with regard to the expected number of tests required for classifying an unseen object is an NP-complete problem. Hancock et al. [43] have proved that finding a minimal decision tree consistent with the training set is NP-Hard, which is also the case of finding the minimal equivalent decision tree for a given decision tree [129], and building the optimal decision tree from decision tables [81]. These papers indicate that growing optimal decision trees (a brute-force approach) is only feasible in very small problems.

Hence, it was necessary the development of heuristics for solving the problem of growing decision trees. In that sense, several approaches which were developed in the last three decades are capable of providing reasonably accurate, if suboptimal, decision trees in a reduced amount of time. Among these approaches, there is a clear preference in the literature for algorithms that rely on a greedy, top-down, recursive partitioning strategy for the growth of the tree (top-down induction).

## 机器学习代写|决策树作业代写decision tree代考|Top-Down Induction

Hunt’s Concept Learning System framework (CLS) [49] is said to be the pioneer work in top-down induction of decision trees. CLS attempts to minimize the cost of classifying an object. Cost, in this context, is referred to two different concepts: the
10
2 Decision-Tree Induction
measurement cost of determining the value of a certain property (attribute) exhibited by the object, and the cost of classifying the object as belonging to class $j$ when it actually belongs to class $k$. At each stage, CLS exploits the space of possible decision trees to a fixed depth, chooses an action to minimize cost in this limited space, then moves one level down in the tree.

In a higher level of abstraction, Hunt’s algorithm can be recursively defined in only two steps. Let $\mathbf{X}{t}$ be the set of training instances associated with node $t$ and $y=\left{y{1}, y_{2}, \ldots, y_{k}\right}$ be the class labels in a $k$-class problem [110]:

1. If all the instances in $\mathbf{X}{t}$ belong to the same class $y{t}$ then $t$ is a leaf node labeled as $y_{t}$
2. If $\mathbf{X}{T}$ contains instances that belong to more than one class, an attribute test condition is selected to partition the instances into smaller subsets. A child node is created for each outcome of the test condition and the instances in $\mathbf{X}{I}$ are distributed to the children based on the outcomes. Recursively apply the algorithm to each child node.

Hunt’s simplified algorithm is the basis for all current top-down decision-tree induction algorithms. Nevertheless, its assumptions are too stringent for practical use. For instance, it would only work if every combination of attribute values is present in the training data, and if the training data is inconsistency-free (each combination has a unique class label).

Hunt’s algorithm was improved in many ways. Its stopping criterion, for example, as expressed in step 1, requires all leaf nodes to be pure (i.e., belonging to the same class). In most practical cases, this constraint leads to enormous decision trees, which tend to suffer from overfitting (an issue discussed later in this chapter). Possible solutions to overcome this problem include prematurely stopping the tree growth when a minimum level of impurity is reached, or performing a pruning step after the tree has been fully grown (more details on other stopping criteria and on pruning in Sects. 2.3.2 and 2.3.3). Another design issue is how to select the attribute test condition to partition the instances into smaller subsets. In Hunt’s original approach, a cost-driven function was responsible for partitioning the tree. Subsequent algorithms such as ID3 [91, 92] and C4.5 [89] make use of information theory based functions for partitioning nodes in purer subsets (more details on Sect. 2.3.1).

An up-to-date algorithmic framework for top-down induction of decision trees is presented in [98], and we reproduce it in Algorithm 1. It contains three procedures: one for growing the tree (treeGrowing), one for pruning the tree (treePruning) and one to combine those two procedures (inducer). The first issue to be discussed is how to select the test condition $f(A)$, i.e., how to select the best combination of attribute(s) and value(s) for splitting nodes.

## 机器学习代写|决策树作业代写decision tree代考|Basic Concepts

• 边必须是有序对(在,在)顶点数，即图必须是有向的；
• 图内不能有环，即图必须是无环的；
• 只有一个节点，称为根，没有边进入；
• 除根外，每个节点都只有一个进入边；
• 有一条唯一的路径——形式的一系列边(在1,在2),(在2,在3),…, (在n−1,在n)- 从根到每个节点；
• 当有来自节点的路径时在到在,在≠在,在是正确的祖先在和在是正确的后裔在. 没有适当后代的节点称为叶子（或终端）。所有其他都称为内部节点（根除外）。

## 机器学习代写|决策树作业代写decision tree代考|Top-Down Induction

Hunt 的概念学习系统框架 (CLS) [49] 据说是自上而下归纳决策树的先驱工作。CLS 试图最小化对象分类的成本。在这种情况下，成本指的是两个不同的概念：确定对象表现出的某个属性（属性）值的
10
2 决策树归纳

1. 如果所有实例在X吨属于同一类是吨然后吨是一个叶节点，标记为是吨
2. 如果X吨包含属于多个类的实例，选择属性测试条件将实例划分为更小的子集。为测试条件的每个结果和其中的实例创建一个子节点X一世根据结果​​分配给孩子们。递归地将算法应用于每个子节点。

Hunt 的简化算法是当前所有自上而下的决策树归纳算法的基础。然而，它的假设对于实际使用来说过于严格。例如，只有当训练数据中存在属性值的每个组合，并且训练数据没有不一致时（每个组合都有一个唯一的类标签），它才会起作用。

Hunt 的算法在很多方面都得到了改进。例如，它的停止标准，如步骤 1 所示，要求所有叶节点都是纯的（即，属于同一类）。在大多数实际情况下，这种约束会导致巨大的决策树，这往往会受到过度拟合的影响（本章稍后会讨论这个问题）。克服此问题的可能解决方案包括在达到最低杂质水平时过早停止树的生长，或在树完全生长后执行修剪步骤（有关其他停止标准和修剪的详细信息，请参阅第 2.3.2 节和2.3.3)。另一个设计问题是如何选择属性测试条件将实例划分为更小的子集。在 Hunt 的原始方法中，成本驱动的函数负责对树进行分区。

[98] 中提出了一种用于自上而下归纳决策树的最新算法框架，我们在算法 1 中重现了它。它包含三个过程：一个用于生长树（treeGrowing），一个用于修剪树（treePruning）和一个结合这两个程序（inducer）。首先要讨论的问题是如何选择测试条件F(一种)，即如何选择属性和值的最佳组合来分割节点。

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## MATLAB代写

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