### 机器学习代写|决策树作业代写decision tree代考|Other Classification Criteria

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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|Other Classification Criteria

In this category, we include all criteria that did not fit in the previously-mentioned categories.

Li and Dubes [62] propose a binary criterion for binary-class problems called permutation statistic. It evaluates the degree of similarity between two vectors, $V_{a_{i}}$ and $y$, and the larger this statistic, the more alike the vectors. Vector $V_{a_{i}}$ is calculated as follows. Let $a_{i}$ be a given numeric attribute with the values $[8.20,7.3,9.35,4.8,7.65,4.33]$ and $N_{x}=6$. Vector $y=[0,0,1,1,0,1]$ holds the corresponding class labels. Now consider a given threshold $\Delta=5.0$. Vector $V_{a_{i}}$ is calculated in two steps: first, attribute $a_{i}$ values are sorted, i.e., $a_{i}=$ $[4.33,4.8,7.3,7.65,8.20,9.35]$, consequently rearranging $y=[1,1,0,0,0,1] ;$ then, $V_{a_{i}}(n)$ takes 0 when $a_{i}(n) \leq \Delta$, and 1 otherwise. Thus, $V_{a_{i}}=[0,0,1,1,1,1]$. The permutation statistic first analyses how many $1-1$ matches $(d)$ vectors $V_{a_{i}}$ and $y$ have. In this particular example, $d=1$. Next, it counts how many l’s there are in $V_{a_{i}}\left(n_{a}\right)$ and in $y\left(n_{y}\right)$. Finally, the permutation statistic can be computed as:
\begin{aligned} \beta^{\text {permutation }}\left(V_{a_{i}}, y\right) &=\sum_{j=0}^{d} \frac{\left(\begin{array}{c} n_{a} \ j \end{array}\right)\left(\begin{array}{c} N_{x}-n_{a} \ n_{y}-j \end{array}\right)}{\left(\begin{array}{c} N_{x} \ n_{y} \end{array}\right)}-\frac{\left(\begin{array}{c} n_{a} \ d \end{array}\right)\left(\begin{array}{c} N_{x}-n_{a} \ n_{y}-d \end{array}\right)}{\left(\begin{array}{c} N_{x} \ n_{g} \end{array}\right)} U \ \left(\begin{array}{c} n \ m \end{array}\right) &=0 \text { if } n<0 \text { or } m<0 \text { or } n<m \ &=\frac{n !}{m !(n-m) !} \text { otherwise } \end{aligned}
where $U$ is a (continuous) random variable distributed uniformly over $[0,1]$.

## 机器学习代写|决策树作业代写decision tree代考|Regression Criteria

All criteria presented so far are dedicated to classification problems. For regression problems, where the target variable $y$ is continuous, a common approach is to calculate the mean squared error (MSE) as a splitting criterion:
$$\operatorname{MSE}\left(a_{i}, \mathbf{X}, y\right)=N_{x}^{-1} \sum_{j=1}^{\left|a_{i}\right|} \sum_{x_{l} \in v_{j}}\left(y\left(x_{l}\right)-\overline{v_{v}}\right)^{2}$$
where $\overline{y_{v}}=N_{v_{i},}^{-1} \sum_{x_{i} \in v_{j}} y\left(x_{l}\right)$. Just as with clustering, we are trying to minimize the within-partition variance. Usually, the sum of squared errors is weighted over each partition according to the estimated probability of an instance belonging to the given partition [12]. Thus, we should rewrite MSE to:
$$w \operatorname{MSE}\left(a_{i}, \mathbf{X}, y\right)=\sum_{j=1}^{\left|a_{l}\right|} p_{v_{j}}, \sum_{x_{l} \in v_{j}}\left(y\left(x_{l}\right)-\overline{y_{v_{j}}}\right)^{2}$$
Another common criterion for regression is the sum of absolute deviations (SAD) [12], or similarly its weighted version given by:
$$w S A D\left(a_{i}, \mathbf{X}, y\right)=\sum_{j=1}^{\left|a_{i}\right|} p_{v j \cdot \bullet} \sum_{x_{l} \in v_{j}} a b s\left(y\left(x_{l}\right)-\operatorname{median}\left(y_{v j}\right)\right)$$
where median $\left(y_{v_{j}}\right)$ is the target attribute’s median of instances belonging to $\mathbf{X}{\mathrm{a}{i}=\mathbf{v}{\text {}}}$. Quinlan [93] proposes the use of the standard deviation reduction (SDR) for his pioneering system of model trees induction, M5. Wang and Witten [124] extend the work of Quinlan in their proposed system M5′, also employing the SDR criterion. It is given by: $$\operatorname{SDR}\left(a{i}, \mathbf{X}, y\right)=\sigma_{X}-\sum_{j=1}^{\left|a_{i}\right|} p_{v_{j}, \bullet} \sigma_{v j}$$
where $\sigma_{X}$ is the standard deviation of instances in $\mathbf{X}$ and $\sigma_{v_{j}}$ the standard deviation of instances in $\mathbf{X}{\mathbf{a}{l}=\mathbf{v}{j}}$. SDR should be maximized, i.e., the weighted sum of standard deviations of each partition should be as small as possible. Thus, partitioning the instance space according to a particular attribute $a{i}$ should provide partitions whose target attribute variance is small (once again we are interested in minimizing the within-partition variance). Observe that minimizing the second term in SDR is equivalent to minimizing wMSE, but in SDR we are using the partition standard deviation $(\sigma)$ as a similarity criterion whereas in wMSE we are using the partition variance $\left(\sigma^{2}\right)$.

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

All criteria presented so far are intended for building univariate splits. Decision trees with multivariate splits (known as oblique, linear or multivariate decision trees) are not so popular as the univariate ones, mainly because they are harder to interpret. Nevertheless, researchers reckon that multivariate splits can improve the performance

of the tree in several data sets, while generating smaller trees $[47,77,98]$. Clearly, there is a tradeoff to consider in allowing multivariate tests: simple tests may result in large trees that are hard to understand, yet multivariate tests may result in small trees with tests hard to understand [121].

A decision tree with multivariate splits is able to produce polygonal (polyhedral) partitions of the attribute space (hyperplanes at an oblique orientation to the attribute axes) whereas univariate trees can only produce hyper-rectangles parallel to the attribute axes. The tests at each node have the form:
$$w_{0}+\sum_{i=1}^{n} w_{i} a_{i}(x) \leq 0$$
where $w_{i}$ is a real-valued coefficient associated to the $i$ th attribute and $w_{0}$ the disturbance coefficient of the test.

CART (Classification and Regression Trees) [12] is one of the first systems that allowed multivariate splits. It employs a hill-climbing strategy with a backward attribute elimination for finding good (albeit suboptimal) linear combinations of attributes in non-terminal nodes. It is a fully-deterministic algorithm with no built-in mechanisms to escape local-optima. Breiman et al. [12] point out that the proposed algorithm has much room for improvement.

Another approach for building oblique decision trees is LMDT (Linear Machine Decision Trees) $[14,119]$, which is an evolution of the perceptron tree method [117]. Each non-terminal node holds a linear machine [83], which is a set of $k$ linear discriminant functions that are used collectively to assign an instance to one of the $k$ existing classes. LMDT uses heuristics to determine when a linear machine has stabilized (since convergence cannot be guaranteed). More specifically, for handling non-linearly separable problems, a method similar to simulated annealing (SA) is used (called thermal training). Draper and Brodley [30] show how LMDT can be altered to induce decision trees that minimize arbitrary misclassification cost functions.

SADT (Simulated Annealing of Decision Trees) [47] is a system that employs SA for finding good coefficient values for attributes in non-terminal nodes of decision trees. First, it places a hyperplane in a canonical location, and then iteratively perturbs the coefficients in small random amounts. At the beginning, when the temperature parameter of the SA is high, practically any perturbation of the coefficients is accepted regardless of the goodness-of-split value (the value of the utilised splitting criterion). As the SA cools down, only perturbations that improve the goodness-of-split are likely to be allowed. Although SADT can eventually escape from local-optima, its efficiency is compromised since it may consider tens of thousands of hyperplanes in a single node during annealing.

## 机器学习代写|决策树作业代写decision tree代考|Other Classification Criteria

Li 和 Dubes [62] 提出了一种二元类问题的二元标准，称为排列统计。它评估两个向量之间的相似程度，在一种一世和是，并且这个统计量越大，向量越相似。向量在一种一世计算如下。让一种一世是具有值的给定数字属性[8.20,7.3,9.35,4.8,7.65,4.33]和ñX=6. 向量是=[0,0,1,1,0,1]持有相应的类标签。现在考虑给定的阈值Δ=5.0. 向量在一种一世计算分两步：首先，属性一种一世值是排序的，即一种一世= [4.33,4.8,7.3,7.65,8.20,9.35]，因此重新排列是=[1,1,0,0,0,1];然后，在一种一世(n)取 0 时一种一世(n)≤Δ, 否则为 1。因此，在一种一世=[0,0,1,1,1,1]. 排列统计量首先分析有多少1−1火柴(d)矢量图在一种一世和是有。在这个特定的例子中，d=1. 接下来，它计算有多少 l 有在一种一世(n一种)并且在是(n是). 最后，排列统计量可以计算为：
b排列 (在一种一世,是)=∑j=0d(n一种 j)(ñX−n一种 n是−j)(ñX n是)−(n一种 d)(ñX−n一种 n是−d)(ñX nG)在 (n 米)=0 如果 n<0 或者 米<0 或者 n<米 =n!米!(n−米)! 除此以外

## 机器学习代写|决策树作业代写decision tree代考|Regression Criteria

MSE⁡(一种一世,X,是)=ñX−1∑j=1|一种一世|∑Xl∈在j(是(Xl)−在在¯)2

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

CART（分类和回归树）[12] 是最早允许多变量拆分的系统之一。它采用具有后向属性消除的爬山策略，以在非终端节点中找到良好（尽管次优）的属性线性组合。它是一种完全确定的算法，没有内置机制来逃避局部最优。布雷曼等人。[12]指出，所提出的算法有很大的改进空间。

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