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统计代写|统计与机器学习作业代写Statistical and Machine Learning代考|Ensemble of Trees
Decision trees are particularly good models, easy to implement, fast to train, and easy to interpret, but they are unstable. In some cases, a change in the class label of one case could result in a completely different structure for the tree, even though it has nearly the same accuracy. This can be problematic when deploying decision trees to support business actions. Even though the overall accuracy can remain similar when the final structure of the decision tree changes, the set of rules based on thresholds will be different because they are based on the tree structure. This might impact the business campaign because it is based on that set of rules.
The decision tree’s instability results from the considerable number of univariate splits and fragmentation of the data. At each split, there are typically many splits on the same predictor variable or different predictor variables that give similar performance. For example, suppose age is split at 45 since it is the most significant split with the predictor variable and the target. However, other splits at 38 or 47 might be almost as significant. A slight change in the data can easily result in an effect that can cascade and create a different tree. A change in the input data can result in a split at 38 years old. Even more problematic, the change in the input data can result in another input variable being more significant to the target such as income. Then, instead of splitting the final tree structure will be quite different, and therefore, the set of rules and thresholds will also be different.
Several methods have been used to take advantage of this decision tree instability to create models that are more powerful, especially in terms of generalizing the final predictive model. One of the most popular methods is to create an ensemble of decision trees.
An ensemble model is the combination of multiple models. The combinations can be formed in these Ways:
- voting on the classifications
- using weighted voting, where some models have more weight
- averaging (weighted or unweighted) the predicted values
There are two types of ensemble of trees: random forests and gradient boosting models.
统计代写|统计与机器学习作业代写Statistical and Machine Learning代考|Gradient Boosting
Another approach to ensemble of decision trees is gradient boosting. Gradient boosting is a weighted linear combination of individual decision trees. The algorithm starts with an initial decision tree and generates the residuals. In the next step, the target is the residuals from the previous decision tree. At each step, the accuracy of the tree is computed, and successive trees are adjusted to accommodate previous inaccuracies. Therefore, the gradient boosting algorithm fits a sequence of trees based on the residuals from the previous trees. The final model also has the predicted values averaged over the decision trees.
Just like the forest models, the gradient boosting model should have improved predictive accuracy because of variance reduction. It is hoped that the final model has low bias and low variance.
A major difference between random forest and gradient boosting is in the way the ensemble of decision trees is created. In forests, each decision tree is created independently. In gradient boosting, the set of decision trees is created in a sequence. This difference can allow random forest models to be trained faster and gradient boosting models to be more accurate. On the other hand, random forests can better generalize, and gradient boosting models are easier to overfit.
Gradient boosting is based on a slightly different approach than random forests, particularly assigned to the perturb and combine method. Boosting is a machine learning ensemble meta-algorithm for primarily reducing variance. The term boosting refers to a family of algorithms that can convert weak learners (in this case, decision trees with large residuals) into strong learners.
Adaptive resampling and combining methods are examples of boosting. They sequentially perturb the training data based on the results of the previous models. Cases that are incorrectly classified are given more weight in subsequent models. For continuous targets, from the second decision tree in objective function which in this model minimizes the residuals.
The gradient boosting model is shown below.
$$
F_M(x)=F_0+\beta_1 T_1(x)+\beta_2 T_1(x)+\cdots+\beta_M T_M(x)
$$
where $\mathrm{M}$ is the number of iterations, or decision trees in the gradient boosting model, $\mathrm{F}_0$ is the initial guess, $\beta$ is the weight for each decision tree in the linear combination, and $T$ is the decision tree model in each iteration.

统计与机器学习代考
统计代写|统计与机器学习作业代写统计和机器学习代考|树的集合
决策树是特别好的模型,易于实现,训练速度快,易于解释,但它们不稳定。在某些情况下,更改一种情况的类标签可能导致树的结构完全不同,尽管它具有几乎相同的准确性。在部署决策树以支持业务操作时,这可能会产生问题。尽管当决策树的最终结构发生变化时,总体精度可以保持相似,但基于阈值的规则集将有所不同,因为它们是基于树结构的。这可能会影响商业活动,因为它是基于那套规则
决策树的不稳定性源于数据的大量单变量分裂和碎片化。在每次拆分时,通常会在相同的预测变量或提供相似性能的不同预测变量上进行多次拆分。例如,假设年龄在45岁分裂,因为这是预测变量和目标变量之间最显著的分裂。然而,其他年龄在38岁或47岁时的差异可能同样显著。对数据的轻微更改很容易导致层叠效果并创建不同的树。输入数据的变化可能导致在38岁时分道扬镳。更有问题的是,输入数据的变化可能导致另一个对目标更重要的输入变量,如收入。那么,最终的树形结构将非常不同,因此,规则和阈值的集合也将不同。
已经使用了几种方法来利用这种决策树的不稳定性来创建更强大的模型,特别是在推广最终预测模型方面。最流行的方法之一是创建决策树的集合
集合模型是多个模型的组合。组合可以通过以下方式形成:
- 对分类进行投票
- 使用加权投票,其中一些模型有更多的权重
- 平均(加权或不加权)预测值
有两种类型的树集合:随机森林和梯度提升模型
统计代写|统计与机器学习作业代写统计与机器学习代考|梯度提升
.
集成决策树的另一种方法是梯度增强。梯度提升是个体决策树的加权线性组合。该算法从初始决策树开始,生成残差。下一步,目标是来自上一个决策树的残差。在每一步,计算树的准确性,并调整后续树以适应之前的不准确性。因此,梯度增强算法基于之前树的残差拟合出一个树序列。最后的模型还有在决策树上平均的预测值。就像森林模型一样,梯度增强模型应该有更好的预测精度,因为方差减少了。希望最终模型具有低偏差和低方差。随机森林和梯度增强的一个主要区别在于决策树集合的创建方式。在森林中,每棵决策树都是独立创建的。在梯度提升中,决策树集是按顺序创建的。这种差异可以使随机森林模型训练得更快,梯度增强模型更准确。另一方面,随机森林可以更好地泛化,梯度增强模型更容易过拟合。
梯度增强基于一种与随机森林略有不同的方法,特别是分配给摄动和组合方法。Boosting是一种主要用于减少方差的机器学习集成元算法。boost一词指的是一组算法,它们可以将弱学习者(在这种情况下,是具有大残差的决策树)转化为强学习者
自适应重采样和组合方法是boost的例子。它们根据之前模型的结果对训练数据进行顺序扰动。分类错误的案例在后续的模型中被赋予更多的权重。对于连续目标,从目标函数的第二决策树开始,该模型使残差最小化。梯度提升模型如下图所示。
$$
F_M(x)=F_0+\beta_1 T_1(x)+\beta_2 T_1(x)+\cdots+\beta_M T_M(x)
$$
其中$\mathrm{M}$为迭代次数,或梯度提升模型中的决策树,$\mathrm{F}_0$为初始猜测,$\beta$为线性组合中每个决策树的权重,$T$为每次迭代中的决策树模型

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