## 计算机代写|机器学习代写machine learning代考|COMP5318

2023年2月6日

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## 计算机代写|机器学习代写machine learning代考|Support vector regression

SVM is designed for classification problems. It can be extended to address regression problems by conducting proper revisions, and the model is called support vector regression or SVR for short. In standard ML models for regression tasks, if the predicted output and the real output are not equal, their difference will be counted in the loss function such as MSE and MAE. Nevertheless, SVR aims to find a line with a certain width to fit the data, such that the error of the examples within the line with certain width is not considered when calculating the value of the loss function, while only the examples outside the line with certain width will be considered. An illustration of SVR where only one feature is considered is shown in Figure 7.7. The line of the SVR model developed to predict the output of the examples is $f(x)=w x+b$, and the width is $2 \epsilon$. Thus, the error of the examples falling within the area bounded by lines $f(x)+\epsilon$ and $f(x)-\epsilon$ will not be accounted into the calculation of the loss function, while the others will be accounted for.
Mathematically, an SVR model can be formulated as follows:
$$\left[M_3\right] \min {w, b} \frac{1}{2}|w|^2+C \sum{i=1}^n l_\epsilon\left(f\left(x_i\right)-y_i\right) .$$
where $C$ is the penalty term, and $l_\epsilon\left(f\left(x_i\right)-y_i\right)$ is the loss function given the value of $\epsilon$ which takes the following form:

Then, two slack variables, $\xi_{i 1}$ and $\xi_{i 2}$, are introduced to linearize $l_\epsilon\left(f\left(x_i\right)-y_i\right)$, and $\operatorname{model}\left[M_3\right]$ is turned to $$\left[M_3^{\prime}\right] \min {w, b, \xi{i 1}, \xi_{22},=1, \ldots, n} \frac{1}{2}|w|^2+C \sum_{i=1}^n\left(\xi_{i 1}+\xi_{i 2}\right)$$
subject to
\begin{aligned} & f\left(x_i\right)-y_i \leq \epsilon+\xi_{i 1}, i=1, \ldots, n, \ & y_i-f\left(x_i\right) \leq \epsilon+\xi_{i 2}, i=1, \ldots, n, \ & \xi_{i 1} \geq 0, i=1, \ldots, n, \ & \xi_{i 2} \geq 0, i=1, \ldots, n . \end{aligned}
Similar to optimization model $\left[M_{\mathrm{1}}^{\prime}\right]$, Lagrange multipliers can be introduced to model $\left[M_3^{\prime}\right]$ so as to obtain its dual problem. Then, the format of an SVR model can be written as
$$f(x)=\sum^n\left(\alpha_{i 1}-\alpha_{i 2}\right) x_i x+b .$$

## 计算机代写|机器学习代写machine learning代考|The structure and basic concepts of an ANN

Suppose we have a data set $D=\left{\left(\mathbf{x}_i, y_i\right), i=1, \ldots, n\right}$, where $\mathbf{x}_i \in R^m$ is the vector of features and $y_i \in R$ is the output. The structure of a typical ANN model to predict $y_i$ (denoted by $\hat{y}_i, i=1, \ldots, n$ ) is presented in Figure 8.1. It contains three layers: an input layer, a hidden layer, and an output layer. Especially, the input layer contains $m$ nodes to receive the $m$ feature values of the examples, which can either be categorical or numerical, in addition to a bias node. The output layer gives the final predicted target $\hat{y}_i$. For a specific problem, the structure of the input and output layers is fixed. In contrast, the hidden layer can be much more flexible: an ANN can have one or more hidden layers, and the number of nodes in one hidden layer is also flexible. In particular, the number of hidden layers and the number of nodes contained by each of them depend on the specific problem and the data structure. In Figure $8.1$ one hidden layer with $K+1$ nodes is contained. Weight is attached to each arrow connecting two nodes in consecutive layers in Figure 8.1. The direction of the arrows shows the data flow direction in an ANN model, which is opposite to the direction of the error flow which will be covered later.

One node in the neural network is also called a neuron. The details of a neuron in Figure $8.1$ are shown in Figure 8.2. A neuron is an elementary unit of an ANN model, which can be viewed as a mathematical function receiving one or more inputs from an example or from the neurons in the preceding layer. For the neuron shown in Figure 8.2, it receives inputs from $a_1, a_2$, and $a_3$ with weights $w_1, w_2$, and $w_3$ attached. The weighted sum of the inputs, which is denoted by $t$, is first calculated by $t=w_1 a_1+w_2 a_2+w_3 a_3+b$, where $b$ is the bias. Then, $t$ is passed to an activation function $f$ to generate the output of this neuron, which is denoted by $z$, i.e., $z=f(t)$. In particular, activation functions deciding whether the neurons should be activated play a fundamental role in ANNs as they introduce nonlinearity (i.e., nonlinear transformation) into the network. Nonlinearity guarantees the universal approximation property of ANNs: “Standard multilayer feedforward networks with as few as one hidden layer using arbitrary squashing functions are capable of approximating any Borel measurable function from one finite dimensional space to another to any desired degree of accuracy, provided sufficiently many hidden units are available” [1]. Popular activation functions are presented in Figures 8.3-8.6.

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|Support vector regression

SVM 是为分类问题而设计的。它可以通过适当的修正扩 展到解决回归问题，该模型称为支持向量回归或简称 SVR。在用于回归任务的标准 ML 模型中，如果预测输 出和实际输出不相等，则它们的差异将计入 MSE 和 MAE 等损失函数中。然而，SVR 的目标是找到一条具有 一定宽度的线来拟合数据，这样在计算损失函数值时不 考虑具有一定宽度的线内的示例的误差，而只考虑线外 的示例将考虑一定的宽度。图 $7.7$ 显示了仅考虑一个特 征的 SVR 的图示。为预测示例输出而开发的 SVR 模型的 线是 $f(x)=w x+b$, 宽度为 $2 \epsilon$. 因此，样本的误差落 在由线包围的区域内 $f(x)+\epsilon$ 和 $f(x)-\epsilon$ 不会被计入 损失函数的计算，而其他的都会被计入。 在数学上，SVR 模型可以表述如下:
$$\left[M_3\right] \min w, b \frac{1}{2}|w|^2+C \sum i=1^n l_\epsilon\left(f\left(x_i\right)-y_i\right)$$

$$\left[M_3^{\prime}\right] \min w, b, \xi i 1, \xi_{22},=1, \ldots, n \frac{1}{2}|w|^2+C \sum_{i=1}^n\left(\xi_{i 1}\right.$$

$$f\left(x_i\right)-y_i \leq \epsilon+\xi_{i 1}, i=1, \ldots, n, \quad y_i-f\left(x_i\right)$$

$$f(x)=\sum^n\left(\alpha_{i 1}-\alpha_{i 2}\right) x_i x+b .$$

## 计算机代写|机器学习代写machine learning代考|The structure and basic concepts of an ANN

$\left.\hat{y}_i, i=1, \ldots, n\right)$ 如图 $8.1$ 所示。它包含三层: 输入 层、隐藏层和输出层。特别地，输入层包含 $m$ 节点接收 $m$ 除了偏差节点之外，示例的特征值可以是分类的或数 字的。输出层给出最终的预测目标 $\hat{y}_i$. 对于一个特定的问 题，输入层和输出层的结构是固定的。相比之下，隐藏 层可以灵活得多: 一个 ANN 可以有一个或多个隐藏层， 并且一个隐藏层中的节点数量也很灵活。特别地，隐藏 层的数量和每个隐藏层包含的节点数量取决于具体的问 题和数据结构。在图中 $8.1$ 一个隐藏层 $K+1$ 包含节 点。在图 $8.1$ 中，权重附加到连接连续层中两个节点的 每个箭头。箭头方向表示 ANN 模型中的数据流向，与后 面将介绍的错误流向相反。

## 有限元方法代写

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## 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。