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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

## 计算机代写|深度学习代写deep learning代考|Multiclass Classification Using Logistic Regression

In SVM, we mainly discussed the binary classification problem, in which a hyperplane is defined to separate two classes. Now, consider Fig. 3.4, where we want to define three hyperplanes that can split the data into multiple categories.
A direct extension of the SVM for the multiple class classifier design problem is to consider all the combinatorial combinations of the hyperplanes. More specifically, a data $x_{i}$ can be on either side of the hyperplane so that given three hyperplanes in Fig. 3.4, one could design a classifier that can potentially classify $2^{3}=8$ classes. Although this approach may reduce the number of hyperplanes for a given number of classes $c$, one of the main technical difficulties of such extension of SVM is that we need to consider all combinatorial combinations of the constraint sets, which is difficult to implement.

A quick remedy for this multi-class classifier design problem is to use the logistic regreession. More specifically, for given $c$-class categories, we define a probability vector $\boldsymbol{q}=\left[q_{1}, \cdots, q_{c}\right]^{\top} \in \mathbb{R}^{c}$, where $q_{i} \in[0,1]$ denotes the probability that a data belongs to the class $i$. Then, by extending (3.9) to vector-valued probabilities for a given dependent variable $\boldsymbol{x} \in \mathbb{R}^{p}$, we have
$$\left[\begin{array}{c} \log \left(\frac{q_{1}}{1-q_{1}}\right) \ \vdots \ \log \left(\frac{q_{c}}{1-q_{c}}\right) \end{array}\right]=\boldsymbol{W}^{\top} \boldsymbol{x}+\boldsymbol{b}$$
where $W \in \mathbb{R}^{p \times c}$ denotes the matrix composed of $c$-normal vectors in the $p$ dimensional spaces, and $\boldsymbol{b} \in \mathbb{R}^{c}$ is the associated bias term. Then, we can easily see that the corresponding probability vector is given by
$$\boldsymbol{p}=\operatorname{Sig}\left(\boldsymbol{W}^{\top} \boldsymbol{x}+\boldsymbol{b}\right)$$

## 计算机代写|深度学习代写deep learning代考|Ridge Regression

Recall that the basic assumption for the linear regression solution in (3.7) is that $\boldsymbol{X}^{\top}$ has full column rank or $\boldsymbol{X}$ has the full row rank. However, when $\boldsymbol{X}^{\top}$ is highdimensional, the columns of $\boldsymbol{X}^{\top}$ can be collinear, which in statistical terms refers to thé êvennt of two (ór multiplé) covariātēs bêing highly linéarly reelateed. Conséquenntly, $\boldsymbol{X}^{\top}$ may not be of full column rank or close to not being the full column rank, and we cannot use the standard linear regression. To deal with this issue, the ridge regression is useful.
Specifically, the following regularized least squares problem is solved:
$$\min {\boldsymbol{\beta}} \ell{\text {ridge }}(\boldsymbol{\beta}),$$
where
$$\ell_{\text {ridge }}(\boldsymbol{\beta}):=\frac{1}{2}\left|\boldsymbol{y}-\boldsymbol{X}^{\top} \boldsymbol{\beta}\right|^{2}+\frac{\lambda}{2}|\boldsymbol{\beta}|^{2},$$
where $\lambda>0$ is the regularization parameter. This type of regularization is often called the Tikhonov regularization. Using Lemma 3.1, we can easily show
$$\left.\frac{\partial \ell_{\text {ridge }{e}}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}}\right|{\boldsymbol{\beta}=\hat{\boldsymbol{\beta}}}=-\boldsymbol{X} \boldsymbol{y}+\boldsymbol{X} \boldsymbol{X}^{\top} \hat{\boldsymbol{\beta}}+\lambda \hat{\boldsymbol{\beta}}=\mathbf{0},$$

# 深度学习代写

## 计算机代写|深度学习代写deep learning代考|Multiclass Classification Using Logistic Regression

$$\left[\log \left(\frac{q_{1}}{1-q_{1}}\right) \vdots \log \left(\frac{q_{c}}{1-q_{c}}\right)\right]=\boldsymbol{W}^{\top} \boldsymbol{x}+\boldsymbol{b}$$
$$\boldsymbol{p}=\operatorname{Sig}\left(\boldsymbol{W}^{\top} \boldsymbol{x}+\boldsymbol{b}\right)$$

## 计算机代写|深度学习代写deep learning代考|Ridge Regression

$$\min \beta \text { lidge ( } \boldsymbol{\beta} \text { ), }$$

$$\ell_{\text {ridge }}(\boldsymbol{\beta}):=\frac{1}{2}\left|\boldsymbol{y}-\boldsymbol{X}^{\top} \boldsymbol{\beta}\right|^{2}+\frac{\lambda}{2}|\boldsymbol{\beta}|^{2}$$

$$\frac{\partial \ell_{\text {ridge }}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} \mid \boldsymbol{\beta}=\hat{\boldsymbol{\beta}}=-\boldsymbol{X} \boldsymbol{y}+\boldsymbol{X} \boldsymbol{X}^{\top} \hat{\boldsymbol{\beta}}+\lambda \hat{\boldsymbol{\beta}}=\mathbf{0},$$

## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师