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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

## 统计代写|回归分析作业代写Regression Analysis代考|Multiple Regression from the Matrix Point of View

In the case of simple regression, you saw that the OLS estimate of slope has a simple form: It is the estimated covariance of the $(X, Y)$ distribution, divided by the estimated variance of the $X$ distribution, or $\hat{\beta}1=\hat{\sigma}{x y} / \hat{\sigma}_x^2$. There is no such simple formula in multiple regression. Instead, you must use matrix algebra, involving matrix multiplication and matrix inverses. If you are unfamiliar with basic matrix algebra, including multiplication, addition, subtraction, transpose, identity matrix, and matrix inverse, you should take some time now to get acquainted with those particular concepts before reading on. (Perhaps you can locate a “matrix algebra for beginners” type of web page.)
Our first use of matrix algebra in regression is to give a concise representation of the regression model. Multiple regression models refer to $n$ observations and $k$ variables, both of which can be in the thousands or even millions. The following matrix form of the model provides a very convenient shorthand to represent all this information.
$$Y=\mathrm{X} \beta+\varepsilon$$
This concise form covers all the $n$ observations and all the $X$ variables ( $k$ of them) in one simple equation. Note that there are boldface non-italic terms and boldface italic terms in the expression. To make the material easier to read, we use the convention that boldface means a matrix, while boldface italic refers to a vector, which is a matrix with a single column. Thus $\boldsymbol{Y}, \boldsymbol{\beta}$, and $\varepsilon$, are vectors (single-column matrices), while $\mathbf{X}$ is a matrix having multiple columns.

To understand this model, consider your data set, which has the structure shown in Table 7.1.

You can relate the matrices in the model $Y=\mathrm{X} \beta+\varepsilon$ easily to the data set shown in Table $7.1$ as follows:
The $\boldsymbol{Y}$ vector is the list of all the $Y_i$ values:
$$\boldsymbol{Y}=\left[\begin{array}{c} Y_1 \ Y_2 \ \vdots \ Y_n \end{array}\right]$$
The $\mathbf{X}$ matrix is the array of all the $X_{i j}$ values, with an additional column of 1 ‘s to account for the intercept $\beta_0$ :
$$\mathbf{X}=\left[\begin{array}{ccccc} 1 & X_{11} & X_{12} & \ldots & X_{1 k} \ 1 & X_{21} & X_{22} & \ldots & X_{2 k} \ \vdots & \vdots & \vdots & \ddots & \vdots \ 1 & X_{n 1} & X_{n 2} & \cdots & X_{n k} \end{array}\right]$$

## 统计代写|回归分析作业代写Regression Analysis代考|The Regression Model in Matrix Form

The model representation $Y=\mathrm{X} \beta+\varepsilon$ is not complete because it states nothing about the assumptions. The following expression is a complete representation of the classical model; notice how simple the model looks when expressed in matrix form.
The classical model in matrix form
$$Y \mid \mathrm{X}=\mathrm{x} \sim \mathrm{N}_n\left(\mathrm{x} \boldsymbol{\beta}, \sigma^2 \mathrm{I}\right)$$
Here, the $\mathbf{X}=\mathbf{x}$ condition refers to a specific realized matrix $\mathbf{x}$ of the random matrix $\mathbf{X}$ and is a simple generalization of the $X=x$ condition we have used repeatedly to its matrix form. The matrix $\mathbf{X}$ contains potentially observable (random) $X$ values, as well as fixed values for any non-random $X$ data. The first column of $\mathbf{X}$ is ordinarily the column of 1’s needed to capture the intercept term $\beta_0$, and this column is not random.

In Appendix A of Chapter 1, we introduced the bivariate normal distribution, which is a distribution of two variables. Here, the symbol ” $\mathrm{N}_n\left(\mathbf{x} \boldsymbol{\beta}, \sigma^2 \mathbf{I}\right)^{\prime \prime}$ refers to a multivariate normal distribution. The ” $n$ ” subscript identifies that it is a distribution of the $n$ variables $Y_1, Y_2, \ldots$, $Y_n$. The $\mathbf{x} \beta$ term refers to the mean vector of the distribution, and the term $\sigma^2 \mathbf{I}$ refers to its covariance matrix (explained in detail below).

All assumptions in the classical regression model are embodied in the concise matrix form of the model: The correct functional specification assumption is embodied in the mean vector $(x \boldsymbol{\beta})$ specification, the constant variance and independence assumptions are implied by specification of $\sigma^2 \mathbf{I}$ as covariance matrix, as will be described below, and the normality assumption is embodied in the multivariate normal specification.

A covariance matrix is a matrix that contains all the variances and covariances among a set of random variables. For example, if $\left(W_1, W_2, W_3\right)$ are jointly distributed random variables, then the covariance matrix of $W=\left(W_1, W_2, W_3\right)$ is given by
$$\operatorname{Cov}(W)=\left[\begin{array}{ccc} \operatorname{Var}\left(W_1\right) & \operatorname{Cov}\left(W_1, W_2\right) & \operatorname{Cov}\left(W_1, W_3\right) \ \operatorname{Cov}\left(W_2, W_1\right) & \operatorname{Var}\left(W_2\right) & \operatorname{Cov}\left(W_2, W_3\right) \ \operatorname{Cov}\left(W_3, W_1\right) & \operatorname{Cov}\left(W_3, W_2\right) & \operatorname{Var}\left(W_3\right) \end{array}\right]$$
Notice that the row/column combination tells you which pair of variables are involved, or which variable is involved in the case of the diagonal elements. Note also that the covariance of a variable with itself is just the variance of that variable, which explains why the variances are on the diagonal of the covariance matrix.

# 回归分析代写

## 统计代写|回归分析作业代写Regression Analysis代考|The Regression Model in Matrix Form

$$Y \mid \mathrm{X}=\mathrm{x} \sim \mathrm{N}_n\left(\mathrm{x} \boldsymbol{\beta}, \sigma^2 \mathrm{I}\right)$$

$$\operatorname{Cov}(W)=\left[\operatorname{Var}\left(W_1\right) \quad \operatorname{Cov}\left(W_1, W_2\right) \quad \operatorname{Cov}\left(W_1, W_3\right) \operatorname{Cov}\left(W_2, W_1\right) \quad \operatorname{Var}\left(W_2\right) \quad \operatorname{Cov}\left(W_2, W_3\right) \operatorname{Cov}\left(W_3, W_1\right) \quad \operatorname{Cov}\left(W_3, W_2\right) \quad \operatorname{Var}\left(W_3\right)\right]$$

## 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