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# 统计代写|线性回归代写Linear Regression代考|STAT311 Residual Plots

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## 统计代写|线性回归代写Linear Regression代考|Residual Plots

Remark 2.3. Residual plots magnify departures from the model while the response plot emphasizes how well the MLR model fits the data.

Since the residuals $r_{i}=\hat{e}_{i}$ are estimators of the errors, the residual plot is used to visualize the conditional distribution $e \mid S P$ of the errors given the sufficient predictor SP $=\boldsymbol{x}^{T} \boldsymbol{\beta}$, where SP is estimated by $\widehat{Y}=\boldsymbol{x}^{T} \hat{\boldsymbol{\beta}}$. For the unimodal MLR model, there should not be any pattern in the residual plot: as a narrow vertical strip is moved from left to right, the behavior of the residuals within the strip should show little change.

Notation. A rule of thumb is a rule that often but not always works well in practice.

Rule of thumb 2.1. If the residual plot would look good after several points have been deleted, and if these deleted points were not gross outliers (points far from the point cloud formed by the bulk of the data), then the residual plot is probably good. Beginners often find too many things wrong with a good model. For practice, use the lregpack function MLRsim to generate several MLR data sets, and make the response and residual plots for these data sets: type MLRsim(nruns $=10)$ in $R$ and right click Stop for each plot (20 times) to generate 10 pairs of response and residual plots. This exercise will help show that the plots can have considerable variability even when the MLR model is good. See Problem $2.30$.

## 统计代写|线性回归代写Linear Regression代考|Other Model Violations

Without loss of generality, $E(e)=0$ for the unimodal MLR model with a constant, in that if $E(\tilde{e})=\mu \neq 0$, then the MLR model can always be written as $Y=\boldsymbol{x}^{T} \boldsymbol{\beta}+e$ where $E(e)=0$ and $E(Y) \equiv E(Y \mid \boldsymbol{x})=\boldsymbol{x}^{T} \boldsymbol{\beta}$. To see this claim notice that
$$\begin{gathered} Y=\tilde{\beta}{1}+x{2} \beta_{2}+\cdots+x_{p} \beta_{p}+\tilde{e}=\tilde{\beta}{1}+E(\tilde{e})+x{2} \beta_{2}+\cdots+x_{p} \beta_{p}+\tilde{e}-E(\tilde{e}) \ =\beta_{1}+x_{2} \beta_{2}+\cdots+x_{p} \beta_{p}+e \end{gathered}$$
where $\beta_{1}=\tilde{\beta}{1}+E(\tilde{e})$ and $e=\tilde{e}-E(\tilde{e})$. For example, if the errors $\tilde{e}{i}$ are iid exponential $(\lambda)$ with $E\left(\tilde{e}{i}\right)=\lambda$, use $e{i}=\tilde{e}_{i}-\lambda$.

For least squares, it is crucial that $\sigma^{2}$ exists. For example, if the $e_{i}$ are iid Cauchy $(0,1)$, then $\sigma^{2}$ does not exist and the least squares estimators tend to perform very poorly.

## 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。