# 统计代写|线性回归分析代写linear regression analysis代考|STAT108

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## 统计代写|线性回归分析代写linear regression analysis代考|Prediction

The estimated mean function can be used to obtain values of the response for given values of the predictor. The two important variants of this problem are prediction and estimation of fitted values. Since prediction is more important, we discuss it first.

In prediction we have a new case, possibly a future value, not one used to estimate parameters, with observed value of the predictor $x_$. We would like to know the value $y_$, the corresponding response, but it has not yet been observed. If we assume that the data used to estimate the mean function are relevant to the new case, then the model fitted to the observed data can be used to predict for the new case. In the heights example, we would probably be willing to apply the fitted mean function to mother-daughter pairs alive in England at the end of the nineteenth century. Whether the prediction would be reasonable for mother-daughter pairs in other countries or in other time periods is much less clear. In Forbes’s problem, we would probably be willing to apply the results for altitudes in the range he studied. Given this additional assumption, a point prediction of $y_$, say $\tilde{y}$, is just
$$\tilde{y}=\hat{\beta}0+\hat{\beta}_1 x$$
$\tilde{y}$ predicts the as yet unobserved $y$. Assuming the model is correct, then the true value of $y_$ is $$y_=\beta_0+\beta_1 x_+e_$$
where $e_{\text {o }}$ is the random error attached to the future value, presumably with variance $\sigma^2$. Thus, even if $\beta_0$ and $\beta_1$ were known exactly, predictions would not match true values perfectly, but would be off by a random amount with standard deviation $\sigma$. In the more usual case where the coefficients are estimated, the prediction error variability will have a second component that arises from the uncertainty in the estimates of the coefficients. Combining these two sources of variation and using Appendix A.4,
$$\operatorname{Var}\left(\tilde{y}* \mid x\right)=\sigma^2+\sigma^2\left(\frac{1}{n}+\frac{\left(x_-\bar{x}\right)^2}{\operatorname{SXX}}\right)$$

## 统计代写|线性回归分析代写linear regression analysis代考|THE COEFFICIENT OF DETERMINATION, R

Ignoring all possible predictors, the best prediction of a response $y$ would simply be the sample average $\bar{y}$ of the values of the response observed in the data. The total sum of squares SYY $=\Sigma\left(y_i-\bar{y}\right)^2$ is the observed total variation of the response, ignoring any and all predictors. The total sum of squares is the sum of squared deviations from the horizontal line illustrated in Figure 2.4.
When we include a predictor, the unexplained variation is given by RSS, the sum of squared deviations from the fitted line, as shown on Figure 2.4. The difference between these sums of squares is called the sum of squares due to regression, SSreg, defined by
$$\text { SSreg }=\text { SYY }- \text { RSS }$$
We can get a computing formula for SSreg by substituting for RSS from (2.8),
$$\text { SSreg }=\text { SYY }-\left(\text { SYY }-\frac{(S Y Y)^2}{S X X}\right)=\frac{(S X Y)^2}{S X X}$$
If both sides of (2.18) are divided by SYY, we get
$$\frac{\text { SSreg }}{\text { SYY }}=1-\frac{\text { RSS }}{\text { SYY }}$$
The left-hand side of (2.20) is the proportion of observed variability in the response explained by regression on the predictor. The right-hand side consists of one minus the remaining unexplained variability. This concept of dividing up the total variability according to whether or not it is explained is of sufficient importance that a special name is given to it. We define $R^2$, the coefficient of determination, to be
$$R^2=\frac{\text { SSreg }}{\mathrm{SYY}}=1-\frac{\mathrm{RSS}}{\mathrm{SYY}}$$

# 线性回归代写

## 统计代写|线性回归分析代写linear regression analysis代考|Prediction

$$\tilde{y}=\hat{\beta}0+\hat{\beta}1 x$$ $\tilde{y}$预测了尚未观察到的$y$。假设模型正确，则$y$的真实值为$$y_=\beta_0+\beta_1 x_+e_$$

$$\operatorname{Var}\left(\tilde{y}* \mid x\right)=\sigma^2+\sigma^2\left(\frac{1}{n}+\frac{\left(x_-\bar{x}\right)^2}{\operatorname{SXX}}\right)$$

## 统计代写|线性回归分析代写linear regression analysis代考|THE COEFFICIENT OF DETERMINATION, R

$$\text { SSreg }=\text { SYY }- \text { RSS }$$

$$\text { SSreg }=\text { SYY }-\left(\text { SYY }-\frac{(S Y Y)^2}{S X X}\right)=\frac{(S X Y)^2}{S X X}$$
(2.18)的两边除以SYY，得到
$$\frac{\text { SSreg }}{\text { SYY }}=1-\frac{\text { RSS }}{\text { SYY }}$$
(2.20)的左侧是通过回归预测器解释的响应中观察到的可变性的比例。右边是1减去剩余的无法解释的变异性。根据是否得到解释来划分总变异性的概念非常重要，因此给它起了一个特殊的名称。我们定义决定系数$R^2$为
$$R^2=\frac{\text { SSreg }}{\mathrm{SYY}}=1-\frac{\mathrm{RSS}}{\mathrm{SYY}}$$

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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。

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